Subgroups — C₂×C₆₇×#3252 #30740153

Order 17152.

1898 conjugacy classes of subgroups.

The subgroup lattice has 592 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×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3252	C₂	1	16	yes	maximal, direct product factor
4	C₂×#479×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#477×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#476×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#470×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#468×C₆₇	C₂	1	2	yes	maximal
9	C₂×#464×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#462×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#461×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
16	#533×C₆₇	—	2	4	no	—
17	#533×C₆₇	C₂²	1	8	yes	complemented
18	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	#529×C₆₇	—	2	4	no	—
21	#529×C₆₇	C₂²	1	8	yes	—
22	C₂×#126×C₆₇	—	2	1	no	—
23	C₂×#125×C₆₇	—	2	1	no	—
24	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#123×C₆₇	—	2	1	no	—
27	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
32	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₆₇	—	2	1	no	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#117×C₆₇	—	2	1	no	—
37	C₂×#117×C₆₇	—	2	1	no	—
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	#479×C₆₇	C₂²	1	16	yes	complemented
40	#477×C₆₇	C₂²	1	8	yes	complemented
41	#476×C₆₇	C₂²	1	8	yes	complemented
42	#470×C₆₇	C₂²	1	16	yes	complemented
43	#468×C₆₇	C₂²	1	16	yes	—
44	#464×C₆₇	C₂²	1	16	yes	complemented
45	#462×C₆₇	C₂²	1	8	yes	complemented
46	#461×C₆₇	C₂²	1	8	yes	complemented
47	C₂×#107×C₆₇	C₂²	1	2	yes	—
48	C₂×#106×C₆₇	C₂²	1	2	yes	—
49	#423×C₆₇	C₂²	1	8	yes	complemented
50	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
51	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
52	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
53	C₂×#103×C₆₇	C₂²	1	2	yes	—
54	C₂×#103×C₆₇	C₂²	1	2	yes	—
55	C₂×#103×C₆₇	C₂²	1	2	yes	—
56	C₂×#102×C₆₇	C₂²	1	2	yes	—
57	C₂²×Q₈×C₆₇	—	2	1	no	—
58	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
60	C₂²×D₄×C₆₇	—	2	1	no	—
61	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	—	2	1	no	—
63	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₆₇	—	4	4	no	—
67	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
68	C₂×SD₁₆×C₆₇	—	4	4	no	—
69	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
70	C₂×M₁₆×C₆₇	—	4	4	no	—
71	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
72	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
73	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₆₇	—	2	2	no	—
75	C₄⋊Q₈×C₆₇	—	2	2	no	—
76	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
77	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
78	#126×C₆₇	—	2	8	no	—
79	#125×C₆₇	—	2	8	no	—
80	#125×C₆₇	—	2	2	no	—
81	#125×C₆₇	C₂³	1	4	yes	—
82	#124×C₆₇	—	2	2	no	—
83	#124×C₆₇	C₂³	1	4	yes	—
84	#123×C₆₇	—	2	8	no	—
85	C₂²⋊Q₈×C₆₇	—	2	8	no	—
86	C₂²⋊Q₈×C₆₇	—	2	2	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
89	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
90	#121×C₆₇	—	2	2	no	—
91	#121×C₆₇	C₂³	1	4	yes	—
92	C₄×Q₈×C₆₇	—	2	8	no	—
93	C₄×Q₈×C₆₇	—	2	2	no	—
94	C₄×Q₈×C₆₇	—	2	4	no	—
95	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
96	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
97	C₄×D₄×C₆₇	—	2	8	no	—
98	C₄×D₄×C₆₇	—	2	2	no	—
99	C₄×D₄×C₆₇	C₂³	1	4	yes	—
100	#117×C₆₇	—	2	8	no	—
101	#117×C₆₇	—	2	8	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
106	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
107	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
108	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
