Subgroups — C₂×#421×C₆₇ #22557331

Order 8576.

772 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#117×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#421×C₆₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#104×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#41×C₆₇	—	2	1	no	—
7	C₂×#41×C₆₇	—	2	2	no	—
8	C₂×#41×C₆₇	—	2	2	no	—
9	C₂×#41×C₆₇	—	2	1	no	—
10	C₂×#41×C₆₇	C₄	1	2	yes	complemented
11	C₂×#41×C₆₇	C₄	1	4	yes	complemented
12	C₂×#41×C₆₇	C₂²	1	2	yes	—
13	C₂×#41×C₆₇	C₂²	1	4	yes	—
14	C₂²×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₄×C₆₇	—	2	1	no	—
16	C₂²×D₄×C₆₇	C₄	1	1	yes	characteristic, complemented
17	C₂³×C₄×C₆₇	—	2	1	no	—
18	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	—	2	2	no	—
20	C₂×M₁₆×C₆₇	—	2	1	no	—
21	C₂×M₁₆×C₆₇	C₂²	1	2	yes	complemented
22	C₂×M₁₆×C₆₇	C₂²	1	4	yes	complemented
23	C₂²×C₈×C₆₇	—	2	1	no	—
24	#117×C₆₇	—	2	2	no	—
25	#117×C₆₇	C₂²	1	4	yes	complemented
26	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
27	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
28	C₂×C₄²×C₆₇	C₂²	1	2	yes	complemented
29	#104×C₆₇	C₂²	1	16	yes	complemented
30	C₂⁴×C₆₇	—	2	1	no	—
31	C₂⁴×C₆₇	D₄	1	1	yes	characteristic
32	#41×C₆₇	—	2	4	no	—
33	#41×C₆₇	—	2	4	no	—
34	#41×C₆₇	—	2	4	no	—
35	#41×C₆₇	—	2	2	no	—
36	#41×C₆₇	—	2	4	no	—
37	#41×C₆₇	—	2	4	no	—
38	#41×C₆₇	—	2	2	no	—
39	#41×C₆₇	C₂×C₄	1	4	yes	—
40	#41×C₆₇	C₂×C₄	1	8	yes	complemented
41	#41×C₆₇	C₂³	1	4	yes	—
42	C₂×Q₈×C₆₇	—	2	1	no	—
43	C₂×Q₈×C₆₇	—	2	2	no	—
44	C₂×Q₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
46	C₂×D₄×C₆₇	—	2	2	no	—
47	C₂×D₄×C₆₇	—	2	4	no	—
48	C₂×D₄×C₆₇	—	2	2	no	—
49	C₂×D₄×C₆₇	—	2	4	no	—
50	C₂×D₄×C₆₇	—	2	2	no	—
51	C₂×D₄×C₆₇	—	2	1	no	—
52	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	—	2	4	no	—
57	C₂²×C₄×C₆₇	—	2	4	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	D₄	1	2	yes	—
65	C₂²×C₄×C₆₇	D₄	1	2	yes	—
66	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	M₁₆×C₆₇	—	2	4	no	—
71	M₁₆×C₆₇	—	2	2	no	—
72	M₁₆×C₆₇	C₂³	1	4	yes	complemented
73	C₂×C₈×C₆₇	—	2	4	no	—
74	C₂×C₈×C₆₇	—	2	2	no	—
75	C₄⋊C₄×C₆₇	—	2	4	no	—
76	C₂²⋊C₄×C₆₇	—	2	4	no	—
77	C₄²×C₆₇	—	2	2	no	—
78	C₄²×C₆₇	C₂³	1	4	yes	complemented
79	C₂³×C₆₇	—	2	2	no	—
80	C₂³×C₆₇	—	2	4	no	—
81	C₂³×C₆₇	—	4	2	no	—
82	C₂³×C₆₇	—	4	4	no	—
83	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₂³×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₂³×C₆₇	C₂²⋊C₄	1	2	yes	—
86	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₆₇	C₂×D₄	1	2	yes	—
88	Q₈×C₆₇	—	2	2	no	—
89	Q₈×C₆₇	—	2	4	no	—
90	Q₈×C₆₇	C₂²×C₄	1	4	yes	—
91	D₄×C₆₇	—	2	4	no	—
92	D₄×C₆₇	—	2	8	no	—
93	D₄×C₆₇	—	2	4	no	—
94	D₄×C₆₇	—	2	2	no	—
95	D₄×C₆₇	—	2	4	no	—
96	D₄×C₆₇	C₂²×C₄	1	4	yes	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	4	no	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	8	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	4	4	no	—
104	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
111	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
122	C₈×C₆₇	—	2	4	no	—
123	C₂²×C₆₇	—	2	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	—	2	2	no	—
126	C₂²×C₆₇	—	2	4	no	—
127	C₂²×C₆₇	—	4	2	no	—
128	C₂²×C₆₇	—	4	2	no	—
129	C₂²×C₆₇	—	4	8	no	—
130	C₂²×C₆₇	—	4	4	no	—
