Subgroups — C₇₁×#7875 #31828747

Order 18176.

670 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₇₁	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1277	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈×C₇₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
6	#432×C₇₁	—	2	4	no	—
7	#432×C₇₁	C₄	1	8	yes	—
8	C₂×C₄⋊C₈×C₇₁	C₂²	1	4	yes	—
9	C₂×C₂²⋊C₈×C₇₁	—	2	2	no	—
10	C₂×#97×C₇₁	—	2	2	no	—
11	C₂×C₄×C₈×C₇₁	—	2	4	no	—
12	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
14	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₇₁	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₇₁	—	2	4	no	—
18	C₂²×C₈×C₇₁	—	2	8	no	—
19	C₂²×C₈×C₇₁	—	2	4	no	—
20	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	D₄	1	4	yes	—
25	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	—	2	4	no	—
28	C₄⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
29	C₂²⋊C₈×C₇₁	—	2	8	no	—
30	#97×C₇₁	—	2	8	no	—
31	C₄×C₈×C₇₁	—	2	4	no	—
32	C₄×C₈×C₇₁	—	2	4	no	—
33	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	—	2	4	no	—
35	C₂²×C₄×C₇₁	—	2	4	no	—
36	C₂²×C₄×C₇₁	—	2	4	no	—
37	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
38	C₂²×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	2	16	no	—
48	C₂×C₈×C₇₁	—	2	4	no	—
49	C₂×C₈×C₇₁	—	2	4	no	—
50	C₂×C₈×C₇₁	—	2	8	no	—
51	C₂×C₈×C₇₁	—	4	8	no	—
52	C₂×C₈×C₇₁	M₁₆	1	8	yes	—
53	C₄²×C₇₁	C₄²	1	2	yes	—
54	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
55	C₄²×C₇₁	C₂²⋊C₄	1	4	yes	—
56	C₄²×C₇₁	C₄⋊C₄	1	4	yes	—
57	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
58	C₄²×C₇₁	C₂×D₄	1	2	yes	—
59	C₂³×C₇₁	—	2	2	no	—
60	C₂³×C₇₁	—	2	2	no	—
61	C₂³×C₇₁	#95	1	2	yes	—
62	C₂³×C₇₁	#108	1	4	yes	—
63	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇₁	—	2	8	no	—
65	C₂×C₄×C₇₁	—	2	2	no	—
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	4	no	—
70	C₂×C₄×C₇₁	—	2	4	no	—
71	C₂×C₄×C₇₁	—	2	8	no	—
72	C₂×C₄×C₇₁	#95	1	2	yes	—
73	C₂×C₄×C₇₁	#95	1	2	yes	—
74	C₂×C₄×C₇₁	#95	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
80	C₂×C₄×C₇₁	#135	1	4	yes	—
81	C₈×C₇₁	—	2	8	no	—
82	C₈×C₇₁	—	4	8	no	—
83	C₂²×C₇₁	—	2	8	no	—
84	C₂²×C₇₁	—	2	2	no	—
85	C₂²×C₇₁	—	2	2	no	—
86	C₂²×C₇₁	#341	1	4	yes	—
87	C₂²×C₇₁	C₂×#95	1	1	yes	characteristic
88	C₂²×C₇₁	#423	1	2	yes	—
89	C₂²×C₇₁	#427	1	2	yes	—
90	C₂²×C₇₁	C₂×#108	1	2	yes	—
91	C₄×C₇₁	—	2	2	no	—
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	—	2	2	no	—
94	C₄×C₇₁	—	2	2	no	—
95	C₄×C₇₁	C₄×M₁₆	1	2	yes	—
96	C₄×C₇₁	#407	1	2	yes	—
97	C₄×C₇₁	#416	1	4	yes	—
98	#7875	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₇₁	—	2	4	no	—
100	C₂×C₇₁	#1568	1	4	yes	—
101	C₂×C₇₁	#1728	1	1	yes	characteristic
102	C₂×C₇₁	#1735	1	1	yes	characteristic
103	C₂×C₇₁	C₂×#341	1	1	yes	characteristic
104	C₂×#432	C₂×C₇₁	1	2	yes	—
105	C₂²×C₄×C₈	C₂×C₇₁	1	1	yes	characteristic
106	#1277	C₂×C₇₁	1	4	yes	complemented
107	C₇₁	#7875	1	1	yes	characteristic, direct product factor, Sylow
108	C₂³×C₈	C₄×C₇₁	1	2	yes	—
109	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
110	#432	—	2	4	no	—
111	#432	C₄×C₇₁	1	8	yes	—
112	C₂×C₄⋊C₈	C₂²×C₇₁	1	4	yes	—
113	C₂×C₂²⋊C₈	—	2	2	no	—
114	C₂×#97	—	2	2	no	—
115	C₂×C₄×C₈	—	2	4	no	—
