Subgroups — C₂×C₆₇×#2620 #30713862

Order 17152.

3566 conjugacy classes of subgroups.

The subgroup lattice has 508 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×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2620	C₂	1	16	yes	maximal, direct product factor
6	C₂×#396×C₆₇	C₂	1	4	yes	maximal, complemented
7	C₂×#394×C₆₇	C₂	1	4	yes	maximal, complemented
8	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#390×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₆₇	C₂²	1	2	yes	—
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	—	2	8	no	—
16	C₂×#124×C₆₇	C₂²	1	8	yes	complemented
17	C₂×C₂²⋊Q₈×C₆₇	—	2	8	no	—
18	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	8	yes	complemented
19	C₂×#121×C₆₇	—	2	8	no	—
20	C₂×#121×C₆₇	C₂²	1	8	yes	—
21	C₂×#117×C₆₇	—	2	8	no	—
22	C₂×#117×C₆₇	C₂²	1	8	yes	complemented
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
31	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
35	#396×C₆₇	C₂²	1	32	yes	complemented
36	#394×C₆₇	C₂²	1	32	yes	complemented
37	#391×C₆₇	C₂²	1	8	yes	complemented
38	#390×C₆₇	C₂²	1	8	yes	complemented
39	#384×C₆₇	C₂²	1	8	yes	complemented
40	C₂×#95×C₆₇	C₂²	1	4	yes	—
41	C₂⁵×C₆₇	C₂³	1	2	yes	—
42	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×Q₈×C₆₇	—	2	4	no	—
44	C₂²×Q₈×C₆₇	—	2	4	no	—
45	C₂²×Q₈×C₆₇	—	2	4	no	—
46	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
47	C₂²×D₄×C₆₇	—	2	8	no	—
48	C₂²×D₄×C₆₇	—	2	4	no	—
49	C₂²×D₄×C₆₇	—	2	4	no	—
50	C₂²×D₄×C₆₇	—	2	4	no	—
51	C₂²×D₄×C₆₇	—	4	8	no	—
52	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
53	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
54	C₂³×C₄×C₆₇	—	2	4	no	—
55	C₂³×C₄×C₆₇	—	2	4	no	—
56	C₂³×C₄×C₆₇	D₄	1	8	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
58	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	#124×C₆₇	—	2	32	no	—
66	C₂²⋊Q₈×C₆₇	—	2	32	no	—
67	#121×C₆₇	—	2	32	no	—
68	#117×C₆₇	—	2	32	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
73	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	4	8	no	—
84	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
89	C₂×C₄²×C₆₇	—	2	4	no	—
90	C₂×C₄²×C₆₇	D₄	1	8	yes	complemented
91	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
92	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
93	#95×C₆₇	C₂³	1	16	yes	—
94	C₂⁴×C₆₇	—	2	2	no	—
95	C₂⁴×C₆₇	—	2	2	no	—
96	C₂⁴×C₆₇	—	2	4	no	—
97	C₂⁴×C₆₇	—	2	4	no	—
98	C₂⁴×C₆₇	—	2	4	no	—
99	C₂⁴×C₆₇	—	2	4	no	—
100	C₂⁴×C₆₇	—	2	4	no	—
101	C₂⁴×C₆₇	—	2	8	no	—
102	C₂⁴×C₆₇	—	2	8	no	—
103	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
104	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
105	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
106	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂×Q₈×C₆₇	—	2	8	no	—
108	C₂×Q₈×C₆₇	—	2	8	no	—
109	C₂×Q₈×C₆₇	—	2	8	no	—
110	C₂×Q₈×C₆₇	—	4	16	no	—
111	C₂×D₄×C₆₇	—	2	16	no	—
112	C₂×D₄×C₆₇	—	2	8	no	—
113	C₂×D₄×C₆₇	—	2	8	no	—
114	C₂×D₄×C₆₇	—	2	8	no	—
115	C₂×D₄×C₆₇	—	4	16	no	—
116	C₂×D₄×C₆₇	—	4	32	no	—
117	C₂×D₄×C₆₇	—	4	16	no	—
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	4	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	32	no	—
123	C₂²×C₄×C₆₇	—	2	8	no	—
124	C₂²×C₄×C₆₇	—	2	8	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	4	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	—	2	8	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	16	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₆₇	#41	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
148	C₄⋊C₄×C₆₇	—	2	16	no	—
149	C₄⋊C₄×C₆₇	—	2	8	no	—
150	C₄⋊C₄×C₆₇	—	2	8	no	—
151	C₄⋊C₄×C₆₇	—	2	16	no	—
152	C₄⋊C₄×C₆₇	—	4	16	no	—
153	C₂²⋊C₄×C₆₇	—	2	16	no	—
154	C₂²⋊C₄×C₆₇	—	2	16	no	—
155	C₂²⋊C₄×C₆₇	—	2	16	no	—
156	C₂²⋊C₄×C₆₇	—	2	32	no	—
157	C₂²⋊C₄×C₆₇	—	4	16	no	—
158	C₄²×C₆₇	—	2	8	no	—
159	C₄²×C₆₇	C₂×D₄	1	16	yes	complemented
160	C₂³×C₆₇	—	2	8	no	—
161	C₂³×C₆₇	—	2	8	no	—
162	C₂³×C₆₇	—	2	8	no	—
163	C₂³×C₆₇	—	2	2	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	2	4	no	—
166	C₂³×C₆₇	—	2	4	no	—
167	C₂³×C₆₇	—	2	4	no	—
168	C₂³×C₆₇	—	2	4	no	—