131	C₂²×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
132	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂²×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₆₇	—	2	4	no	—
137	C₄×C₆₇	—	4	4	no	—
138	C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
139	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
142	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	—	2	2	no	—
146	C₂×C₆₇	—	2	1	no	—
147	C₂×C₆₇	—	2	2	no	—
148	C₂×C₆₇	—	4	4	no	—
149	C₂×C₆₇	#421	1	2	yes	direct product factor
150	C₂×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×#421	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₆₇	C₂×#421	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×#41	C₂×C₆₇	1	1	yes	characteristic
154	C₂²×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
155	C₂×#117	C₂×C₆₇	1	1	yes	characteristic, complemented
156	#421	C₂×C₆₇	1	8	yes	direct product factor
157	C₂×#104	C₂×C₆₇	1	4	yes	complemented
158	C₂×#41	—	2	1	no	—
159	C₂×#41	—	2	2	no	—
160	C₂×#41	—	2	1	no	—
161	C₂×#41	—	2	2	no	—
162	C₂×#41	C₄×C₆₇	1	2	yes	complemented
163	C₂×#41	C₄×C₆₇	1	4	yes	complemented
164	C₂×#41	C₂²×C₆₇	1	2	yes	—
165	C₂×#41	C₂²×C₆₇	1	4	yes	—
166	C₂²×Q₈	C₄×C₆₇	1	1	yes	characteristic, complemented
167	C₂²×D₄	—	2	1	no	—
168	C₂²×D₄	C₄×C₆₇	1	1	yes	characteristic, complemented
169	C₂³×C₄	—	2	1	no	—
170	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	C₂²×C₆₇	1	2	yes	complemented
174	C₂×M₁₆	C₂²×C₆₇	1	4	yes	complemented
175	C₂²×C₈	—	2	1	no	—
176	#117	—	2	2	no	—
177	#117	C₂²×C₆₇	1	4	yes	complemented
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₂²⋊C₄	—	2	1	no	—
180	C₂×C₄²	C₂²×C₆₇	1	2	yes	complemented
181	#104	C₂²×C₆₇	1	16	yes	complemented
182	C₂⁴	—	2	1	no	—
183	C₂⁴	D₄×C₆₇	1	1	yes	characteristic
184	#41	—	2	4	no	—
185	#41	—	2	4	no	—
186	#41	—	2	4	no	—
187	#41	—	2	2	no	—
188	#41	—	2	4	no	—
189	#41	—	2	4	no	—
190	#41	—	2	2	no	—
191	#41	C₂×C₄×C₆₇	1	4	yes	—
192	#41	C₂×C₄×C₆₇	1	8	yes	complemented
193	#41	C₂³×C₆₇	1	4	yes	—
194	C₂×Q₈	—	2	1	no	—
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	C₂×C₄×C₆₇	1	2	yes	—
197	C₂×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	4	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	C₂×C₄×C₆₇	1	2	yes	—
205	C₂×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×C₆₇	1	2	yes	—
215	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
216	C₂²×C₄	D₄×C₆₇	1	2	yes	—
217	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
218	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
219	C₂²×C₄	D₄×C₆₇	1	2	yes	—
220	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	2	no	—
224	M₁₆	C₂³×C₆₇	1	4	yes	complemented
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	2	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	C₂³×C₆₇	1	4	yes	complemented
231	C₂³	—	2	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	4	4	no	—
235	C₂³	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
236	C₂³	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
237	C₂³	C₂²⋊C₄×C₆₇	1	2	yes	—
238	C₂³	C₂×D₄×C₆₇	1	1	yes	characteristic
239	C₂³	C₂×D₄×C₆₇	1	2	yes	—
240	Q₈	—	2	2	no	—
241	Q₈	—	2	4	no	—
242	Q₈	C₂²×C₄×C₆₇	1	4	yes	—
243	D₄	—	2	4	no	—
244	D₄	—	2	8	no	—
