Subgroups — C₂×C₆₇×#2427 #30713948

Order 17152.

3894 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#121×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2427	C₂	1	16	yes	maximal, direct product factor
6	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#394×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#389×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#386×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×#380×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
15	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
18	C₂×#124×C₆₇	—	2	8	no	—
19	C₂×#124×C₆₇	C₂²	1	8	yes	complemented
20	C₂×C₂²⋊Q₈×C₆₇	—	2	8	no	—
21	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
22	C₂×#121×C₆₇	—	2	8	no	—
23	C₂×#121×C₆₇	C₂²	1	8	yes	—
24	C₂×C₄×D₄×C₆₇	—	2	8	no	—
25	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	complemented
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
33	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
36	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
37	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	#397×C₆₇	C₂²	1	16	yes	complemented
40	#394×C₆₇	C₂²	1	16	yes	complemented
41	#391×C₆₇	C₂²	1	8	yes	complemented
42	#389×C₆₇	C₂²	1	8	yes	complemented
43	#386×C₆₇	C₂²	1	16	yes	complemented
44	#380×C₆₇	C₂²	1	16	yes	complemented
45	C₄×C₂²⋊C₄×C₆₇	C₂²	1	8	yes	complemented
46	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂×#95×C₆₇	C₂²	1	2	yes	—
50	C₂×#95×C₆₇	C₂²	1	2	yes	—
51	C₂⁵×C₆₇	C₂³	1	2	yes	—
52	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×Q₈×C₆₇	—	2	4	no	—
54	C₂²×Q₈×C₆₇	—	2	4	no	—
55	C₂²×Q₈×C₆₇	—	2	4	no	—
56	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
57	C₂²×D₄×C₆₇	—	2	8	no	—
58	C₂²×D₄×C₆₇	—	2	8	no	—
59	C₂²×D₄×C₆₇	—	2	4	no	—
60	C₂²×D₄×C₆₇	—	2	4	no	—
61	C₂²×D₄×C₆₇	—	4	4	no	—
62	C₂²×D₄×C₆₇	D₄	1	8	yes	—
63	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
64	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
65	C₂³×C₄×C₆₇	—	2	8	no	—
66	C₂³×C₄×C₆₇	—	2	4	no	—
67	C₂³×C₄×C₆₇	—	2	4	no	—
68	C₂³×C₄×C₆₇	—	2	8	no	—
69	C₂³×C₄×C₆₇	—	2	4	no	—
70	C₂³×C₄×C₆₇	—	4	4	no	—
71	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
78	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
79	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
80	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
82	#124×C₆₇	—	2	32	no	—
83	C₂²⋊Q₈×C₆₇	—	2	32	no	—
84	#121×C₆₇	—	2	32	no	—
85	C₄×D₄×C₆₇	—	2	32	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
88	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
89	C₂×C₄⋊C₄×C₆₇	D₄	1	8	yes	complemented
90	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
100	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
101	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
103	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
104	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
105	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
106	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
108	C₂×C₄²×C₆₇	—	2	4	no	—
109	C₂×C₄²×C₆₇	—	2	4	no	—
110	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
111	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
112	#95×C₆₇	C₂³	1	4	yes	—
113	#95×C₆₇	C₂³	1	4	yes	—
114	#95×C₆₇	C₂³	1	8	yes	—
115	#95×C₆₇	C₂³	1	8	yes	—
116	#95×C₆₇	C₂³	1	8	yes	—
117	C₂⁴×C₆₇	—	2	2	no	—
118	C₂⁴×C₆₇	—	2	4	no	—
119	C₂⁴×C₆₇	—	2	2	no	—
120	C₂⁴×C₆₇	—	2	2	no	—
121	C₂⁴×C₆₇	—	2	2	no	—
122	C₂⁴×C₆₇	—	2	4	no	—
123	C₂⁴×C₆₇	—	2	4	no	—
124	C₂⁴×C₆₇	—	2	4	no	—
125	C₂⁴×C₆₇	—	2	4	no	—
126	C₂⁴×C₆₇	—	2	2	no	—
127	C₂⁴×C₆₇	—	2	4	no	—
128	C₂⁴×C₆₇	—	2	2	no	—
129	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
