Subgroups — C₂×#22587115 #31966513

Order 18176.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986023	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22587115	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17985752	C₂	1	4	yes	maximal, complemented
5	C₂×#431×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11765849	—	2	1	no	—
7	C₂³×C₇₁⋊C₈	—	2	1	no	—
8	#17986023	—	2	2	no	—
9	#17986023	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
12	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
13	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
16	#17985752	C₂²	1	16	yes	complemented
17	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
18	#431×C₇₁	C₄	1	4	yes	—
19	#431×C₇₁	C₂²	1	4	yes	—
20	C₂×C₄⋊C₈×C₇₁	C₄	1	2	yes	—
21	C₂×C₂²⋊C₈×C₇₁	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
23	C₂×#11765849	—	2	4	no	—
24	C₂×#11765849	—	4	4	no	—
25	C₂²×C₇₁⋊C₈	—	2	4	no	—
26	C₂²×C₇₁⋊C₈	—	2	2	no	—
27	C₂²×C₇₁⋊C₈	—	2	2	no	—
28	C₂²×C₇₁⋊C₈	—	4	4	no	—
29	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
30	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
31	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
32	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	C₂³	1	4	yes	—
34	C₄⋊(C₇₁⋊C₈)	C₂³	1	4	yes	—
35	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₁	Q₈	1	1	yes	characteristic
38	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	D₄	1	2	yes	—
43	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	Q₈	1	2	yes	—
45	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
48	C₂²⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₇₁	C₂³	1	4	yes	—
51	#11765849	—	4	8	no	—
52	C₂×C₇₁⋊C₈	—	2	4	no	—
53	C₂×C₇₁⋊C₈	—	2	4	no	—
54	C₂×C₇₁⋊C₈	—	4	4	no	—
55	C₂×C₇₁⋊C₈	—	4	8	no	—
56	C₂×C₇₁⋊C₈	—	4	4	no	—
57	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	—	2	4	no	—
59	C₂²×C₄×C₇₁	—	4	2	no	—
60	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	D₈	1	4	yes	—
67	C₂²×C₄×C₇₁	Q₁₆	1	4	yes	—
68	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₇₁	—	2	8	no	—
75	C₂×C₈×C₇₁	C₂²×C₄	1	8	yes	—
76	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
77	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
81	C₄²×C₇₁	C₂×D₄	1	2	yes	—
82	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
86	C₂³×C₇₁	—	2	2	no	—
87	C₂³×C₇₁	—	2	2	no	—
88	C₂³×C₇₁	#107	1	4	yes	—
89	C₂³×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	—	2	2	no	—
92	C₂×C₄×C₇₁	—	2	2	no	—
93	C₂×C₄×C₇₁	—	2	2	no	—
94	C₂×C₄×C₇₁	—	2	4	no	—
95	C₂×C₄×C₇₁	—	2	4	no	—
96	C₂×C₄×C₇₁	—	4	2	no	—
97	C₂×C₄×C₇₁	—	4	2	no	—
98	C₂×C₄×C₇₁	#107	1	8	yes	—
99	C₂×C₄×C₇₁	#107	1	4	yes	—
100	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₁	#131	1	4	yes	—
107	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₇₁	C₂×D₈	1	4	yes	—
109	C₂×C₄×C₇₁	C₂×Q₁₆	1	4	yes	—
110	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
114	C₈×C₇₁	—	2	8	no	—
115	C₇₁⋊C₈	—	4	4	no	—
116	C₇₁⋊C₈	—	4	4	no	—
117	C₂²×C₇₁	—	2	8	no	—
118	C₂²×C₇₁	—	2	2	no	—
119	C₂²×C₇₁	—	2	2	no	—
120	C₂²×C₇₁	#424	1	2	yes	—
121	C₂²×C₇₁	C₂×#107	1	4	yes	—
122	C₂²×C₇₁	C₂×#107	1	2	yes	—
123	C₂²×C₇₁	#430	1	2	yes	—
124	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₄×C₇₁	—	2	2	no	—
126	C₄×C₇₁	—	2	2	no	—
127	C₄×C₇₁	—	2	1	no	—
128	C₄×C₇₁	—	2	1	no	—
129	C₄×C₇₁	#424	1	2	yes	—
130	C₄×C₇₁	C₂×#107	1	2	yes	—
131	C₄×C₇₁	C₂×#107	1	4	yes	—
132	C₄×C₇₁	C₂×#131	1	2	yes	—
133	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
134	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
135	C₂×#1571	—	71	1	no	maximal, Sylow
136	C₂×C₇₁	—	2	4	no	—
137	C₂×C₇₁	#1571	1	4	yes	complemented
138	C₂×C₇₁	C₂×#424	1	1	yes	characteristic
139	C₂×C₇₁	C₂²×#107	1	1	yes	characteristic
140	C₂×C₇₁	C₂×#430	1	1	yes	characteristic
141	C₂×#431	D₇₁	1	1	yes	characteristic
142	C₂×#423	—	71	1	no	—
143	C₂²×C₄⋊C₈	—	71	1	no	—
144	#1571	—	71	8	no	—
145	C₂×#335	—	71	4	no	—
146	C₇₁	C₂×#1571	1	1	yes	characteristic, complemented, Sylow
147	C₂²×M₁₆	—	142	1	no	—
148	C₂³×C₈	—	142	1	no	—
149	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
150	#431	Dic₇₁	1	4	yes	—
151	#431	C₂×D₇₁	1	4	yes	—
152	#423	—	71	4	no	—
153	#423	—	142	2	no	—
154	C₂×C₄⋊C₈	—	71	2	no	—
155	C₂×C₄⋊C₈	—	71	2	no	—
156	C₂×C₄⋊C₈	—	71	4	no	—
