Subgroups — C₆₇×#34947 #30714198

Order 17152.

2522 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2566	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2565	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2564	C₂	1	4	yes	maximal, complemented
8	C₆₇×#2563	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2417	C₂	1	4	yes	maximal, complemented
11	C₆₇×#2396	C₂	1	4	yes	maximal, complemented
12	C₄×C₂²⋊Q₈×C₆₇	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2297	C₂	1	2	yes	maximal
14	C₆₇×#2285	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
17	#533×C₆₇	—	2	8	no	—
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#125×C₆₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
22	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
24	C₄×#41×C₆₇	—	2	8	no	—
25	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#117×C₆₇	C₂²	1	2	yes	—
35	C₂×#117×C₆₇	C₂²	1	2	yes	—
36	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
38	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
39	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
40	#400×C₆₇	C₂²	1	4	yes	—
41	#399×C₆₇	C₂²	1	4	yes	—
42	#398×C₆₇	C₂²	1	8	yes	—
43	#396×C₆₇	C₂²	1	8	yes	—
44	#395×C₆₇	C₂²	1	4	yes	—
45	#393×C₆₇	C₂²	1	4	yes	—
46	#391×C₆₇	C₂²	1	2	yes	—
47	#391×C₆₇	C₂²	1	2	yes	—
48	#391×C₆₇	C₂²	1	4	yes	—
49	#390×C₆₇	C₂²	1	2	yes	—
50	#390×C₆₇	C₂²	1	2	yes	—
51	#389×C₆₇	C₂²	1	4	yes	—
52	#387×C₆₇	C₂²	1	8	yes	—
53	#386×C₆₇	C₂²	1	8	yes	—
54	#385×C₆₇	C₂²	1	8	yes	—
55	#384×C₆₇	C₂²	1	2	yes	—
56	#384×C₆₇	C₂²	1	2	yes	—
57	#382×C₆₇	C₂²	1	2	yes	—
58	#382×C₆₇	C₂²	1	2	yes	—
59	#380×C₆₇	C₂²	1	8	yes	—
60	#380×C₆₇	C₂²	1	4	yes	—
61	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
62	C₄×C₄⋊C₄×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	2	yes	—
66	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	—
67	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	—
68	C₂×#95×C₆₇	C₂²	1	2	yes	—
69	C₂×#41×C₆₇	—	2	4	no	—
70	C₂×#41×C₆₇	—	2	4	no	—
71	C₂×#41×C₆₇	Q₈	1	8	yes	complemented
72	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
78	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
81	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
82	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊Q₈×C₆₇	—	2	4	no	—
85	C₄⋊Q₈×C₆₇	—	2	4	no	—
86	#125×C₆₇	—	2	8	no	—
87	C₂²⋊Q₈×C₆₇	—	2	8	no	—
88	C₂²⋊Q₈×C₆₇	—	2	8	no	—
89	C₂²⋊Q₈×C₆₇	—	2	16	no	—
90	C₄×Q₈×C₆₇	—	2	4	no	—
91	C₄×Q₈×C₆₇	—	2	4	no	—
92	C₄×Q₈×C₆₇	—	2	8	no	—
93	C₄×Q₈×C₆₇	—	2	8	no	—
94	C₄×D₄×C₆₇	—	2	8	no	—
95	C₄×D₄×C₆₇	—	2	8	no	—
96	C₄×D₄×C₆₇	—	2	4	no	—
97	C₄×D₄×C₆₇	—	2	4	no	—
98	#117×C₆₇	—	2	4	no	—
99	#117×C₆₇	—	2	8	no	—
100	#117×C₆₇	—	2	8	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
102	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
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	2	yes	—
110	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	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	8	yes	—
115	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
116	C₂×C₄⋊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	2	yes	—
120	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
122	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
123	C₂×C₄²×C₆₇	—	2	8	no	—
124	C₂×C₄²×C₆₇	—	2	8	no	—
125	C₂×C₄²×C₆₇	D₄	1	8	yes	—
126	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
127	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
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	4	yes	—
132	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
133	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
137	#95×C₆₇	C₂³	1	4	yes	—
138	#95×C₆₇	C₂³	1	2	yes	—
139	#95×C₆₇	C₂³	1	2	yes	—
140	#95×C₆₇	C₂³	1	8	yes	—
141	#95×C₆₇	C₂³	1	8	yes	—
142	#95×C₆₇	C₂³	1	8	yes	—
143	#95×C₆₇	C₂³	1	8	yes	—
144	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
145	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
146	#41×C₆₇	—	2	32	no	—
147	C₂×Q₈×C₆₇	—	2	2	no	—
