Subgroups — C₇₁×#23801 #31844673

Order 18176.

986 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₁₆×C₇₁	C₂	1	4	yes	maximal
2	C₄×C₄⋊Q₈×C₇₁	C₂	1	2	yes	maximal
3	C₇₁×#2095	C₂	1	4	yes	maximal
4	C₇₁×#2016	C₂	1	4	yes	maximal
5	C₇₁×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₇₁	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₇₁	C₂²	1	4	yes	—
9	C₄×Q₁₆×C₇₁	—	2	16	no	—
10	C₂×#107×C₇₁	C₂²	1	4	yes	—
11	C₂×#103×C₇₁	C₂²	1	8	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
13	#391×C₇₁	C₂²	1	4	yes	—
14	#389×C₇₁	—	2	4	no	—
15	#384×C₇₁	C₂²	1	2	yes	—
16	C₄×C₄⋊C₄×C₇₁	—	2	4	no	—
17	C₄³×C₇₁	C₂²	1	1	yes	characteristic
18	#336×C₇₁	C₂²	1	4	yes	—
19	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
20	C₂×Q₁₆×C₇₁	—	2	16	no	—
21	C₂×Q₁₆×C₇₁	—	2	8	no	—
22	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
23	C₄⋊Q₈×C₇₁	—	2	8	no	—
24	C₄×Q₈×C₇₁	—	2	16	no	—
25	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
26	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
28	C₂×C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
29	C₂×C₄²×C₇₁	—	2	4	no	—
30	C₂×C₄²×C₇₁	D₄	1	2	yes	—
31	C₂×C₄²×C₇₁	D₄	1	2	yes	—
32	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	#107×C₇₁	—	2	8	no	—
35	#103×C₇₁	—	2	16	no	—
36	C₄×C₈×C₇₁	D₄	1	8	yes	—
37	#95×C₇₁	—	2	8	no	—
38	C₂×Q₈×C₇₁	—	2	4	no	—
39	C₂×Q₈×C₇₁	—	2	8	no	—
40	C₂×Q₈×C₇₁	—	2	8	no	—
41	C₂×Q₈×C₇₁	#41	1	8	yes	—
42	C₂²×C₄×C₇₁	—	2	4	no	—
43	C₂²×C₄×C₇₁	—	2	4	no	—
44	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
48	Q₁₆×C₇₁	—	2	32	no	—
49	C₂×C₈×C₇₁	—	2	8	no	—
50	C₂×C₈×C₇₁	C₂×D₄	1	8	yes	—
51	C₄⋊C₄×C₇₁	—	2	16	no	—
52	C₄⋊C₄×C₇₁	—	2	8	no	—
53	C₄⋊C₄×C₇₁	—	4	8	no	—
54	C₄⋊C₄×C₇₁	—	4	8	no	—
55	C₄²×C₇₁	—	2	4	no	—
56	C₄²×C₇₁	—	4	2	no	—
57	C₄²×C₇₁	—	4	8	no	—
58	C₄²×C₇₁	Q₁₆	1	8	yes	complemented
59	C₄²×C₇₁	C₂×D₄	1	4	yes	—
60	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
61	Q₈×C₇₁	—	2	16	no	—
62	Q₈×C₇₁	—	4	8	no	—
63	C₂×C₄×C₇₁	—	2	4	no	—
64	C₂×C₄×C₇₁	—	2	4	no	—
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	8	no	—
67	C₂×C₄×C₇₁	—	4	4	no	—
68	C₂×C₄×C₇₁	—	4	4	no	—
69	C₂×C₄×C₇₁	—	4	4	no	—
70	C₂×C₄×C₇₁	—	4	8	no	—
71	C₂×C₄×C₇₁	#127	1	2	yes	—
72	C₂×C₄×C₇₁	#127	1	2	yes	—
73	C₂×C₄×C₇₁	C₂×Q₁₆	1	8	yes	—
74	C₂×C₄×C₇₁	C₂×Q₁₆	1	4	yes	—
75	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₇₁	C₂×#41	1	4	yes	—
77	C₈×C₇₁	—	2	8	no	—
78	C₂²×C₇₁	#525	1	2	yes	—
79	C₂²×C₇₁	C₂×#127	1	1	yes	characteristic
80	C₂²×C₇₁	C₂²×Q₁₆	1	2	yes	—
81	C₂²×C₇₁	#575	1	2	yes	—
82	C₄×C₇₁	—	2	8	no	—
83	C₄×C₇₁	—	2	2	no	—
84	C₄×C₇₁	—	4	8	no	—
85	C₄×C₇₁	—	8	4	no	—
86	C₄×C₇₁	#494	1	4	yes	—
87	C₄×C₇₁	#525	1	4	yes	—
88	#23801	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂×C₇₁	#2397	1	1	yes	characteristic
90	C₂×C₇₁	#3104	1	4	yes	—
91	C₂×C₇₁	C₂×#494	1	1	yes	characteristic
92	C₂×C₇₁	#3150	1	1	yes	characteristic
93	C₂×C₄×Q₁₆	C₂×C₇₁	1	4	yes	—
94	C₄×C₄⋊Q₈	C₂×C₇₁	1	2	yes	—
95	#2095	C₂×C₇₁	1	4	yes	—
96	#2016	C₂×C₇₁	1	4	yes	—
97	#1744	C₂×C₇₁	1	1	yes	characteristic
