Subgroups — C₂×C₆₇×#2995 #30739513

Order 17152.

1882 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2995	C₂	1	16	yes	maximal, direct product factor
3	C₂×#446×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#435×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#433×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×#432×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#424×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
15	#529×C₆₇	C₄	1	8	yes	—
16	#529×C₆₇	C₂²	1	8	yes	—
17	C₂×#126×C₆₇	C₄	1	2	yes	—
18	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
20	C₂×#123×C₆₇	C₄	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
22	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
26	C₂×#117×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
28	#446×C₆₇	C₂²	1	16	yes	complemented
29	#436×C₆₇	C₂²	1	16	yes	complemented
30	#435×C₆₇	C₂²	1	16	yes	complemented
31	#435×C₆₇	C₂²	1	16	yes	complemented
32	#433×C₆₇	C₂²	1	16	yes	complemented
33	#432×C₆₇	C₂²	1	8	yes	complemented
34	#424×C₆₇	C₂²	1	16	yes	complemented
35	#423×C₆₇	C₂²	1	8	yes	complemented
36	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
38	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
41	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
43	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
45	C₂×#97×C₆₇	C₂²	1	2	yes	—
46	C₂×#97×C₆₇	C₂²	1	2	yes	—
47	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
48	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
55	C₂×M₁₆×C₆₇	—	4	2	no	—
56	C₂×M₁₆×C₆₇	—	4	4	no	—
57	C₂×M₁₆×C₆₇	—	4	2	no	—
58	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
59	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
60	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
61	C₂²×C₈×C₆₇	—	4	4	no	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
67	#126×C₆₇	C₂×C₄	1	16	yes	—
68	#125×C₆₇	C₂×C₄	1	8	yes	—
69	#124×C₆₇	C₂×C₄	1	8	yes	—
70	#123×C₆₇	C₂×C₄	1	16	yes	—
71	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
72	#121×C₆₇	C₂×C₄	1	8	yes	—
73	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
74	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
75	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
76	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
77	C₄×D₄×C₆₇	C₂³	1	4	yes	—
78	C₄×D₄×C₆₇	C₂³	1	8	yes	—
79	#117×C₆₇	C₂×C₄	1	8	yes	—
80	#117×C₆₇	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
83	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
85	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
87	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
92	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
95	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
99	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
100	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
101	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
102	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
103	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
104	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
105	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
106	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
107	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
108	#97×C₆₇	C₂³	1	8	yes	—
109	#97×C₆₇	C₂³	1	8	yes	—
110	C₄×C₈×C₆₇	C₂³	1	8	yes	—
111	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
112	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×Q₈×C₆₇	—	4	2	no	—
114	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
115	C₂×D₄×C₆₇	—	4	2	no	—
116	C₂×D₄×C₆₇	—	4	4	no	—
117	C₂×D₄×C₆₇	C₂²×C₄	1	8	yes	—
118	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	4	2	no	—
124	C₂²×C₄×C₆₇	—	4	2	no	—
125	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
128	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
133	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	M₁₆×C₆₇	—	4	4	no	—
141	M₁₆×C₆₇	—	4	8	no	—
142	M₁₆×C₆₇	—	4	4	no	—
143	C₂×C₈×C₆₇	—	2	4	no	—
144	C₂×C₈×C₆₇	—	2	4	no	—
145	C₂×C₈×C₆₇	—	2	4	no	—
146	C₂×C₈×C₆₇	—	2	4	no	—
147	C₂×C₈×C₆₇	—	4	8	no	—
148	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
149	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
150	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
151	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
152	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
153	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
154	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
155	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
158	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
159	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
160	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
161	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
162	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
163	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
164	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
165	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
166	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
167	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
168	C₄²×C₆₇	C₂⁴	1	2	yes	—
169	C₄²×C₆₇	C₂⁴	1	2	yes	—
170	C₄²×C₆₇	C₂⁴	1	4	yes	—
171	C₂³×C₆₇	—	2	4	no	—
172	C₂³×C₆₇	—	2	2	no	—
173	C₂³×C₆₇	—	2	2	no	—
174	C₂³×C₆₇	—	2	2	no	—
175	C₂³×C₆₇	—	2	4	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
178	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
179	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
180	Q₈×C₆₇	—	4	4	no	—
181	D₄×C₆₇	—	4	4	no	—
182	D₄×C₆₇	—	4	8	no	—
183	C₂×C₄×C₆₇	—	2	4	no	—
184	C₂×C₄×C₆₇	—	2	2	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	8	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	4	4	no	—
198	C₂×C₄×C₆₇	—	4	4	no	—
199	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
200	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
201	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
202	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
203	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
204	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
205	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
206	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
207	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
209	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
210	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