110	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
111	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
115	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
118	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
119	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
120	C₂×C₄²×C₆₇	—	2	1	no	—
121	C₂×C₄²×C₆₇	—	2	2	no	—
122	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
123	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₆₇	D₄	1	4	yes	—
125	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
128	#107×C₆₇	C₂³	1	8	yes	—
129	#106×C₆₇	C₂³	1	8	yes	—
130	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
131	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
132	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
133	#103×C₆₇	C₂³	1	8	yes	—
134	#103×C₆₇	C₂³	1	8	yes	—
135	#103×C₆₇	C₂³	1	8	yes	—
136	#102×C₆₇	C₂³	1	8	yes	—
137	C₂⁴×C₆₇	—	2	1	no	—
138	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	2	2	no	—
141	C₂×Q₈×C₆₇	—	2	4	no	—
142	C₂×Q₈×C₆₇	—	2	4	no	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	2	2	no	—
145	C₂×Q₈×C₆₇	—	2	4	no	—
146	C₂×Q₈×C₆₇	—	4	4	no	—
147	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
148	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
149	C₂×D₄×C₆₇	—	2	4	no	—
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	2	no	—
152	C₂×D₄×C₆₇	—	2	2	no	—
153	C₂×D₄×C₆₇	—	4	4	no	—
154	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₆₇	—	2	4	no	—
156	C₂²×C₄×C₆₇	—	2	2	no	—
157	C₂²×C₄×C₆₇	—	2	2	no	—
158	C₂²×C₄×C₆₇	—	2	1	no	—
159	C₂²×C₄×C₆₇	—	2	2	no	—
160	C₂²×C₄×C₆₇	—	2	1	no	—
161	C₂²×C₄×C₆₇	—	2	2	no	—
162	C₂²×C₄×C₆₇	—	2	1	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	4	2	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₆₇	—	4	8	no	—
177	SD₁₆×C₆₇	—	4	8	no	—
178	M₁₆×C₆₇	—	4	8	no	—
179	C₂×C₈×C₆₇	—	2	4	no	—
180	C₂×C₈×C₆₇	—	2	4	no	—
181	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
182	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	4	no	—
185	C₄⋊C₄×C₆₇	—	2	4	no	—
186	C₄⋊C₄×C₆₇	—	2	4	no	—
187	C₄⋊C₄×C₆₇	—	2	4	no	—
188	C₄⋊C₄×C₆₇	—	2	4	no	—
189	C₄⋊C₄×C₆₇	—	2	8	no	—
190	C₄⋊C₄×C₆₇	—	2	8	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	1	no	—
193	C₄⋊C₄×C₆₇	—	2	1	no	—
194	C₄⋊C₄×C₆₇	—	2	2	no	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
197	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
198	C₂²⋊C₄×C₆₇	—	2	8	no	—
199	C₂²⋊C₄×C₆₇	—	2	4	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₄²×C₆₇	—	2	4	no	—
205	C₄²×C₆₇	—	2	8	no	—
206	C₄²×C₆₇	C₂×D₄	1	4	yes	—
207	C₄²×C₆₇	C₂×D₄	1	2	yes	—
208	C₄²×C₆₇	C₂×D₄	1	2	yes	—
209	C₄²×C₆₇	C₂×D₄	1	4	yes	—
210	C₄²×C₆₇	C₂⁴	1	2	yes	—
211	C₄²×C₆₇	C₂⁴	1	2	yes	—
212	C₂³×C₆₇	—	2	2	no	—
213	C₂³×C₆₇	—	2	2	no	—
214	C₂³×C₆₇	—	2	2	no	—
215	C₂³×C₆₇	—	2	2	no	—
216	C₂³×C₆₇	—	4	2	no	—
217	C₂³×C₆₇	—	4	2	no	—
218	C₂³×C₆₇	—	4	2	no	—
219	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
221	Q₈×C₆₇	—	2	4	no	—
222	Q₈×C₆₇	—	2	2	no	—
223	Q₈×C₆₇	—	2	2	no	—
224	Q₈×C₆₇	—	2	4	no	—
225	Q₈×C₆₇	—	4	8	no	—
226	Q₈×C₆₇	—	4	2	no	—
227	Q₈×C₆₇	—	4	4	no	—
228	D₄×C₆₇	—	2	2	no	—
229	D₄×C₆₇	—	2	2	no	—
230	D₄×C₆₇	—	4	8	no	—
231	D₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	2	2	no	—
233	C₂×C₄×C₆₇	—	2	2	no	—
234	C₂×C₄×C₆₇	—	2	2	no	—
235	C₂×C₄×C₆₇	—	2	2	no	—
236	C₂×C₄×C₆₇	—	2	4	no	—
237	C₂×C₄×C₆₇	—	2	8	no	—
238	C₂×C₄×C₆₇	—	2	2	no	—
239	C₂×C₄×C₆₇	—	2	2	no	—
240	C₂×C₄×C₆₇	—	2	2	no	—
241	C₂×C₄×C₆₇	—	2	4	no	—
242	C₂×C₄×C₆₇	—	2	2	no	—
243	C₂×C₄×C₆₇	—	2	4	no	—
244	C₂×C₄×C₆₇	—	2	2	no	—
245	C₂×C₄×C₆₇	—	4	4	no	—
246	C₂×C₄×C₆₇	—	4	2	no	—
247	C₂×C₄×C₆₇	—	4	4	no	—
248	C₂×C₄×C₆₇	—	4	2	no	—