130	C₂⁴×C₆₇	#41	1	4	yes	—
131	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
132	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
133	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
134	C₂×Q₈×C₆₇	—	2	8	no	—
135	C₂×Q₈×C₆₇	—	2	8	no	—
136	C₂×Q₈×C₆₇	—	2	8	no	—
137	C₂×Q₈×C₆₇	—	4	16	no	—
138	C₂×D₄×C₆₇	—	2	16	no	—
139	C₂×D₄×C₆₇	—	2	16	no	—
140	C₂×D₄×C₆₇	—	2	32	no	—
141	C₂×D₄×C₆₇	—	2	8	no	—
142	C₂×D₄×C₆₇	—	2	8	no	—
143	C₂×D₄×C₆₇	—	4	8	no	—
144	C₂×D₄×C₆₇	—	4	32	no	—
145	C₂×D₄×C₆₇	C₂×D₄	1	16	yes	—
146	C₂²×C₄×C₆₇	—	2	2	no	—
147	C₂²×C₄×C₆₇	—	2	4	no	—
148	C₂²×C₄×C₆₇	—	2	4	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	2	no	—
153	C₂²×C₄×C₆₇	—	2	2	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	4	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	2	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	2	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	2	no	—
165	C₂²×C₄×C₆₇	—	2	4	no	—
166	C₂²×C₄×C₆₇	—	2	2	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	16	no	—
169	C₂²×C₄×C₆₇	—	2	8	no	—
170	C₂²×C₄×C₆₇	—	2	8	no	—
171	C₂²×C₄×C₆₇	—	2	16	no	—
172	C₂²×C₄×C₆₇	—	2	8	no	—
173	C₂²×C₄×C₆₇	—	4	8	no	—
174	C₂²×C₄×C₆₇	—	4	32	no	—
175	C₂²×C₄×C₆₇	—	4	16	no	—
176	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
177	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
178	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
179	C₂²×C₄×C₆₇	#41	1	4	yes	—
180	C₂²×C₄×C₆₇	#41	1	4	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	4	yes	—
185	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
187	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
190	C₄⋊C₄×C₆₇	—	2	8	no	—
191	C₄⋊C₄×C₆₇	—	2	16	no	—
192	C₄⋊C₄×C₆₇	—	4	8	no	—
193	C₄⋊C₄×C₆₇	C₂×D₄	1	16	yes	complemented
194	C₂²⋊C₄×C₆₇	—	2	16	no	—
195	C₂²⋊C₄×C₆₇	—	2	16	no	—
196	C₂²⋊C₄×C₆₇	—	2	16	no	—
197	C₂²⋊C₄×C₆₇	—	2	8	no	—
198	C₂²⋊C₄×C₆₇	—	2	16	no	—
199	C₂²⋊C₄×C₆₇	—	2	16	no	—
200	C₂²⋊C₄×C₆₇	—	2	8	no	—
201	C₄²×C₆₇	—	2	8	no	—
202	C₄²×C₆₇	—	2	8	no	—
203	C₂³×C₆₇	—	2	16	no	—
204	C₂³×C₆₇	—	2	8	no	—
205	C₂³×C₆₇	—	2	8	no	—
206	C₂³×C₆₇	—	2	16	no	—
207	C₂³×C₆₇	—	2	16	no	—
208	C₂³×C₆₇	—	2	8	no	—
209	C₂³×C₆₇	—	2	2	no	—
210	C₂³×C₆₇	—	2	4	no	—
211	C₂³×C₆₇	—	2	2	no	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	2	4	no	—
215	C₂³×C₆₇	—	2	4	no	—
216	C₂³×C₆₇	—	2	4	no	—
217	C₂³×C₆₇	—	2	4	no	—
218	C₂³×C₆₇	—	2	2	no	—
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	—	2	2	no	—
221	C₂³×C₆₇	—	4	4	no	—
222	C₂³×C₆₇	—	4	8	no	—
223	C₂³×C₆₇	—	4	4	no	—
224	C₂³×C₆₇	—	4	8	no	—
225	C₂³×C₆₇	—	4	8	no	—
226	C₂³×C₆₇	—	4	4	no	—
227	C₂³×C₆₇	—	4	4	no	—
228	C₂³×C₆₇	—	4	8	no	—
229	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
230	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
231	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
232	C₂³×C₆₇	C₂×#41	1	4	yes	—
233	C₂³×C₆₇	C₂×#41	1	2	yes	—
234	C₂³×C₆₇	C₂×#41	1	2	yes	—
235	C₂³×C₆₇	C₂×#41	1	2	yes	—
236	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
237	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
238	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
239	Q₈×C₆₇	—	4	16	no	—
240	D₄×C₆₇	—	2	32	no	—
241	D₄×C₆₇	—	4	32	no	—
242	C₂×C₄×C₆₇	—	2	8	no	—
243	C₂×C₄×C₆₇	—	2	16	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	2	4	no	—
246	C₂×C₄×C₆₇	—	2	4	no	—
247	C₂×C₄×C₆₇	—	2	2	no	—
248	C₂×C₄×C₆₇	—	2	2	no	—
249	C₂×C₄×C₆₇	—	2	2	no	—
250	C₂×C₄×C₆₇	—	2	2	no	—
251	C₂×C₄×C₆₇	—	2	2	no	—
252	C₂×C₄×C₆₇	—	2	4	no	—
253	C₂×C₄×C₆₇	—	2	4	no	—
254	C₂×C₄×C₆₇	—	2	4	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	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	2	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	2	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	4	32	no	—
