Subgroups — C₇₁×#23200 #31844072

Order 18176.

878 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2950	C₂	1	4	yes	maximal
3	C₇₁×#1979	C₂	1	4	yes	maximal
4	C₇₁×#1886	C₂	1	4	yes	maximal
5	C₇₁×#1772	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₇₁	C₂²	1	4	yes	—
7	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
8	#519×C₇₁	C₂²	1	4	yes	—
9	#494×C₇₁	C₄	1	8	yes	complemented
10	#441×C₇₁	—	2	8	no	—
11	C₄×Q₁₆×C₇₁	—	2	8	no	—
12	C₂×#108×C₇₁	C₂²	1	2	yes	—
13	#427×C₇₁	C₂²	1	2	yes	—
14	#419×C₇₁	C₂²	1	8	yes	—
15	C₂×#101×C₇₁	C₂²	1	4	yes	—
16	#405×C₇₁	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#339×C₇₁	C₂²	1	4	yes	—
19	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
20	C₂×Q₁₆×C₇₁	—	2	4	no	—
21	C₂×Q₁₆×C₇₁	—	2	4	no	—
22	C₂×Q₁₆×C₇₁	—	2	8	no	—
23	C₂×Q₁₆×C₇₁	C₂×C₄	1	16	yes	—
24	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
25	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
26	C₄⋊Q₈×C₇₁	—	4	2	no	—
27	C₄⋊Q₈×C₇₁	C₂×C₄	1	8	yes	—
28	C₄×Q₈×C₇₁	—	2	16	no	—
29	#117×C₇₁	—	4	4	no	—
30	#117×C₇₁	C₂³	1	4	yes	—
31	C₂×C₄⋊C₄×C₇₁	—	4	2	no	—
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	#108×C₇₁	—	2	4	no	—
34	#107×C₇₁	D₄	1	4	yes	—
35	#106×C₇₁	D₄	1	4	yes	—
36	#103×C₇₁	—	2	16	no	—
37	#101×C₇₁	—	4	4	no	—
38	#97×C₇₁	—	2	4	no	—
39	C₄×C₈×C₇₁	—	2	4	no	—
40	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
41	C₂×Q₈×C₇₁	—	2	4	no	—
42	C₂×Q₈×C₇₁	—	2	8	no	—
43	C₂×Q₈×C₇₁	—	2	8	no	—
44	C₂×Q₈×C₇₁	C₂²×C₄	1	8	yes	—
45	C₂²×C₄×C₇₁	—	2	1	no	—
46	C₂²×C₄×C₇₁	—	2	4	no	—
47	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
48	Q₁₆×C₇₁	—	2	16	no	—
49	Q₁₆×C₇₁	—	2	16	no	—
50	M₁₆×C₇₁	—	2	4	no	—
51	M₁₆×C₇₁	—	2	4	no	—
52	C₂×C₈×C₇₁	—	2	4	no	—
53	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
54	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₇₁	#41	1	4	yes	—
57	C₄⋊C₄×C₇₁	—	4	8	no	—
58	C₄⋊C₄×C₇₁	—	4	8	no	—
59	C₄⋊C₄×C₇₁	C₂×D₄	1	8	yes	—
60	C₂²⋊C₄×C₇₁	—	2	4	no	—
61	C₄²×C₇₁	—	2	4	no	—
62	C₄²×C₇₁	—	2	1	no	—
63	C₄²×C₇₁	—	4	8	no	—
64	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
65	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
66	Q₈×C₇₁	—	2	16	no	—
67	Q₈×C₇₁	—	4	8	no	—
68	C₂×C₄×C₇₁	—	2	4	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	8	no	—
71	C₂×C₄×C₇₁	—	2	8	no	—
72	C₂×C₄×C₇₁	—	4	4	no	—
73	C₂×C₄×C₇₁	—	4	4	no	—
74	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
76	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
77	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
78	C₈×C₇₁	—	4	4	no	—
79	C₈×C₇₁	C₄×D₄	1	4	yes	—
80	C₈×C₇₁	C₄×D₄	1	4	yes	—
81	C₂²×C₇₁	—	2	2	no	—
82	C₂²×C₇₁	#518	1	1	yes	characteristic
83	C₂²×C₇₁	D₄²	1	1	yes	characteristic
84	C₂²×C₇₁	#559	1	1	yes	characteristic
85	C₄×C₇₁	—	2	4	no	—
86	C₄×C₇₁	—	4	8	no	—
87	C₄×C₇₁	—	4	8	no	—
88	C₄×C₇₁	C₂×C₄×D₄	1	2	yes	—
89	C₄×C₇₁	#550	1	2	yes	—
90	#23200	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	#2366	1	1	yes	characteristic
94	C₂×C₇₁	#3293	1	2	yes	—
95	C₂×#494	C₂×C₇₁	1	1	yes	characteristic
96	#2950	C₂×C₇₁	1	4	yes	—
97	#1979	C₂×C₇₁	1	4	yes	—
98	#1886	C₂×C₇₁	1	4	yes	—
99	#1772	C₂×C₇₁	1	2	yes	—
