Subgroups — C₂×C₆₇×#2599 #30714115

Order 17152.

2766 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×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2599	C₂	1	16	yes	maximal, direct product factor
4	C₂×#398×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#397×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#393×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#388×C₆₇	C₂	1	2	yes	maximal
9	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#387×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₂×#385×C₆₇	C₂	1	2	yes	maximal
12	C₂×#383×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
17	C₂×#117×C₆₇	—	2	8	no	—
18	C₂×#117×C₆₇	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#398×C₆₇	C₂²	1	8	yes	complemented
33	#397×C₆₇	C₂²	1	8	yes	—
34	#395×C₆₇	C₂²	1	8	yes	complemented
35	#393×C₆₇	C₂²	1	8	yes	—
36	#388×C₆₇	C₂²	1	16	yes	—
37	#387×C₆₇	C₂²	1	16	yes	complemented
38	#387×C₆₇	C₂²	1	16	yes	complemented
39	#385×C₆₇	C₂²	1	16	yes	—
40	#383×C₆₇	C₂²	1	8	yes	complemented
41	C₂×#95×C₆₇	C₂²	1	2	yes	—
42	C₂×#95×C₆₇	C₂²	1	2	yes	—
43	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
44	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×#95×C₆₇	C₂²	1	2	yes	—
46	C₂×#95×C₆₇	C₂²	1	2	yes	—
47	C₂×#95×C₆₇	C₂²	1	2	yes	—
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₆₇	—	2	4	no	—
51	C₂²×Q₈×C₆₇	—	2	8	no	—
52	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
53	C₂³×C₄×C₆₇	—	2	4	no	—
54	C₂³×C₄×C₆₇	—	2	4	no	—
55	C₂³×C₄×C₆₇	—	2	4	no	—
56	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₆₇	—	2	32	no	—
68	#117×C₆₇	—	2	32	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
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	4	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
85	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
89	C₂×C₄²×C₆₇	—	2	4	no	—
90	C₂×C₄²×C₆₇	—	2	8	no	—
91	C₂×C₄²×C₆₇	D₄	1	8	yes	—
92	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
94	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
95	#95×C₆₇	C₂³	1	8	yes	—
96	#95×C₆₇	C₂³	1	8	yes	—
97	#95×C₆₇	C₂³	1	4	yes	—
98	#95×C₆₇	C₂³	1	4	yes	—
99	#95×C₆₇	C₂³	1	8	yes	—
100	#95×C₆₇	C₂³	1	8	yes	—
101	#95×C₆₇	C₂³	1	8	yes	—
102	#95×C₆₇	C₂³	1	8	yes	—
103	C₂⁴×C₆₇	—	2	2	no	—
104	C₂⁴×C₆₇	—	2	4	no	—
105	C₂⁴×C₆₇	—	2	2	no	—
106	C₂⁴×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	#41	1	4	yes	—
108	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
109	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
110	C₂×Q₈×C₆₇	—	2	32	no	—
111	C₂×Q₈×C₆₇	—	2	8	no	—
112	C₂×Q₈×C₆₇	—	2	16	no	—
113	C₂²×C₄×C₆₇	—	2	32	no	—
114	C₂²×C₄×C₆₇	—	2	4	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	2	8	no	—
117	C₂²×C₄×C₆₇	—	2	2	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	4	no	—
123	C₂²×C₄×C₆₇	—	2	8	no	—
124	C₂²×C₄×C₆₇	—	2	8	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	2	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	4	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	4	no	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
141	C₂²×C₄×C₆₇	#41	1	8	yes	—
142	C₂²×C₄×C₆₇	#41	1	4	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
149	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
153	C₄⋊C₄×C₆₇	—	2	8	no	—
154	C₄⋊C₄×C₆₇	—	2	8	no	—
155	C₄⋊C₄×C₆₇	—	2	16	no	—
156	C₄⋊C₄×C₆₇	—	2	8	no	—
157	C₄⋊C₄×C₆₇	—	2	16	no	—
158	C₄⋊C₄×C₆₇	—	2	16	no	—
159	C₂²⋊C₄×C₆₇	—	2	16	no	—
160	C₂²⋊C₄×C₆₇	—	2	8	no	—
161	C₂²⋊C₄×C₆₇	—	2	8	no	—
162	C₂²⋊C₄×C₆₇	—	2	16	no	—
163	C₄²×C₆₇	—	2	8	no	—
164	C₄²×C₆₇	—	2	16	no	—
165	C₄²×C₆₇	C₂×D₄	1	16	yes	—
166	C₂³×C₆₇	—	2	8	no	—
167	C₂³×C₆₇	—	2	16	no	—
168	C₂³×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	—	2	4	no	—
