Subgroups — C₆₇×#20422 #30699673

Order 17152.

1458 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#568×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2873	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1765	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1764	C₂	1	4	yes	maximal, complemented
8	C₂×#341×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#1733	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
11	#568×C₆₇	—	2	4	no	—
12	#568×C₆₇	C₄	1	8	yes	complemented
13	C₂×#131×C₆₇	—	2	2	no	—
14	C₂×#131×C₆₇	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₆₇	—	2	2	no	—
16	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	#421×C₆₇	C₄	1	8	yes	complemented
23	C₂×#104×C₆₇	C₂²	1	4	yes	—
24	#416×C₆₇	C₄	1	4	yes	complemented
25	#416×C₆₇	C₂²	1	2	yes	—
26	#416×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₄	1	2	yes	complemented
28	C₂×#102×C₆₇	C₄	1	2	yes	complemented
29	#405×C₆₇	—	2	4	no	—
30	#405×C₆₇	—	2	2	no	—
31	#405×C₆₇	C₂²	1	4	yes	complemented
32	C₄×M₁₆×C₆₇	—	2	2	no	—
33	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
34	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
35	#341×C₆₇	C₂²	1	2	yes	complemented
36	#341×C₆₇	C₂²	1	2	yes	complemented
37	#339×C₆₇	C₂²	1	4	yes	complemented
38	C₂×#41×C₆₇	—	2	1	no	—
39	C₂×#41×C₆₇	—	2	2	no	—
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
43	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
44	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×#41×C₆₇	C₂³	1	2	yes	—
46	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₆₇	—	2	1	no	—
48	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂³×C₄×C₆₇	—	2	1	no	—
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	#131×C₆₇	—	2	16	no	—
52	#131×C₆₇	—	2	8	no	—
53	#131×C₆₇	C₂×C₄	1	16	yes	complemented
54	C₂×M₁₆×C₆₇	—	2	1	no	—
55	C₂×M₁₆×C₆₇	—	2	2	no	—
56	C₂×M₁₆×C₆₇	—	2	1	no	—
57	C₂×M₁₆×C₆₇	—	2	2	no	—
58	C₂×M₁₆×C₆₇	—	2	16	no	—
59	C₂×M₁₆×C₆₇	—	2	4	no	—
60	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
61	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
64	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
65	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
67	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
68	C₂²×C₈×C₆₇	—	2	2	no	—
69	C₂²×C₈×C₆₇	—	2	1	no	—
70	C₂²×C₈×C₆₇	—	2	1	no	—
71	C₂²×C₈×C₆₇	—	2	2	no	—
72	C₂²×C₈×C₆₇	—	2	4	no	—
73	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
74	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
75	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
78	#117×C₆₇	—	2	2	no	—
79	#117×C₆₇	—	2	1	no	—
80	#117×C₆₇	—	2	1	no	—
81	#117×C₆₇	C₂×C₄	1	2	yes	—
82	#117×C₆₇	C₂×C₄	1	4	yes	—
83	#117×C₆₇	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
88	C₂×C₄²×C₆₇	—	2	1	no	—
89	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
90	#104×C₆₇	C₂×C₄	1	16	yes	complemented
91	#103×C₆₇	C₂×C₄	1	4	yes	—
92	#103×C₆₇	C₂×C₄	1	4	yes	—
93	#102×C₆₇	C₂×C₄	1	4	yes	—
94	#102×C₆₇	C₂×C₄	1	4	yes	—
95	#97×C₆₇	—	2	4	no	—
96	#97×C₆₇	—	2	4	no	—
97	C₄×C₈×C₆₇	—	2	4	no	—
98	C₄×C₈×C₆₇	—	2	4	no	—
99	C₂⁴×C₆₇	—	2	1	no	—
100	C₂⁴×C₆₇	#41	1	1	yes	characteristic
101	#41×C₆₇	—	2	4	no	—
102	#41×C₆₇	—	2	4	no	—
103	#41×C₆₇	—	2	4	no	—
104	#41×C₆₇	—	2	4	no	—
105	#41×C₆₇	—	2	4	no	—
106	#41×C₆₇	—	2	2	no	—
107	#41×C₆₇	—	2	2	no	—
108	#41×C₆₇	C₄²	1	8	yes	—
109	#41×C₆₇	C₂²×C₄	1	4	yes	—
110	#41×C₆₇	C₂²×C₄	1	4	yes	—
111	C₂×Q₈×C₆₇	—	2	1	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
114	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂×D₄×C₆₇	—	2	4	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	2	4	no	—
118	C₂×D₄×C₆₇	—	2	2	no	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	1	no	—
121	C₂×D₄×C₆₇	C₄²	1	4	yes	—
122	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	4	no	—
125	C₂²×C₄×C₆₇	—	2	2	no	—
126	C₂²×C₄×C₆₇	—	2	4	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	#41	1	2	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	M₁₆×C₆₇	—	2	4	no	—
141	M₁₆×C₆₇	—	2	4	no	—
142	M₁₆×C₆₇	—	2	2	no	—
143	M₁₆×C₆₇	—	2	2	no	—
144	M₁₆×C₆₇	—	2	16	no	—
145	M₁₆×C₆₇	C₂²⋊C₄	1	8	yes	—
146	M₁₆×C₆₇	C₂²⋊C₄	1	8	yes	—
147	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
