Subgroups — C₆₇×#20420 #30699671

Order 17152.

1474 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#421×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2896	C₂	1	1	yes	maximal, characteristic
4	C₂×#405×C₆₇	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	C₆₇×#1738	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	C₂×#131×C₆₇	—	2	4	no	—
12	C₂×#131×C₆₇	—	2	1	no	—
13	C₂×#131×C₆₇	—	2	1	no	—
14	C₂×#131×C₆₇	C₄	1	8	yes	complemented
15	C₂×#131×C₆₇	C₂²	1	2	yes	—
16	C₂×#131×C₆₇	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₆₇	—	2	1	no	—
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₆₇	—	2	1	no	—
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
25	#421×C₆₇	C₄	1	8	yes	complemented
26	C₂×#104×C₆₇	C₂²	1	4	yes	—
27	#419×C₆₇	C₄	1	2	yes	complemented
28	#418×C₆₇	C₄	1	2	yes	complemented
29	#417×C₆₇	C₄	1	4	yes	complemented
30	#416×C₆₇	C₂²	1	4	yes	—
31	#405×C₆₇	—	2	1	no	—
32	#405×C₆₇	—	2	4	no	—
33	#405×C₆₇	—	2	1	no	—
34	#405×C₆₇	C₂²	1	2	yes	complemented
35	#405×C₆₇	C₂²	1	2	yes	complemented
36	C₂×#97×C₆₇	—	2	1	no	—
37	C₂×#97×C₆₇	C₂²	1	2	yes	complemented
38	C₂×C₄×C₈×C₆₇	—	2	1	no	—
39	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	complemented
40	#341×C₆₇	C₂²	1	4	yes	complemented
41	#339×C₆₇	C₂²	1	4	yes	complemented
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂×#41×C₆₇	—	2	2	no	—
46	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
47	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×#41×C₆₇	C₂³	1	2	yes	—
50	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₄×C₆₇	—	2	1	no	—
52	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₆₇	—	2	1	no	—
54	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	#131×C₆₇	—	2	8	no	—
56	#131×C₆₇	—	2	8	no	—
57	#131×C₆₇	—	2	4	no	—
58	#131×C₆₇	—	2	4	no	—
59	#131×C₆₇	C₂×C₄	1	8	yes	complemented
60	#131×C₆₇	C₂×C₄	1	8	yes	complemented
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	—	2	1	no	—
63	C₂×M₁₆×C₆₇	—	2	1	no	—
64	C₂×M₁₆×C₆₇	—	2	2	no	—
65	C₂×M₁₆×C₆₇	—	2	2	no	—
66	C₂×M₁₆×C₆₇	—	2	2	no	—
67	C₂×M₁₆×C₆₇	—	2	8	no	—
68	C₂×M₁₆×C₆₇	—	2	2	no	—
69	C₂×M₁₆×C₆₇	—	2	2	no	—
70	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
71	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
72	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
73	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
74	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	—	2	1	no	—
77	C₂²×C₈×C₆₇	—	2	1	no	—
78	C₂²×C₈×C₆₇	—	2	8	no	—
79	C₂²×C₈×C₆₇	—	2	2	no	—
80	C₂²×C₈×C₆₇	—	2	2	no	—
81	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
82	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
83	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
84	C₂²×C₈×C₆₇	D₄	1	4	yes	—
85	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
86	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
87	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
88	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
90	#117×C₆₇	—	2	1	no	—
91	#117×C₆₇	—	2	2	no	—
92	#117×C₆₇	—	2	1	no	—
93	#117×C₆₇	C₂×C₄	1	2	yes	—
94	#117×C₆₇	C₂×C₄	1	4	yes	—
95	#117×C₆₇	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	—	2	1	no	—
101	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
102	#104×C₆₇	C₂×C₄	1	16	yes	complemented
103	#103×C₆₇	C₂×C₄	1	8	yes	—
104	#102×C₆₇	C₂×C₄	1	8	yes	—
105	#97×C₆₇	—	2	2	no	—
106	#97×C₆₇	—	2	2	no	—
107	#97×C₆₇	—	2	4	no	—
108	C₄×C₈×C₆₇	—	2	2	no	—
109	C₄×C₈×C₆₇	—	2	4	no	—
110	C₄×C₈×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	1	no	—
112	C₂⁴×C₆₇	#41	1	1	yes	characteristic
113	#41×C₆₇	—	2	4	no	—
114	#41×C₆₇	—	2	4	no	—
115	#41×C₆₇	—	2	4	no	—
116	#41×C₆₇	—	2	4	no	—
117	#41×C₆₇	—	2	4	no	—
118	#41×C₆₇	—	2	2	no	—
119	#41×C₆₇	—	2	2	no	—
120	#41×C₆₇	C₄²	1	8	yes	—
121	#41×C₆₇	C₂²×C₄	1	4	yes	—
122	#41×C₆₇	C₂²×C₄	1	4	yes	—
123	C₂×Q₈×C₆₇	—	2	1	no	—
124	C₂×Q₈×C₆₇	—	2	2	no	—
125	C₂×Q₈×C₆₇	C₄²	1	4	yes	—