98	C₇₁	#23801	1	1	yes	characteristic, direct product factor, Sylow
99	C₂²×Q₁₆	C₂²×C₇₁	1	4	yes	—
100	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
101	C₂×C₄×Q₈	C₂²×C₇₁	1	4	yes	—
102	C₄×Q₁₆	—	2	16	no	—
103	C₂×#107	C₂²×C₇₁	1	4	yes	—
104	C₂×#103	C₂²×C₇₁	1	8	yes	—
105	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
106	#391	C₂²×C₇₁	1	4	yes	—
107	#389	—	2	4	no	—
108	#384	C₂²×C₇₁	1	2	yes	—
109	C₄×C₄⋊C₄	—	2	4	no	—
110	C₄³	C₂²×C₇₁	1	1	yes	characteristic
111	#336	C₂²×C₇₁	1	4	yes	—
112	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
113	C₂×Q₁₆	—	2	16	no	—
114	C₂×Q₁₆	—	2	8	no	—
115	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
116	C₄⋊Q₈	—	2	8	no	—
117	C₄×Q₈	—	2	16	no	—
118	C₂×C₄⋊C₄	—	2	4	no	—
119	C₂×C₄⋊C₄	—	2	4	no	—
120	C₂×C₄⋊C₄	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂³×C₇₁	1	4	yes	—
122	C₂×C₄²	—	2	4	no	—
123	C₂×C₄²	D₄×C₇₁	1	2	yes	—
124	C₂×C₄²	D₄×C₇₁	1	2	yes	—
125	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
126	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
127	#107	—	2	8	no	—
128	#103	—	2	16	no	—
129	C₄×C₈	D₄×C₇₁	1	8	yes	—
130	#95	—	2	8	no	—
131	C₂×Q₈	—	2	4	no	—
132	C₂×Q₈	—	2	8	no	—
133	C₂×Q₈	—	2	8	no	—
134	C₂×Q₈	#41×C₇₁	1	8	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×D₄×C₇₁	1	4	yes	—
139	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
140	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
141	Q₁₆	—	2	32	no	—
142	C₂×C₈	—	2	8	no	—
143	C₂×C₈	C₂×D₄×C₇₁	1	8	yes	—
144	C₄⋊C₄	—	2	16	no	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	4	8	no	—
147	C₄⋊C₄	—	4	8	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	4	2	no	—
150	C₄²	—	4	8	no	—
151	C₄²	Q₁₆×C₇₁	1	8	yes	complemented
152	C₄²	C₂×D₄×C₇₁	1	4	yes	—
153	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
154	Q₈	—	2	16	no	—
155	Q₈	—	4	8	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	#127×C₇₁	1	2	yes	—
165	C₂×C₄	#127×C₇₁	1	2	yes	—
166	C₂×C₄	C₂×Q₁₆×C₇₁	1	8	yes	—
167	C₂×C₄	C₂×Q₁₆×C₇₁	1	4	yes	—
168	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
169	C₂×C₄	C₂×#41×C₇₁	1	4	yes	—
170	C₈	—	2	8	no	—
171	C₂²	#525×C₇₁	1	2	yes	—
172	C₂²	C₂×#127×C₇₁	1	1	yes	characteristic
173	C₂²	C₂²×Q₁₆×C₇₁	1	2	yes	—
174	C₂²	#575×C₇₁	1	2	yes	—
175	C₄	—	2	8	no	—
176	C₄	—	2	2	no	—
177	C₄	—	4	8	no	—
178	C₄	—	8	4	no	—
179	C₄	#494×C₇₁	1	4	yes	—
180	C₄	#525×C₇₁	1	4	yes	—
181	C₂	C₇₁×#2397	1	1	yes	characteristic
182	C₂	C₇₁×#3104	1	4	yes	—
183	C₂	C₂×#494×C₇₁	1	1	yes	characteristic
184	C₂	C₇₁×#3150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle, prime radical	C₂³×C₇₁	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₁ ◁ C₄³×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
58. C₄²×C₇₁	Q₁₆	semidirect
88. #23801	C₇₁	direct (N × H)
98. C₇₁	#23801	direct (N × H)
151. C₄²	Q₁₆×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₁₆×C₇₁	2
C₄×C₄⋊Q₈×C₇₁	2
C₇₁×#2095	2
C₇₁×#2016	2
C₇₁×#1744	2
#23801	71

Subgroups — C71×#23801 #31844673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23801 #31844673