148	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
149	C₂×C₈×C₆₇	—	2	4	no	—
150	C₂×C₈×C₆₇	—	2	4	no	—
151	C₂×C₈×C₆₇	—	2	4	no	—
152	C₂×C₈×C₆₇	—	2	4	no	—
153	C₂×C₈×C₆₇	—	2	1	no	—
154	C₂×C₈×C₆₇	—	2	1	no	—
155	C₂×C₈×C₆₇	—	2	2	no	—
156	C₂×C₈×C₆₇	—	2	2	no	—
157	C₂×C₈×C₆₇	—	2	1	no	—
158	C₂×C₈×C₆₇	—	2	1	no	—
159	C₂×C₈×C₆₇	—	2	16	no	—
160	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
161	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
163	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
164	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
165	C₄⋊C₄×C₆₇	—	2	4	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
168	C₂²⋊C₄×C₆₇	—	2	2	no	—
169	C₂²⋊C₄×C₆₇	—	2	4	no	—
170	C₂²⋊C₄×C₆₇	—	2	2	no	—
171	C₄²×C₆₇	—	2	2	no	—
172	C₄²×C₆₇	—	2	2	no	—
173	C₄²×C₆₇	—	2	2	no	—
174	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	4	2	no	—
178	C₂³×C₆₇	—	8	2	no	—
179	C₂³×C₆₇	#117	1	2	yes	—
180	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
181	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
182	C₂³×C₆₇	C₄×D₄	1	2	yes	—
183	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
184	Q₈×C₆₇	—	2	2	no	—
185	Q₈×C₆₇	—	2	4	no	—
186	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
187	D₄×C₆₇	—	2	4	no	—
188	D₄×C₆₇	—	2	4	no	—
189	D₄×C₆₇	—	2	8	no	—
190	D₄×C₆₇	—	2	2	no	—
191	D₄×C₆₇	—	2	4	no	—
192	D₄×C₆₇	C₂×C₄²	1	4	yes	—
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	8	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	4	4	no	—
201	C₂×C₄×C₆₇	—	8	2	no	—
202	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
203	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
204	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
206	C₂×C₄×C₆₇	#117	1	2	yes	—
207	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
208	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
210	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
211	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
213	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
214	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
216	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
220	C₈×C₆₇	—	2	4	no	—
221	C₈×C₆₇	—	2	2	no	—
222	C₈×C₆₇	—	2	2	no	—
223	C₈×C₆₇	—	2	4	no	—
224	C₈×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
225	C₂²×C₆₇	—	2	2	no	—
226	C₂²×C₆₇	—	2	2	no	—
227	C₂²×C₆₇	—	2	2	no	—
228	C₂²×C₆₇	—	2	4	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	8	4	no	—
233	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
234	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
235	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
236	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
237	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
238	C₄×C₆₇	—	2	4	no	—
239	C₄×C₆₇	—	4	4	no	—
240	C₄×C₆₇	—	4	4	no	—
241	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
242	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
243	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
244	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
245	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
246	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
247	#20422	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂×C₆₇	—	2	1	no	—
249	C₂×C₆₇	—	2	2	no	—
250	C₂×C₆₇	—	2	1	no	—
251	C₂×C₆₇	—	2	2	no	—
252	C₂×C₆₇	—	4	4	no	—
253	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₆₇	#2957	1	2	yes	—
255	C₂×#568	C₂×C₆₇	1	1	yes	characteristic
256	C₂×#421	C₂×C₆₇	1	1	yes	characteristic
257	C₂×#416	C₂×C₆₇	1	1	yes	characteristic
258	#2873	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
260	#1765	C₂×C₆₇	1	4	yes	complemented
261	#1764	C₂×C₆₇	1	4	yes	complemented
262	C₂×#341	C₂×C₆₇	1	1	yes	characteristic, complemented
263	#1733	C₂×C₆₇	1	1	yes	characteristic, complemented
264	C₆₇	#20422	1	1	yes	characteristic, direct product factor, Sylow
265	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
266	#568	—	2	4	no	—
267	#568	C₄×C₆₇	1	8	yes	complemented
268	C₂×#131	—	2	2	no	—
269	C₂×#131	C₂²×C₆₇	1	4	yes	—
270	C₂²×M₁₆	—	2	2	no	—
271	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
272	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