267	C₂×C₄×C₆₇	—	4	4	no	—
268	C₂×C₄×C₆₇	—	4	8	no	—
269	C₂×C₄×C₆₇	—	4	4	no	—
270	C₂×C₄×C₆₇	—	4	8	no	—
271	C₂×C₄×C₆₇	—	4	4	no	—
272	C₂×C₄×C₆₇	—	4	4	no	—
273	C₂×C₄×C₆₇	—	4	16	no	—
274	C₂×C₄×C₆₇	#121	1	16	yes	—
275	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
276	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
277	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
278	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
279	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
280	C₂²×C₆₇	—	2	8	no	—
281	C₂²×C₆₇	—	2	4	no	—
282	C₂²×C₆₇	—	2	4	no	—
283	C₂²×C₆₇	—	2	8	no	—
284	C₂²×C₆₇	—	2	8	no	—
285	C₂²×C₆₇	—	2	4	no	—
286	C₂²×C₆₇	—	4	2	no	—
287	C₂²×C₆₇	—	4	4	no	—
288	C₂²×C₆₇	—	4	32	no	—
289	C₂²×C₆₇	—	4	16	no	—
290	C₂²×C₆₇	—	4	2	no	—
291	C₂²×C₆₇	—	4	4	no	—
292	C₂²×C₆₇	—	4	4	no	—
293	C₂²×C₆₇	—	4	2	no	—
294	C₂²×C₆₇	—	4	2	no	—
295	C₂²×C₆₇	—	4	4	no	—
296	C₂²×C₆₇	C₂×#121	1	4	yes	—
297	C₂²×C₆₇	#529	1	4	yes	—
298	C₂²×C₆₇	#542	1	4	yes	—
299	C₂²×C₆₇	#546	1	4	yes	—
300	C₂²×C₆₇	#546	1	4	yes	—
301	C₂²×C₆₇	#547	1	4	yes	—
302	C₂²×C₆₇	#548	1	4	yes	—
303	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
304	C₂²×C₆₇	C₂³×D₄	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	C₂×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
310	C₄×C₆₇	—	2	4	no	—
311	C₄×C₆₇	—	2	8	no	—
312	C₄×C₆₇	—	4	2	no	—
313	C₄×C₆₇	—	4	4	no	—
314	C₄×C₆₇	—	4	2	no	—
315	C₄×C₆₇	—	4	4	no	—
316	C₄×C₆₇	—	4	2	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	C₂×#121	1	8	yes	—
319	C₂×#2427	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
320	C₂×C₆₇	—	4	8	no	—
321	C₂×C₆₇	—	4	4	no	—
322	C₂×C₆₇	#2427	1	8	yes	direct product factor
323	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
324	C₂×C₆₇	C₂×#529	1	1	yes	characteristic
325	C₂×C₆₇	C₂×#542	1	1	yes	characteristic
326	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
327	C₂×C₆₇	C₂×#546	1	1	yes	characteristic
328	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
329	C₂×C₆₇	C₂×#548	1	1	yes	characteristic
330	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
331	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
332	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
333	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
334	#2427	C₂×C₆₇	1	16	yes	direct product factor
335	C₂×#397	C₂×C₆₇	1	2	yes	complemented
336	C₂×#394	C₂×C₆₇	1	2	yes	complemented
337	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
338	C₂×#389	C₂×C₆₇	1	1	yes	characteristic, complemented
339	C₂×#386	C₂×C₆₇	1	2	yes	complemented
340	C₂×#380	C₂×C₆₇	1	2	yes	complemented
341	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
342	C₆₇	C₂×#2427	1	1	yes	characteristic, direct product factor, Sylow
343	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
344	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
345	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
346	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
347	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
348	C₂×#124	—	2	8	no	—
349	C₂×#124	C₂²×C₆₇	1	8	yes	complemented
350	C₂×C₂²⋊Q₈	—	2	8	no	—
351	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
352	C₂×#121	—	2	8	no	—
353	C₂×#121	C₂²×C₆₇	1	8	yes	—
354	C₂×C₄×D₄	—	2	8	no	—
355	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	complemented
356	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
357	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
358	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
359	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
360	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
