Subgroups — C₆₇×#22532 #30701783

Order 17152.

1364 conjugacy classes of subgroups.

The subgroup lattice has 484 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×#446×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1955	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1954	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1797	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1749	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₆₇	C₄	1	1	yes	characteristic
10	C₄×#41×C₆₇	—	2	8	no	—
11	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₆₇	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	C₄	1	2	yes	—
18	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
21	#446×C₆₇	—	2	8	no	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	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	4	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×#97×C₆₇	C₂²	1	4	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	#336×C₆₇	C₂²	1	1	yes	characteristic
31	#336×C₆₇	C₂²	1	1	yes	characteristic
32	#336×C₆₇	C₂²	1	2	yes	—
33	C₂×#41×C₆₇	—	2	4	no	—
34	C₂×#41×C₆₇	—	2	4	no	—
35	C₂×#41×C₆₇	—	2	4	no	—
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	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₄×Q₈×C₆₇	—	2	4	no	—
49	C₄×Q₈×C₆₇	D₄	1	8	yes	—
50	C₄×D₄×C₆₇	—	2	8	no	—
51	C₄×D₄×C₆₇	—	2	4	no	—
52	C₄×D₄×C₆₇	—	2	4	no	—
53	C₄×D₄×C₆₇	—	2	8	no	—
54	#117×C₆₇	—	2	8	no	—
55	#117×C₆₇	—	2	4	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₆₇	—	2	4	no	—
64	C₂×C₄²×C₆₇	—	2	8	no	—
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₄⋊C₈×C₆₇	—	2	8	no	—
74	C₂²⋊C₈×C₆₇	—	2	4	no	—
75	C₂²⋊C₈×C₆₇	—	2	8	no	—
76	C₂²⋊C₈×C₆₇	—	4	4	no	—
77	C₂²⋊C₈×C₆₇	—	4	2	no	—
78	#97×C₆₇	—	2	8	no	—
79	C₄×C₈×C₆₇	—	2	4	no	—
80	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	#41×C₆₇	—	2	32	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
86	C₂×D₄×C₆₇	—	2	4	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	2	4	no	—
89	C₂×D₄×C₆₇	—	2	2	no	—
90	C₂×D₄×C₆₇	—	2	2	no	—
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	4	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	2	2	no	—
98	C₂²×C₄×C₆₇	—	2	4	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	2	4	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	4	no	—
105	C₂²×C₄×C₆₇	—	2	1	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	4	2	no	—
110	C₂²×C₄×C₆₇	—	4	1	no	—
111	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
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	2	no	—
124	C₂×C₈×C₆₇	—	2	4	no	—
125	C₂×C₈×C₆₇	—	2	2	no	—
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	4	no	—
129	C₂×C₈×C₆₇	#41	1	4	yes	complemented
130	C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	2	no	—
132	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
133	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
134	C₂²⋊C₄×C₆₇	—	2	4	no	—
135	C₂²⋊C₄×C₆₇	—	2	4	no	—
136	C₂²⋊C₄×C₆₇	—	2	4	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	2	4	no	—
139	C₄²×C₆₇	—	2	4	no	—
140	C₄²×C₆₇	—	2	2	no	—
141	C₄²×C₆₇	—	2	2	no	—
142	C₄²×C₆₇	C₂×D₄	1	4	yes	—
143	C₄²×C₆₇	C₂×D₄	1	8	yes	—
144	C₂³×C₆₇	—	2	2	no	—
145	C₂³×C₆₇	—	2	1	no	—
146	C₂³×C₆₇	—	2	1	no	—
147	C₂³×C₆₇	—	2	2	no	—
148	C₂³×C₆₇	—	2	1	no	—
149	C₂³×C₆₇	—	2	2	no	—
150	C₂³×C₆₇	—	2	2	no	—
151	C₂³×C₆₇	—	2	1	no	—
152	C₂³×C₆₇	—	2	1	no	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	C₂×M₁₆	1	4	yes	—
157	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
158	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
159	Q₈×C₆₇	—	2	8	no	—
160	D₄×C₆₇	—	2	16	no	—
161	D₄×C₆₇	—	2	8	no	—
162	C₂×C₄×C₆₇	—	2	16	no	—
163	C₂×C₄×C₆₇	—	2	1	no	—
164	C₂×C₄×C₆₇	—	2	1	no	—
165	C₂×C₄×C₆₇	—	2	1	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	2	no	—
173	C₂×C₄×C₆₇	—	2	8	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	4	no	—
181	C₂×C₄×C₆₇	—	4	8	no	—
182	C₂×C₄×C₆₇	—	4	2	no	—
183	C₂×C₄×C₆₇	—	4	4	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	4	no	—
186	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
188	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
189	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
191	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
192	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
193	C₂×C₄×C₆₇	#131	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
196	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₆₇	—	2	4	no	—
199	C₈×C₆₇	—	4	2	no	—