273	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
274	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
275	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
276	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
277	#421	C₄×C₆₇	1	8	yes	complemented
278	C₂×#104	C₂²×C₆₇	1	4	yes	—
279	#416	C₄×C₆₇	1	4	yes	complemented
280	#416	C₂²×C₆₇	1	2	yes	—
281	#416	C₂²×C₆₇	1	2	yes	—
282	C₂×#103	C₄×C₆₇	1	2	yes	complemented
283	C₂×#102	C₄×C₆₇	1	2	yes	complemented
284	#405	—	2	4	no	—
285	#405	—	2	2	no	—
286	#405	C₂²×C₆₇	1	4	yes	complemented
287	C₄×M₁₆	—	2	2	no	—
288	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
289	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
290	#341	C₂²×C₆₇	1	2	yes	complemented
291	#341	C₂²×C₆₇	1	2	yes	complemented
292	#339	C₂²×C₆₇	1	4	yes	complemented
293	C₂×#41	—	2	1	no	—
294	C₂×#41	—	2	2	no	—
295	C₂×#41	—	2	1	no	—
296	C₂×#41	—	2	2	no	—
297	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
298	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
299	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
300	C₂×#41	C₂³×C₆₇	1	2	yes	—
301	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
304	C₂³×C₄	—	2	1	no	—
305	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
306	#131	—	2	16	no	—
307	#131	—	2	8	no	—
308	#131	C₂×C₄×C₆₇	1	16	yes	complemented
309	C₂×M₁₆	—	2	1	no	—
310	C₂×M₁₆	—	2	2	no	—
311	C₂×M₁₆	—	2	1	no	—
312	C₂×M₁₆	—	2	2	no	—
313	C₂×M₁₆	—	2	16	no	—
314	C₂×M₁₆	—	2	4	no	—
315	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
316	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
317	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
318	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
319	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
320	C₂×M₁₆	D₄×C₆₇	1	4	yes	—
321	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
322	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
323	C₂²×C₈	—	2	2	no	—
324	C₂²×C₈	—	2	1	no	—
325	C₂²×C₈	—	2	1	no	—
326	C₂²×C₈	—	2	2	no	—
327	C₂²×C₈	—	2	4	no	—
328	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
329	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
330	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
333	#117	—	2	1	no	—
334	#117	—	2	2	no	—
335	#117	—	2	1	no	—
336	#117	C₂×C₄×C₆₇	1	2	yes	—
337	#117	C₂×C₄×C₆₇	1	4	yes	—
338	#117	C₂³×C₆₇	1	2	yes	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₂²⋊C₄	—	2	1	no	—
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
345	#104	C₂×C₄×C₆₇	1	16	yes	complemented
346	#103	C₂×C₄×C₆₇	1	4	yes	—
347	#103	C₂×C₄×C₆₇	1	4	yes	—
348	#102	C₂×C₄×C₆₇	1	4	yes	—
349	#102	C₂×C₄×C₆₇	1	4	yes	—
350	#97	—	2	4	no	—
351	#97	—	2	4	no	—
352	C₄×C₈	—	2	4	no	—
353	C₄×C₈	—	2	4	no	—
354	C₂⁴	—	2	1	no	—
355	C₂⁴	#41×C₆₇	1	1	yes	characteristic
356	#41	—	2	4	no	—
357	#41	—	2	4	no	—
358	#41	—	2	4	no	—
359	#41	—	2	4	no	—
360	#41	—	2	4	no	—
361	#41	—	2	2	no	—
362	#41	—	2	2	no	—
363	#41	C₄²×C₆₇	1	8	yes	—
364	#41	C₂²×C₄×C₆₇	1	4	yes	—
365	#41	C₂²×C₄×C₆₇	1	4	yes	—
366	C₂×Q₈	—	2	1	no	—
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	C₄²×C₆₇	1	4	yes	—
369	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	4	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	C₄²×C₆₇	1	4	yes	—
377	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
386	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
387	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
388	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
392	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
393	C₂²×C₄	#41×C₆₇	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
395	M₁₆	—	2	4	no	—
396	M₁₆	—	2	4	no	—
397	M₁₆	—	2	2	no	—
398	M₁₆	—	2	2	no	—
399	M₁₆	—	2	16	no	—
400	M₁₆	C₂²⋊C₄×C₆₇	1	8	yes	—
401	M₁₆	C₂²⋊C₄×C₆₇	1	8	yes	—
402	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
403	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
404	C₂×C₈	—	2	4	no	—
405	C₂×C₈	—	2	4	no	—
406	C₂×C₈	—	2	4	no	—
407	C₂×C₈	—	2	4	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₂×C₈	—	2	16	no	—
415	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
416	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
417	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
