Subgroups — C₂×C₇₁×#3391 #31882259

Order 18176.

2582 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₇₁	C₂	1	2	yes	maximal, complemented
2	C₇₁×#3391	C₂	1	16	yes	maximal, direct product factor
3	C₂×#493×C₇₁	C₂	1	3	yes	maximal, complemented
4	C₂×#459×C₇₁	C₂	1	6	yes	maximal, complemented
5	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₈×C₇₁	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₇₁	—	2	2	no	—
8	C₂²×D₈×C₇₁	C₂²	1	3	yes	—
9	C₂²×D₈×C₇₁	C₂²	1	6	yes	—
10	#550×C₇₁	—	2	8	no	—
11	#550×C₇₁	C₂²	1	16	yes	complemented
12	C₂×#127×C₇₁	C₂²	1	6	yes	—
13	C₂×#121×C₇₁	—	2	6	no	—
14	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₇₁	C₂²	1	6	yes	—
16	#493×C₇₁	C₂²	1	24	yes	complemented
17	#459×C₇₁	C₂²	1	48	yes	complemented
18	C₈×Q₈×C₇₁	C₂²	1	8	yes	complemented
19	C₄×D₈×C₇₁	C₂²	1	24	yes	complemented
20	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
22	C₂×#102×C₇₁	C₂²	1	6	yes	—
23	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
24	C₂×#41×C₇₁	—	2	16	no	—
25	C₂×#41×C₇₁	—	4	16	no	—
26	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₇₁	—	2	6	no	—
28	C₂²×D₄×C₇₁	—	2	6	no	—
29	C₂²×D₄×C₇₁	C₂³	1	6	yes	—
30	C₂³×C₄×C₇₁	—	2	6	no	—
31	C₂×D₈×C₇₁	—	2	12	no	—
32	C₂×D₈×C₇₁	—	4	12	no	—
33	C₂×D₈×C₇₁	C₂³	1	12	yes	—
34	C₂×D₈×C₇₁	C₂³	1	24	yes	—
35	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
36	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
37	#127×C₇₁	—	2	12	no	—
38	#127×C₇₁	C₂³	1	24	yes	—
39	#121×C₇₁	—	2	48	no	—
40	C₄×Q₈×C₇₁	D₄	1	4	yes	—
41	C₄×Q₈×C₇₁	C₂³	1	4	yes	—
42	C₄×D₄×C₇₁	—	2	12	no	—
43	C₄×D₄×C₇₁	C₂³	1	24	yes	—
44	C₂×C₄⋊C₄×C₇₁	D₄	1	3	yes	—
45	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
46	C₂×C₂²⋊C₄×C₇₁	—	2	6	no	—
47	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
48	#107×C₇₁	C₂³	1	4	yes	—
49	C₄⋊C₈×C₇₁	C₂³	1	12	yes	—
50	#102×C₇₁	C₂³	1	24	yes	—
51	C₄×C₈×C₇₁	C₂³	1	12	yes	—
52	C₂⁴×C₇₁	—	2	6	no	—
53	#41×C₇₁	—	4	64	no	—
54	C₂×Q₈×C₇₁	D₈	1	8	yes	complemented
55	C₂×Q₈×C₇₁	C₂×D₄	1	4	yes	—
56	C₂×D₄×C₇₁	—	2	6	no	—
57	C₂×D₄×C₇₁	—	2	24	no	—
58	C₂×D₄×C₇₁	—	2	24	no	—
59	C₂×D₄×C₇₁	—	2	12	no	—
60	C₂×D₄×C₇₁	—	2	12	no	—
61	C₂×D₄×C₇₁	—	4	24	no	—
62	C₂×D₄×C₇₁	—	8	12	no	—
63	C₂×D₄×C₇₁	C₂⁴	1	12	yes	—
64	C₂²×C₄×C₇₁	—	2	2	no	—
65	C₂²×C₄×C₇₁	—	2	24	no	—
66	C₂²×C₄×C₇₁	—	4	24	no	—
67	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
68	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