361	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
362	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
363	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
364	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
365	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
366	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
367	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
368	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
369	#397	C₂²×C₆₇	1	16	yes	complemented
370	#394	C₂²×C₆₇	1	16	yes	complemented
371	#391	C₂²×C₆₇	1	8	yes	complemented
372	#389	C₂²×C₆₇	1	8	yes	complemented
373	#386	C₂²×C₆₇	1	16	yes	complemented
374	#380	C₂²×C₆₇	1	16	yes	complemented
375	C₄×C₂²⋊C₄	C₂²×C₆₇	1	8	yes	complemented
376	C₂×#95	C₂²×C₆₇	1	2	yes	—
377	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
378	C₂×#95	C₂²×C₆₇	1	2	yes	—
379	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
380	C₂×#95	C₂²×C₆₇	1	2	yes	—
381	C₂⁵	C₂³×C₆₇	1	2	yes	—
382	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
383	C₂²×Q₈	—	2	4	no	—
384	C₂²×Q₈	—	2	4	no	—
385	C₂²×Q₈	—	2	4	no	—
386	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
387	C₂²×D₄	—	2	8	no	—
388	C₂²×D₄	—	2	8	no	—
389	C₂²×D₄	—	2	4	no	—
390	C₂²×D₄	—	2	4	no	—
391	C₂²×D₄	—	4	4	no	—
392	C₂²×D₄	D₄×C₆₇	1	8	yes	—
393	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
394	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
395	C₂³×C₄	—	2	8	no	—
396	C₂³×C₄	—	2	4	no	—
397	C₂³×C₄	—	2	4	no	—
398	C₂³×C₄	—	2	8	no	—
399	C₂³×C₄	—	2	4	no	—
400	C₂³×C₄	—	4	4	no	—
401	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
402	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
403	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
404	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
405	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
406	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
407	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
408	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
409	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
410	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
411	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
412	#124	—	2	32	no	—
413	C₂²⋊Q₈	—	2	32	no	—
414	#121	—	2	32	no	—
415	C₄×D₄	—	2	32	no	—
416	C₂×C₄⋊C₄	—	2	4	no	—
417	C₂×C₄⋊C₄	—	2	8	no	—
418	C₂×C₄⋊C₄	—	4	4	no	—
419	C₂×C₄⋊C₄	D₄×C₆₇	1	8	yes	complemented
420	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
422	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
423	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
424	C₂×C₂²⋊C₄	—	2	8	no	—
425	C₂×C₂²⋊C₄	—	2	8	no	—
426	C₂×C₂²⋊C₄	—	2	8	no	—
427	C₂×C₂²⋊C₄	—	2	4	no	—
428	C₂×C₂²⋊C₄	—	2	8	no	—
429	C₂×C₂²⋊C₄	—	2	8	no	—
430	C₂×C₂²⋊C₄	—	2	4	no	—
431	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
434	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
435	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
436	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
437	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
438	C₂×C₄²	—	2	4	no	—
439	C₂×C₄²	—	2	4	no	—
440	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
441	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
442	#95	C₂³×C₆₇	1	4	yes	—
443	#95	C₂³×C₆₇	1	4	yes	—
444	#95	C₂³×C₆₇	1	8	yes	—
445	#95	C₂³×C₆₇	1	8	yes	—
446	#95	C₂³×C₆₇	1	8	yes	—
447	C₂⁴	—	2	2	no	—
448	C₂⁴	—	2	4	no	—
449	C₂⁴	—	2	2	no	—
450	C₂⁴	—	2	2	no	—
451	C₂⁴	—	2	2	no	—
452	C₂⁴	—	2	4	no	—
453	C₂⁴	—	2	4	no	—
454	C₂⁴	—	2	4	no	—
455	C₂⁴	—	2	4	no	—
456	C₂⁴	—	2	2	no	—
457	C₂⁴	—	2	4	no	—
458	C₂⁴	—	2	2	no	—
459	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