126	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
127	C₂×D₄×C₆₇	—	2	4	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	4	no	—
130	C₂×D₄×C₆₇	—	2	2	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	1	no	—
133	C₂×D₄×C₆₇	C₄²	1	4	yes	—
134	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	4	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	4	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	2	2	no	—
142	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
149	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	#41	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	M₁₆×C₆₇	—	2	4	no	—
153	M₁₆×C₆₇	—	2	4	no	—
154	M₁₆×C₆₇	—	2	4	no	—
155	M₁₆×C₆₇	—	2	4	no	—
156	M₁₆×C₆₇	—	2	2	no	—
157	M₁₆×C₆₇	—	2	2	no	—
158	M₁₆×C₆₇	—	2	8	no	—
159	M₁₆×C₆₇	—	2	8	no	—
160	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
161	M₁₆×C₆₇	C₂²×C₄	1	4	yes	—
162	C₂×C₈×C₆₇	—	2	4	no	—
163	C₂×C₈×C₆₇	—	2	4	no	—
164	C₂×C₈×C₆₇	—	2	2	no	—
165	C₂×C₈×C₆₇	—	2	2	no	—
166	C₂×C₈×C₆₇	—	2	2	no	—
167	C₂×C₈×C₆₇	—	2	2	no	—
168	C₂×C₈×C₆₇	—	2	8	no	—
169	C₂×C₈×C₆₇	—	2	8	no	—
170	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
171	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
172	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
173	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
174	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
175	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
176	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
177	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
181	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₄×C₆₇	—	2	4	no	—
184	C₄⋊C₄×C₆₇	—	2	2	no	—
185	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
186	C₂²⋊C₄×C₆₇	—	2	2	no	—
187	C₂²⋊C₄×C₆₇	—	2	4	no	—
188	C₂²⋊C₄×C₆₇	—	2	2	no	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	—	2	2	no	—
192	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
193	C₂³×C₆₇	—	2	2	no	—
194	C₂³×C₆₇	—	2	4	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	4	no	—
197	C₂³×C₆₇	#117	1	2	yes	—
198	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
199	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂³×C₆₇	C₄×D₄	1	2	yes	—
201	C₂³×C₆₇	C₂×#41	1	1	yes	characteristic
202	Q₈×C₆₇	—	2	2	no	—
203	Q₈×C₆₇	—	2	4	no	—
204	Q₈×C₆₇	C₂×C₄²	1	4	yes	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	—	2	4	no	—
207	D₄×C₆₇	—	2	8	no	—
208	D₄×C₆₇	—	2	2	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	C₂×C₄²	1	4	yes	—
211	C₂×C₄×C₆₇	—	2	2	no	—
212	C₂×C₄×C₆₇	—	2	4	no	—
213	C₂×C₄×C₆₇	—	2	4	no	—
214	C₂×C₄×C₆₇	—	2	4	no	—
215	C₂×C₄×C₆₇	—	2	8	no	—
216	C₂×C₄×C₆₇	—	2	4	no	—
217	C₂×C₄×C₆₇	—	2	4	no	—
218	C₂×C₄×C₆₇	—	4	4	no	—
219	C₂×C₄×C₆₇	—	4	4	no	—
220	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
221	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
222	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂×C₄×C₆₇	#117	1	2	yes	—
225	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
226	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
227	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
228	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
229	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
230	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
232	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
233	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
238	C₈×C₆₇	—	2	4	no	—
239	C₈×C₆₇	—	2	4	no	—
240	C₈×C₆₇	—	2	4	no	—
241	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
242	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
243	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
244	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
245	C₂²×C₆₇	—	2	2	no	—
246	C₂²×C₆₇	—	2	2	no	—
247	C₂²×C₆₇	—	2	2	no	—
248	C₂²×C₆₇	—	2	4	no	—
249	C₂²×C₆₇	—	4	2	no	—
250	C₂²×C₆₇	—	4	2	no	—
251	C₂²×C₆₇	—	4	8	no	—
252	C₂²×C₆₇	—	4	4	no	—