69	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
70	D₈×C₇₁	—	2	24	no	—
71	D₈×C₇₁	—	4	24	no	—
72	C₂×C₈×C₇₁	—	2	6	no	—
73	C₂×C₈×C₇₁	#41	1	4	yes	complemented
74	C₂×C₈×C₇₁	C₂⁴	1	6	yes	—
75	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
76	C₄⋊C₄×C₇₁	—	2	2	no	—
77	C₄⋊C₄×C₇₁	C₂×D₄	1	12	yes	—
78	C₄⋊C₄×C₇₁	C₂⁴	1	4	yes	—
79	C₂²⋊C₄×C₇₁	—	2	24	no	—
80	C₄²×C₇₁	C₂×D₄	1	6	yes	—
81	C₄²×C₇₁	C₂⁴	1	6	yes	—
82	C₂³×C₇₁	—	2	12	no	—
83	C₂³×C₇₁	—	4	12	no	—
84	C₂³×C₇₁	—	4	12	no	—
85	C₂³×C₇₁	—	4	12	no	—
86	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
87	Q₈×C₇₁	C₂×D₈	1	16	yes	complemented
88	D₄×C₇₁	—	2	12	no	—
89	D₄×C₇₁	—	2	12	no	—
90	D₄×C₇₁	—	4	12	no	—
91	D₄×C₇₁	—	4	48	no	—
92	D₄×C₇₁	—	8	24	no	—
93	C₂×C₄×C₇₁	—	2	6	no	—
94	C₂×C₄×C₇₁	—	2	4	no	—
95	C₂×C₄×C₇₁	—	4	4	no	—
96	C₂×C₄×C₇₁	—	4	48	no	—
97	C₂×C₄×C₇₁	C₂×D₈	1	12	yes	—
98	C₂×C₄×C₇₁	C₂²×D₄	1	6	yes	—
99	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂²×D₄	1	6	yes	—
101	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
102	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
103	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
104	C₈×C₇₁	—	2	6	no	—
105	C₈×C₇₁	C₂×#41	1	4	yes	complemented
106	C₂²×C₇₁	—	4	12	no	—
107	C₂²×C₇₁	—	4	12	no	—
108	C₂²×C₇₁	—	4	12	no	—
109	C₂²×C₇₁	—	8	24	no	—
110	C₂²×C₇₁	#550	1	2	yes	—
111	C₂²×C₇₁	C₂²×D₈	1	2	yes	—
112	C₂²×C₇₁	#576	1	2	yes	—
113	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
114	C₄×C₇₁	—	2	6	no	—
115	C₄×C₇₁	—	4	4	no	—
116	C₄×C₇₁	#550	1	2	yes	—
117	C₄×C₇₁	C₂²×D₈	1	12	yes	—
118	C₄×C₇₁	C₂²×#41	1	1	yes	characteristic
119	C₄×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
120	C₂×#3391	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₇₁	—	8	12	no	—
122	C₂×C₇₁	#3391	1	4	yes	direct product factor
123	C₂×C₇₁	C₂×#550	1	1	yes	characteristic
124	C₂×C₇₁	C₂³×D₈	1	1	yes	characteristic
125	C₂×C₇₁	C₂×#576	1	1	yes	characteristic
126	C₂×#550	C₂×C₇₁	1	2	yes	complemented
127	#3391	C₂×C₇₁	1	16	yes	direct product factor
128	C₂×#493	C₂×C₇₁	1	3	yes	complemented
129	C₂×#459	C₂×C₇₁	1	6	yes	complemented
130	C₂×C₈×Q₈	C₂×C₇₁	1	1	yes	characteristic, complemented
131	C₂×C₄×D₈	C₂×C₇₁	1	3	yes	complemented
132	C₇₁	C₂×#3391	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²×#41	—	2	2	no	—
134	C₂²×D₈	C₂²×C₇₁	1	3	yes	—
135	C₂²×D₈	C₂²×C₇₁	1	6	yes	—
136	#550	—	2	8	no	—
137	#550	C₂²×C₇₁	1	16	yes	complemented
138	C₂×#127	C₂²×C₇₁	1	6	yes	—
139	C₂×#121	—	2	6	no	—
