Subgroups — C₆₇×#22469 #30701720

Order 17152.

1500 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1984	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1869	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1817	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#336×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₆₇	C₂²	1	4	yes	—
11	C₄×#41×C₆₇	—	2	8	no	—
12	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
19	C₂×#117×C₆₇	C₄	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
22	C₈×D₄×C₆₇	—	2	8	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
28	#336×C₆₇	C₂²	1	1	yes	characteristic
29	#336×C₆₇	C₂²	1	1	yes	characteristic
30	#336×C₆₇	C₂²	1	2	yes	—
31	#336×C₆₇	C₂²	1	4	yes	—
32	#336×C₆₇	C₂²	1	4	yes	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	—	2	4	no	—
35	C₂×#41×C₆₇	C₈	1	8	yes	complemented
36	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₆₇	—	2	8	no	—
46	C₂²×C₈×C₆₇	—	2	8	no	—
47	C₂²×C₈×C₆₇	—	4	4	no	—
48	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄×Q₈×C₆₇	—	2	4	no	—
52	C₄×Q₈×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	—	2	4	no	—
54	C₄×D₄×C₆₇	—	2	8	no	—
55	C₄×D₄×C₆₇	—	2	8	no	—
56	C₄×D₄×C₆₇	D₄	1	8	yes	—
57	#117×C₆₇	—	2	4	no	—
58	#117×C₆₇	—	2	8	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₆₇	—	2	8	no	—
67	C₂×C₄²×C₆₇	—	2	4	no	—
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₈×C₆₇	—	2	4	no	—
76	C₂²⋊C₈×C₆₇	—	2	8	no	—
77	C₂²⋊C₈×C₆₇	—	2	4	no	—
78	C₂²⋊C₈×C₆₇	—	4	2	no	—
79	C₄×C₈×C₆₇	—	2	4	no	—
80	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
82	#41×C₆₇	—	2	32	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	complemented
86	C₂×D₄×C₆₇	—	2	4	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
89	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
90	C₂×D₄×C₆₇	C₂×C₈	1	8	yes	complemented
91	C₂×D₄×C₆₇	M₁₆	1	4	yes	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	4	no	—
100	C₂²×C₄×C₆₇	—	2	2	no	—
101	C₂²×C₄×C₆₇	—	2	2	no	—
102	C₂²×C₄×C₆₇	—	2	4	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	1	no	—
107	C₂²×C₄×C₆₇	—	4	1	no	—
108	C₂²×C₄×C₆₇	—	4	2	no	—
109	C₂²×C₄×C₆₇	C₂×C₈	1	8	yes	complemented
110	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	complemented
111	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
120	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
121	C₂×C₈×C₆₇	—	2	2	no	—
122	C₂×C₈×C₆₇	—	2	2	no	—
123	C₂×C₈×C₆₇	—	2	4	no	—
124	C₂×C₈×C₆₇	—	2	2	no	—
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	4	no	—
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	4	16	no	—
130	C₂×C₈×C₆₇	#41	1	4	yes	complemented
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	2	no	—
134	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
135	C₂²⋊C₄×C₆₇	—	2	4	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₂²⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
138	C₄²×C₆₇	—	2	4	no	—
139	C₄²×C₆₇	—	2	2	no	—
140	C₄²×C₆₇	—	2	4	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	—	2	2	no	—
143	C₄²×C₆₇	—	2	4	no	—
144	C₄²×C₆₇	C₂×D₄	1	4	yes	—
145	C₂³×C₆₇	—	2	2	no	—
146	C₂³×C₆₇	—	2	2	no	—
147	C₂³×C₆₇	—	2	2	no	—
148	C₂³×C₆₇	—	2	2	no	—
149	C₂³×C₆₇	—	2	1	no	—
150	C₂³×C₆₇	—	2	1	no	—
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	1	no	—
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
156	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
157	C₂³×C₆₇	C₂²×C₈	1	4	yes	—
158	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
159	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
160	Q₈×C₆₇	—	2	8	no	—
161	D₄×C₆₇	—	2	16	no	—
162	D₄×C₆₇	C₂²⋊C₈	1	16	yes	complemented
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	2	2	no	—
167	C₂×C₄×C₆₇	—	2	2	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	1	no	—
173	C₂×C₄×C₆₇	—	2	16	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	2	8	no	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	1	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	8	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
188	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
189	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
191	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
192	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
193	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
194	C₂×C₄×C₆₇	#131	1	2	yes	—
195	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
197	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
199	C₈×C₆₇	—	2	4	no	—
200	C₈×C₆₇	—	4	4	no	—
201	C₈×C₆₇	—	4	2	no	—
202	C₂²×C₆₇	—	2	4	no	—
203	C₂²×C₆₇	—	2	4	no	—
204	C₂²×C₆₇	—	2	4	no	—
205	C₂²×C₆₇	—	2	4	no	—
206	C₂²×C₆₇	—	2	4	no	—
207	C₂²×C₆₇	—	2	4	no	—
208	C₂²×C₆₇	—	2	2	no	—