148	C₂×Q₈×C₆₇	—	2	4	no	—
149	C₂×Q₈×C₆₇	—	2	4	no	—
150	C₂×Q₈×C₆₇	—	2	2	no	—
151	C₂×Q₈×C₆₇	—	2	8	no	—
152	C₂×Q₈×C₆₇	—	4	4	no	—
153	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
154	C₂×D₄×C₆₇	—	2	4	no	—
155	C₂×D₄×C₆₇	—	2	4	no	—
156	C₂×D₄×C₆₇	—	2	2	no	—
157	C₂×D₄×C₆₇	—	2	2	no	—
158	C₂×D₄×C₆₇	C₂×Q₈	1	8	yes	—
159	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
160	C₂²×C₄×C₆₇	—	2	4	no	—
161	C₂²×C₄×C₆₇	—	2	4	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	8	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	2	4	no	—
166	C₂²×C₄×C₆₇	—	2	8	no	—
167	C₂²×C₄×C₆₇	—	2	8	no	—
168	C₂²×C₄×C₆₇	—	4	4	no	—
169	C₂²×C₄×C₆₇	—	4	2	no	—
170	C₂²×C₄×C₆₇	—	4	4	no	—
171	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
175	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
176	C₂²×C₄×C₆₇	#41	1	4	yes	—
177	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₆₇	C₂⁴	1	2	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	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	2	no	—
193	C₄⋊C₄×C₆₇	—	2	2	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	—	2	4	no	—
196	C₄⋊C₄×C₆₇	—	2	8	no	—
197	C₄⋊C₄×C₆₇	—	2	8	no	—
198	C₄⋊C₄×C₆₇	—	2	4	no	—
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	2	8	no	—
201	C₄⋊C₄×C₆₇	—	2	4	no	—
202	C₄⋊C₄×C₆₇	—	2	16	no	—
203	C₄⋊C₄×C₆₇	—	4	8	no	—
204	C₄⋊C₄×C₆₇	#41	1	8	yes	complemented
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₂²⋊C₄×C₆₇	—	2	8	no	—
207	C₂²⋊C₄×C₆₇	—	2	4	no	—
208	C₂²⋊C₄×C₆₇	—	2	4	no	—
209	C₂²⋊C₄×C₆₇	—	2	16	no	—
210	C₂²⋊C₄×C₆₇	—	4	8	no	—
211	C₄²×C₆₇	—	2	4	no	—
212	C₄²×C₆₇	—	2	4	no	—
213	C₄²×C₆₇	—	2	4	no	—
214	C₄²×C₆₇	—	2	8	no	—
215	C₄²×C₆₇	—	2	4	no	—
216	C₄²×C₆₇	—	2	8	no	—
217	C₄²×C₆₇	—	2	4	no	—
218	C₄²×C₆₇	C₂×D₄	1	4	yes	—
219	C₄²×C₆₇	C₂×D₄	1	4	yes	—
220	C₄²×C₆₇	C₂×D₄	1	4	yes	—
221	C₄²×C₆₇	C₂×D₄	1	4	yes	—
222	C₂³×C₆₇	—	2	2	no	—
223	C₂³×C₆₇	—	2	2	no	—
224	C₂³×C₆₇	—	2	2	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	—	2	2	no	—
227	C₂³×C₆₇	—	2	4	no	—
228	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
229	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
230	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
231	C₂³×C₆₇	C₂×#41	1	2	yes	—
232	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
233	Q₈×C₆₇	—	2	8	no	—
234	Q₈×C₆₇	—	4	8	no	—
235	Q₈×C₆₇	—	4	8	no	—
236	D₄×C₆₇	—	2	16	no	—
237	D₄×C₆₇	—	2	8	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	2	no	—
242	C₂×C₄×C₆₇	—	2	4	no	—
243	C₂×C₄×C₆₇	—	2	2	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	16	no	—
246	C₂×C₄×C₆₇	—	2	8	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	4	no	—
250	C₂×C₄×C₆₇	—	2	4	no	—
251	C₂×C₄×C₆₇	—	2	4	no	—
252	C₂×C₄×C₆₇	—	2	8	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	no	—
255	C₂×C₄×C₆₇	—	2	4	no	—
256	C₂×C₄×C₆₇	—	2	8	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	4	4	no	—
259	C₂×C₄×C₆₇	—	4	16	no	—
260	C₂×C₄×C₆₇	—	4	8	no	—
261	C₂×C₄×C₆₇	—	4	8	no	—
262	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	complemented
263	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
264	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
265	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
266	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
270	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
272	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
273	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
274	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
275	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
276	C₂²×C₆₇	—	2	2	no	—
277	C₂²×C₆₇	—	2	2	no	—
278	C₂²×C₆₇	—	2	4	no	—
279	C₂²×C₆₇	—	2	4	no	—
280	C₂²×C₆₇	—	2	4	no	—
281	C₂²×C₆₇	—	2	8	no	—
282	C₂²×C₆₇	—	4	2	no	—
283	C₂²×C₆₇	—	4	2	no	—
284	C₂²×C₆₇	—	4	4	no	—
285	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
286	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
287	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
288	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
