Subgroups — C₆₇×#34945 #30714196

Order 17152.

1812 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2573	C₂	1	4	yes	maximal, complemented
2	C₆₇×#2571	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2547	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2538	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2514	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2494	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2447	C₂	1	2	yes	maximal
9	C₆₇×#2421	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2401	C₂	1	4	yes	maximal, complemented
11	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#124×C₆₇	C₂	1	2	yes	maximal
13	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2295	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2283	C₂	1	2	yes	maximal, complemented
16	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄²×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₂²	1	2	yes	—
20	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
29	#401×C₆₇	C₂²	1	4	yes	—
30	#400×C₆₇	C₂²	1	4	yes	—
31	#400×C₆₇	C₂²	1	4	yes	—
32	#400×C₆₇	C₂²	1	4	yes	—
33	#397×C₆₇	C₂²	1	4	yes	—
34	#397×C₆₇	C₂²	1	4	yes	—
35	#397×C₆₇	C₂²	1	4	yes	—
36	#396×C₆₇	C₂²	1	4	yes	—
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	2	yes	—
39	#391×C₆₇	C₂²	1	2	yes	—
40	#391×C₆₇	C₂²	1	2	yes	—
41	#391×C₆₇	C₂²	1	2	yes	—
42	#390×C₆₇	C₂²	1	2	yes	—
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#388×C₆₇	C₂²	1	4	yes	—
49	#388×C₆₇	C₂²	1	4	yes	—
50	#387×C₆₇	C₂²	1	4	yes	—
51	#387×C₆₇	C₂²	1	4	yes	—
52	#387×C₆₇	C₂²	1	4	yes	—
53	#385×C₆₇	C₂²	1	4	yes	—
54	#384×C₆₇	C₂²	1	1	yes	characteristic
55	#383×C₆₇	C₂²	1	4	yes	—
56	#382×C₆₇	C₂²	1	2	yes	—
57	#382×C₆₇	C₂²	1	2	yes	—
58	#382×C₆₇	C₂²	1	2	yes	—
59	#381×C₆₇	C₂²	1	4	yes	—
60	#380×C₆₇	C₂²	1	2	yes	—
61	#380×C₆₇	C₂²	1	2	yes	—
62	#380×C₆₇	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
64	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
66	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
67	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
68	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
69	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
71	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
72	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
74	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
75	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
76	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
77	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
78	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
79	C₄×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
80	#376×C₆₇	C₂²	1	4	yes	—
81	C₄³×C₆₇	C₂²	1	1	yes	characteristic
82	C₄³×C₆₇	C₂²	1	2	yes	—
83	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
84	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
87	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
88	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
89	C₄⋊Q₈×C₆₇	—	2	4	no	—
90	#124×C₆₇	—	2	8	no	—
91	C₂²⋊Q₈×C₆₇	—	2	8	no	—
92	C₄×Q₈×C₆₇	—	2	4	no	—
93	C₄×Q₈×C₆₇	—	2	4	no	—
94	C₄×Q₈×C₆₇	—	2	8	no	—
95	C₄×Q₈×C₆₇	—	2	4	no	—
96	C₄×D₄×C₆₇	—	2	4	no	—
97	C₄×D₄×C₆₇	—	2	4	no	—
98	C₄×D₄×C₆₇	—	2	4	no	—
99	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
123	C₂×C₄²×C₆₇	—	2	8	no	—
124	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
129	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
130	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
134	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
138	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
141	#95×C₆₇	C₂³	1	4	yes	—
142	#95×C₆₇	C₂³	1	4	yes	—
143	#95×C₆₇	C₂³	1	4	yes	—
144	#95×C₆₇	C₂³	1	4	yes	—
145	#95×C₆₇	C₂³	1	2	yes	—
146	#95×C₆₇	C₂³	1	2	yes	—
147	#95×C₆₇	C₂³	1	2	yes	—
148	#95×C₆₇	C₂³	1	4	yes	—
149	#95×C₆₇	C₂³	1	4	yes	—
150	#95×C₆₇	C₂³	1	4	yes	—
151	#95×C₆₇	C₂³	1	4	yes	—
152	#95×C₆₇	C₂³	1	2	yes	—
153	#95×C₆₇	C₂³	1	2	yes	—
154	#95×C₆₇	C₂³	1	2	yes	—
155	#95×C₆₇	C₂³	1	4	yes	—
156	#95×C₆₇	C₂³	1	4	yes	—
157	#95×C₆₇	C₂³	1	4	yes	—
158	#95×C₆₇	C₂³	1	4	yes	—
159	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
160	C₂×Q₈×C₆₇	—	2	2	no	—
161	C₂×Q₈×C₆₇	—	2	2	no	—