249	C₂×C₄×C₆₇	—	4	2	no	—
250	C₂×C₄×C₆₇	—	4	4	no	—
251	C₂×C₄×C₆₇	—	4	8	no	—
252	C₂×C₄×C₆₇	#137	1	4	yes	—
253	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
260	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
261	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
262	C₈×C₆₇	—	2	4	no	—
263	C₈×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	2	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	—	4	2	no	—
269	C₂²×C₆₇	—	4	2	no	—
270	C₂²×C₆₇	—	4	2	no	—
271	C₂²×C₆₇	—	8	4	no	—
272	C₂²×C₆₇	#537	1	2	yes	—
273	C₂²×C₆₇	C₂×#137	1	2	yes	—
274	C₂²×C₆₇	#575	1	2	yes	—
275	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
276	C₄×C₆₇	—	2	2	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	2	2	no	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	4	2	no	—
281	C₄×C₆₇	—	4	4	no	—
282	C₄×C₆₇	—	4	2	no	—
283	C₄×C₆₇	—	4	2	no	—
284	C₄×C₆₇	—	4	4	no	—
285	C₄×C₆₇	#537	1	2	yes	—
286	C₄×C₆₇	C₂×#137	1	4	yes	—
287	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
288	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
289	C₂×#3252	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
290	C₂×C₆₇	—	4	2	no	—
291	C₂×C₆₇	—	8	2	no	—
292	C₂×C₆₇	#3252	1	4	yes	direct product factor
293	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
294	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
295	C₂×C₆₇	C₂×#575	1	1	yes	characteristic
296	C₂×#533	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
298	#3252	C₂×C₆₇	1	16	yes	direct product factor
299	C₂×#479	C₂×C₆₇	1	2	yes	complemented
300	C₂×#477	C₂×C₆₇	1	1	yes	characteristic, complemented
301	C₂×#476	C₂×C₆₇	1	1	yes	characteristic, complemented
302	C₂×#470	C₂×C₆₇	1	2	yes	complemented
303	C₂×#468	C₂×C₆₇	1	2	yes	—
304	C₂×#464	C₂×C₆₇	1	2	yes	complemented
305	C₂×#462	C₂×C₆₇	1	1	yes	characteristic, complemented
306	C₂×#461	C₂×C₆₇	1	1	yes	characteristic, complemented
307	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
308	C₆₇	C₂×#3252	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	—
310	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	—
311	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
312	#533	—	2	4	no	—
313	#533	C₂²×C₆₇	1	8	yes	complemented
314	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
315	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
316	#529	—	2	4	no	—
317	#529	C₂²×C₆₇	1	8	yes	—
318	C₂×#126	—	2	1	no	—
319	C₂×#125	—	2	1	no	—
320	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
321	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#123	—	2	1	no	—
323	C₂×C₂²⋊Q₈	—	2	1	no	—
324	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
326	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
327	C₂×C₄×Q₈	—	2	1	no	—
328	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
329	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
330	C₂×C₄×D₄	—	2	1	no	—
331	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#117	—	2	1	no	—
333	C₂×#117	—	2	1	no	—
334	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
335	#479	C₂²×C₆₇	1	16	yes	complemented
336	#477	C₂²×C₆₇	1	8	yes	complemented
337	#476	C₂²×C₆₇	1	8	yes	complemented
338	#470	C₂²×C₆₇	1	16	yes	complemented
339	#468	C₂²×C₆₇	1	16	yes	—
340	#464	C₂²×C₆₇	1	16	yes	complemented
341	#462	C₂²×C₆₇	1	8	yes	complemented
342	#461	C₂²×C₆₇	1	8	yes	complemented
343	C₂×#107	C₂²×C₆₇	1	2	yes	—
344	C₂×#106	C₂²×C₆₇	1	2	yes	—
345	#423	C₂²×C₆₇	1	8	yes	complemented
346	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
347	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
348	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
349	C₂×#103	C₂²×C₆₇	1	2	yes	—
350	C₂×#103	C₂²×C₆₇	1	2	yes	—