418	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
419	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
423	C₂²⋊C₄	—	2	2	no	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	2	no	—
429	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	8	2	no	—
434	C₂³	#117×C₆₇	1	2	yes	—
435	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
436	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
437	C₂³	C₄×D₄×C₆₇	1	2	yes	—
438	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
439	Q₈	—	2	2	no	—
440	Q₈	—	2	4	no	—
441	Q₈	C₂×C₄²×C₆₇	1	4	yes	—
442	D₄	—	2	4	no	—
443	D₄	—	2	4	no	—
444	D₄	—	2	8	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	4	no	—
447	D₄	C₂×C₄²×C₆₇	1	4	yes	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	8	2	no	—
457	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
458	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
459	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
461	C₂×C₄	#117×C₆₇	1	2	yes	—
462	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
463	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
464	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
465	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
466	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
467	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
468	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
469	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
470	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
471	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
472	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
475	C₈	—	2	4	no	—
476	C₈	—	2	2	no	—
477	C₈	—	2	2	no	—
478	C₈	—	2	4	no	—
479	C₈	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	4	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	8	4	no	—
488	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
489	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
490	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
491	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
492	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
493	C₄	—	2	4	no	—
494	C₄	—	4	4	no	—
495	C₄	—	4	4	no	—
496	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
497	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
498	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
499	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
500	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
501	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
502	C₂	—	2	1	no	—
503	C₂	—	2	2	no	—
504	C₂	—	2	1	no	—
505	C₂	—	2	2	no	—
506	C₂	—	4	4	no	—
507	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
508	C₂	C₆₇×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₆₇×#2873	C₂	semidirect
5. C₂×#405×C₆₇	C₂	semidirect
6. C₆₇×#1765	C₂	semidirect
7. C₆₇×#1764	C₂	semidirect
8. C₂×#341×C₆₇	C₂	semidirect
9. C₆₇×#1733	C₂	semidirect
12. #568×C₆₇	C₄	semidirect
22. #421×C₆₇	C₄	semidirect
24. #416×C₆₇	C₄	semidirect
27. C₂×#103×C₆₇	C₄	semidirect
28. C₂×#102×C₆₇	C₄	semidirect
31. #405×C₆₇	C₂²	semidirect
33. C₂×#97×C₆₇	C₂²	semidirect
34. C₂×C₄×C₈×C₆₇	C₂²	semidirect
35. #341×C₆₇	C₂²	semidirect
36. #341×C₆₇	C₂²	semidirect
37. #339×C₆₇	C₂²	semidirect
53. #131×C₆₇	C₂×C₄	semidirect
90. #104×C₆₇	C₂×C₄	semidirect
247. #20422	C₆₇	direct (N × H)
258. #2873	C₂×C₆₇	semidirect
259. C₂×#405	C₂×C₆₇	semidirect
260. #1765	C₂×C₆₇	semidirect
261. #1764	C₂×C₆₇	semidirect
262. C₂×#341	C₂×C₆₇	semidirect
263. #1733	C₂×C₆₇	semidirect
264. C₆₇	#20422	direct (N × H)
267. #568	C₄×C₆₇	semidirect
277. #421	C₄×C₆₇	semidirect
279. #416	C₄×C₆₇	semidirect
282. C₂×#103	C₄×C₆₇	semidirect
283. C₂×#102	C₄×C₆₇	semidirect
286. #405	C₂²×C₆₇	semidirect
288. C₂×#97	C₂²×C₆₇	semidirect
289. C₂×C₄×C₈	C₂²×C₆₇	semidirect
290. #341	C₂²×C₆₇	semidirect
291. #341	C₂²×C₆₇	semidirect
292. #339	C₂²×C₆₇	semidirect
308. #131	C₂×C₄×C₆₇	semidirect
345. #104	C₂×C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#568×C₆₇	2
C₂×#421×C₆₇	2
C₂×#416×C₆₇	2
C₆₇×#2873	2
C₂×#405×C₆₇	2
C₆₇×#1765	2
C₆₇×#1764	2
C₂×#341×C₆₇	2
C₆₇×#1733	2
#20422	67

Subgroups — C67×#20422 #30699673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20422 #30699673