170	C₂³×C₆₇	—	2	2	no	—
171	C₂³×C₆₇	—	2	4	no	—
172	C₂³×C₆₇	—	4	4	no	—
173	C₂³×C₆₇	—	4	4	no	—
174	C₂³×C₆₇	—	4	8	no	—
175	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
176	C₂³×C₆₇	C₂×#41	1	4	yes	—
177	C₂³×C₆₇	C₂×#41	1	2	yes	—
178	C₂³×C₆₇	C₂×#41	1	4	yes	—
179	C₂³×C₆₇	C₂×#41	1	2	yes	—
180	C₂³×C₆₇	2₋¹⁺⁴	1	4	yes	—
181	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
182	Q₈×C₆₇	—	2	32	no	—
183	C₂×C₄×C₆₇	—	2	32	no	—
184	C₂×C₄×C₆₇	—	2	8	no	—
185	C₂×C₄×C₆₇	—	2	8	no	—
186	C₂×C₄×C₆₇	—	2	8	no	—
187	C₂×C₄×C₆₇	—	2	16	no	—
188	C₂×C₄×C₆₇	—	2	8	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	2	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	2	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	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	4	4	no	—
211	C₂×C₄×C₆₇	—	4	8	no	—
212	C₂×C₄×C₆₇	—	4	4	no	—
213	C₂×C₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	4	8	no	—
215	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
216	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
219	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
220	C₂²×C₆₇	—	2	4	no	—
221	C₂²×C₆₇	—	2	8	no	—
222	C₂²×C₆₇	—	4	16	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	4	4	no	—
226	C₂²×C₆₇	#525	1	4	yes	—
227	C₂²×C₆₇	#525	1	4	yes	—
228	C₂²×C₆₇	#536	1	4	yes	—
229	C₂²×C₆₇	#552	1	8	yes	—
230	C₂²×C₆₇	#556	1	8	yes	—
231	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
232	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
233	C₂²×C₆₇	C₂²×#41	1	2	yes	—
234	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
235	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
236	C₄×C₆₇	—	2	4	no	—
237	C₄×C₆₇	—	2	4	no	—
238	C₄×C₆₇	—	2	4	no	—
239	C₄×C₆₇	—	2	8	no	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	4	2	no	—
242	C₄×C₆₇	—	4	4	no	—
243	C₄×C₆₇	—	4	2	no	—
244	C₄×C₆₇	—	4	4	no	—
245	C₄×C₆₇	—	4	4	no	—
246	C₂×#2599	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂×C₆₇	—	4	4	no	—
248	C₂×C₆₇	#2599	1	8	yes	direct product factor
249	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
250	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
251	C₂×C₆₇	C₂×#536	1	1	yes	characteristic
252	C₂×C₆₇	C₂×#552	1	2	yes	—
253	C₂×C₆₇	C₂×#556	1	2	yes	—
254	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
255	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic
256	#2599	C₂×C₆₇	1	16	yes	direct product factor
257	C₂×#398	C₂×C₆₇	1	1	yes	characteristic, complemented
258	C₂×#397	C₂×C₆₇	1	1	yes	characteristic
259	C₂×#395	C₂×C₆₇	1	1	yes	characteristic, complemented
260	C₂×#393	C₂×C₆₇	1	1	yes	characteristic
261	C₂×#388	C₂×C₆₇	1	2	yes	—
262	C₂×#387	C₂×C₆₇	1	2	yes	complemented
263	C₂×#387	C₂×C₆₇	1	2	yes	complemented
264	C₂×#385	C₂×C₆₇	1	2	yes	—
265	C₂×#383	C₂×C₆₇	1	1	yes	characteristic, complemented
266	C₆₇	C₂×#2599	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
268	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×C₄×Q₈	—	2	8	no	—
270	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
271	C₂×#117	—	2	8	no	—
272	C₂×#117	C₂²×C₆₇	1	8	yes	—
273	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
274	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
275	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
276	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
277	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
278	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
279	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
280	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
281	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
282	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
