Subgroups — C₂×C₆₇×#2538 #30714177

Order 17152.

3614 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2538	C₂	1	16	yes	maximal, direct product factor
5	C₂×#396×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#390×C₆₇	C₂	1	2	yes	maximal
7	C₂×#389×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#380×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₆₇	—	2	16	no	—
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	16	yes	complemented
15	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
17	C₂×C₄×D₄×C₆₇	—	2	8	no	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
29	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
31	#396×C₆₇	C₂²	1	32	yes	complemented
32	#390×C₆₇	C₂²	1	16	yes	—
33	#389×C₆₇	C₂²	1	16	yes	complemented
34	#384×C₆₇	C₂²	1	8	yes	complemented
35	#380×C₆₇	C₂²	1	16	yes	complemented
36	C₂×#95×C₆₇	C₂²	1	4	yes	—
37	C₂⁵×C₆₇	C₂³	1	2	yes	—
38	C₂²×Q₈×C₆₇	—	2	4	no	—
39	C₂²×Q₈×C₆₇	—	2	4	no	—
40	C₂²×Q₈×C₆₇	D₄	1	8	yes	complemented
41	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
42	C₂²×D₄×C₆₇	—	2	4	no	—
43	C₂²×D₄×C₆₇	—	2	4	no	—
44	C₂²×D₄×C₆₇	Q₈	1	8	yes	complemented
45	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
46	C₂³×C₄×C₆₇	—	2	8	no	—
47	C₂³×C₄×C₆₇	—	2	8	no	—
48	C₂³×C₄×C₆₇	—	2	8	no	—
49	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
50	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂²⋊Q₈×C₆₇	—	2	64	no	—
58	C₄×Q₈×C₆₇	—	2	32	no	—
59	C₄×D₄×C₆₇	—	2	32	no	—
60	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
61	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	16	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
66	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
77	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
79	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
80	C₂×C₄²×C₆₇	—	2	8	no	—
81	C₂×C₄²×C₆₇	D₄	1	8	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
84	#95×C₆₇	C₂³	1	16	yes	—
85	C₂⁴×C₆₇	—	2	4	no	—
86	C₂⁴×C₆₇	—	2	4	no	—
87	C₂⁴×C₆₇	—	2	4	no	—
88	C₂⁴×C₆₇	—	2	4	no	—
89	C₂⁴×C₆₇	—	2	4	no	—
90	C₂⁴×C₆₇	—	2	4	no	—
91	C₂⁴×C₆₇	C₂×Q₈	1	8	yes	—
92	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
93	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₂×Q₈×C₆₇	—	2	32	no	—
95	C₂×Q₈×C₆₇	—	2	8	no	—
96	C₂×Q₈×C₆₇	—	2	8	no	—
97	C₂×Q₈×C₆₇	C₂×D₄	1	16	yes	complemented
98	C₂×D₄×C₆₇	—	2	32	no	—
99	C₂×D₄×C₆₇	—	2	8	no	—
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	C₂×Q₈	1	16	yes	complemented
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	4	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	4	no	—
107	C₂²×C₄×C₆₇	—	2	8	no	—
108	C₂²×C₄×C₆₇	—	2	4	no	—
109	C₂²×C₄×C₆₇	—	2	4	no	—
110	C₂²×C₄×C₆₇	—	2	8	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	2	8	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	16	no	—
116	C₂²×C₄×C₆₇	—	2	16	no	—
117	C₂²×C₄×C₆₇	—	2	16	no	—
118	C₂²×C₄×C₆₇	—	4	32	no	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
122	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
125	C₂²×C₄×C₆₇	#41	1	4	yes	—
126	C₂²×C₄×C₆₇	#41	1	4	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
129	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
133	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
134	C₄⋊C₄×C₆₇	—	2	8	no	—
135	C₄⋊C₄×C₆₇	—	2	16	no	—
136	C₄⋊C₄×C₆₇	—	2	8	no	—
137	C₄⋊C₄×C₆₇	—	2	16	no	—
138	C₄⋊C₄×C₆₇	—	2	32	no	—
139	C₄⋊C₄×C₆₇	—	2	16	no	—
140	C₄⋊C₄×C₆₇	—	4	8	no	—
141	C₂²⋊C₄×C₆₇	—	2	16	no	—
142	C₂²⋊C₄×C₆₇	—	2	32	no	—
143	C₂²⋊C₄×C₆₇	—	4	16	no	—
144	C₄²×C₆₇	—	2	16	no	—
145	C₄²×C₆₇	C₂×D₄	1	16	yes	—
146	C₂³×C₆₇	—	2	16	no	—
147	C₂³×C₆₇	—	2	16	no	—
148	C₂³×C₆₇	—	2	16	no	—
149	C₂³×C₆₇	—	2	4	no	—
150	C₂³×C₆₇	—	2	4	no	—
151	C₂³×C₆₇	—	2	4	no	—
152	C₂³×C₆₇	—	2	4	no	—
153	C₂³×C₆₇	—	2	4	no	—
154	C₂³×C₆₇	—	2	4	no	—
155	C₂³×C₆₇	—	4	8	no	—
156	C₂³×C₆₇	—	4	8	no	—
157	C₂³×C₆₇	—	4	8	no	—
158	C₂³×C₆₇	—	4	8	no	—
159	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
160	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
161	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
162	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
163	C₂³×C₆₇	C₂×#41	1	2	yes	—
164	C₂³×C₆₇	C₂×#41	1	2	yes	—
165	C₂³×C₆₇	H(5,𝔽₂)	1	2	yes	—