209	C₂²×C₆₇	—	2	2	no	—
210	C₂²×C₆₇	—	2	2	no	—
211	C₂²×C₆₇	—	4	1	no	—
212	C₂²×C₆₇	—	4	1	no	—
213	C₂²×C₆₇	—	4	1	no	—
214	C₂²×C₆₇	—	4	1	no	—
215	C₂²×C₆₇	C₂×C₂²⋊C₈	1	8	yes	—
216	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
218	C₂²×C₆₇	#520	1	1	yes	characteristic
219	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
220	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
222	C₂²×C₆₇	#568	1	1	yes	characteristic
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	4	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	2	no	—
228	C₄×C₆₇	—	4	1	no	—
229	C₄×C₆₇	—	4	2	no	—
230	C₄×C₆₇	C₂×C₂²⋊C₈	1	4	yes	—
231	C₄×C₆₇	#546	1	2	yes	—
232	C₄×C₆₇	#547	1	2	yes	—
233	#22469	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂×C₆₇	—	2	8	no	—
235	C₂×C₆₇	—	4	2	no	—
236	C₂×C₆₇	#2339	1	1	yes	characteristic
237	C₂×C₆₇	C₂²×C₂²⋊C₈	1	1	yes	characteristic
238	C₂×C₆₇	#2880	1	1	yes	characteristic
239	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
240	C₂×C₆₇	#2970	1	1	yes	characteristic
241	C₂×C₆₇	#2988	1	1	yes	characteristic
242	C₂×C₆₇	#2990	1	1	yes	characteristic
243	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
244	C₂×C₈×D₄	C₂×C₆₇	1	2	yes	complemented
245	#1984	C₂×C₆₇	1	2	yes	complemented
246	#1869	C₂×C₆₇	1	4	yes	complemented
247	#1817	C₂×C₆₇	1	2	yes	complemented
248	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
249	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₂×#336	C₂×C₆₇	1	2	yes	complemented
251	C₆₇	#22469	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic
253	C₂³×C₈	C₂²×C₆₇	1	4	yes	—
254	C₄×#41	—	2	8	no	—
255	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
256	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
259	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
260	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
261	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
262	C₂×#117	C₄×C₆₇	1	2	yes	—
263	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
264	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
265	C₈×D₄	—	2	8	no	—
266	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
267	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
268	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
269	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
270	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
271	#336	C₂²×C₆₇	1	2	yes	—
272	#336	C₂²×C₆₇	1	1	yes	characteristic
273	#336	C₂²×C₆₇	1	4	yes	—
274	#336	C₂²×C₆₇	1	4	yes	—
275	#336	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#41	—	2	4	no	—
277	C₂×#41	—	2	4	no	—
278	C₂×#41	C₈×C₆₇	1	8	yes	complemented
279	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
280	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
281	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
282	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
283	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
284	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
285	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
286	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
288	C₂²×C₈	—	2	8	no	—
289	C₂²×C₈	—	2	8	no	—
290	C₂²×C₈	—	4	4	no	—
291	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
292	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
293	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
294	C₄×Q₈	—	2	4	no	—
295	C₄×Q₈	—	2	4	no	—
296	C₄×D₄	—	2	4	no	—
297	C₄×D₄	—	2	8	no	—
298	C₄×D₄	—	2	8	no	—
299	C₄×D₄	D₄×C₆₇	1	8	yes	—
300	#117	—	2	4	no	—
301	#117	—	2	8	no	—
302	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
304	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
305	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
309	C₂×C₄²	—	2	8	no	—
310	C₂×C₄²	—	2	4	no	—
311	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
312	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
313	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
314	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
315	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
316	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
317	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
318	C₄⋊C₈	—	2	4	no	—
319	C₂²⋊C₈	—	2	8	no	—
320	C₂²⋊C₈	—	2	4	no	—
321	C₂²⋊C₈	—	4	2	no	—
322	C₄×C₈	—	2	4	no	—
323	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
324	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
325	#41	—	2	32	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	C₂×C₈×C₆₇	1	4	yes	complemented
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
332	C₂×D₄	C₂×C₈×C₆₇	1	4	yes	—
333	C₂×D₄	C₂×C₈×C₆₇	1	8	yes	complemented