351	C₂×#103	C₂²×C₆₇	1	2	yes	—
352	C₂×#102	C₂²×C₆₇	1	2	yes	—
353	C₂²×Q₈	—	2	1	no	—
354	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
355	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
356	C₂²×D₄	—	2	1	no	—
357	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
358	C₂³×C₄	—	2	1	no	—
359	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
360	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
361	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
362	C₂×Q₁₆	—	4	4	no	—
363	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
364	C₂×SD₁₆	—	4	4	no	—
365	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
366	C₂×M₁₆	—	4	4	no	—
367	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
368	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
369	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
370	C₄⋊Q₈	—	2	2	no	—
371	C₄⋊Q₈	—	2	2	no	—
372	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
373	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
374	#126	—	2	8	no	—
375	#125	—	2	8	no	—
376	#125	—	2	2	no	—
377	#125	C₂³×C₆₇	1	4	yes	—
378	#124	—	2	2	no	—
379	#124	C₂³×C₆₇	1	4	yes	—
380	#123	—	2	8	no	—
381	C₂²⋊Q₈	—	2	8	no	—
382	C₂²⋊Q₈	—	2	2	no	—
383	C₂²⋊Q₈	—	2	4	no	—
384	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
385	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
386	#121	—	2	2	no	—
387	#121	C₂³×C₆₇	1	4	yes	—
388	C₄×Q₈	—	2	8	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	2	4	no	—
391	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
392	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
393	C₄×D₄	—	2	8	no	—
394	C₄×D₄	—	2	2	no	—
395	C₄×D₄	C₂³×C₆₇	1	4	yes	—
396	#117	—	2	8	no	—
397	#117	—	2	8	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	—	2	2	no	—
405	C₂×C₄⋊C₄	—	2	2	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
410	C₂×C₂²⋊C₄	—	2	2	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₄²	—	2	1	no	—
417	C₂×C₄²	—	2	2	no	—
418	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
419	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
420	C₂×C₄²	D₄×C₆₇	1	4	yes	—
421	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
424	#107	C₂³×C₆₇	1	8	yes	—
425	#106	C₂³×C₆₇	1	8	yes	—
426	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
427	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
428	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
429	#103	C₂³×C₆₇	1	8	yes	—
430	#103	C₂³×C₆₇	1	8	yes	—
431	#103	C₂³×C₆₇	1	8	yes	—
432	#102	C₂³×C₆₇	1	8	yes	—
433	C₂⁴	—	2	1	no	—
434	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	4	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	—	2	2	no	—
441	C₂×Q₈	—	2	4	no	—
442	C₂×Q₈	—	4	4	no	—
443	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
444	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	4	4	no	—
450	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	2	no	—
453	C₂²×C₄	—	2	2	no	—
454	C₂²×C₄	—	2	1	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	4	2	no	—
461	C₂²×C₄	—	4	4	no	—
462	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
465	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
467	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
468	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
469	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
472	Q₁₆	—	4	8	no	—
473	SD₁₆	—	4	8	no	—
474	M₁₆	—	4	8	no	—
475	C₂×C₈	—	2	4	no	—
476	C₂×C₈	—	2	4	no	—
477	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
478	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	8	no	—
486	C₄⋊C₄	—	2	8	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