166	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
167	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
168	Q₈×C₆₇	—	2	32	no	—
169	D₄×C₆₇	—	2	32	no	—
170	C₂×C₄×C₆₇	—	2	16	no	—
171	C₂×C₄×C₆₇	—	2	8	no	—
172	C₂×C₄×C₆₇	—	2	4	no	—
173	C₂×C₄×C₆₇	—	2	4	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	4	no	—
177	C₂×C₄×C₆₇	—	2	8	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	4	no	—
180	C₂×C₄×C₆₇	—	2	8	no	—
181	C₂×C₄×C₆₇	—	2	4	no	—
182	C₂×C₄×C₆₇	—	2	4	no	—
183	C₂×C₄×C₆₇	—	2	8	no	—
184	C₂×C₄×C₆₇	—	2	4	no	—
185	C₂×C₄×C₆₇	—	4	8	no	—
186	C₂×C₄×C₆₇	—	4	8	no	—
187	C₂×C₄×C₆₇	—	4	16	no	—
188	C₂×C₄×C₆₇	—	4	32	no	—
189	C₂×C₄×C₆₇	#121	1	16	yes	—
190	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	complemented
191	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
192	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
193	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
196	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
197	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
198	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
199	C₂²×C₆₇	—	2	8	no	—
200	C₂²×C₆₇	—	2	8	no	—
201	C₂²×C₆₇	—	2	8	no	—
202	C₂²×C₆₇	—	4	4	no	—
203	C₂²×C₆₇	—	4	4	no	—
204	C₂²×C₆₇	—	4	4	no	—
205	C₂²×C₆₇	—	4	4	no	—
206	C₂²×C₆₇	—	4	32	no	—
207	C₂²×C₆₇	C₂×#121	1	4	yes	—
208	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
209	C₂²×C₆₇	#537	1	4	yes	—
210	C₂²×C₆₇	D₄×Q₈	1	4	yes	—
211	C₂²×C₆₇	#550	1	4	yes	—
212	C₂²×C₆₇	#554	1	4	yes	—
213	C₂²×C₆₇	#556	1	4	yes	—
214	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
215	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
216	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
217	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
218	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
221	C₄×C₆₇	—	2	8	no	—
222	C₄×C₆₇	—	2	4	no	—
223	C₄×C₆₇	—	4	4	no	—
224	C₄×C₆₇	—	4	4	no	—
225	C₄×C₆₇	—	4	8	no	—
226	C₄×C₆₇	C₂×#121	1	8	yes	—
227	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	complemented
228	C₂×#2538	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂×C₆₇	—	4	8	no	—
230	C₂×C₆₇	#2538	1	8	yes	direct product factor
231	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
232	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
234	C₂×C₆₇	C₂×D₄×Q₈	1	1	yes	characteristic
235	C₂×C₆₇	C₂×#550	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#554	1	1	yes	characteristic
237	C₂×C₆₇	C₂×#556	1	1	yes	characteristic
238	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	complemented
239	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
240	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
241	#2538	C₂×C₆₇	1	16	yes	direct product factor
242	C₂×#396	C₂×C₆₇	1	4	yes	complemented
243	C₂×#390	C₂×C₆₇	1	2	yes	—
244	C₂×#389	C₂×C₆₇	1	2	yes	complemented
245	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
246	C₂×#380	C₂×C₆₇	1	2	yes	complemented
247	C₆₇	C₂×#2538	1	1	yes	characteristic, direct product factor, Sylow
248	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
249	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
250	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
251	C₂×C₂²⋊Q₈	—	2	16	no	—
252	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	16	yes	complemented
253	C₂×C₄×Q₈	—	2	8	no	—
254	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
255	C₂×C₄×D₄	—	2	8	no	—
256	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
257	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
258	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
259	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
261	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
262	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
263	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
264	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
265	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
266	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
267	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
268	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
269	#396	C₂²×C₆₇	1	32	yes	complemented
270	#390	C₂²×C₆₇	1	16	yes	—
271	#389	C₂²×C₆₇	1	16	yes	complemented