253	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
254	C₂²×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
255	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
256	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
257	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
258	C₄×C₆₇	—	2	4	no	—
259	C₄×C₆₇	—	4	4	no	—
260	C₄×C₆₇	—	4	4	no	—
261	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
262	C₄×C₆₇	C₄×C₂²⋊C₄	1	2	yes	—
263	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
264	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
265	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
266	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
267	#20420	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
268	C₂×C₆₇	—	2	1	no	—
269	C₂×C₆₇	—	2	2	no	—
270	C₂×C₆₇	—	2	1	no	—
271	C₂×C₆₇	—	2	2	no	—
272	C₂×C₆₇	—	4	4	no	—
273	C₂×C₆₇	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₆₇	#2955	1	2	yes	—
275	C₂²×#131	C₂×C₆₇	1	1	yes	characteristic
276	C₂×#421	C₂×C₆₇	1	1	yes	characteristic
277	#2896	C₂×C₆₇	1	1	yes	characteristic
278	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
279	C₂×#405	C₂×C₆₇	1	1	yes	characteristic, complemented
280	#1765	C₂×C₆₇	1	4	yes	complemented
281	#1764	C₂×C₆₇	1	4	yes	complemented
282	#1738	C₂×C₆₇	1	1	yes	characteristic, complemented
283	#1733	C₂×C₆₇	1	1	yes	characteristic, complemented
284	C₆₇	#20420	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
286	C₂×#131	—	2	4	no	—
287	C₂×#131	—	2	1	no	—
288	C₂×#131	—	2	1	no	—
289	C₂×#131	C₄×C₆₇	1	8	yes	complemented
290	C₂×#131	C₂²×C₆₇	1	2	yes	—
291	C₂×#131	C₂²×C₆₇	1	2	yes	—
292	C₂²×M₁₆	—	2	1	no	—
293	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
294	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
295	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
296	C₂³×C₈	—	2	1	no	—
297	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
298	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
299	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
300	#421	C₄×C₆₇	1	8	yes	complemented
301	C₂×#104	C₂²×C₆₇	1	4	yes	—
302	#419	C₄×C₆₇	1	2	yes	complemented
303	#418	C₄×C₆₇	1	2	yes	complemented
304	#417	C₄×C₆₇	1	4	yes	complemented
305	#416	C₂²×C₆₇	1	4	yes	—
306	#405	—	2	1	no	—
307	#405	—	2	1	no	—
308	#405	—	2	4	no	—
309	#405	C₂²×C₆₇	1	2	yes	complemented
310	#405	C₂²×C₆₇	1	2	yes	complemented
311	C₂×#97	—	2	1	no	—
312	C₂×#97	C₂²×C₆₇	1	2	yes	complemented
313	C₂×C₄×C₈	—	2	1	no	—
314	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	complemented
315	#341	C₂²×C₆₇	1	4	yes	complemented
316	#339	C₂²×C₆₇	1	4	yes	complemented
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	2	1	no	—
319	C₂×#41	—	2	1	no	—
320	C₂×#41	—	2	2	no	—
321	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
322	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
323	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
324	C₂×#41	C₂³×C₆₇	1	2	yes	—
325	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
326	C₂²×D₄	—	2	1	no	—
327	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
328	C₂³×C₄	—	2	1	no	—
329	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
330	#131	—	2	8	no	—
331	#131	—	2	8	no	—
332	#131	—	2	4	no	—
333	#131	—	2	4	no	—
334	#131	C₂×C₄×C₆₇	1	8	yes	complemented
335	#131	C₂×C₄×C₆₇	1	8	yes	complemented
336	C₂×M₁₆	—	2	2	no	—
337	C₂×M₁₆	—	2	1	no	—
338	C₂×M₁₆	—	2	1	no	—
339	C₂×M₁₆	—	2	2	no	—
340	C₂×M₁₆	—	2	2	no	—
341	C₂×M₁₆	—	2	2	no	—
342	C₂×M₁₆	—	2	8	no	—
343	C₂×M₁₆	—	2	2	no	—
344	C₂×M₁₆	—	2	2	no	—
345	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
346	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
347	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
348	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
349	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
350	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
351	C₂²×C₈	—	2	1	no	—