492	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
493	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
494	C₂²⋊C₄	—	2	8	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₄²	—	2	4	no	—
501	C₄²	—	2	8	no	—
502	C₄²	C₂×D₄×C₆₇	1	4	yes	—
503	C₄²	C₂×D₄×C₆₇	1	2	yes	—
504	C₄²	C₂×D₄×C₆₇	1	2	yes	—
505	C₄²	C₂×D₄×C₆₇	1	4	yes	—
506	C₄²	C₂⁴×C₆₇	1	2	yes	—
507	C₄²	C₂⁴×C₆₇	1	2	yes	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	—	4	2	no	—
514	C₂³	—	4	2	no	—
515	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
516	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
517	Q₈	—	2	4	no	—
518	Q₈	—	2	2	no	—
519	Q₈	—	2	2	no	—
520	Q₈	—	2	4	no	—
521	Q₈	—	4	8	no	—
522	Q₈	—	4	2	no	—
523	Q₈	—	4	4	no	—
524	D₄	—	2	2	no	—
525	D₄	—	2	2	no	—
526	D₄	—	4	8	no	—
527	D₄	—	4	2	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	8	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	4	4	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	4	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	—	4	8	no	—
548	C₂×C₄	#137×C₆₇	1	4	yes	—
549	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
555	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
556	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
557	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
558	C₈	—	2	4	no	—
559	C₈	—	2	4	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	8	4	no	—
568	C₂²	#537×C₆₇	1	2	yes	—
569	C₂²	C₂×#137×C₆₇	1	2	yes	—
570	C₂²	#575×C₆₇	1	2	yes	—
571	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	2	no	—
578	C₄	—	4	4	no	—
579	C₄	—	4	2	no	—
580	C₄	—	4	4	no	—
581	C₄	#537×C₆₇	1	2	yes	—
582	C₄	C₂×#137×C₆₇	1	4	yes	—
583	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
584	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
585	C₂	—	4	2	no	—
586	C₂	—	8	2	no	—
587	C₂	C₆₇×#3252	1	4	yes	direct product factor
588	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
589	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
590	C₂	C₂×#575×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×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×#3252	C₂	direct (N × H)
4. C₂×#479×C₆₇	C₂	semidirect
5. C₂×#477×C₆₇	C₂	semidirect
6. C₂×#476×C₆₇	C₂	semidirect
7. C₂×#470×C₆₇	C₂	semidirect
9. C₂×#464×C₆₇	C₂	semidirect
10. C₂×#462×C₆₇	C₂	semidirect
11. C₂×#461×C₆₇	C₂	semidirect
12. C₂×#423×C₆₇	C₂	semidirect
17. #533×C₆₇	C₂²	semidirect
39. #479×C₆₇	C₂²	semidirect
40. #477×C₆₇	C₂²	semidirect
41. #476×C₆₇	C₂²	semidirect
42. #470×C₆₇	C₂²	semidirect
44. #464×C₆₇	C₂²	semidirect
45. #462×C₆₇	C₂²	semidirect
46. #461×C₆₇	C₂²	semidirect
49. #423×C₆₇	C₂²	semidirect
289. C₂×#3252	C₆₇	direct (N × H)
292. C₂×C₆₇	#3252	direct (N × H)
296. C₂×#533	C₂×C₆₇	semidirect
298. #3252	C₂×C₆₇	direct (N × H)
299. C₂×#479	C₂×C₆₇	semidirect
300. C₂×#477	C₂×C₆₇	semidirect
301. C₂×#476	C₂×C₆₇	semidirect
302. C₂×#470	C₂×C₆₇	semidirect
304. C₂×#464	C₂×C₆₇	semidirect
305. C₂×#462	C₂×C₆₇	semidirect
306. C₂×#461	C₂×C₆₇	semidirect
307. C₂×#423	C₂×C₆₇	semidirect
308. C₆₇	C₂×#3252	direct (N × H)
313. #533	C₂²×C₆₇	semidirect
335. #479	C₂²×C₆₇	semidirect
336. #477	C₂²×C₆₇	semidirect
337. #476	C₂²×C₆₇	semidirect
338. #470	C₂²×C₆₇	semidirect
340. #464	C₂²×C₆₇	semidirect
341. #462	C₂²×C₆₇	semidirect
342. #461	C₂²×C₆₇	semidirect
345. #423	C₂²×C₆₇	semidirect
587. C₂	C₆₇×#3252	direct (N × H)

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×#529×C₆₇	2
C₆₇×#3252	2
C₂×#479×C₆₇	2
C₂×#477×C₆₇	2
C₂×#476×C₆₇	2
C₂×#470×C₆₇	2
C₂×#468×C₆₇	2
C₂×#464×C₆₇	2
C₂×#462×C₆₇	2
C₂×#461×C₆₇	2
C₂×#423×C₆₇	2
C₂×#3252	67

Subgroups — C2×C67×#3252 #30740153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3252 #30740153