272	#384	C₂²×C₆₇	1	8	yes	complemented
273	#380	C₂²×C₆₇	1	16	yes	complemented
274	C₂×#95	C₂²×C₆₇	1	4	yes	—
275	C₂⁵	C₂³×C₆₇	1	2	yes	—
276	C₂²×Q₈	—	2	4	no	—
277	C₂²×Q₈	—	2	4	no	—
278	C₂²×Q₈	D₄×C₆₇	1	8	yes	complemented
279	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
280	C₂²×D₄	—	2	4	no	—
281	C₂²×D₄	—	2	4	no	—
282	C₂²×D₄	Q₈×C₆₇	1	8	yes	complemented
283	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
284	C₂³×C₄	—	2	8	no	—
285	C₂³×C₄	—	2	8	no	—
286	C₂³×C₄	—	2	8	no	—
287	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
288	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
289	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
290	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
291	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
292	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
293	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
295	C₂²⋊Q₈	—	2	64	no	—
296	C₄×Q₈	—	2	32	no	—
297	C₄×D₄	—	2	32	no	—
298	C₂×C₄⋊C₄	—	2	4	no	—
299	C₂×C₄⋊C₄	—	2	8	no	—
300	C₂×C₄⋊C₄	—	2	4	no	—
301	C₂×C₄⋊C₄	—	2	8	no	—
302	C₂×C₄⋊C₄	—	2	16	no	—
303	C₂×C₄⋊C₄	—	2	8	no	—
304	C₂×C₄⋊C₄	—	4	4	no	—
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
308	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
309	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
310	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
311	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
312	C₂×C₂²⋊C₄	—	2	8	no	—
313	C₂×C₂²⋊C₄	—	2	16	no	—
314	C₂×C₂²⋊C₄	—	4	8	no	—
315	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
316	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
317	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
318	C₂×C₄²	—	2	8	no	—
319	C₂×C₄²	D₄×C₆₇	1	8	yes	—
320	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
321	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
322	#95	C₂³×C₆₇	1	16	yes	—
323	C₂⁴	—	2	4	no	—
324	C₂⁴	—	2	4	no	—
325	C₂⁴	—	2	4	no	—
326	C₂⁴	—	2	4	no	—
327	C₂⁴	—	2	4	no	—
328	C₂⁴	—	2	4	no	—
329	C₂⁴	C₂×Q₈×C₆₇	1	8	yes	—
330	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
331	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
332	C₂×Q₈	—	2	32	no	—
333	C₂×Q₈	—	2	8	no	—
334	C₂×Q₈	—	2	8	no	—
335	C₂×Q₈	C₂×D₄×C₆₇	1	16	yes	complemented
336	C₂×D₄	—	2	32	no	—
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	C₂×Q₈×C₆₇	1	16	yes	complemented
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	8	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	16	no	—
354	C₂²×C₄	—	2	16	no	—
355	C₂²×C₄	—	2	16	no	—
356	C₂²×C₄	—	4	32	no	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
358	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
359	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
360	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
361	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
362	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
363	C₂²×C₄	#41×C₆₇	1	4	yes	—
364	C₂²×C₄	#41×C₆₇	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
367	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
368	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
369	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
370	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
372	C₄⋊C₄	—	2	8	no	—
373	C₄⋊C₄	—	2	16	no	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	16	no	—
376	C₄⋊C₄	—	2	32	no	—
377	C₄⋊C₄	—	2	16	no	—
378	C₄⋊C₄	—	4	8	no	—
379	C₂²⋊C₄	—	2	16	no	—
380	C₂²⋊C₄	—	2	32	no	—
381	C₂²⋊C₄	—	4	16	no	—
382	C₄²	—	2	16	no	—
383	C₄²	C₂×D₄×C₆₇	1	16	yes	—
384	C₂³	—	2	16	no	—
385	C₂³	—	2	16	no	—
386	C₂³	—	2	16	no	—
387	C₂³	—	2	4	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	4	no	—
391	C₂³	—	2	4	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	4	8	no	—
394	C₂³	—	4	8	no	—
395	C₂³	—	4	8	no	—
396	C₂³	—	4	8	no	—
397	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
398	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
399	C₂³	C₂²×Q₈×C₆₇	1	8	yes	—
400	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
401	C₂³	C₂×#41×C₆₇	1	2	yes	—
402	C₂³	C₂×#41×C₆₇	1	2	yes	—
403	C₂³	H(5,𝔽₂)×C₆₇	1	2	yes	—
404	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