211	C₈×C₆₇	—	2	4	no	—
212	C₈×C₆₇	—	2	4	no	—
213	C₈×C₆₇	—	2	4	no	—
214	C₈×C₆₇	—	2	4	no	—
215	C₂²×C₆₇	—	2	4	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	2	2	no	—
219	C₂²×C₆₇	—	2	4	no	—
220	C₂²×C₆₇	—	2	4	no	—
221	C₂²×C₆₇	—	4	4	no	—
222	C₂²×C₆₇	—	4	8	no	—
223	C₂²×C₆₇	#520	1	2	yes	—
224	C₂²×C₆₇	#568	1	2	yes	—
225	C₂²×C₆₇	#568	1	2	yes	—
226	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
227	C₄×C₆₇	—	2	4	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	2	2	no	—
231	C₄×C₆₇	—	2	2	no	—
232	C₄×C₆₇	—	2	2	no	—
233	C₄×C₆₇	—	2	4	no	—
234	C₄×C₆₇	—	2	4	no	—
235	C₄×C₆₇	#520	1	2	yes	—
236	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
237	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
238	C₂×#2995	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂×C₆₇	—	4	2	no	—
240	C₂×C₆₇	—	4	4	no	—
241	C₂×C₆₇	#2995	1	4	yes	direct product factor
242	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
243	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
244	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
245	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
246	#2995	C₂×C₆₇	1	16	yes	direct product factor
247	C₂×#446	C₂×C₆₇	1	2	yes	complemented
248	C₂×#436	C₂×C₆₇	1	2	yes	complemented
249	C₂×#435	C₂×C₆₇	1	2	yes	complemented
250	C₂×#435	C₂×C₆₇	1	2	yes	complemented
251	C₂×#433	C₂×C₆₇	1	2	yes	complemented
252	C₂×#432	C₂×C₆₇	1	1	yes	characteristic, complemented
253	C₂×#424	C₂×C₆₇	1	2	yes	complemented
254	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
255	C₆₇	C₂×#2995	1	1	yes	characteristic, direct product factor, Sylow
256	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
257	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
258	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
259	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
260	#529	C₄×C₆₇	1	8	yes	—
261	#529	C₂²×C₆₇	1	8	yes	—
262	C₂×#126	C₄×C₆₇	1	2	yes	—
263	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
264	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
265	C₂×#123	C₄×C₆₇	1	2	yes	—
266	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
267	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
268	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
270	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
271	C₂×#117	C₂²×C₆₇	1	2	yes	—
272	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
273	#446	C₂²×C₆₇	1	16	yes	complemented
274	#436	C₂²×C₆₇	1	16	yes	complemented
275	#435	C₂²×C₆₇	1	16	yes	complemented
276	#435	C₂²×C₆₇	1	16	yes	complemented
277	#433	C₂²×C₆₇	1	16	yes	complemented
278	#432	C₂²×C₆₇	1	8	yes	complemented
279	#424	C₂²×C₆₇	1	16	yes	complemented
280	#423	C₂²×C₆₇	1	8	yes	complemented
281	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
282	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
283	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
285	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
286	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
287	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
288	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
289	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
290	C₂×#97	C₂²×C₆₇	1	2	yes	—
291	C₂×#97	C₂²×C₆₇	1	2	yes	—
292	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
293	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
294	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
295	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
296	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
297	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
299	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
300	C₂×M₁₆	—	4	2	no	—
301	C₂×M₁₆	—	4	4	no	—
302	C₂×M₁₆	—	4	2	no	—
303	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
304	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
305	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
306	C₂²×C₈	—	4	4	no	—
307	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
308	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
309	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
310	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
311	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
312	#126	C₂×C₄×C₆₇	1	16	yes	—
313	#125	C₂×C₄×C₆₇	1	8	yes	—
314	#124	C₂×C₄×C₆₇	1	8	yes	—
315	#123	C₂×C₄×C₆₇	1	16	yes	—
316	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
317	#121	C₂×C₄×C₆₇	1	8	yes	—
318	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
319	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
320	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
321	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
322	C₄×D₄	C₂³×C₆₇	1	4	yes	—
323	C₄×D₄	C₂³×C₆₇	1	8	yes	—
324	#117	C₂×C₄×C₆₇	1	8	yes	—
325	#117	C₂³×C₆₇	1	8	yes	—
326	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
327	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
328	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
329	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
330	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
331	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
332	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
333	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
334	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
336	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
337	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
338	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
339	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
340	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
342	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
343	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
344	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
345	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
346	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
347	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
348	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
349	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
350	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
351	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
352	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
353	#97	C₂³×C₆₇	1	8	yes	—
354	#97	C₂³×C₆₇	1	8	yes	—
355	C₄×C₈	C₂³×C₆₇	1	8	yes	—
356	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
