Subgroups — C₆₇×#22533 #30701784

Order 17152.

1108 conjugacy classes of subgroups.

The subgroup lattice has 512 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	C₂×#446×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1955	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1955	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1954	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1798	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1750	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1749	C₂	1	1	yes	maximal, characteristic, complemented
10	#529×C₆₇	—	2	8	no	—
11	C₂×#126×C₆₇	C₄	1	2	yes	—
12	C₂×#125×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
14	C₂×#123×C₆₇	C₄	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	#446×C₆₇	—	2	8	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
32	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
33	C₂×#97×C₆₇	C₂²	1	2	yes	—
34	C₂×#97×C₆₇	C₂²	1	2	yes	—
35	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
36	#336×C₆₇	C₂²	1	1	yes	characteristic
37	#336×C₆₇	C₂²	1	1	yes	characteristic
38	#336×C₆₇	C₂²	1	2	yes	—
39	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
41	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	#126×C₆₇	—	2	8	no	—
51	#125×C₆₇	—	2	4	no	—
52	#124×C₆₇	—	2	4	no	—
53	#123×C₆₇	—	2	8	no	—
54	C₂²⋊Q₈×C₆₇	—	2	4	no	—
55	#121×C₆₇	—	2	4	no	—
56	C₄×Q₈×C₆₇	D₄	1	8	yes	—
57	C₄×D₄×C₆₇	—	2	4	no	—
58	C₄×D₄×C₆₇	—	2	4	no	—
59	C₄×D₄×C₆₇	—	2	4	no	—
60	#117×C₆₇	—	2	4	no	—
61	#117×C₆₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
72	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₆₇	—	2	4	no	—
74	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
79	C₄⋊C₈×C₆₇	—	2	4	no	—
80	C₄⋊C₈×C₆₇	—	2	4	no	—
81	C₄⋊C₈×C₆₇	—	2	4	no	—
82	C₂²⋊C₈×C₆₇	—	2	4	no	—
83	C₂²⋊C₈×C₆₇	—	2	4	no	—
84	C₂²⋊C₈×C₆₇	—	2	4	no	—
85	C₂²⋊C₈×C₆₇	—	4	1	no	—
86	C₂²⋊C₈×C₆₇	—	4	1	no	—
87	C₂²⋊C₈×C₆₇	—	4	2	no	—
88	C₂²⋊C₈×C₆₇	—	4	2	no	—
89	#97×C₆₇	—	2	4	no	—
90	#97×C₆₇	—	2	4	no	—
91	C₄×C₈×C₆₇	—	2	4	no	—
92	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×Q₈×C₆₇	—	4	1	no	—
95	C₂×Q₈×C₆₇	—	4	1	no	—
96	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
97	C₂×D₄×C₆₇	—	2	4	no	—
98	C₂×D₄×C₆₇	—	2	4	no	—
99	C₂×D₄×C₆₇	—	2	2	no	—
100	C₂×D₄×C₆₇	—	4	1	no	—
101	C₂×D₄×C₆₇	—	4	1	no	—
102	C₂×D₄×C₆₇	—	4	2	no	—
103	C₂²×C₄×C₆₇	—	2	2	no	—
104	C₂²×C₄×C₆₇	—	2	2	no	—
105	C₂²×C₄×C₆₇	—	2	2	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	2	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	2	no	—
111	C₂²×C₄×C₆₇	—	2	2	no	—
112	C₂²×C₄×C₆₇	—	2	1	no	—
113	C₂²×C₄×C₆₇	—	2	2	no	—
114	C₂²×C₄×C₆₇	—	2	1	no	—
115	C₂²×C₄×C₆₇	—	2	2	no	—
116	C₂²×C₄×C₆₇	—	4	1	no	—
117	C₂²×C₄×C₆₇	—	4	1	no	—
118	C₂²×C₄×C₆₇	—	4	1	no	—
119	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
121	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂×C₈×C₆₇	—	2	2	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	2	no	—
132	C₂×C₈×C₆₇	—	2	2	no	—
133	C₂×C₈×C₆₇	—	2	2	no	—
134	C₂×C₈×C₆₇	—	2	2	no	—
135	C₂×C₈×C₆₇	—	2	2	no	—
136	C₂×C₈×C₆₇	—	2	2	no	—
137	C₂×C₈×C₆₇	—	2	2	no	—
138	C₂×C₈×C₆₇	—	2	2	no	—
139	C₂×C₈×C₆₇	#41	1	4	yes	—
140	C₄⋊C₄×C₆₇	—	2	4	no	—
141	C₄⋊C₄×C₆₇	—	2	4	no	—
142	C₄⋊C₄×C₆₇	—	2	2	no	—
143	C₄⋊C₄×C₆₇	—	2	4	no	—
144	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
145	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
146	C₂²⋊C₄×C₆₇	—	2	4	no	—
147	C₂²⋊C₄×C₆₇	—	2	4	no	—
148	C₂²⋊C₄×C₆₇	—	2	4	no	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	4	no	—
151	C₄²×C₆₇	—	2	2	no	—
152	C₄²×C₆₇	—	2	2	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	C₂×D₄	1	4	yes	—
155	C₄²×C₆₇	C₂×D₄	1	4	yes	—
156	C₄²×C₆₇	C₂×D₄	1	4	yes	—
157	C₂³×C₆₇	—	2	1	no	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	—	2	1	no	—
162	C₂³×C₆₇	—	2	1	no	—
163	C₂³×C₆₇	—	2	2	no	—
164	C₂³×C₆₇	—	2	2	no	—
165	C₂³×C₆₇	—	2	2	no	—
166	C₂³×C₆₇	—	2	2	no	—
167	C₂³×C₆₇	—	2	2	no	—
168	C₂³×C₆₇	—	2	2	no	—
169	C₂³×C₆₇	—	2	2	no	—
170	C₂³×C₆₇	#131	1	2	yes	—
171	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
172	Q₈×C₆₇	—	4	4	no	—
173	D₄×C₆₇	—	4	8	no	—
174	D₄×C₆₇	—	4	4	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	1	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	4	4	no	—
196	C₂×C₄×C₆₇	—	4	4	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	2	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
204	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
206	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
207	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
208	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