162	C₂×Q₈×C₆₇	—	2	4	no	—
163	C₂×Q₈×C₆₇	—	2	4	no	—
164	C₂×Q₈×C₆₇	—	2	4	no	—
165	C₂×Q₈×C₆₇	#41	1	4	yes	—
166	C₂×D₄×C₆₇	—	2	2	no	—
167	C₂×D₄×C₆₇	—	2	2	no	—
168	C₂×D₄×C₆₇	#41	1	4	yes	—
169	C₂²×C₄×C₆₇	—	2	4	no	—
170	C₂²×C₄×C₆₇	—	2	4	no	—
171	C₂²×C₄×C₆₇	—	2	4	no	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	4	no	—
175	C₂²×C₄×C₆₇	—	2	4	no	—
176	C₂²×C₄×C₆₇	—	2	4	no	—
177	C₂²×C₄×C₆₇	—	4	2	no	—
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	—	2	2	no	—
196	C₄⋊C₄×C₆₇	—	2	2	no	—
197	C₄⋊C₄×C₆₇	—	2	4	no	—
198	C₄⋊C₄×C₆₇	—	2	4	no	—
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	2	2	no	—
201	C₄⋊C₄×C₆₇	—	2	2	no	—
202	C₄⋊C₄×C₆₇	—	2	2	no	—
203	C₄⋊C₄×C₆₇	—	2	4	no	—
204	C₄⋊C₄×C₆₇	—	2	8	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	8	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	—	2	4	no	—
209	C₄⋊C₄×C₆₇	—	4	2	no	—
210	C₄⋊C₄×C₆₇	—	4	4	no	—
211	C₂²⋊C₄×C₆₇	—	2	4	no	—
212	C₂²⋊C₄×C₆₇	—	2	8	no	—
213	C₂²⋊C₄×C₆₇	—	2	8	no	—
214	C₂²⋊C₄×C₆₇	—	2	4	no	—
215	C₂²⋊C₄×C₆₇	—	2	4	no	—
216	C₂²⋊C₄×C₆₇	—	2	8	no	—
217	C₂²⋊C₄×C₆₇	—	4	4	no	—
218	C₄²×C₆₇	—	2	4	no	—
219	C₄²×C₆₇	—	2	4	no	—
220	C₄²×C₆₇	—	2	4	no	—
221	C₄²×C₆₇	—	2	2	no	—
222	C₄²×C₆₇	—	2	4	no	—
223	C₄²×C₆₇	—	2	2	no	—
224	C₄²×C₆₇	—	2	4	no	—
225	C₄²×C₆₇	—	2	4	no	—
226	C₄²×C₆₇	—	2	8	no	—
227	C₄²×C₆₇	—	2	4	no	—
228	C₄²×C₆₇	—	2	2	no	—
229	C₄²×C₆₇	—	2	4	no	—
230	C₄²×C₆₇	—	4	1	no	—
231	C₄²×C₆₇	—	4	2	no	—
232	C₄²×C₆₇	#41	1	4	yes	—
233	C₄²×C₆₇	#41	1	4	yes	—
234	C₄²×C₆₇	#41	1	4	yes	—
235	C₂³×C₆₇	—	2	2	no	—
236	C₂³×C₆₇	—	2	2	no	—
237	C₂³×C₆₇	—	2	2	no	—
238	C₂³×C₆₇	—	2	2	no	—
239	C₂³×C₆₇	—	2	2	no	—
240	C₂³×C₆₇	—	2	2	no	—
241	C₂³×C₆₇	C₂×#41	1	4	yes	—
242	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
243	Q₈×C₆₇	—	2	8	no	—
244	Q₈×C₆₇	—	4	8	no	—
245	D₄×C₆₇	—	2	8	no	—
246	C₂×C₄×C₆₇	—	2	1	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	1	no	—
250	C₂×C₄×C₆₇	—	2	1	no	—
251	C₂×C₄×C₆₇	—	2	1	no	—
252	C₂×C₄×C₆₇	—	2	1	no	—
253	C₂×C₄×C₆₇	—	2	1	no	—
254	C₂×C₄×C₆₇	—	2	2	no	—
255	C₂×C₄×C₆₇	—	2	2	no	—
256	C₂×C₄×C₆₇	—	2	2	no	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	1	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	4	no	—
263	C₂×C₄×C₆₇	—	2	2	no	—
264	C₂×C₄×C₆₇	—	2	1	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	2	2	no	—
270	C₂×C₄×C₆₇	—	2	2	no	—
271	C₂×C₄×C₆₇	—	2	2	no	—
272	C₂×C₄×C₆₇	—	2	4	no	—
273	C₂×C₄×C₆₇	—	2	2	no	—
274	C₂×C₄×C₆₇	—	2	1	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	2	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	4	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	4	8	no	—
282	C₂×C₄×C₆₇	—	4	8	no	—
283	C₂×C₄×C₆₇	—	4	8	no	—
284	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
285	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
286	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
287	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
288	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
289	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
290	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
291	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
292	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
293	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
294	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
295	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
296	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
297	C₂²×C₆₇	—	2	4	no	—
298	C₂²×C₆₇	—	2	4	no	—
299	C₂²×C₆₇	—	2	4	no	—
300	C₂²×C₆₇	—	4	2	no	—
301	C₂²×C₆₇	—	4	2	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	4	2	no	—
304	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
305	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
306	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
307	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
308	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
309	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
310	C₂²×C₆₇	#585	1	1	yes	characteristic
311	C₄×C₆₇	—	2	4	no	—
312	C₄×C₆₇	—	2	2	no	—
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	2	2	no	—
315	C₄×C₆₇	—	4	2	no	—
316	C₄×C₆₇	—	4	4	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	—	4	2	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	—	4	1	no	—