200	C₈×C₆₇	—	4	4	no	—
201	C₂²×C₆₇	—	2	4	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	—	2	4	no	—
205	C₂²×C₆₇	—	2	2	no	—
206	C₂²×C₆₇	—	2	4	no	—
207	C₂²×C₆₇	—	4	1	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	4	1	no	—
210	C₂²×C₆₇	—	4	2	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	1	no	—
213	C₂²×C₆₇	—	4	2	no	—
214	C₂²×C₆₇	—	4	1	no	—
215	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
216	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
217	C₂²×C₆₇	#520	1	1	yes	characteristic
218	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
219	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
221	C₂²×C₆₇	#568	1	1	yes	characteristic
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	2	2	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	2	4	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	—	4	1	no	—
229	C₄×C₆₇	#407	1	4	yes	—
230	C₄×C₆₇	#548	1	2	yes	—
231	C₄×C₆₇	#550	1	2	yes	—
232	#22532	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂×C₆₇	—	4	2	no	—
234	C₂×C₆₇	—	4	4	no	—
235	C₂×C₆₇	#2341	1	1	yes	characteristic
236	C₂×C₆₇	C₂×#407	1	1	yes	characteristic
237	C₂×C₆₇	#2880	1	1	yes	characteristic
238	C₂×C₆₇	#2965	1	1	yes	characteristic
239	C₂×C₆₇	#2970	1	1	yes	characteristic
240	C₂×C₆₇	#2989	1	1	yes	characteristic
241	C₂×C₆₇	#2990	1	1	yes	characteristic
242	C₂×C₄×#41	C₂×C₆₇	1	1	yes	characteristic
243	C₂×#446	C₂×C₆₇	1	2	yes	complemented
244	#1955	C₂×C₆₇	1	4	yes	complemented
245	#1954	C₂×C₆₇	1	2	yes	complemented
246	#1797	C₂×C₆₇	1	2	yes	complemented
247	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#1749	C₂×C₆₇	1	2	yes	complemented
249	C₄×C₂²⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₆₇	#22532	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²×#41	C₄×C₆₇	1	1	yes	characteristic
252	C₄×#41	—	2	8	no	—
253	C₂×C₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×C₄×D₄	C₄×C₆₇	1	2	yes	—
256	C₂×C₄×D₄	C₄×C₆₇	1	1	yes	characteristic
257	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
258	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
259	C₂×#117	C₄×C₆₇	1	2	yes	—
260	C₂×#117	C₄×C₆₇	1	1	yes	characteristic
261	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
262	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
263	#446	—	2	8	no	—
264	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
265	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
266	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
267	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
268	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	—
269	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
270	C₂×#97	C₂²×C₆₇	1	4	yes	—
271	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
272	#336	C₂²×C₆₇	1	1	yes	characteristic
273	#336	C₂²×C₆₇	1	2	yes	—
274	#336	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#41	—	2	4	no	—
276	C₂×#41	—	2	4	no	—
277	C₂×#41	—	2	4	no	—
278	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
279	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
280	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
281	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
282	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
283	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
284	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
286	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
288	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
289	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
290	C₄×Q₈	—	2	4	no	—
291	C₄×Q₈	D₄×C₆₇	1	8	yes	—
292	C₄×D₄	—	2	8	no	—
293	C₄×D₄	—	2	4	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	2	8	no	—
296	#117	—	2	8	no	—
297	#117	—	2	4	no	—
298	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
299	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
301	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
302	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
303	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
304	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
305	C₂×C₄²	—	2	4	no	—
306	C₂×C₄²	—	2	8	no	—
307	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
309	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
310	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
314	C₄⋊C₈	—	2	4	no	—
315	C₄⋊C₈	—	2	8	no	—
316	C₂²⋊C₈	—	2	8	no	—
317	C₂²⋊C₈	—	2	4	no	—
318	C₂²⋊C₈	—	4	4	no	—
319	C₂²⋊C₈	—	4	2	no	—
320	#97	—	2	8	no	—