289	C₂²×C₆₇	C₂²×#41	1	2	yes	—
290	C₂²×C₆₇	#585	1	2	yes	—
291	C₄×C₆₇	—	2	4	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	2	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	2	8	no	—
296	C₄×C₆₇	—	4	2	no	—
297	C₄×C₆₇	—	4	4	no	—
298	C₄×C₆₇	—	4	2	no	—
299	C₄×C₆₇	—	4	4	no	—
300	C₄×C₆₇	—	4	4	no	—
301	C₄×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
302	C₄×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
303	C₄×C₆₇	#532	1	4	yes	—
304	#34947	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂×C₆₇	—	2	4	no	—
306	C₂×C₆₇	—	4	4	no	—
307	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
308	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
309	C₂×C₆₇	#3449	1	1	yes	characteristic
310	C₂×C₆₇	#3475	1	2	yes	—
311	C₂×C₆₇	Q₈×#41	1	2	yes	—
312	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
313	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
314	#2568	C₂×C₆₇	1	1	yes	characteristic, complemented
315	#2567	C₂×C₆₇	1	1	yes	characteristic, complemented
316	#2566	C₂×C₆₇	1	2	yes	complemented
317	#2565	C₂×C₆₇	1	2	yes	complemented
318	#2564	C₂×C₆₇	1	4	yes	complemented
319	#2563	C₂×C₆₇	1	1	yes	characteristic, complemented
320	#2542	C₂×C₆₇	1	1	yes	characteristic, complemented
321	#2417	C₂×C₆₇	1	4	yes	complemented
322	#2396	C₂×C₆₇	1	4	yes	complemented
323	C₄×C₂²⋊Q₈	C₂×C₆₇	1	4	yes	complemented
324	#2297	C₂×C₆₇	1	2	yes	—
325	#2285	C₂×C₆₇	1	2	yes	complemented
326	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
327	C₆₇	#34947	1	1	yes	characteristic, direct product factor, Sylow
328	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
329	#533	—	2	8	no	—
330	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
331	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#125	C₂²×C₆₇	1	2	yes	—
333	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
334	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
335	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
336	C₄×#41	—	2	8	no	—
337	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
338	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
339	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
340	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
341	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
342	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
343	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
344	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
345	C₂×#117	C₂²×C₆₇	1	2	yes	—
346	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
347	C₂×#117	C₂²×C₆₇	1	2	yes	—
348	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
349	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
350	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
351	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
352	#400	C₂²×C₆₇	1	4	yes	—
353	#399	C₂²×C₆₇	1	4	yes	—
354	#398	C₂²×C₆₇	1	8	yes	—
355	#396	C₂²×C₆₇	1	8	yes	—
356	#395	C₂²×C₆₇	1	4	yes	—
357	#393	C₂²×C₆₇	1	4	yes	—
358	#391	C₂²×C₆₇	1	2	yes	—
359	#391	C₂²×C₆₇	1	2	yes	—
360	#391	C₂²×C₆₇	1	4	yes	—
361	#390	C₂²×C₆₇	1	2	yes	—
362	#390	C₂²×C₆₇	1	2	yes	—
363	#389	C₂²×C₆₇	1	4	yes	—
364	#387	C₂²×C₆₇	1	8	yes	—
365	#386	C₂²×C₆₇	1	8	yes	—
366	#385	C₂²×C₆₇	1	8	yes	—
367	#384	C₂²×C₆₇	1	2	yes	—
368	#384	C₂²×C₆₇	1	2	yes	—
369	#382	C₂²×C₆₇	1	2	yes	—
370	#382	C₂²×C₆₇	1	2	yes	—
371	#380	C₂²×C₆₇	1	4	yes	—
372	#380	C₂²×C₆₇	1	8	yes	—
373	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
374	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
375	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
376	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
377	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	—
378	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
379	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	—
380	C₂×#95	C₂²×C₆₇	1	2	yes	—
381	C₂×#41	—	2	4	no	—
382	C₂×#41	—	2	4	no	—
383	C₂×#41	Q₈×C₆₇	1	8	yes	complemented