323	C₄×C₆₇	—	4	2	no	—
324	C₄×C₆₇	#529	1	4	yes	—
325	C₄×C₆₇	#529	1	4	yes	—
326	C₄×C₆₇	#556	1	4	yes	—
327	#34945	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
328	C₂×C₆₇	—	4	4	no	—
329	C₂×C₆₇	C₂×#529	1	1	yes	characteristic
330	C₂×C₆₇	C₂×#529	1	1	yes	characteristic
331	C₂×C₆₇	#3461	1	1	yes	characteristic
332	C₂×C₆₇	C₂×#556	1	1	yes	characteristic
333	C₂×C₆₇	#3475	1	1	yes	characteristic
334	C₂×C₆₇	#3480	1	1	yes	characteristic
335	C₂×C₆₇	#3482	1	1	yes	characteristic
336	#2573	C₂×C₆₇	1	4	yes	complemented
337	#2571	C₂×C₆₇	1	1	yes	characteristic, complemented
338	#2568	C₂×C₆₇	1	1	yes	characteristic, complemented
339	#2547	C₂×C₆₇	1	1	yes	characteristic
340	#2538	C₂×C₆₇	1	1	yes	characteristic
341	#2514	C₂×C₆₇	1	2	yes	complemented
342	#2494	C₂×C₆₇	1	2	yes	complemented
343	#2447	C₂×C₆₇	1	2	yes	—
344	#2421	C₂×C₆₇	1	2	yes	complemented
345	#2401	C₂×C₆₇	1	4	yes	complemented
346	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
347	C₄×#124	C₂×C₆₇	1	2	yes	—
348	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
349	#2295	C₂×C₆₇	1	2	yes	complemented
350	#2283	C₂×C₆₇	1	2	yes	complemented
351	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
352	C₄²×D₄	C₂×C₆₇	1	1	yes	characteristic
353	C₆₇	#34945	1	1	yes	characteristic, direct product factor, Sylow
354	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
355	C₂×#124	C₂²×C₆₇	1	2	yes	—
356	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
357	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
358	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
359	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
360	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
361	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
362	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
363	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
364	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
365	#401	C₂²×C₆₇	1	4	yes	—
366	#400	C₂²×C₆₇	1	4	yes	—
367	#400	C₂²×C₆₇	1	4	yes	—
368	#400	C₂²×C₆₇	1	4	yes	—
369	#397	C₂²×C₆₇	1	4	yes	—
370	#397	C₂²×C₆₇	1	4	yes	—
371	#397	C₂²×C₆₇	1	4	yes	—
372	#396	C₂²×C₆₇	1	4	yes	—
373	#391	C₂²×C₆₇	1	2	yes	—
374	#391	C₂²×C₆₇	1	2	yes	—
375	#391	C₂²×C₆₇	1	2	yes	—
376	#391	C₂²×C₆₇	1	2	yes	—
377	#391	C₂²×C₆₇	1	2	yes	—
378	#390	C₂²×C₆₇	1	2	yes	—
379	#389	C₂²×C₆₇	1	2	yes	—
380	#389	C₂²×C₆₇	1	2	yes	—
381	#389	C₂²×C₆₇	1	2	yes	—
382	#389	C₂²×C₆₇	1	2	yes	—
383	#389	C₂²×C₆₇	1	2	yes	—
384	#388	C₂²×C₆₇	1	4	yes	—
385	#388	C₂²×C₆₇	1	4	yes	—
386	#387	C₂²×C₆₇	1	4	yes	—
387	#387	C₂²×C₆₇	1	4	yes	—
388	#387	C₂²×C₆₇	1	4	yes	—
389	#385	C₂²×C₆₇	1	4	yes	—
390	#384	C₂²×C₆₇	1	1	yes	characteristic
391	#383	C₂²×C₆₇	1	4	yes	—
392	#382	C₂²×C₆₇	1	2	yes	—
393	#382	C₂²×C₆₇	1	2	yes	—
394	#382	C₂²×C₆₇	1	2	yes	—
395	#381	C₂²×C₆₇	1	4	yes	—
396	#380	C₂²×C₆₇	1	2	yes	—
397	#380	C₂²×C₆₇	1	2	yes	—
398	#380	C₂²×C₆₇	1	2	yes	—
399	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
400	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
401	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
402	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
403	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
404	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
405	C₄×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
406	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
407	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
408	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
409	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
410	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
411	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
412	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
413	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
414	C₄×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
415	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
416	#376	C₂²×C₆₇	1	4	yes	—
417	C₄³	C₂²×C₆₇	1	1	yes	characteristic
418	C₄³	C₂²×C₆₇	1	2	yes	—
419	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
420	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
421	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