334	C₂×D₄	M₁₆×C₆₇	1	4	yes	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	4	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	4	1	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	C₂×C₈×C₆₇	1	8	yes	complemented
353	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	complemented
354	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
355	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
356	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
357	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
360	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
363	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	4	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	4	no	—
370	C₂×C₈	—	2	4	no	—
371	C₂×C₈	—	2	2	no	—
372	C₂×C₈	—	4	16	no	—
373	C₂×C₈	#41×C₆₇	1	4	yes	complemented
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	4	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
378	C₂²⋊C₄	—	2	4	no	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
381	C₄²	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	4	no	—
384	C₄²	—	2	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	4	no	—
387	C₄²	C₂×D₄×C₆₇	1	4	yes	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
399	C₂³	C₂²×C₈×C₆₇	1	2	yes	—
400	C₂³	C₂²×C₈×C₆₇	1	4	yes	—
401	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
402	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
403	Q₈	—	2	8	no	—
404	D₄	—	2	16	no	—
405	D₄	C₂²⋊C₈×C₆₇	1	16	yes	complemented
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	16	no	—
417	C₂×C₄	—	2	8	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	4	8	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
431	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
432	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
433	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
434	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
435	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
436	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
437	C₂×C₄	#131×C₆₇	1	2	yes	—
438	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
439	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
440	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
441	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
442	C₈	—	2	4	no	—
443	C₈	—	4	4	no	—
444	C₈	—	4	2	no	—
445	C₂²	—	2	4	no	—
446	C₂²	—	2	4	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	4	1	no	—
458	C₂²	C₂×C₂²⋊C₈×C₆₇	1	8	yes	—
459	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
460	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
461	C₂²	#520×C₆₇	1	1	yes	characteristic
462	C₂²	C₂³×C₈×C₆₇	1	1	yes	characteristic
463	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
464	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
465	C₂²	#568×C₆₇	1	1	yes	characteristic
466	C₄	—	2	4	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	4	no	—
469	C₄	—	2	2	no	—
470	C₄	—	2	2	no	—
471	C₄	—	4	1	no	—
472	C₄	—	4	2	no	—
473	C₄	C₂×C₂²⋊C₈×C₆₇	1	4	yes	—
474	C₄	#546×C₆₇	1	2	yes	—
475	C₄	#547×C₆₇	1	2	yes	—
476	C₂	—	2	8	no	—
477	C₂	—	4	2	no	—
478	C₂	C₆₇×#2339	1	1	yes	characteristic
479	C₂	C₂²×C₂²⋊C₈×C₆₇	1	1	yes	characteristic
480	C₂	C₆₇×#2880	1	1	yes	characteristic
481	C₂	C₈×#41×C₆₇	1	1	yes	characteristic
482	C₂	C₆₇×#2970	1	1	yes	characteristic
483	C₂	C₆₇×#2988	1	1	yes	characteristic
484	C₂	C₆₇×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂²	C₂³×C₈×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×#41×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. C₂×C₈×D₄×C₆₇	C₂	semidirect
3. C₆₇×#1984	C₂	semidirect
4. C₆₇×#1869	C₂	semidirect
5. C₆₇×#1817	C₂	semidirect
6. C₆₇×#1797	C₂	semidirect
7. C₄×C₂²⋊C₈×C₆₇	C₂	semidirect
8. C₂×#336×C₆₇	C₂	semidirect
35. C₂×#41×C₆₇	C₈	semidirect
85. C₂×Q₈×C₆₇	C₂×C₈	semidirect
90. C₂×D₄×C₆₇	C₂×C₈	semidirect
109. C₂²×C₄×C₆₇	C₂×C₈	semidirect
110. C₂²×C₄×C₆₇	C₂×C₈	semidirect
130. C₂×C₈×C₆₇	#41	semidirect
162. D₄×C₆₇	C₂²⋊C₈	semidirect
233. #22469	C₆₇	direct (N × H)
244. C₂×C₈×D₄	C₂×C₆₇	semidirect
245. #1984	C₂×C₆₇	semidirect
246. #1869	C₂×C₆₇	semidirect
247. #1817	C₂×C₆₇	semidirect
248. #1797	C₂×C₆₇	semidirect
249. C₄×C₂²⋊C₈	C₂×C₆₇	semidirect
250. C₂×#336	C₂×C₆₇	semidirect
251. C₆₇	#22469	direct (N × H)
278. C₂×#41	C₈×C₆₇	semidirect
328. C₂×Q₈	C₂×C₈×C₆₇	semidirect
333. C₂×D₄	C₂×C₈×C₆₇	semidirect
352. C₂²×C₄	C₂×C₈×C₆₇	semidirect
353. C₂²×C₄	C₂×C₈×C₆₇	semidirect
373. C₂×C₈	#41×C₆₇	semidirect
405. D₄	C₂²⋊C₈×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×#41×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₆₇×#1984	2
C₆₇×#1869	2
C₆₇×#1817	2
C₆₇×#1797	2
C₄×C₂²⋊C₈×C₆₇	2
C₂×#336×C₆₇	2
#22469	67

Subgroups — C67×#22469 #30701720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22469 #30701720