352	C₂²×C₈	—	2	1	no	—
353	C₂²×C₈	—	2	8	no	—
354	C₂²×C₈	—	2	2	no	—
355	C₂²×C₈	—	2	2	no	—
356	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
357	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₈	D₄×C₆₇	1	4	yes	—
362	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
363	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
365	#117	—	2	1	no	—
366	#117	—	2	2	no	—
367	#117	—	2	1	no	—
368	#117	C₂×C₄×C₆₇	1	2	yes	—
369	#117	C₂×C₄×C₆₇	1	4	yes	—
370	#117	C₂³×C₆₇	1	2	yes	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
373	C₂×C₂²⋊C₄	—	2	1	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₄²	—	2	1	no	—
376	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
377	#104	C₂×C₄×C₆₇	1	16	yes	complemented
378	#103	C₂×C₄×C₆₇	1	8	yes	—
379	#102	C₂×C₄×C₆₇	1	8	yes	—
380	#97	—	2	2	no	—
381	#97	—	2	2	no	—
382	#97	—	2	4	no	—
383	C₄×C₈	—	2	2	no	—
384	C₄×C₈	—	2	2	no	—
385	C₄×C₈	—	2	4	no	—
386	C₂⁴	—	2	1	no	—
387	C₂⁴	#41×C₆₇	1	1	yes	characteristic
388	#41	—	2	4	no	—
389	#41	—	2	4	no	—
390	#41	—	2	4	no	—
391	#41	—	2	4	no	—
392	#41	—	2	4	no	—
393	#41	—	2	2	no	—
394	#41	—	2	2	no	—
395	#41	C₄²×C₆₇	1	8	yes	—
396	#41	C₂²×C₄×C₆₇	1	4	yes	—
397	#41	C₂²×C₄×C₆₇	1	4	yes	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	2	no	—
400	C₂×Q₈	C₄²×C₆₇	1	4	yes	—
401	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	2	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	C₄²×C₆₇	1	4	yes	—
409	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	2	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	2	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
418	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
419	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
420	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
421	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
423	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
424	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
425	C₂²×C₄	#41×C₆₇	1	2	yes	—
426	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
427	M₁₆	—	2	4	no	—
428	M₁₆	—	2	4	no	—
429	M₁₆	—	2	4	no	—
430	M₁₆	—	2	4	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	2	2	no	—
433	M₁₆	—	2	8	no	—
434	M₁₆	—	2	8	no	—
435	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
436	M₁₆	C₂²×C₄×C₆₇	1	4	yes	—
437	C₂×C₈	—	2	4	no	—
438	C₂×C₈	—	2	4	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	2	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	8	no	—
444	C₂×C₈	—	2	8	no	—
445	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
446	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
447	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
448	C₂×C₈	C₂²⋊C₄×C₆₇	1	4	yes	—
449	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
450	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
451	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
452	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
454	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
455	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
456	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
457	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₄²	—	2	2	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	2	2	no	—
467	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	4	no	—
470	C₂³	—	4	2	no	—
471	C₂³	—	4	4	no	—
472	C₂³	#117×C₆₇	1	2	yes	—
473	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
474	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
475	C₂³	C₄×D₄×C₆₇	1	2	yes	—
476	C₂³	C₂×#41×C₆₇	1	1	yes	characteristic
477	Q₈	—	2	2	no	—
478	Q₈	—	2	4	no	—
479	Q₈	C₂×C₄²×C₆₇	1	4	yes	—
480	D₄	—	2	4	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	8	no	—
483	D₄	—	2	2	no	—
484	D₄	—	2	4	no	—
485	D₄	C₂×C₄²×C₆₇	1	4	yes	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	4	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	4	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	4	4	no	—