405	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
406	Q₈	—	2	32	no	—
407	D₄	—	2	32	no	—
408	C₂×C₄	—	2	16	no	—
409	C₂×C₄	—	2	8	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	8	no	—
422	C₂×C₄	—	2	4	no	—
423	C₂×C₄	—	4	8	no	—
424	C₂×C₄	—	4	8	no	—
425	C₂×C₄	—	4	16	no	—
426	C₂×C₄	—	4	32	no	—
427	C₂×C₄	#121×C₆₇	1	16	yes	—
428	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	complemented
429	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
430	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
431	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
432	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
433	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
434	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
435	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
436	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
437	C₂²	—	2	8	no	—
438	C₂²	—	2	8	no	—
439	C₂²	—	2	8	no	—
440	C₂²	—	4	32	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	4	no	—
445	C₂²	C₂×#121×C₆₇	1	4	yes	—
446	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
447	C₂²	#537×C₆₇	1	4	yes	—
448	C₂²	D₄×Q₈×C₆₇	1	4	yes	—
449	C₂²	#550×C₆₇	1	4	yes	—
450	C₂²	#554×C₆₇	1	4	yes	—
451	C₂²	#556×C₆₇	1	4	yes	—
452	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
453	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
454	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
455	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
456	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
457	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
458	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
459	C₄	—	2	8	no	—
460	C₄	—	2	4	no	—
461	C₄	—	4	4	no	—
462	C₄	—	4	4	no	—
463	C₄	—	4	8	no	—
464	C₄	C₂×#121×C₆₇	1	8	yes	—
465	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	complemented
466	C₂	—	4	8	no	—
467	C₂	C₆₇×#2538	1	8	yes	direct product factor
468	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
469	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
470	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
471	C₂	C₂×D₄×Q₈×C₆₇	1	1	yes	characteristic
472	C₂	C₂×#550×C₆₇	1	1	yes	characteristic
473	C₂	C₂×#554×C₆₇	1	1	yes	characteristic
474	C₂	C₂×#556×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
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×C₂²⋊Q₈×C₆₇	C₂	semidirect
2. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
4. C₆₇×#2538	C₂	direct (N × H)
5. C₂×#396×C₆₇	C₂	semidirect
7. C₂×#389×C₆₇	C₂	semidirect
8. C₂×#384×C₆₇	C₂	semidirect
9. C₂×#380×C₆₇	C₂	semidirect
14. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
16. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
31. #396×C₆₇	C₂²	semidirect
33. #389×C₆₇	C₂²	semidirect
34. #384×C₆₇	C₂²	semidirect
35. #380×C₆₇	C₂²	semidirect
40. C₂²×Q₈×C₆₇	D₄	semidirect
44. C₂²×D₄×C₆₇	Q₈	semidirect
97. C₂×Q₈×C₆₇	C₂×D₄	semidirect
101. C₂×D₄×C₆₇	C₂×Q₈	semidirect
190. C₂×C₄×C₆₇	C₂²⋊Q₈	semidirect
227. C₄×C₆₇	C₂×C₂²⋊Q₈	semidirect
228. C₂×#2538	C₆₇	direct (N × H)
230. C₂×C₆₇	#2538	direct (N × H)
238. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
239. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
241. #2538	C₂×C₆₇	direct (N × H)
242. C₂×#396	C₂×C₆₇	semidirect
244. C₂×#389	C₂×C₆₇	semidirect
245. C₂×#384	C₂×C₆₇	semidirect
246. C₂×#380	C₂×C₆₇	semidirect
247. C₆₇	C₂×#2538	direct (N × H)
252. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
254. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
269. #396	C₂²×C₆₇	semidirect
271. #389	C₂²×C₆₇	semidirect
272. #384	C₂²×C₆₇	semidirect
273. #380	C₂²×C₆₇	semidirect
278. C₂²×Q₈	D₄×C₆₇	semidirect
282. C₂²×D₄	Q₈×C₆₇	semidirect
335. C₂×Q₈	C₂×D₄×C₆₇	semidirect
339. C₂×D₄	C₂×Q₈×C₆₇	semidirect
428. C₂×C₄	C₂²⋊Q₈×C₆₇	semidirect
465. C₄	C₂×C₂²⋊Q₈×C₆₇	semidirect
467. C₂	C₆₇×#2538	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×C₂²⋊Q₈×C₆₇	2
C₂²×C₄×Q₈×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2538	2
C₂×#396×C₆₇	2
C₂×#390×C₆₇	2
C₂×#389×C₆₇	2
C₂×#384×C₆₇	2
C₂×#380×C₆₇	2
C₂×#2538	67

Subgroups — C2×C67×#2538 #30714177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2538 #30714177