Subgroups — C₂×C₆₇×#2228 #30712873

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2228	C₂	1	8	yes	maximal, direct product factor
4	C₂×#367×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄⋊C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#357×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#357×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#125×C₆₇	—	2	1	no	—
11	C₂×C₄×D₄×C₆₇	—	2	1	no	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#499×C₆₇	—	2	2	no	—
15	#499×C₆₇	C₂²	1	4	yes	complemented
16	C₄×D₈×C₆₇	—	2	2	no	—
17	C₄×D₈×C₆₇	C₂²	1	4	yes	—
18	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₆₇	—	2	1	no	—
22	C₂×#102×C₆₇	—	2	1	no	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	#367×C₆₇	C₂²	1	4	yes	complemented
25	#366×C₆₇	C₂²	1	4	yes	complemented
26	C₄⋊C₁₆×C₆₇	C₂²	1	4	yes	complemented
27	#357×C₆₇	C₂²	1	4	yes	—
28	#357×C₆₇	C₂²	1	4	yes	—
29	C₂²×D₄×C₆₇	—	2	1	no	—
30	C₂³×C₄×C₆₇	—	4	1	no	—
31	C₂×D₈×C₆₇	—	2	1	no	—
32	C₂×D₈×C₆₇	—	2	2	no	—
33	C₂×D₈×C₆₇	Q₈	1	4	yes	—
34	C₂×D₈×C₆₇	C₂³	1	2	yes	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	#125×C₆₇	—	2	4	no	—
38	#125×C₆₇	—	4	2	no	—
39	C₄×D₄×C₆₇	—	2	4	no	—
40	C₄×D₄×C₆₇	—	4	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
46	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
47	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	—	2	2	no	—
49	C₂×C₁₆×C₆₇	—	2	2	no	—
50	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
51	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
52	#107×C₆₇	—	2	1	no	—
53	#107×C₆₇	—	2	1	no	—
54	#107×C₆₇	—	2	1	no	—
55	#107×C₆₇	C₂³	1	2	yes	—
56	#107×C₆₇	C₂³	1	2	yes	—
57	#107×C₆₇	C₂³	1	2	yes	—
58	#106×C₆₇	—	2	4	no	—
59	#102×C₆₇	—	2	4	no	—
60	C₄×C₈×C₆₇	D₄	1	2	yes	—
61	C₄×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂⁴×C₆₇	—	4	1	no	—
63	C₂×D₄×C₆₇	—	2	4	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	—	4	1	no	—
66	C₂×D₄×C₆₇	—	4	2	no	—
67	C₂²×C₄×C₆₇	—	4	1	no	—
68	C₂²×C₄×C₆₇	—	4	1	no	—
69	C₂²×C₄×C₆₇	—	4	1	no	—
70	C₂²×C₄×C₆₇	—	4	4	no	—
71	C₂²×C₄×C₆₇	—	8	4	no	—
72	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	D₈×C₆₇	—	2	4	no	—
76	D₈×C₆₇	—	2	2	no	—
77	D₈×C₆₇	C₂×Q₈	1	4	yes	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₆₇	#41	1	2	yes	—
83	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₄⋊C₄×C₆₇	—	4	1	no	—
88	C₄⋊C₄×C₆₇	—	4	1	no	—
89	C₄⋊C₄×C₆₇	—	4	1	no	—
90	C₄⋊C₄×C₆₇	—	4	4	no	—
91	C₄⋊C₄×C₆₇	—	4	4	no	—
92	C₂²⋊C₄×C₆₇	—	4	4	no	—
93	C₄²×C₆₇	D₈	1	2	yes	—
94	C₄²×C₆₇	C₂×D₄	1	2	yes	—
95	C₁₆×C₆₇	—	2	2	no	—
96	C₁₆×C₆₇	—	2	2	no	—
97	C₂³×C₆₇	—	4	2	no	—
98	C₂³×C₆₇	—	4	4	no	—
99	C₂³×C₆₇	—	8	2	no	—
100	C₂³×C₆₇	—	8	2	no	—
101	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
102	D₄×C₆₇	—	2	4	no	—
103	D₄×C₆₇	—	4	2	no	—
104	D₄×C₆₇	—	4	4	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	2	no	—
107	C₂×C₄×C₆₇	—	4	2	no	—
108	C₂×C₄×C₆₇	—	4	2	no	—
109	C₂×C₄×C₆₇	—	4	2	no	—
110	C₂×C₄×C₆₇	—	8	8	no	—
111	C₂×C₄×C₆₇	—	8	2	no	—
112	C₂×C₄×C₆₇	—	8	2	no	—
113	C₂×C₄×C₆₇	—	8	2	no	—
114	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₆₇	#137	1	2	yes	—
119	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₆₇	—	2	2	no	—
121	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
122	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
123	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
124	C₂²×C₆₇	—	4	4	no	—
125	C₂²×C₆₇	—	8	2	no	—
126	C₂²×C₆₇	—	8	2	no	—
127	C₂²×C₆₇	—	8	8	no	—
128	C₂²×C₆₇	#474	1	2	yes	—
129	C₂²×C₆₇	#508	1	2	yes	—
130	C₂²×C₆₇	#509	1	2	yes	—
131	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
132	C₄×C₆₇	—	2	2	no	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	8	2	no	—
135	C₄×C₆₇	—	8	2	no	—
136	C₄×C₆₇	—	8	2	no	—
137	C₄×C₆₇	#474	1	2	yes	—
138	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
140	C₂×#2228	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂×C₆₇	—	8	4	no	—
142	C₂×C₆₇	#2228	1	4	yes	direct product factor
143	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
144	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
145	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
146	C₂×#499	C₂×C₆₇	1	1	yes	characteristic, complemented
147	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
148	#2228	C₂×C₆₇	1	8	yes	direct product factor
149	C₂×#367	C₂×C₆₇	1	1	yes	characteristic, complemented
150	C₂×#366	C₂×C₆₇	1	1	yes	characteristic, complemented