321	C₄×C₈	—	2	4	no	—
322	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
323	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
324	#41	—	2	32	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	M₁₆×C₆₇	1	4	yes	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	4	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	2	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
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	1	yes	characteristic
357	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
358	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
359	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	4	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	4	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	4	no	—
371	C₂×C₈	#41×C₆₇	1	4	yes	complemented
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
375	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	4	no	—
378	C₂²⋊C₄	—	2	4	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	2	4	no	—
381	C₄²	—	2	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	C₂×D₄×C₆₇	1	4	yes	—
385	C₄²	C₂×D₄×C₆₇	1	8	yes	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	1	no	—
388	C₂³	—	2	1	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	1	no	—
398	C₂³	C₂×M₁₆×C₆₇	1	4	yes	—
399	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
400	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
401	Q₈	—	2	8	no	—
402	D₄	—	2	16	no	—
403	D₄	—	2	8	no	—
404	C₂×C₄	—	2	16	no	—
405	C₂×C₄	—	2	1	no	—
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	1	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	8	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	4	no	—
428	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
429	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
430	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
431	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
432	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
433	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
434	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
435	C₂×C₄	#131×C₆₇	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
437	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
438	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
439	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
440	C₈	—	2	4	no	—
441	C₈	—	4	2	no	—
442	C₈	—	4	4	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	4	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	4	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	1	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	1	no	—
457	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
458	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
459	C₂²	#520×C₆₇	1	1	yes	characteristic
460	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
461	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
462	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
463	C₂²	#568×C₆₇	1	1	yes	characteristic
464	C₄	—	2	2	no	—
465	C₄	—	2	4	no	—
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	4	no	—
469	C₄	—	4	2	no	—
470	C₄	—	4	1	no	—
471	C₄	#407×C₆₇	1	4	yes	—
472	C₄	#548×C₆₇	1	2	yes	—
473	C₄	#550×C₆₇	1	2	yes	—
474	C₂	—	4	2	no	—
475	C₂	—	4	4	no	—
476	C₂	C₆₇×#2341	1	1	yes	characteristic
477	C₂	C₂×#407×C₆₇	1	1	yes	characteristic
478	C₂	C₆₇×#2880	1	1	yes	characteristic
479	C₂	C₆₇×#2965	1	1	yes	characteristic
480	C₂	C₆₇×#2970	1	1	yes	characteristic
481	C₂	C₆₇×#2989	1	1	yes	characteristic
482	C₂	C₆₇×#2990	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂²×#41×C₆₇	C₄
LCS	G ▷ C₂³ ▷ 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×#446×C₆₇	C₂	semidirect
3. C₆₇×#1955	C₂	semidirect
4. C₆₇×#1954	C₂	semidirect
5. C₆₇×#1797	C₂	semidirect
6. C₆₇×#1797	C₂	semidirect
7. C₆₇×#1749	C₂	semidirect
8. C₄×C₂²⋊C₈×C₆₇	C₂	semidirect
129. C₂×C₈×C₆₇	#41	semidirect
232. #22532	C₆₇	direct (N × H)
243. C₂×#446	C₂×C₆₇	semidirect
244. #1955	C₂×C₆₇	semidirect
245. #1954	C₂×C₆₇	semidirect
246. #1797	C₂×C₆₇	semidirect
247. #1797	C₂×C₆₇	semidirect
248. #1749	C₂×C₆₇	semidirect
249. C₄×C₂²⋊C₈	C₂×C₆₇	semidirect
250. C₆₇	#22532	direct (N × H)
371. C₂×C₈	#41×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×#446×C₆₇	2
C₆₇×#1955	2
C₆₇×#1954	2
C₆₇×#1797	2
C₆₇×#1749	2
C₄×C₂²⋊C₈×C₆₇	2
#22532	67

Subgroups — C67×#22532 #30701783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22532 #30701783