100	C₇₁	#23200	1	1	yes	characteristic, direct product factor, Sylow
101	C₂²×Q₁₆	C₂²×C₇₁	1	4	yes	—
102	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
103	#519	C₂²×C₇₁	1	4	yes	—
104	#494	C₄×C₇₁	1	8	yes	complemented
105	#441	—	2	8	no	—
106	C₄×Q₁₆	—	2	8	no	—
107	C₂×#108	C₂²×C₇₁	1	2	yes	—
108	#427	C₂²×C₇₁	1	2	yes	—
109	#419	C₂²×C₇₁	1	8	yes	—
110	C₂×#101	C₂²×C₇₁	1	4	yes	—
111	#405	C₂²×C₇₁	1	4	yes	—
112	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
113	#339	C₂²×C₇₁	1	4	yes	—
114	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
115	C₂×Q₁₆	—	2	4	no	—
116	C₂×Q₁₆	—	2	4	no	—
117	C₂×Q₁₆	—	2	8	no	—
118	C₂×Q₁₆	C₂×C₄×C₇₁	1	16	yes	—
119	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
120	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
121	C₄⋊Q₈	—	4	2	no	—
122	C₄⋊Q₈	C₂×C₄×C₇₁	1	8	yes	—
123	C₄×Q₈	—	2	16	no	—
124	#117	—	4	4	no	—
125	#117	C₂³×C₇₁	1	4	yes	—
126	C₂×C₄⋊C₄	—	4	2	no	—
127	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
128	#108	—	2	4	no	—
129	#107	D₄×C₇₁	1	4	yes	—
130	#106	D₄×C₇₁	1	4	yes	—
131	#103	—	2	16	no	—
132	#101	—	4	4	no	—
133	#97	—	2	4	no	—
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	C₂×C₄×C₇₁	1	4	yes	—
136	C₂×Q₈	—	2	4	no	—
137	C₂×Q₈	—	2	8	no	—
138	C₂×Q₈	—	2	8	no	—
139	C₂×Q₈	C₂²×C₄×C₇₁	1	8	yes	—
140	C₂²×C₄	—	2	1	no	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
143	Q₁₆	—	2	16	no	—
144	Q₁₆	—	2	16	no	—
145	M₁₆	—	2	4	no	—
146	M₁₆	—	2	4	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	C₂²×C₄×C₇₁	1	4	yes	—
149	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
150	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
151	C₂×C₈	#41×C₇₁	1	4	yes	—
152	C₄⋊C₄	—	4	8	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₄⋊C₄	C₂×D₄×C₇₁	1	8	yes	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₄²	—	2	4	no	—
157	C₄²	—	2	1	no	—
158	C₄²	—	4	8	no	—
159	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
160	C₂³	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
161	Q₈	—	2	16	no	—
162	Q₈	—	4	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	2	8	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
170	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
171	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
172	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
173	C₈	—	4	4	no	—
174	C₈	C₄×D₄×C₇₁	1	4	yes	—
175	C₈	C₄×D₄×C₇₁	1	4	yes	—
176	C₂²	—	2	2	no	—
177	C₂²	#518×C₇₁	1	1	yes	characteristic
178	C₂²	D₄²×C₇₁	1	1	yes	characteristic
179	C₂²	#559×C₇₁	1	1	yes	characteristic
180	C₄	—	2	4	no	—
181	C₄	—	4	8	no	—
182	C₄	—	4	8	no	—
183	C₄	C₂×C₄×D₄×C₇₁	1	2	yes	—
184	C₄	#550×C₇₁	1	2	yes	—
185	C₂	—	2	1	no	—
186	C₂	—	2	1	no	—
187	C₂	C₇₁×#2366	1	1	yes	characteristic
188	C₂	C₇₁×#3293	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₂³×C₇₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. #494×C₇₁	C₄	semidirect
90. #23200	C₇₁	direct (N × H)
100. C₇₁	#23200	direct (N × H)
104. #494	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₇₁	2
C₇₁×#2950	2
C₇₁×#1979	2
C₇₁×#1886	2
C₇₁×#1772	2
#23200	71

Subgroups — C71×#23200 #31844072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23200 #31844072