283	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
284	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
285	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
286	#398	C₂²×C₆₇	1	8	yes	complemented
287	#397	C₂²×C₆₇	1	8	yes	—
288	#395	C₂²×C₆₇	1	8	yes	complemented
289	#393	C₂²×C₆₇	1	8	yes	—
290	#388	C₂²×C₆₇	1	16	yes	—
291	#387	C₂²×C₆₇	1	16	yes	complemented
292	#387	C₂²×C₆₇	1	16	yes	complemented
293	#385	C₂²×C₆₇	1	16	yes	—
294	#383	C₂²×C₆₇	1	8	yes	complemented
295	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
296	C₂×#95	C₂²×C₆₇	1	2	yes	—
297	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×#95	C₂²×C₆₇	1	2	yes	—
299	C₂×#95	C₂²×C₆₇	1	2	yes	—
300	C₂×#95	C₂²×C₆₇	1	2	yes	—
301	C₂×#95	C₂²×C₆₇	1	2	yes	—
302	C₂×#95	C₂²×C₆₇	1	2	yes	—
303	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
304	C₂²×Q₈	—	2	4	no	—
305	C₂²×Q₈	—	2	8	no	—
306	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
307	C₂³×C₄	—	2	4	no	—
308	C₂³×C₄	—	2	4	no	—
309	C₂³×C₄	—	2	4	no	—
310	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
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	1	yes	characteristic
315	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
316	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
317	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
318	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
320	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
321	C₄×Q₈	—	2	32	no	—
322	#117	—	2	32	no	—
323	C₂×C₄⋊C₄	—	2	4	no	—
324	C₂×C₄⋊C₄	—	2	4	no	—
325	C₂×C₄⋊C₄	—	2	8	no	—
326	C₂×C₄⋊C₄	—	2	4	no	—
327	C₂×C₄⋊C₄	—	2	8	no	—
328	C₂×C₄⋊C₄	—	2	8	no	—
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	4	yes	—
333	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
334	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
335	C₂×C₂²⋊C₄	—	2	8	no	—
336	C₂×C₂²⋊C₄	—	2	4	no	—
337	C₂×C₂²⋊C₄	—	2	4	no	—
338	C₂×C₂²⋊C₄	—	2	8	no	—
339	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
341	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
342	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
343	C₂×C₄²	—	2	4	no	—
344	C₂×C₄²	—	2	8	no	—
345	C₂×C₄²	D₄×C₆₇	1	8	yes	—
346	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
347	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
348	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
349	#95	C₂³×C₆₇	1	8	yes	—
350	#95	C₂³×C₆₇	1	8	yes	—
351	#95	C₂³×C₆₇	1	4	yes	—
352	#95	C₂³×C₆₇	1	4	yes	—
353	#95	C₂³×C₆₇	1	8	yes	—
354	#95	C₂³×C₆₇	1	8	yes	—
355	#95	C₂³×C₆₇	1	8	yes	—
356	#95	C₂³×C₆₇	1	8	yes	—
357	C₂⁴	—	2	2	no	—
358	C₂⁴	—	2	4	no	—
359	C₂⁴	—	2	2	no	—
360	C₂⁴	—	2	4	no	—
361	C₂⁴	#41×C₆₇	1	4	yes	—
362	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
363	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
364	C₂×Q₈	—	2	32	no	—
365	C₂×Q₈	—	2	8	no	—
366	C₂×Q₈	—	2	16	no	—
367	C₂²×C₄	—	2	32	no	—
368	C₂²×C₄	—	2	4	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	8	no	—
371	C₂²×C₄	—	2	2	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	4	no	—
377	C₂²×C₄	—	2	8	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	2	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	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
393	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
394	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
395	C₂²×C₄	#41×C₆₇	1	8	yes	—
396	C₂²×C₄	#41×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	2	yes	—
400	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
401	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
404	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
407	C₄⋊C₄	—	2	8	no	—
408	C₄⋊C₄	—	2	8	no	—
409	C₄⋊C₄	—	2	16	no	—