169	C₂³×C₆₇	—	2	4	no	—
170	C₂³×C₆₇	—	2	8	no	—
171	C₂³×C₆₇	—	2	8	no	—
172	C₂³×C₆₇	—	4	8	no	—
173	C₂³×C₆₇	—	4	8	no	—
174	C₂³×C₆₇	—	4	8	no	—
175	C₂³×C₆₇	—	4	8	no	—
176	C₂³×C₆₇	—	4	8	no	—
177	C₂³×C₆₇	—	4	16	no	—
178	C₂³×C₆₇	—	4	16	no	—
179	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
180	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
181	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
182	C₂³×C₆₇	C₂×#41	1	2	yes	—
183	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
184	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
185	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
186	Q₈×C₆₇	—	4	16	no	—
187	D₄×C₆₇	—	4	32	no	—
188	D₄×C₆₇	—	4	16	no	—
189	C₂×C₄×C₆₇	—	2	8	no	—
190	C₂×C₄×C₆₇	—	2	16	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	32	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	4	no	—
202	C₂×C₄×C₆₇	—	2	4	no	—
203	C₂×C₄×C₆₇	—	2	8	no	—
204	C₂×C₄×C₆₇	—	2	8	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	4	8	no	—
207	C₂×C₄×C₆₇	—	4	8	no	—
208	C₂×C₄×C₆₇	—	4	8	no	—
209	C₂×C₄×C₆₇	—	4	8	no	—
210	C₂×C₄×C₆₇	#121	1	16	yes	—
211	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
212	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
213	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
214	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
215	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
216	C₂²×C₆₇	—	2	4	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	2	4	no	—
219	C₂²×C₆₇	—	4	16	no	—
220	C₂²×C₆₇	—	4	4	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	4	4	no	—
223	C₂²×C₆₇	—	4	4	no	—
224	C₂²×C₆₇	—	4	4	no	—
225	C₂²×C₆₇	—	4	8	no	—
226	C₂²×C₆₇	—	4	8	no	—
227	C₂²×C₆₇	—	8	16	no	—
228	C₂²×C₆₇	C₂×#121	1	4	yes	—
229	C₂²×C₆₇	#535	1	4	yes	—
230	C₂²×C₆₇	#536	1	4	yes	—
231	C₂²×C₆₇	#537	1	8	yes	—
232	C₂²×C₆₇	#542	1	8	yes	—
233	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
234	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
235	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
236	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
237	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
238	C₄×C₆₇	—	2	4	no	—
239	C₄×C₆₇	—	2	8	no	—
240	C₄×C₆₇	—	4	4	no	—
241	C₄×C₆₇	—	4	4	no	—
242	C₄×C₆₇	—	4	4	no	—
243	C₄×C₆₇	—	4	4	no	—
244	C₄×C₆₇	C₂×#121	1	8	yes	—
245	C₂×#2620	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
246	C₂×C₆₇	—	4	4	no	—
247	C₂×C₆₇	—	8	4	no	—
248	C₂×C₆₇	#2620	1	8	yes	direct product factor
249	C₂×C₆₇	C₂²×#121	1	1	yes	characteristic
250	C₂×C₆₇	C₂×#535	1	1	yes	characteristic
251	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
252	C₂×C₆₇	C₂×#537	1	2	yes	—
253	C₂×C₆₇	C₂×#542	1	2	yes	—
254	C₂²×#124	C₂×C₆₇	1	1	yes	characteristic, complemented
255	C₂²×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
256	C₂²×#121	C₂×C₆₇	1	1	yes	characteristic
257	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
258	#2620	C₂×C₆₇	1	16	yes	direct product factor
259	C₂×#396	C₂×C₆₇	1	4	yes	complemented
260	C₂×#394	C₂×C₆₇	1	4	yes	complemented
261	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₂×#390	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₂×#384	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₆₇	C₂×#2620	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
266	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
267	C₂³×D₄	C₂²×C₆₇	1	2	yes	—
268	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×#124	—	2	8	no	—
270	C₂×#124	C₂²×C₆₇	1	8	yes	complemented
271	C₂×C₂²⋊Q₈	—	2	8	no	—
272	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	8	yes	complemented
273	C₂×#121	—	2	8	no	—
274	C₂×#121	C₂²×C₆₇	1	8	yes	—
275	C₂×#117	—	2	8	no	—
276	C₂×#117	C₂²×C₆₇	1	8	yes	complemented
277	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
278	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
279	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
280	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
281	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
282	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