384	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
385	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
386	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
387	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
388	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
390	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
391	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
392	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
393	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
394	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
396	C₄⋊Q₈	—	2	4	no	—
397	C₄⋊Q₈	—	2	4	no	—
398	#125	—	2	8	no	—
399	C₂²⋊Q₈	—	2	8	no	—
400	C₂²⋊Q₈	—	2	8	no	—
401	C₂²⋊Q₈	—	2	16	no	—
402	C₄×Q₈	—	2	4	no	—
403	C₄×Q₈	—	2	4	no	—
404	C₄×Q₈	—	2	8	no	—
405	C₄×Q₈	—	2	8	no	—
406	C₄×D₄	—	2	8	no	—
407	C₄×D₄	—	2	8	no	—
408	C₄×D₄	—	2	4	no	—
409	C₄×D₄	—	2	4	no	—
410	#117	—	2	4	no	—
411	#117	—	2	8	no	—
412	#117	—	2	8	no	—
413	C₂×C₄⋊C₄	—	2	8	no	—
414	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
424	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
425	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
426	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
427	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
428	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
435	C₂×C₄²	—	2	8	no	—
436	C₂×C₄²	—	2	8	no	—
437	C₂×C₄²	D₄×C₆₇	1	8	yes	—
438	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
439	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
440	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
441	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
442	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
443	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
444	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
445	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
449	#95	C₂³×C₆₇	1	4	yes	—
450	#95	C₂³×C₆₇	1	2	yes	—
451	#95	C₂³×C₆₇	1	2	yes	—
452	#95	C₂³×C₆₇	1	8	yes	—
453	#95	C₂³×C₆₇	1	8	yes	—
454	#95	C₂³×C₆₇	1	8	yes	—
455	#95	C₂³×C₆₇	1	8	yes	—
456	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
457	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
458	#41	—	2	32	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	4	no	—
461	C₂×Q₈	—	2	4	no	—
462	C₂×Q₈	—	2	2	no	—
463	C₂×Q₈	—	2	8	no	—
464	C₂×Q₈	—	4	4	no	—
465	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	C₂×Q₈×C₆₇	1	8	yes	—
471	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
472	C₂²×C₄	—	2	4	no	—
473	C₂²×C₄	—	2	4	no	—
474	C₂²×C₄	—	2	4	no	—
475	C₂²×C₄	—	2	8	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	8	no	—
479	C₂²×C₄	—	2	8	no	—
480	C₂²×C₄	—	4	4	no	—
481	C₂²×C₄	—	4	2	no	—
482	C₂²×C₄	—	4	4	no	—
483	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
484	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
485	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
486	C₂²×C₄	C₂×Q₈×C₆₇	1	8	yes	—
487	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
488	C₂²×C₄	#41×C₆₇	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
494	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
495	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
500	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
501	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	4	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	8	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	4	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	8	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	16	no	—
515	C₄⋊C₄	—	4	8	no	—
516	C₄⋊C₄	#41×C₆₇	1	8	yes	complemented
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	8	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	—	2	16	no	—
522	C₂²⋊C₄	—	4	8	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	8	no	—
527	C₄²	—	2	4	no	—
528	C₄²	—	2	8	no	—
529	C₄²	—	2	4	no	—
530	C₄²	C₂×D₄×C₆₇	1	4	yes	—
531	C₄²	C₂×D₄×C₆₇	1	4	yes	—
532	C₄²	C₂×D₄×C₆₇	1	4	yes	—