245	D₄	—	2	4	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	4	no	—
248	D₄	C₂²×C₄×C₆₇	1	4	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
257	C₂×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
259	C₂×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
260	C₂×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
261	C₂×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
262	C₂×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
263	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×C₆₇	1	2	yes	—
265	C₂×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₄	C₂²×C₄×C₆₇	1	2	yes	—
267	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
268	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
269	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
271	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
272	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
273	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	8	no	—
282	C₂²	—	4	4	no	—
283	C₂²	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
284	C₂²	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
285	C₂²	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
286	C₂²	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
287	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
291	C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
292	C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
293	C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
294	C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
295	C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	#421×C₆₇	1	2	yes	direct product factor
302	C₂	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂²×#41×C₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₁₆×C₆₇	C₂	semidirect
3. C₂×#117×C₆₇	C₂	semidirect
4. #421×C₆₇	C₂	direct (N × H)
5. C₂×#104×C₆₇	C₂	semidirect
10. C₂×#41×C₆₇	C₄	semidirect
11. C₂×#41×C₆₇	C₄	semidirect
14. C₂²×Q₈×C₆₇	C₄	semidirect
16. C₂²×D₄×C₆₇	C₄	semidirect
21. C₂×M₁₆×C₆₇	C₂²	semidirect
22. C₂×M₁₆×C₆₇	C₂²	semidirect
25. #117×C₆₇	C₂²	semidirect
28. C₂×C₄²×C₆₇	C₂²	semidirect
29. #104×C₆₇	C₂²	semidirect
40. #41×C₆₇	C₂×C₄	semidirect
45. C₂×Q₈×C₆₇	C₂×C₄	semidirect
53. C₂×D₄×C₆₇	C₂×C₄	semidirect
72. M₁₆×C₆₇	C₂³	semidirect
78. C₄²×C₆₇	C₂³	semidirect
149. C₂×C₆₇	#421	direct (N × H)
151. C₂×#421	C₆₇	direct (N × H)
152. C₆₇	C₂×#421	direct (N × H)
154. C₂²×M₁₆	C₂×C₆₇	semidirect
155. C₂×#117	C₂×C₆₇	semidirect
156. #421	C₂×C₆₇	direct (N × H)
157. C₂×#104	C₂×C₆₇	semidirect
162. C₂×#41	C₄×C₆₇	semidirect
163. C₂×#41	C₄×C₆₇	semidirect
166. C₂²×Q₈	C₄×C₆₇	semidirect
168. C₂²×D₄	C₄×C₆₇	semidirect
173. C₂×M₁₆	C₂²×C₆₇	semidirect
174. C₂×M₁₆	C₂²×C₆₇	semidirect
177. #117	C₂²×C₆₇	semidirect
180. C₂×C₄²	C₂²×C₆₇	semidirect
181. #104	C₂²×C₆₇	semidirect
192. #41	C₂×C₄×C₆₇	semidirect
197. C₂×Q₈	C₂×C₄×C₆₇	semidirect
205. C₂×D₄	C₂×C₄×C₆₇	semidirect
224. M₁₆	C₂³×C₆₇	semidirect
230. C₄²	C₂³×C₆₇	semidirect
301. C₂	#421×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₆₇	2
C₂²×M₁₆×C₆₇	2
C₂×#117×C₆₇	2
#421×C₆₇	2
C₂×#104×C₆₇	2
C₂×#421	67

Subgroups — C2×#421×C67 #22557331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#421×C₆₇ #22557331