422	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
423	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
424	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
425	C₄⋊Q₈	—	2	4	no	—
426	#124	—	2	8	no	—
427	C₂²⋊Q₈	—	2	8	no	—
428	C₄×Q₈	—	2	4	no	—
429	C₄×Q₈	—	2	4	no	—
430	C₄×Q₈	—	2	8	no	—
431	C₄×Q₈	—	2	4	no	—
432	C₄×D₄	—	2	4	no	—
433	C₄×D₄	—	2	4	no	—
434	C₄×D₄	—	2	4	no	—
435	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
436	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
437	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
438	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
439	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
443	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
445	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
448	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
450	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
452	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
453	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
454	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
455	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
456	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
457	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
459	C₂×C₄²	—	2	8	no	—
460	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
461	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
462	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
463	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
464	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
465	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
466	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
467	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
468	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
469	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
470	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
471	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
472	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
474	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
475	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
476	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
477	#95	C₂³×C₆₇	1	4	yes	—
478	#95	C₂³×C₆₇	1	4	yes	—
479	#95	C₂³×C₆₇	1	4	yes	—
480	#95	C₂³×C₆₇	1	4	yes	—
481	#95	C₂³×C₆₇	1	2	yes	—
482	#95	C₂³×C₆₇	1	2	yes	—
483	#95	C₂³×C₆₇	1	2	yes	—
484	#95	C₂³×C₆₇	1	4	yes	—
485	#95	C₂³×C₆₇	1	4	yes	—
486	#95	C₂³×C₆₇	1	4	yes	—
487	#95	C₂³×C₆₇	1	4	yes	—
488	#95	C₂³×C₆₇	1	2	yes	—
489	#95	C₂³×C₆₇	1	2	yes	—
490	#95	C₂³×C₆₇	1	2	yes	—
491	#95	C₂³×C₆₇	1	4	yes	—
492	#95	C₂³×C₆₇	1	4	yes	—
493	#95	C₂³×C₆₇	1	4	yes	—
494	#95	C₂³×C₆₇	1	4	yes	—
495	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
496	C₂×Q₈	—	2	2	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	4	no	—
499	C₂×Q₈	—	2	4	no	—
500	C₂×Q₈	—	2	4	no	—
501	C₂×Q₈	#41×C₆₇	1	4	yes	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	#41×C₆₇	1	4	yes	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	4	no	—
507	C₂²×C₄	—	2	4	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	4	no	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	4	no	—
513	C₂²×C₄	—	4	2	no	—
514	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
515	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
517	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
518	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
521	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
523	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
524	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
525	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
526	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
527	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
528	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	2	no	—
532	C₄⋊C₄	—	2	2	no	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	8	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	8	no	—
543	C₄⋊C₄	—	2	4	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	4	2	no	—
546	C₄⋊C₄	—	4	4	no	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₂²⋊C₄	—	2	8	no	—