116	C₂×C₄×C₈	C₄×C₇₁	1	2	yes	—
117	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
118	C₂³×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
119	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
120	C₂³×C₄	Q₈×C₇₁	1	1	yes	characteristic
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	2	8	no	—
123	C₂²×C₈	—	2	4	no	—
124	C₂²×C₈	C₂×C₄×C₇₁	1	4	yes	—
125	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
126	C₂×C₄²	C₂×C₄×C₇₁	1	4	yes	—
127	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
128	C₂×C₄²	D₄×C₇₁	1	4	yes	—
129	C₂×C₄²	Q₈×C₇₁	1	1	yes	characteristic
130	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
131	C₄⋊C₈	—	2	4	no	—
132	C₄⋊C₈	C₂×C₄×C₇₁	1	8	yes	—
133	C₂²⋊C₈	—	2	8	no	—
134	#97	—	2	8	no	—
135	C₄×C₈	—	2	4	no	—
136	C₄×C₈	—	2	4	no	—
137	C₂⁴	C₄⋊C₄×C₇₁	1	1	yes	characteristic
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₄²×C₇₁	1	2	yes	—
142	C₂²×C₄	C₂²⋊C₄×C₇₁	1	4	yes	—
143	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
144	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
146	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊C₄×C₇₁	1	4	yes	—
148	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
150	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
151	C₂×C₈	—	2	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	—	4	8	no	—
156	C₂×C₈	M₁₆×C₇₁	1	8	yes	—
157	C₄²	C₄²×C₇₁	1	2	yes	—
158	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
159	C₄²	C₂²⋊C₄×C₇₁	1	4	yes	—
160	C₄²	C₄⋊C₄×C₇₁	1	4	yes	—
161	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
162	C₄²	C₂×D₄×C₇₁	1	2	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	#95×C₇₁	1	2	yes	—
166	C₂³	#108×C₇₁	1	4	yes	—
167	C₂³	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	#95×C₇₁	1	2	yes	—
177	C₂×C₄	#95×C₇₁	1	2	yes	—
178	C₂×C₄	#95×C₇₁	1	2	yes	—
179	C₂×C₄	C₂×C₄²×C₇₁	1	1	yes	characteristic
180	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
181	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
182	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	2	yes	—
183	C₂×C₄	C₂×M₁₆×C₇₁	1	4	yes	—
184	C₂×C₄	#135×C₇₁	1	4	yes	—
185	C₈	—	2	8	no	—
186	C₈	—	4	8	no	—
187	C₂²	—	2	8	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	2	no	—
190	C₂²	#341×C₇₁	1	4	yes	—
191	C₂²	C₂×#95×C₇₁	1	1	yes	characteristic
192	C₂²	#423×C₇₁	1	2	yes	—
193	C₂²	#427×C₇₁	1	2	yes	—
194	C₂²	C₂×#108×C₇₁	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	C₄×M₁₆×C₇₁	1	2	yes	—
200	C₄	#407×C₇₁	1	2	yes	—
201	C₄	#416×C₇₁	1	4	yes	—
202	C₂	—	2	4	no	—
203	C₂	C₇₁×#1568	1	4	yes	—
204	C₂	C₇₁×#1728	1	1	yes	characteristic
205	C₂	C₇₁×#1735	1	1	yes	characteristic
206	C₂	C₂×#341×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂³×C₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₁ ◁ C₂²×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₁×#1277	C₂	semidirect
98. #7875	C₇₁	direct (N × H)
106. #1277	C₂×C₇₁	semidirect
107. C₇₁	#7875	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×#432×C₇₁	2
C₂²×C₄×C₈×C₇₁	2
C₇₁×#1277	2
#7875	71

Subgroups — C71×#7875 #31828747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7875 #31828747