209	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
211	C₈×C₆₇	—	2	4	no	—
212	C₈×C₆₇	—	4	2	no	—
213	C₈×C₆₇	—	4	2	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	2	2	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	—	4	1	no	—
219	C₂²×C₆₇	—	4	1	no	—
220	C₂²×C₆₇	—	4	1	no	—
221	C₂²×C₆₇	—	4	1	no	—
222	C₂²×C₆₇	—	4	2	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	4	2	no	—
226	C₂²×C₆₇	—	4	2	no	—
227	C₂²×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	—	4	2	no	—
229	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
230	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
231	C₂²×C₆₇	#520	1	1	yes	characteristic
232	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
233	C₂²×C₆₇	#568	1	1	yes	characteristic
234	C₂²×C₆₇	#568	1	1	yes	characteristic
235	C₂²×C₆₇	#568	1	1	yes	characteristic
236	C₄×C₆₇	—	2	2	no	—
237	C₄×C₆₇	—	2	4	no	—
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	2	4	no	—
240	C₄×C₆₇	—	4	1	no	—
241	C₄×C₆₇	—	4	2	no	—
242	C₄×C₆₇	—	4	1	no	—
243	C₄×C₆₇	—	4	1	no	—
244	C₄×C₆₇	#548	1	2	yes	—
245	C₄×C₆₇	#550	1	2	yes	—
246	#22533	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂×C₆₇	—	4	2	no	—
248	C₂×C₆₇	—	8	2	no	—
249	C₂×C₆₇	#2341	1	1	yes	characteristic
250	C₂×C₆₇	#2879	1	1	yes	characteristic
251	C₂×C₆₇	#2880	1	1	yes	characteristic
252	C₂×C₆₇	#2967	1	1	yes	characteristic
253	C₂×C₆₇	#2968	1	1	yes	characteristic
254	C₂×C₆₇	#2992	1	1	yes	characteristic
255	C₂×C₆₇	#2995	1	1	yes	characteristic
256	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
257	C₂×#446	C₂×C₆₇	1	2	yes	complemented
258	#1955	C₂×C₆₇	1	2	yes	complemented
259	#1955	C₂×C₆₇	1	2	yes	complemented
260	#1954	C₂×C₆₇	1	2	yes	complemented
261	#1798	C₂×C₆₇	1	2	yes	complemented
262	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#1750	C₂×C₆₇	1	2	yes	complemented
264	#1749	C₂×C₆₇	1	1	yes	characteristic, complemented
265	C₆₇	#22533	1	1	yes	characteristic, direct product factor, Sylow
266	#529	—	2	8	no	—
267	C₂×#126	C₄×C₆₇	1	2	yes	—
268	C₂×#125	C₄×C₆₇	1	1	yes	characteristic
269	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
270	C₂×#123	C₄×C₆₇	1	2	yes	—
271	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
272	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
273	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
275	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#117	C₂²×C₆₇	1	2	yes	—
277	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
278	#446	—	2	8	no	—
279	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
280	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
281	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
282	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
283	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
284	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
285	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×C₂²⋊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×#97	C₂²×C₆₇	1	2	yes	—
290	C₂×#97	C₂²×C₆₇	1	2	yes	—
291	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
292	#336	C₂²×C₆₇	1	2	yes	—
293	#336	C₂²×C₆₇	1	1	yes	characteristic
294	#336	C₂²×C₆₇	1	1	yes	characteristic
295	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
296	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
297	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
298	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
299	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
300	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
301	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
303	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
304	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
305	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
306	#126	—	2	8	no	—
307	#125	—	2	4	no	—
308	#124	—	2	4	no	—
309	#123	—	2	8	no	—
310	C₂²⋊Q₈	—	2	4	no	—
311	#121	—	2	4	no	—
312	C₄×Q₈	D₄×C₆₇	1	8	yes	—
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	—	2	4	no	—
315	C₄×D₄	—	2	4	no	—
316	#117	—	2	4	no	—
317	#117	—	2	4	no	—
318	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
319	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
320	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
321	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
322	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
324	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
325	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
326	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
327	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