140	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
141	C₂×C₄×D₄	C₂²×C₇₁	1	6	yes	—
142	#493	C₂²×C₇₁	1	24	yes	complemented
143	#459	C₂²×C₇₁	1	48	yes	complemented
144	C₈×Q₈	C₂²×C₇₁	1	8	yes	complemented
145	C₄×D₈	C₂²×C₇₁	1	24	yes	complemented
146	C₂×#107	C₂²×C₇₁	1	1	yes	characteristic
147	C₂×C₄⋊C₈	C₂²×C₇₁	1	3	yes	—
148	C₂×#102	C₂²×C₇₁	1	6	yes	—
149	C₂×C₄×C₈	C₂²×C₇₁	1	3	yes	—
150	C₂×#41	—	2	16	no	—
151	C₂×#41	—	4	16	no	—
152	C₂²×Q₈	D₄×C₇₁	1	1	yes	characteristic
153	C₂²×D₄	—	2	6	no	—
154	C₂²×D₄	—	2	6	no	—
155	C₂²×D₄	C₂³×C₇₁	1	6	yes	—
156	C₂³×C₄	—	2	6	no	—
157	C₂×D₈	—	2	12	no	—
158	C₂×D₈	—	4	12	no	—
159	C₂×D₈	C₂³×C₇₁	1	12	yes	—
160	C₂×D₈	C₂³×C₇₁	1	24	yes	—
161	C₂²×C₈	C₂³×C₇₁	1	3	yes	—
162	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
163	#127	—	2	12	no	—
164	#127	C₂³×C₇₁	1	24	yes	—
165	#121	—	2	48	no	—
166	C₄×Q₈	D₄×C₇₁	1	4	yes	—
167	C₄×Q₈	C₂³×C₇₁	1	4	yes	—
168	C₄×D₄	—	2	12	no	—
169	C₄×D₄	C₂³×C₇₁	1	24	yes	—
170	C₂×C₄⋊C₄	D₄×C₇₁	1	3	yes	—
171	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
172	C₂×C₂²⋊C₄	—	2	6	no	—
173	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
174	#107	C₂³×C₇₁	1	4	yes	—
175	C₄⋊C₈	C₂³×C₇₁	1	12	yes	—
176	#102	C₂³×C₇₁	1	24	yes	—
177	C₄×C₈	C₂³×C₇₁	1	12	yes	—
178	C₂⁴	—	2	6	no	—
179	#41	—	4	64	no	—
180	C₂×Q₈	D₈×C₇₁	1	8	yes	complemented
181	C₂×Q₈	C₂×D₄×C₇₁	1	4	yes	—
182	C₂×D₄	—	2	6	no	—
183	C₂×D₄	—	2	24	no	—
184	C₂×D₄	—	2	24	no	—
185	C₂×D₄	—	2	12	no	—
186	C₂×D₄	—	2	12	no	—
187	C₂×D₄	—	4	24	no	—
188	C₂×D₄	—	8	12	no	—
189	C₂×D₄	C₂⁴×C₇₁	1	12	yes	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	24	no	—
192	C₂²×C₄	—	4	24	no	—
193	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
194	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
195	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
196	D₈	—	2	24	no	—
197	D₈	—	4	24	no	—
198	C₂×C₈	—	2	6	no	—
199	C₂×C₈	#41×C₇₁	1	4	yes	complemented
200	C₂×C₈	C₂⁴×C₇₁	1	6	yes	—
201	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	C₂×D₄×C₇₁	1	12	yes	—
204	C₄⋊C₄	C₂⁴×C₇₁	1	4	yes	—
205	C₂²⋊C₄	—	2	24	no	—
206	C₄²	C₂×D₄×C₇₁	1	6	yes	—
207	C₄²	C₂⁴×C₇₁	1	6	yes	—
208	C₂³	—	2	12	no	—
209	C₂³	—	4	12	no	—
210	C₂³	—	4	12	no	—
211	C₂³	—	4	12	no	—
212	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
213	Q₈	C₂×D₈×C₇₁	1	16	yes	complemented
214	D₄	—	2	12	no	—
215	D₄	—	2	12	no	—
216	D₄	—	4	12	no	—
217	D₄	—	4	48	no	—
218	D₄	—	8	24	no	—