151	C₂×C₄⋊C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
152	C₂×#357	C₂×C₆₇	1	1	yes	characteristic
153	C₂×#357	C₂×C₆₇	1	1	yes	characteristic
154	C₆₇	C₂×#2228	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
156	C₂×#125	—	2	1	no	—
157	C₂×C₄×D₄	—	2	1	no	—
158	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
159	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
160	#499	—	2	2	no	—
161	#499	C₂²×C₆₇	1	4	yes	complemented
162	C₄×D₈	—	2	2	no	—
163	C₄×D₈	C₂²×C₆₇	1	4	yes	—
164	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
166	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
167	C₂×#106	—	2	1	no	—
168	C₂×#102	—	2	1	no	—
169	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
170	#367	C₂²×C₆₇	1	4	yes	complemented
171	#366	C₂²×C₆₇	1	4	yes	complemented
172	C₄⋊C₁₆	C₂²×C₆₇	1	4	yes	complemented
173	#357	C₂²×C₆₇	1	4	yes	—
174	#357	C₂²×C₆₇	1	4	yes	—
175	C₂²×D₄	—	2	1	no	—
176	C₂³×C₄	—	4	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	Q₈×C₆₇	1	4	yes	—
180	C₂×D₈	C₂³×C₆₇	1	2	yes	—
181	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
183	#125	—	2	4	no	—
184	#125	—	4	2	no	—
185	C₄×D₄	—	2	4	no	—
186	C₄×D₄	—	4	2	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	4	1	no	—
191	C₂×C₄⋊C₄	—	4	1	no	—
192	C₂×C₂²⋊C₄	—	4	1	no	—
193	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
197	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
198	#107	—	2	1	no	—
199	#107	—	2	1	no	—
200	#107	—	2	1	no	—
201	#107	C₂³×C₆₇	1	2	yes	—
202	#107	C₂³×C₆₇	1	2	yes	—
203	#107	C₂³×C₆₇	1	2	yes	—
204	#106	—	2	4	no	—
205	#102	—	2	4	no	—
206	C₄×C₈	D₄×C₆₇	1	2	yes	—
207	C₄×C₈	C₂³×C₆₇	1	2	yes	—
208	C₂⁴	—	4	1	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	—	8	4	no	—
218	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
219	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
221	D₈	—	2	4	no	—
222	D₈	—	2	2	no	—
223	D₈	C₂×Q₈×C₆₇	1	4	yes	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
227	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
228	C₂×C₈	#41×C₆₇	1	2	yes	—
229	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	4	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	D₈×C₆₇	1	2	yes	—
240	C₄²	C₂×D₄×C₆₇	1	2	yes	—
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	—	8	2	no	—
258	C₂×C₄	—	8	2	no	—
259	C₂×C₄	—	8	2	no	—
260	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
261	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
262	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
263	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
264	C₂×C₄	#137×C₆₇	1	2	yes	—
265	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	C₂²⋊Q₈×C₆₇	1	2	yes	—
268	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
269	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
270	C₂²	—	4	4	no	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	8	no	—
274	C₂²	#474×C₆₇	1	2	yes	—
275	C₂²	#508×C₆₇	1	2	yes	—
276	C₂²	#509×C₆₇	1	2	yes	—
277	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	#474×C₆₇	1	2	yes	—
284	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
285	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
286	C₂	—	8	4	no	—
287	C₂	C₆₇×#2228	1	4	yes	direct product factor
288	C₂	C₂×#474×C₆₇	1	1	yes	characteristic
289	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
290	C₂	C₂×#509×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#499×C₆₇	C₂	semidirect
3. C₆₇×#2228	C₂	direct (N × H)
4. C₂×#367×C₆₇	C₂	semidirect
5. C₂×#366×C₆₇	C₂	semidirect
6. C₂×C₄⋊C₁₆×C₆₇	C₂	semidirect
15. #499×C₆₇	C₂²	semidirect
24. #367×C₆₇	C₂²	semidirect
25. #366×C₆₇	C₂²	semidirect
26. C₄⋊C₁₆×C₆₇	C₂²	semidirect
140. C₂×#2228	C₆₇	direct (N × H)
142. C₂×C₆₇	#2228	direct (N × H)
146. C₂×#499	C₂×C₆₇	semidirect
148. #2228	C₂×C₆₇	direct (N × H)
149. C₂×#367	C₂×C₆₇	semidirect
150. C₂×#366	C₂×C₆₇	semidirect
151. C₂×C₄⋊C₁₆	C₂×C₆₇	semidirect
154. C₆₇	C₂×#2228	direct (N × H)
161. #499	C₂²×C₆₇	semidirect
170. #367	C₂²×C₆₇	semidirect
171. #366	C₂²×C₆₇	semidirect
172. C₄⋊C₁₆	C₂²×C₆₇	semidirect
287. C₂	C₆₇×#2228	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×#499×C₆₇	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2228	2
C₂×#367×C₆₇	2
C₂×#366×C₆₇	2
C₂×C₄⋊C₁₆×C₆₇	2
C₂×#357×C₆₇	2
C₂×#2228	67

Subgroups — C2×C67×#2228 #30712873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2228 #30712873