460	C₂⁴	#41×C₆₇	1	4	yes	—
461	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
462	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
463	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
464	C₂×Q₈	—	2	8	no	—
465	C₂×Q₈	—	2	8	no	—
466	C₂×Q₈	—	2	8	no	—
467	C₂×Q₈	—	4	16	no	—
468	C₂×D₄	—	2	16	no	—
469	C₂×D₄	—	2	16	no	—
470	C₂×D₄	—	2	32	no	—
471	C₂×D₄	—	2	8	no	—
472	C₂×D₄	—	2	8	no	—
473	C₂×D₄	—	4	8	no	—
474	C₂×D₄	—	4	32	no	—
475	C₂×D₄	C₂×D₄×C₆₇	1	16	yes	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	4	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	4	no	—
486	C₂²×C₄	—	2	4	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	2	2	no	—
495	C₂²×C₄	—	2	4	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	16	no	—
499	C₂²×C₄	—	2	8	no	—
500	C₂²×C₄	—	2	8	no	—
501	C₂²×C₄	—	2	16	no	—
502	C₂²×C₄	—	2	8	no	—
503	C₂²×C₄	—	4	8	no	—
504	C₂²×C₄	—	4	32	no	—
505	C₂²×C₄	—	4	16	no	—
506	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
507	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
509	C₂²×C₄	#41×C₆₇	1	4	yes	—
510	C₂²×C₄	#41×C₆₇	1	4	yes	—
511	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
514	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
515	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
516	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
517	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
518	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
519	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
520	C₄⋊C₄	—	2	8	no	—
521	C₄⋊C₄	—	2	16	no	—
522	C₄⋊C₄	—	4	8	no	—
523	C₄⋊C₄	C₂×D₄×C₆₇	1	16	yes	complemented
524	C₂²⋊C₄	—	2	16	no	—
525	C₂²⋊C₄	—	2	16	no	—
526	C₂²⋊C₄	—	2	16	no	—
527	C₂²⋊C₄	—	2	8	no	—
528	C₂²⋊C₄	—	2	16	no	—
529	C₂²⋊C₄	—	2	16	no	—
530	C₂²⋊C₄	—	2	8	no	—
531	C₄²	—	2	8	no	—
532	C₄²	—	2	8	no	—
533	C₂³	—	2	16	no	—
534	C₂³	—	2	8	no	—
535	C₂³	—	2	8	no	—
536	C₂³	—	2	16	no	—
537	C₂³	—	2	16	no	—
538	C₂³	—	2	8	no	—
539	C₂³	—	2	2	no	—
540	C₂³	—	2	4	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	4	no	—
545	C₂³	—	2	4	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	2	4	no	—
548	C₂³	—	2	2	no	—
549	C₂³	—	2	4	no	—
550	C₂³	—	2	2	no	—
551	C₂³	—	4	4	no	—
552	C₂³	—	4	8	no	—
553	C₂³	—	4	4	no	—
554	C₂³	—	4	8	no	—
555	C₂³	—	4	8	no	—
556	C₂³	—	4	4	no	—
557	C₂³	—	4	4	no	—
558	C₂³	—	4	8	no	—
559	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
560	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
561	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
562	C₂³	C₂×#41×C₆₇	1	4	yes	—
563	C₂³	C₂×#41×C₆₇	1	2	yes	—
564	C₂³	C₂×#41×C₆₇	1	2	yes	—
565	C₂³	C₂×#41×C₆₇	1	2	yes	—
566	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
567	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
568	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
569	Q₈	—	4	16	no	—
570	D₄	—	2	32	no	—
571	D₄	—	4	32	no	—
572	C₂×C₄	—	2	8	no	—
573	C₂×C₄	—	2	16	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	4	no	—
591	C₂×C₄	—	2	4	no	—
592	C₂×C₄	—	2	2	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	4	32	no	—
597	C₂×C₄	—	4	4	no	—
598	C₂×C₄	—	4	8	no	—
599	C₂×C₄	—	4	8	no	—
600	C₂×C₄	—	4	4	no	—
601	C₂×C₄	—	4	4	no	—
602	C₂×C₄	—	4	4	no	—
603	C₂×C₄	—	4	16	no	—
604	C₂×C₄	#121×C₆₇	1	16	yes	—
605	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
606	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
607	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
608	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
609	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
610	C₂²	—	2	8	no	—
611	C₂²	—	2	4	no	—
612	C₂²	—	2	4	no	—
613	C₂²	—	2	8	no	—
614	C₂²	—	2	8	no	—
615	C₂²	—	2	4	no	—
616	C₂²	—	4	2	no	—