219	C₂×C₄	—	2	6	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	48	no	—
223	C₂×C₄	C₂×D₈×C₇₁	1	12	yes	—
224	C₂×C₄	C₂²×D₄×C₇₁	1	6	yes	—
225	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
226	C₂×C₄	C₂²×D₄×C₇₁	1	6	yes	—
227	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
228	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
229	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
230	C₈	—	2	6	no	—
231	C₈	C₂×#41×C₇₁	1	4	yes	complemented
232	C₂²	—	4	12	no	—
233	C₂²	—	4	12	no	—
234	C₂²	—	4	12	no	—
235	C₂²	—	8	24	no	—
236	C₂²	#550×C₇₁	1	2	yes	—
237	C₂²	C₂²×D₈×C₇₁	1	2	yes	—
238	C₂²	#576×C₇₁	1	2	yes	—
239	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
240	C₄	—	2	6	no	—
241	C₄	—	4	4	no	—
242	C₄	#550×C₇₁	1	2	yes	—
243	C₄	C₂²×D₈×C₇₁	1	12	yes	—
244	C₄	C₂²×#41×C₇₁	1	1	yes	characteristic
245	C₄	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
246	C₂	—	8	12	no	—
247	C₂	C₇₁×#3391	1	4	yes	direct product factor
248	C₂	C₂×#550×C₇₁	1	1	yes	characteristic
249	C₂	C₂³×D₈×C₇₁	1	1	yes	characteristic
250	C₂	C₂×#576×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄×Q₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#550×C₇₁	C₂	semidirect
2. C₇₁×#3391	C₂	direct (N × H)
3. C₂×#493×C₇₁	C₂	semidirect
4. C₂×#459×C₇₁	C₂	semidirect
5. C₂×C₈×Q₈×C₇₁	C₂	semidirect
6. C₂×C₄×D₈×C₇₁	C₂	semidirect
11. #550×C₇₁	C₂²	semidirect
16. #493×C₇₁	C₂²	semidirect
17. #459×C₇₁	C₂²	semidirect
18. C₈×Q₈×C₇₁	C₂²	semidirect
19. C₄×D₈×C₇₁	C₂²	semidirect
54. C₂×Q₈×C₇₁	D₈	semidirect
73. C₂×C₈×C₇₁	#41	semidirect
87. Q₈×C₇₁	C₂×D₈	semidirect
105. C₈×C₇₁	C₂×#41	semidirect
120. C₂×#3391	C₇₁	direct (N × H)
122. C₂×C₇₁	#3391	direct (N × H)
126. C₂×#550	C₂×C₇₁	semidirect
127. #3391	C₂×C₇₁	direct (N × H)
128. C₂×#493	C₂×C₇₁	semidirect
129. C₂×#459	C₂×C₇₁	semidirect
130. C₂×C₈×Q₈	C₂×C₇₁	semidirect
131. C₂×C₄×D₈	C₂×C₇₁	semidirect
132. C₇₁	C₂×#3391	direct (N × H)
137. #550	C₂²×C₇₁	semidirect
142. #493	C₂²×C₇₁	semidirect
143. #459	C₂²×C₇₁	semidirect
144. C₈×Q₈	C₂²×C₇₁	semidirect
145. C₄×D₈	C₂²×C₇₁	semidirect
180. C₂×Q₈	D₈×C₇₁	semidirect
199. C₂×C₈	#41×C₇₁	semidirect
213. Q₈	C₂×D₈×C₇₁	semidirect
231. C₈	C₂×#41×C₇₁	semidirect
247. C₂	C₇₁×#3391	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×#550×C₇₁	2
C₇₁×#3391	2
C₂×#493×C₇₁	2
C₂×#459×C₇₁	2
C₂×C₈×Q₈×C₇₁	2
C₂×C₄×D₈×C₇₁	2
C₂×#3391	71

Subgroups — C2×C71×#3391 #31882259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#3391 #31882259