157	C₂×C₄⋊C₈	—	142	2	no	—
158	C₂×C₄⋊C₈	—	142	1	no	—
159	C₂×C₄⋊C₈	—	142	1	no	—
160	C₂×C₄⋊C₈	Dic₇₁	1	2	yes	—
161	C₂×C₂²⋊C₈	Dic₇₁	1	2	yes	—
162	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
163	#335	—	71	16	no	—
164	C₂³×C₄	Dic₁₄₂	1	1	yes	characteristic
165	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
167	C₂×M₁₆	—	284	4	no	—
168	C₂×M₁₆	—	142	4	no	—
169	C₂²×C₈	—	284	4	no	—
170	C₂²×C₈	—	142	4	no	—
171	C₂²×C₈	—	142	2	no	—
172	C₂²×C₈	—	142	2	no	—
173	C₂²×C₈	C₂×Dic₇₁	1	4	yes	—
174	C₂×C₄²	Dic₁₄₂	1	1	yes	characteristic
175	C₂×C₄²	Dic₁₄₂	1	2	yes	—
176	C₂×C₄²	D₂₈₄	1	2	yes	—
177	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
178	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
179	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
180	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
181	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
182	C₄⋊C₈	—	71	4	no	—
183	C₄⋊C₈	—	71	4	no	—
184	C₄⋊C₈	—	142	2	no	—
185	C₄⋊C₈	—	142	4	no	—
186	C₄⋊C₈	—	142	2	no	—
187	C₄⋊C₈	—	142	4	no	—
188	C₄⋊C₈	C₂×Dic₇₁	1	8	yes	—
189	C₂²⋊C₈	C₂×Dic₇₁	1	8	yes	—
190	C₄×C₈	C₂×Dic₇₁	1	4	yes	—
191	C₄×C₈	C₂²×D₇₁	1	4	yes	—
192	C₂⁴	C₄⋊Dic₇₁	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
197	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
198	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
199	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
201	C₂²×C₄	D₅₆₈	1	4	yes	—
202	C₂²×C₄	Dic₂₈₄	1	4	yes	—
203	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₂₈₄	1	2	yes	—
206	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
208	C₂²×C₄	C₂×Dic₁₄₂	1	2	yes	—
209	M₁₆	—	284	8	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	284	4	no	—
212	C₂×C₈	—	284	8	no	—
213	C₂×C₈	—	284	4	no	—
214	C₂×C₈	—	142	4	no	—
215	C₂×C₈	—	142	4	no	—
216	C₂×C₈	C₂²×Dic₇₁	1	8	yes	—
217	C₄²	C₄⋊Dic₇₁	1	2	yes	—
218	C₄²	C₄⋊Dic₇₁	1	2	yes	—
219	C₄²	C₄⋊Dic₇₁	1	2	yes	—
220	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
221	C₄²	C₂²×Dic₇₁	1	2	yes	—
222	C₄²	C₂×D₂₈₄	1	2	yes	—
223	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
224	C₄²	C₂×Dic₁₄₂	1	2	yes	—
225	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
226	C₄²	C₂³×D₇₁	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#14556062	1	4	yes	—
230	C₂³	C₂×C₄⋊Dic₇₁	1	2	yes	—
231	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#14556062	1	8	yes	—
240	C₂×C₄	#14556062	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊Dic₇₁	1	2	yes	—
246	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
247	C₂×C₄	#14556135	1	4	yes	—
248	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
249	C₂×C₄	C₂×D₅₆₈	1	4	yes	—
250	C₂×C₄	C₂×Dic₂₈₄	1	4	yes	—
251	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
252	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
254	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	284	4	no	—
257	C₈	—	284	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	8	no	—
261	C₂²	#17986026	1	2	yes	—
262	C₂²	C₂×#14556062	1	4	yes	—
263	C₂²	C₂×#14556062	1	2	yes	—
264	C₂²	#17986048	1	2	yes	—
265	C₂²	C₂²×C₄⋊Dic₇₁	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	#17986026	1	2	yes	—
271	C₄	C₂×#14556062	1	4	yes	—
272	C₄	C₂×#14556062	1	2	yes	—
273	C₄	C₂×#14556135	1	2	yes	—
274	C₄	C₂²×D₅₆₈	1	1	yes	characteristic
275	C₄	C₂²×Dic₂₈₄	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	#22587115	1	4	yes	direct product factor
278	C₂	C₂×#17986026	1	1	yes	characteristic
279	C₂	C₂²×#14556062	1	1	yes	characteristic
280	C₂	C₂×#17986048	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₂×#431×C₇₁	C₂
socle	C₂³×C₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₁	C₄⋊C₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	C₂×#431	D₇₁
hypocenter	C₇₁	C₂×#1571
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22587115	C₂	direct (N × H)
4. C₂×#17985752	C₂	semidirect
16. #17985752	C₂²	semidirect
137. C₂×C₇₁	#1571	semidirect
146. C₇₁	C₂×#1571	semidirect
277. C₂	#22587115	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×#17986023	2
C₂²×C₄⋊(C₇₁⋊C₈)	2
#22587115	2
C₂×#17985752	2
C₂×#431×C₇₁	2
C₂×#1571	71

Subgroups — C2×#22587115 #31966513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22587115 #31966513