617	C₂²	—	4	4	no	—
618	C₂²	—	4	32	no	—
619	C₂²	—	4	16	no	—
620	C₂²	—	4	2	no	—
621	C₂²	—	4	4	no	—
622	C₂²	—	4	4	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	4	no	—
626	C₂²	C₂×#121×C₆₇	1	4	yes	—
627	C₂²	#529×C₆₇	1	4	yes	—
628	C₂²	#542×C₆₇	1	4	yes	—
629	C₂²	#546×C₆₇	1	4	yes	—
630	C₂²	#546×C₆₇	1	4	yes	—
631	C₂²	#547×C₆₇	1	4	yes	—
632	C₂²	#548×C₆₇	1	4	yes	—
633	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
634	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
635	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
636	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
637	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
638	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
639	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
640	C₄	—	2	4	no	—
641	C₄	—	2	8	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	4	no	—
644	C₄	—	4	4	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	2	no	—
648	C₄	C₂×#121×C₆₇	1	8	yes	—
649	C₂	—	4	8	no	—
650	C₂	—	4	4	no	—
651	C₂	C₆₇×#2427	1	8	yes	direct product factor
652	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
653	C₂	C₂×#529×C₆₇	1	1	yes	characteristic
654	C₂	C₂×#542×C₆₇	1	1	yes	characteristic
655	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
656	C₂	C₂×#546×C₆₇	1	1	yes	characteristic
657	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
658	C₂	C₂×#548×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×#121×C₆₇	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#124×C₆₇	C₂	semidirect
2. C₂²×C₂²⋊Q₈×C₆₇	C₂	semidirect
4. C₂²×C₄×D₄×C₆₇	C₂	semidirect
5. C₆₇×#2427	C₂	direct (N × H)
6. C₂×#397×C₆₇	C₂	semidirect
7. C₂×#394×C₆₇	C₂	semidirect
8. C₂×#391×C₆₇	C₂	semidirect
9. C₂×#389×C₆₇	C₂	semidirect
10. C₂×#386×C₆₇	C₂	semidirect
11. C₂×#380×C₆₇	C₂	semidirect
12. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
19. C₂×#124×C₆₇	C₂²	semidirect
21. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
25. C₂×C₄×D₄×C₆₇	C₂²	semidirect
39. #397×C₆₇	C₂²	semidirect
40. #394×C₆₇	C₂²	semidirect
41. #391×C₆₇	C₂²	semidirect
42. #389×C₆₇	C₂²	semidirect
43. #386×C₆₇	C₂²	semidirect
44. #380×C₆₇	C₂²	semidirect
45. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
89. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
193. C₄⋊C₄×C₆₇	C₂×D₄	semidirect
319. C₂×#2427	C₆₇	direct (N × H)
322. C₂×C₆₇	#2427	direct (N × H)
330. C₂²×#124	C₂×C₆₇	semidirect
331. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
333. C₂²×C₄×D₄	C₂×C₆₇	semidirect
334. #2427	C₂×C₆₇	direct (N × H)
335. C₂×#397	C₂×C₆₇	semidirect
336. C₂×#394	C₂×C₆₇	semidirect
337. C₂×#391	C₂×C₆₇	semidirect
338. C₂×#389	C₂×C₆₇	semidirect
339. C₂×#386	C₂×C₆₇	semidirect
340. C₂×#380	C₂×C₆₇	semidirect
341. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
342. C₆₇	C₂×#2427	direct (N × H)
349. C₂×#124	C₂²×C₆₇	semidirect
351. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
355. C₂×C₄×D₄	C₂²×C₆₇	semidirect
369. #397	C₂²×C₆₇	semidirect
370. #394	C₂²×C₆₇	semidirect
371. #391	C₂²×C₆₇	semidirect
372. #389	C₂²×C₆₇	semidirect
373. #386	C₂²×C₆₇	semidirect
374. #380	C₂²×C₆₇	semidirect
375. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
419. C₂×C₄⋊C₄	D₄×C₆₇	semidirect
523. C₄⋊C₄	C₂×D₄×C₆₇	semidirect
651. C₂	C₆₇×#2427	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#124×C₆₇	2
C₂²×C₂²⋊Q₈×C₆₇	2
C₂²×#121×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2427	2
C₂×#397×C₆₇	2
C₂×#394×C₆₇	2
C₂×#391×C₆₇	2
C₂×#389×C₆₇	2
C₂×#386×C₆₇	2
C₂×#380×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2427	67

Subgroups — C2×C67×#2427 #30713948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2427 #30713948