549	C₂²⋊C₄	—	2	8	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	8	no	—
553	C₂²⋊C₄	—	4	4	no	—
554	C₄²	—	2	4	no	—
555	C₄²	—	2	4	no	—
556	C₄²	—	2	4	no	—
557	C₄²	—	2	2	no	—
558	C₄²	—	2	4	no	—
559	C₄²	—	2	2	no	—
560	C₄²	—	2	4	no	—
561	C₄²	—	2	4	no	—
562	C₄²	—	2	8	no	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	2	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	4	1	no	—
567	C₄²	—	4	2	no	—
568	C₄²	#41×C₆₇	1	4	yes	—
569	C₄²	#41×C₆₇	1	4	yes	—
570	C₄²	#41×C₆₇	1	4	yes	—
571	C₂³	—	2	2	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	2	2	no	—
574	C₂³	—	2	2	no	—
575	C₂³	—	2	2	no	—
576	C₂³	—	2	2	no	—
577	C₂³	C₂×#41×C₆₇	1	4	yes	—
578	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
579	Q₈	—	2	8	no	—
580	Q₈	—	4	8	no	—
581	D₄	—	2	8	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	2	4	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	4	8	no	—
618	C₂×C₄	—	4	8	no	—
619	C₂×C₄	—	4	8	no	—
620	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
621	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
622	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
623	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
624	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
625	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
626	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
627	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
628	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
629	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
630	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
631	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
632	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
633	C₂²	—	2	4	no	—
634	C₂²	—	2	4	no	—
635	C₂²	—	2	4	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	4	2	no	—
640	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
641	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
642	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
643	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
644	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
645	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
646	C₂²	#585×C₆₇	1	1	yes	characteristic
647	C₄	—	2	4	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	2	2	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	4	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	2	no	—
657	C₄	—	4	4	no	—
658	C₄	—	4	1	no	—
659	C₄	—	4	2	no	—
660	C₄	#529×C₆₇	1	4	yes	—
661	C₄	#529×C₆₇	1	4	yes	—
662	C₄	#556×C₆₇	1	4	yes	—
663	C₂	—	4	4	no	—
664	C₂	C₂×#529×C₆₇	1	1	yes	characteristic
665	C₂	C₂×#529×C₆₇	1	1	yes	characteristic
666	C₂	C₆₇×#3461	1	1	yes	characteristic
667	C₂	C₂×#556×C₆₇	1	1	yes	characteristic
668	C₂	C₆₇×#3475	1	1	yes	characteristic
669	C₂	C₆₇×#3480	1	1	yes	characteristic
670	C₂	C₆₇×#3482	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₆₇×#2573	C₂	semidirect
2. C₆₇×#2571	C₂	semidirect
3. C₆₇×#2568	C₂	semidirect
6. C₆₇×#2514	C₂	semidirect
7. C₆₇×#2494	C₂	semidirect
9. C₆₇×#2421	C₂	semidirect
10. C₆₇×#2401	C₂	semidirect
11. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
13. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
14. C₆₇×#2295	C₂	semidirect
15. C₆₇×#2283	C₂	semidirect
16. C₄²×Q₈×C₆₇	C₂	semidirect
327. #34945	C₆₇	direct (N × H)
336. #2573	C₂×C₆₇	semidirect
337. #2571	C₂×C₆₇	semidirect
338. #2568	C₂×C₆₇	semidirect
341. #2514	C₂×C₆₇	semidirect
342. #2494	C₂×C₆₇	semidirect
344. #2421	C₂×C₆₇	semidirect
345. #2401	C₂×C₆₇	semidirect
346. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
348. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
349. #2295	C₂×C₆₇	semidirect
350. #2283	C₂×C₆₇	semidirect
351. C₄²×Q₈	C₂×C₆₇	semidirect
353. C₆₇	#34945	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₆₇×#2573	2
C₆₇×#2571	2
C₆₇×#2568	2
C₆₇×#2547	2
C₆₇×#2538	2
C₆₇×#2514	2
C₆₇×#2494	2
C₆₇×#2447	2
C₆₇×#2421	2
C₆₇×#2401	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#124×C₆₇	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#2295	2
C₆₇×#2283	2
C₄²×Q₈×C₆₇	2
C₄²×D₄×C₆₇	2
#34945	67

Subgroups — C67×#34945 #30714196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34945 #30714196