283	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
284	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
285	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
286	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
287	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
288	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
289	#396	C₂²×C₆₇	1	32	yes	complemented
290	#394	C₂²×C₆₇	1	32	yes	complemented
291	#391	C₂²×C₆₇	1	8	yes	complemented
292	#390	C₂²×C₆₇	1	8	yes	complemented
293	#384	C₂²×C₆₇	1	8	yes	complemented
294	C₂×#95	C₂²×C₆₇	1	4	yes	—
295	C₂⁵	C₂³×C₆₇	1	2	yes	—
296	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
297	C₂²×Q₈	—	2	4	no	—
298	C₂²×Q₈	—	2	4	no	—
299	C₂²×Q₈	—	2	4	no	—
300	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
301	C₂²×D₄	—	2	8	no	—
302	C₂²×D₄	—	2	4	no	—
303	C₂²×D₄	—	2	4	no	—
304	C₂²×D₄	—	2	4	no	—
305	C₂²×D₄	—	4	8	no	—
306	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
307	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
308	C₂³×C₄	—	2	4	no	—
309	C₂³×C₄	—	2	4	no	—
310	C₂³×C₄	D₄×C₆₇	1	8	yes	—
311	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
312	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
313	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
314	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
315	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
316	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
318	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
319	#124	—	2	32	no	—
320	C₂²⋊Q₈	—	2	32	no	—
321	#121	—	2	32	no	—
322	#117	—	2	32	no	—
323	C₂×C₄⋊C₄	—	2	8	no	—
324	C₂×C₄⋊C₄	—	2	4	no	—
325	C₂×C₄⋊C₄	—	2	4	no	—
326	C₂×C₄⋊C₄	—	2	8	no	—
327	C₂×C₄⋊C₄	—	4	8	no	—
328	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
329	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
331	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
332	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	—
333	C₂×C₂²⋊C₄	—	2	8	no	—
334	C₂×C₂²⋊C₄	—	2	8	no	—
335	C₂×C₂²⋊C₄	—	2	8	no	—
336	C₂×C₂²⋊C₄	—	2	16	no	—
337	C₂×C₂²⋊C₄	—	4	8	no	—
338	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
343	C₂×C₄²	—	2	4	no	—
344	C₂×C₄²	D₄×C₆₇	1	8	yes	complemented
345	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
346	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
347	#95	C₂³×C₆₇	1	16	yes	—
348	C₂⁴	—	2	2	no	—
349	C₂⁴	—	2	2	no	—
350	C₂⁴	—	2	4	no	—
351	C₂⁴	—	2	4	no	—
352	C₂⁴	—	2	4	no	—
353	C₂⁴	—	2	4	no	—
354	C₂⁴	—	2	4	no	—
355	C₂⁴	—	2	8	no	—
356	C₂⁴	—	2	8	no	—
357	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
358	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
359	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
360	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
361	C₂×Q₈	—	2	8	no	—
362	C₂×Q₈	—	2	8	no	—
363	C₂×Q₈	—	2	8	no	—
364	C₂×Q₈	—	4	16	no	—
365	C₂×D₄	—	2	16	no	—
366	C₂×D₄	—	2	8	no	—
367	C₂×D₄	—	2	8	no	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	4	16	no	—
370	C₂×D₄	—	4	32	no	—
371	C₂×D₄	—	4	16	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	32	no	—
377	C₂²×C₄	—	2	8	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	2	8	no	—
387	C₂²×C₄	—	2	8	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
390	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
391	C₂²×C₄	C₂×D₄×C₆₇	1	16	yes	—
392	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
394	C₂²×C₄	#41×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
397	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
398	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
400	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
402	C₄⋊C₄	—	2	16	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	8	no	—
405	C₄⋊C₄	—	2	16	no	—
406	C₄⋊C₄	—	4	16	no	—
407	C₂²⋊C₄	—	2	16	no	—
408	C₂²⋊C₄	—	2	16	no	—
409	C₂²⋊C₄	—	2	16	no	—
410	C₂²⋊C₄	—	2	32	no	—
411	C₂²⋊C₄	—	4	16	no	—
412	C₄²	—	2	8	no	—
413	C₄²	C₂×D₄×C₆₇	1	16	yes	complemented
414	C₂³	—	2	8	no	—
415	C₂³	—	2	8	no	—