357	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
358	C₂×Q₈	—	4	2	no	—
359	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
360	C₂×D₄	—	4	2	no	—
361	C₂×D₄	—	4	4	no	—
362	C₂×D₄	C₂²×C₄×C₆₇	1	8	yes	—
363	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
364	C₂²×C₄	—	2	4	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	4	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	4	2	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
371	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
372	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
373	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
374	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
375	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
376	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
377	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
378	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
379	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
380	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
381	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
382	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
383	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
384	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
385	M₁₆	—	4	4	no	—
386	M₁₆	—	4	8	no	—
387	M₁₆	—	4	4	no	—
388	C₂×C₈	—	2	4	no	—
389	C₂×C₈	—	2	4	no	—
390	C₂×C₈	—	2	4	no	—
391	C₂×C₈	—	2	4	no	—
392	C₂×C₈	—	4	8	no	—
393	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
394	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
395	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
396	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
397	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
398	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
399	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
400	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
401	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
402	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
403	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
404	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
405	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
406	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
407	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
408	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
409	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
410	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
411	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
412	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
413	C₄²	C₂⁴×C₆₇	1	4	yes	—
414	C₄²	C₂⁴×C₆₇	1	2	yes	—
415	C₄²	C₂⁴×C₆₇	1	2	yes	—
416	C₂³	—	2	4	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	4	no	—
421	C₂³	—	2	4	no	—
422	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
423	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
424	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
425	Q₈	—	4	4	no	—
426	D₄	—	4	4	no	—
427	D₄	—	4	8	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	4	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	4	no	—
438	C₂×C₄	—	2	4	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	4	no	—
442	C₂×C₄	—	4	4	no	—
443	C₂×C₄	—	4	4	no	—
444	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
445	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
446	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
447	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
448	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
449	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
450	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
451	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
452	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
453	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
454	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
455	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
456	C₈	—	2	4	no	—
457	C₈	—	2	4	no	—
458	C₈	—	2	4	no	—
459	C₈	—	2	4	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	4	no	—
463	C₂²	—	2	4	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	4	4	no	—
467	C₂²	—	4	8	no	—
468	C₂²	#520×C₆₇	1	2	yes	—
469	C₂²	#568×C₆₇	1	2	yes	—
470	C₂²	#568×C₆₇	1	2	yes	—
471	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
472	C₄	—	2	2	no	—
473	C₄	—	2	2	no	—
474	C₄	—	2	2	no	—
475	C₄	—	2	2	no	—
476	C₄	—	2	4	no	—
477	C₄	—	2	4	no	—
478	C₄	—	2	4	no	—
479	C₄	—	2	2	no	—
480	C₄	#520×C₆₇	1	2	yes	—
481	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
482	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
483	C₂	—	4	2	no	—
484	C₂	—	4	4	no	—
485	C₂	C₆₇×#2995	1	4	yes	direct product factor
486	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
487	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
488	C₂	C₂×#568×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂×#121×C₆₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2995	C₂	direct (N × H)
3. C₂×#446×C₆₇	C₂	semidirect
4. C₂×#436×C₆₇	C₂	semidirect
5. C₂×#435×C₆₇	C₂	semidirect
6. C₂×#435×C₆₇	C₂	semidirect
7. C₂×#433×C₆₇	C₂	semidirect
8. C₂×#432×C₆₇	C₂	semidirect
9. C₂×#424×C₆₇	C₂	semidirect
10. C₂×#423×C₆₇	C₂	semidirect
28. #446×C₆₇	C₂²	semidirect
29. #436×C₆₇	C₂²	semidirect
30. #435×C₆₇	C₂²	semidirect
31. #435×C₆₇	C₂²	semidirect
32. #433×C₆₇	C₂²	semidirect
33. #432×C₆₇	C₂²	semidirect
34. #424×C₆₇	C₂²	semidirect
35. #423×C₆₇	C₂²	semidirect
238. C₂×#2995	C₆₇	direct (N × H)
241. C₂×C₆₇	#2995	direct (N × H)
246. #2995	C₂×C₆₇	direct (N × H)
247. C₂×#446	C₂×C₆₇	semidirect
248. C₂×#436	C₂×C₆₇	semidirect
249. C₂×#435	C₂×C₆₇	semidirect
250. C₂×#435	C₂×C₆₇	semidirect
251. C₂×#433	C₂×C₆₇	semidirect
252. C₂×#432	C₂×C₆₇	semidirect
253. C₂×#424	C₂×C₆₇	semidirect
254. C₂×#423	C₂×C₆₇	semidirect
255. C₆₇	C₂×#2995	direct (N × H)
273. #446	C₂²×C₆₇	semidirect
274. #436	C₂²×C₆₇	semidirect
275. #435	C₂²×C₆₇	semidirect
276. #435	C₂²×C₆₇	semidirect
277. #433	C₂²×C₆₇	semidirect
278. #432	C₂²×C₆₇	semidirect
279. #424	C₂²×C₆₇	semidirect
280. #423	C₂²×C₆₇	semidirect
485. C₂	C₆₇×#2995	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×#529×C₆₇	2
C₆₇×#2995	2
C₂×#446×C₆₇	2
C₂×#436×C₆₇	2
C₂×#435×C₆₇	2
C₂×#433×C₆₇	2
C₂×#432×C₆₇	2
C₂×#424×C₆₇	2
C₂×#423×C₆₇	2
C₂×#2995	67

Subgroups — C2×C67×#2995 #30739513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2995 #30739513