494	C₂×C₄	—	4	4	no	—
495	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
496	C₂×C₄	C₂×C₄²×C₆₇	1	2	yes	—
497	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
498	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
499	C₂×C₄	#117×C₆₇	1	2	yes	—
500	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
501	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
502	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
503	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
504	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
505	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
506	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
507	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
508	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
509	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
510	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
513	C₈	—	2	4	no	—
514	C₈	—	2	4	no	—
515	C₈	—	2	4	no	—
516	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
517	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
518	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
519	C₈	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
520	C₂²	—	2	2	no	—
521	C₂²	—	2	2	no	—
522	C₂²	—	2	2	no	—
523	C₂²	—	2	4	no	—
524	C₂²	—	4	2	no	—
525	C₂²	—	4	2	no	—
526	C₂²	—	4	8	no	—
527	C₂²	—	4	4	no	—
528	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
529	C₂²	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
530	C₂²	C₂×#117×C₆₇	1	1	yes	characteristic
531	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
532	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
533	C₄	—	2	4	no	—
534	C₄	—	4	4	no	—
535	C₄	—	4	4	no	—
536	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
537	C₄	C₄×C₂²⋊C₄×C₆₇	1	2	yes	—
538	C₄	C₂²×C₄²×C₆₇	1	1	yes	characteristic
539	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
540	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
541	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
542	C₂	—	2	1	no	—
543	C₂	—	2	2	no	—
544	C₂	—	2	1	no	—
545	C₂	—	2	2	no	—
546	C₂	—	4	4	no	—
547	C₂	C₂×C₄×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
548	C₂	C₆₇×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₆₇	C₂×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×C₈×C₆₇ ◁ C₂³×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×#405×C₆₇	C₂	semidirect
5. C₂×#405×C₆₇	C₂	semidirect
6. C₆₇×#1765	C₂	semidirect
7. C₆₇×#1764	C₂	semidirect
8. C₆₇×#1738	C₂	semidirect
9. C₆₇×#1733	C₂	semidirect
14. C₂×#131×C₆₇	C₄	semidirect
25. #421×C₆₇	C₄	semidirect
27. #419×C₆₇	C₄	semidirect
28. #418×C₆₇	C₄	semidirect
29. #417×C₆₇	C₄	semidirect
34. #405×C₆₇	C₂²	semidirect
35. #405×C₆₇	C₂²	semidirect
37. C₂×#97×C₆₇	C₂²	semidirect
39. C₂×C₄×C₈×C₆₇	C₂²	semidirect
40. #341×C₆₇	C₂²	semidirect
41. #339×C₆₇	C₂²	semidirect
59. #131×C₆₇	C₂×C₄	semidirect
60. #131×C₆₇	C₂×C₄	semidirect
102. #104×C₆₇	C₂×C₄	semidirect
267. #20420	C₆₇	direct (N × H)
278. C₂×#405	C₂×C₆₇	semidirect
279. C₂×#405	C₂×C₆₇	semidirect
280. #1765	C₂×C₆₇	semidirect
281. #1764	C₂×C₆₇	semidirect
282. #1738	C₂×C₆₇	semidirect
283. #1733	C₂×C₆₇	semidirect
284. C₆₇	#20420	direct (N × H)
289. C₂×#131	C₄×C₆₇	semidirect
300. #421	C₄×C₆₇	semidirect
302. #419	C₄×C₆₇	semidirect
303. #418	C₄×C₆₇	semidirect
304. #417	C₄×C₆₇	semidirect
309. #405	C₂²×C₆₇	semidirect
310. #405	C₂²×C₆₇	semidirect
312. C₂×#97	C₂²×C₆₇	semidirect
314. C₂×C₄×C₈	C₂²×C₆₇	semidirect
315. #341	C₂²×C₆₇	semidirect
316. #339	C₂²×C₆₇	semidirect
334. #131	C₂×C₄×C₆₇	semidirect
335. #131	C₂×C₄×C₆₇	semidirect
377. #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
C₂²×#131×C₆₇	2
C₂×#421×C₆₇	2
C₆₇×#2896	2
C₂×#405×C₆₇	2
C₆₇×#1765	2
C₆₇×#1764	2
C₆₇×#1738	2
C₆₇×#1733	2
#20420	67

Subgroups — C67×#20420 #30699671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20420 #30699671