410	C₄⋊C₄	—	2	8	no	—
411	C₄⋊C₄	—	2	16	no	—
412	C₄⋊C₄	—	2	16	no	—
413	C₂²⋊C₄	—	2	16	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	16	no	—
417	C₄²	—	2	8	no	—
418	C₄²	—	2	16	no	—
419	C₄²	C₂×D₄×C₆₇	1	16	yes	—
420	C₂³	—	2	8	no	—
421	C₂³	—	2	16	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	4	4	no	—
428	C₂³	—	4	8	no	—
429	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
430	C₂³	C₂×#41×C₆₇	1	4	yes	—
431	C₂³	C₂×#41×C₆₇	1	2	yes	—
432	C₂³	C₂×#41×C₆₇	1	4	yes	—
433	C₂³	C₂×#41×C₆₇	1	2	yes	—
434	C₂³	2₋¹⁺⁴×C₆₇	1	4	yes	—
435	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
436	Q₈	—	2	32	no	—
437	C₂×C₄	—	2	32	no	—
438	C₂×C₄	—	2	8	no	—
439	C₂×C₄	—	2	8	no	—
440	C₂×C₄	—	2	8	no	—
441	C₂×C₄	—	2	16	no	—
442	C₂×C₄	—	2	8	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	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	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	2	4	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
470	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
471	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
472	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
473	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
474	C₂²	—	2	4	no	—
475	C₂²	—	2	8	no	—
476	C₂²	—	4	16	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	4	no	—
480	C₂²	#525×C₆₇	1	4	yes	—
481	C₂²	#525×C₆₇	1	4	yes	—
482	C₂²	#536×C₆₇	1	4	yes	—
483	C₂²	#552×C₆₇	1	8	yes	—
484	C₂²	#556×C₆₇	1	8	yes	—
485	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
486	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
487	C₂²	C₂²×#41×C₆₇	1	2	yes	—
488	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
489	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
490	C₄	—	2	4	no	—
491	C₄	—	2	4	no	—
492	C₄	—	2	4	no	—
493	C₄	—	2	8	no	—
494	C₄	—	2	4	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	4	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	4	no	—
499	C₄	—	4	4	no	—
500	C₂	—	4	4	no	—
501	C₂	C₆₇×#2599	1	8	yes	direct product factor
502	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
503	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
504	C₂	C₂×#536×C₆₇	1	1	yes	characteristic
505	C₂	C₂×#552×C₆₇	1	2	yes	—
506	C₂	C₂×#556×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁵×C₆₇	C₂³
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×C₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2599	C₂	direct (N × H)
4. C₂×#398×C₆₇	C₂	semidirect
6. C₂×#395×C₆₇	C₂	semidirect
9. C₂×#387×C₆₇	C₂	semidirect
10. C₂×#387×C₆₇	C₂	semidirect
12. C₂×#383×C₆₇	C₂	semidirect
16. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
32. #398×C₆₇	C₂²	semidirect
34. #395×C₆₇	C₂²	semidirect
37. #387×C₆₇	C₂²	semidirect
38. #387×C₆₇	C₂²	semidirect
40. #383×C₆₇	C₂²	semidirect
246. C₂×#2599	C₆₇	direct (N × H)
248. C₂×C₆₇	#2599	direct (N × H)
254. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
256. #2599	C₂×C₆₇	direct (N × H)
257. C₂×#398	C₂×C₆₇	semidirect
259. C₂×#395	C₂×C₆₇	semidirect
262. C₂×#387	C₂×C₆₇	semidirect
263. C₂×#387	C₂×C₆₇	semidirect
265. C₂×#383	C₂×C₆₇	semidirect
266. C₆₇	C₂×#2599	direct (N × H)
270. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
286. #398	C₂²×C₆₇	semidirect
288. #395	C₂²×C₆₇	semidirect
291. #387	C₂²×C₆₇	semidirect
292. #387	C₂²×C₆₇	semidirect
294. #383	C₂²×C₆₇	semidirect
501. C₂	C₆₇×#2599	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×C₄×Q₈×C₆₇	2
C₂²×#117×C₆₇	2
C₆₇×#2599	2
C₂×#398×C₆₇	2
C₂×#397×C₆₇	2
C₂×#395×C₆₇	2
C₂×#393×C₆₇	2
C₂×#388×C₆₇	2
C₂×#387×C₆₇	2
C₂×#385×C₆₇	2
C₂×#383×C₆₇	2
C₂×#2599	67

Subgroups — C2×C67×#2599 #30714115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2599 #30714115