416	C₂³	—	2	8	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	4	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	2	4	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	8	no	—
425	C₂³	—	2	8	no	—
426	C₂³	—	4	8	no	—
427	C₂³	—	4	8	no	—
428	C₂³	—	4	8	no	—
429	C₂³	—	4	8	no	—
430	C₂³	—	4	8	no	—
431	C₂³	—	4	16	no	—
432	C₂³	—	4	16	no	—
433	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
434	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
435	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
436	C₂³	C₂×#41×C₆₇	1	2	yes	—
437	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
438	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
439	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
440	Q₈	—	4	16	no	—
441	D₄	—	4	32	no	—
442	D₄	—	4	16	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	16	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	32	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	8	no	—
458	C₂×C₄	—	2	8	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	8	no	—
462	C₂×C₄	—	4	8	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	#121×C₆₇	1	16	yes	—
465	C₂×C₄	C₂²×D₄×C₆₇	1	8	yes	—
466	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
467	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
468	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
469	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
470	C₂²	—	2	4	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	4	no	—
473	C₂²	—	4	16	no	—
474	C₂²	—	4	4	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	8	no	—
480	C₂²	—	4	8	no	—
481	C₂²	—	8	16	no	—
482	C₂²	C₂×#121×C₆₇	1	4	yes	—
483	C₂²	#535×C₆₇	1	4	yes	—
484	C₂²	#536×C₆₇	1	4	yes	—
485	C₂²	#537×C₆₇	1	8	yes	—
486	C₂²	#542×C₆₇	1	8	yes	—
487	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
488	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
489	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
490	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
491	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
492	C₄	—	2	4	no	—
493	C₄	—	2	8	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	4	no	—
498	C₄	C₂×#121×C₆₇	1	8	yes	—
499	C₂	—	4	4	no	—
500	C₂	—	8	4	no	—
501	C₂	C₆₇×#2620	1	8	yes	direct product factor
502	C₂	C₂²×#121×C₆₇	1	1	yes	characteristic
503	C₂	C₂×#535×C₆₇	1	1	yes	characteristic
504	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
505	C₂	C₂×#537×C₆₇	1	2	yes	—
506	C₂	C₂×#542×C₆₇	1	2	yes	—

	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×#117×C₆₇	C₂	semidirect
5. C₆₇×#2620	C₂	direct (N × H)
6. C₂×#396×C₆₇	C₂	semidirect
7. C₂×#394×C₆₇	C₂	semidirect
8. C₂×#391×C₆₇	C₂	semidirect
9. C₂×#390×C₆₇	C₂	semidirect
10. C₂×#384×C₆₇	C₂	semidirect
16. C₂×#124×C₆₇	C₂²	semidirect
18. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
22. C₂×#117×C₆₇	C₂²	semidirect
35. #396×C₆₇	C₂²	semidirect
36. #394×C₆₇	C₂²	semidirect
37. #391×C₆₇	C₂²	semidirect
38. #390×C₆₇	C₂²	semidirect
39. #384×C₆₇	C₂²	semidirect
90. C₂×C₄²×C₆₇	D₄	semidirect
159. C₄²×C₆₇	C₂×D₄	semidirect
245. C₂×#2620	C₆₇	direct (N × H)
248. C₂×C₆₇	#2620	direct (N × H)
254. C₂²×#124	C₂×C₆₇	semidirect
255. C₂²×C₂²⋊Q₈	C₂×C₆₇	semidirect
257. C₂²×#117	C₂×C₆₇	semidirect
258. #2620	C₂×C₆₇	direct (N × H)
259. C₂×#396	C₂×C₆₇	semidirect
260. C₂×#394	C₂×C₆₇	semidirect
261. C₂×#391	C₂×C₆₇	semidirect
262. C₂×#390	C₂×C₆₇	semidirect
263. C₂×#384	C₂×C₆₇	semidirect
264. C₆₇	C₂×#2620	direct (N × H)
270. C₂×#124	C₂²×C₆₇	semidirect
272. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
276. C₂×#117	C₂²×C₆₇	semidirect
289. #396	C₂²×C₆₇	semidirect
290. #394	C₂²×C₆₇	semidirect
291. #391	C₂²×C₆₇	semidirect
292. #390	C₂²×C₆₇	semidirect
293. #384	C₂²×C₆₇	semidirect
344. C₂×C₄²	D₄×C₆₇	semidirect
413. C₄²	C₂×D₄×C₆₇	semidirect
501. C₂	C₆₇×#2620	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×#117×C₆₇	2
C₆₇×#2620	2
C₂×#396×C₆₇	2
C₂×#394×C₆₇	2
C₂×#391×C₆₇	2
C₂×#390×C₆₇	2
C₂×#384×C₆₇	2
C₂×#2620	67

Subgroups — C2×C67×#2620 #30713862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2620 #30713862