328	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
329	C₂×C₄²	—	2	4	no	—
330	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
331	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
333	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
334	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
335	C₄⋊C₈	—	2	4	no	—
336	C₄⋊C₈	—	2	4	no	—
337	C₄⋊C₈	—	2	4	no	—
338	C₂²⋊C₈	—	2	4	no	—
339	C₂²⋊C₈	—	2	4	no	—
340	C₂²⋊C₈	—	2	4	no	—
341	C₂²⋊C₈	—	4	1	no	—
342	C₂²⋊C₈	—	4	1	no	—
343	C₂²⋊C₈	—	4	2	no	—
344	C₂²⋊C₈	—	4	2	no	—
345	#97	—	2	4	no	—
346	#97	—	2	4	no	—
347	C₄×C₈	—	2	4	no	—
348	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
349	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
350	C₂×Q₈	—	4	1	no	—
351	C₂×Q₈	—	4	1	no	—
352	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	4	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	4	1	no	—
373	C₂²×C₄	—	4	1	no	—
374	C₂²×C₄	—	4	1	no	—
375	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
376	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
377	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
378	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
379	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
380	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
381	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
382	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
383	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	#41×C₆₇	1	4	yes	—
396	C₄⋊C₄	—	2	4	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
401	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
402	C₂²⋊C₄	—	2	4	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	2	4	no	—
406	C₂²⋊C₄	—	2	4	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	C₂×D₄×C₆₇	1	4	yes	—
411	C₄²	C₂×D₄×C₆₇	1	4	yes	—
412	C₄²	C₂×D₄×C₆₇	1	4	yes	—
413	C₂³	—	2	1	no	—
414	C₂³	—	2	1	no	—
415	C₂³	—	2	1	no	—
416	C₂³	—	2	1	no	—
417	C₂³	—	2	1	no	—
418	C₂³	—	2	1	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	2	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	2	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	2	no	—
426	C₂³	#131×C₆₇	1	2	yes	—
427	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
428	Q₈	—	4	4	no	—
429	D₄	—	4	8	no	—
430	D₄	—	4	4	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	4	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
461	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
462	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
463	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
464	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
465	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
466	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
467	C₈	—	2	4	no	—
468	C₈	—	4	2	no	—
469	C₈	—	4	2	no	—
470	C₈	—	4	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	1	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	4	1	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	2	no	—
480	C₂²	—	4	2	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
486	C₂²	C₄×#41×C₆₇	1	1	yes	characteristic
487	C₂²	#520×C₆₇	1	1	yes	characteristic
488	C₂²	C₂×#131×C₆₇	1	1	yes	characteristic
489	C₂²	#568×C₆₇	1	1	yes	characteristic
490	C₂²	#568×C₆₇	1	1	yes	characteristic
491	C₂²	#568×C₆₇	1	1	yes	characteristic
492	C₄	—	2	2	no	—
493	C₄	—	2	4	no	—
494	C₄	—	2	2	no	—
495	C₄	—	2	4	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	1	no	—
499	C₄	—	4	1	no	—
500	C₄	#548×C₆₇	1	2	yes	—
501	C₄	#550×C₆₇	1	2	yes	—
502	C₂	—	4	2	no	—
503	C₂	—	8	2	no	—
504	C₂	C₆₇×#2341	1	1	yes	characteristic
505	C₂	C₆₇×#2879	1	1	yes	characteristic
506	C₂	C₆₇×#2880	1	1	yes	characteristic
507	C₂	C₆₇×#2967	1	1	yes	characteristic
508	C₂	C₆₇×#2968	1	1	yes	characteristic
509	C₂	C₆₇×#2992	1	1	yes	characteristic
510	C₂	C₆₇×#2995	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	C₂×#121×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×#1955	C₂	semidirect
5. C₆₇×#1954	C₂	semidirect
6. C₆₇×#1798	C₂	semidirect
7. C₆₇×#1797	C₂	semidirect
8. C₆₇×#1750	C₂	semidirect
9. C₆₇×#1749	C₂	semidirect
246. #22533	C₆₇	direct (N × H)
257. C₂×#446	C₂×C₆₇	semidirect
258. #1955	C₂×C₆₇	semidirect
259. #1955	C₂×C₆₇	semidirect
260. #1954	C₂×C₆₇	semidirect
261. #1798	C₂×C₆₇	semidirect
262. #1797	C₂×C₆₇	semidirect
263. #1750	C₂×C₆₇	semidirect
264. #1749	C₂×C₆₇	semidirect
265. C₆₇	#22533	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
C₂×#446×C₆₇	2
C₆₇×#1955	2
C₆₇×#1954	2
C₆₇×#1798	2
C₆₇×#1797	2
C₆₇×#1750	2
C₆₇×#1749	2
#22533	67

Subgroups — C67×#22533 #30701784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22533 #30701784