533	C₄²	C₂×D₄×C₆₇	1	4	yes	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	2	no	—
539	C₂³	—	2	4	no	—
540	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
541	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
542	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
543	C₂³	C₂×#41×C₆₇	1	2	yes	—
544	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
545	Q₈	—	2	8	no	—
546	Q₈	—	4	8	no	—
547	Q₈	—	4	8	no	—
548	D₄	—	2	16	no	—
549	D₄	—	2	8	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	2	16	no	—
558	C₂×C₄	—	2	8	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	8	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	8	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	16	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	complemented
575	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
576	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
577	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
581	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
582	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
583	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
584	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
585	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
586	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
587	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
588	C₂²	—	2	2	no	—
589	C₂²	—	2	2	no	—
590	C₂²	—	2	4	no	—
591	C₂²	—	2	4	no	—
592	C₂²	—	2	4	no	—
593	C₂²	—	2	8	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	4	no	—
597	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
598	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
599	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
600	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
601	C₂²	C₂²×#41×C₆₇	1	2	yes	—
602	C₂²	#585×C₆₇	1	2	yes	—
603	C₄	—	2	4	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	8	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	4	no	—
612	C₄	—	4	4	no	—
613	C₄	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
614	C₄	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
615	C₄	#532×C₆₇	1	4	yes	—
616	C₂	—	2	4	no	—
617	C₂	—	4	4	no	—
618	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
619	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
620	C₂	C₆₇×#3449	1	1	yes	characteristic
621	C₂	C₆₇×#3475	1	2	yes	—
622	C₂	Q₈×#41×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂⁵
prime radical	C₂²×#41×C₆₇	C₂²
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. C₂×#533×C₆₇	C₂	semidirect
3. C₆₇×#2568	C₂	semidirect
4. C₆₇×#2567	C₂	semidirect
5. C₆₇×#2566	C₂	semidirect
6. C₆₇×#2565	C₂	semidirect
7. C₆₇×#2564	C₂	semidirect
8. C₆₇×#2563	C₂	semidirect
9. C₆₇×#2542	C₂	semidirect
10. C₆₇×#2417	C₂	semidirect
11. C₆₇×#2396	C₂	semidirect
12. C₄×C₂²⋊Q₈×C₆₇	C₂	semidirect
14. C₆₇×#2285	C₂	semidirect
15. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
71. C₂×#41×C₆₇	Q₈	semidirect
204. C₄⋊C₄×C₆₇	#41	semidirect
262. C₂×C₄×C₆₇	C₂²⋊Q₈	semidirect
304. #34947	C₆₇	direct (N × H)
312. C₂×#533	C₂×C₆₇	semidirect
314. #2568	C₂×C₆₇	semidirect
315. #2567	C₂×C₆₇	semidirect
316. #2566	C₂×C₆₇	semidirect
317. #2565	C₂×C₆₇	semidirect
318. #2564	C₂×C₆₇	semidirect
319. #2563	C₂×C₆₇	semidirect
320. #2542	C₂×C₆₇	semidirect
321. #2417	C₂×C₆₇	semidirect
322. #2396	C₂×C₆₇	semidirect
323. C₄×C₂²⋊Q₈	C₂×C₆₇	semidirect
325. #2285	C₂×C₆₇	semidirect
326. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
327. C₆₇	#34947	direct (N × H)
383. C₂×#41	Q₈×C₆₇	semidirect
516. C₄⋊C₄	#41×C₆₇	semidirect
574. C₂×C₄	C₂²⋊Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₆₇	2
C₂×C₄×#41×C₆₇	2
C₆₇×#2568	2
C₆₇×#2567	2
C₆₇×#2566	2
C₆₇×#2565	2
C₆₇×#2564	2
C₆₇×#2563	2
C₆₇×#2542	2
C₆₇×#2417	2
C₆₇×#2396	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#2297	2
C₆₇×#2285	2
C₂×C₄×C₄⋊C₄×C₆₇	2
#34947	67

Subgroups — C67×#34947 #30714198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34947 #30714198