Subgroups — C₇₁×#23197 #31844069

Order 18176.

880 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3150	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2957	C₂	1	4	yes	maximal, complemented
3	C₇₁×#1976	C₂	1	4	yes	maximal, complemented
4	C₇₁×#1884	C₂	1	4	yes	maximal, complemented
5	C₇₁×#1773	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₇₁	C₂²	1	4	yes	—
7	#568×C₇₁	C₂²	1	4	yes	—
8	#532×C₇₁	C₂²	1	2	yes	—
9	#497×C₇₁	C₄	1	4	yes	complemented
10	#496×C₇₁	C₄	1	4	yes	complemented
11	#444×C₇₁	—	2	8	no	—
12	#443×C₇₁	—	2	8	no	—
13	C₂×#108×C₇₁	C₂²	1	2	yes	—
14	#427×C₇₁	C₂²	1	2	yes	—
15	#421×C₇₁	C₂²	1	8	yes	—
16	#410×C₇₁	C₂²	1	4	yes	—
17	#405×C₇₁	C₂²	1	4	yes	—
18	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
19	#339×C₇₁	C₂²	1	4	yes	—
20	C₂×#41×C₇₁	C₂³	1	4	yes	—
21	#135×C₇₁	—	2	4	no	—
22	#135×C₇₁	—	2	4	no	—
23	#135×C₇₁	—	2	4	no	—
24	#135×C₇₁	—	2	4	no	—
25	C₂×Q₁₆×C₇₁	C₂×C₄	1	4	yes	—
26	C₂×SD₁₆×C₇₁	C₂×C₄	1	8	yes	—
27	C₂×D₈×C₇₁	C₂×C₄	1	4	yes	—
28	#131×C₇₁	—	2	16	no	—
29	C₂×M₁₆×C₇₁	—	4	4	no	—
30	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
32	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
33	#127×C₇₁	C₂×C₄	1	2	yes	—
34	#124×C₇₁	C₂×C₄	1	4	yes	—
35	#121×C₇₁	—	4	2	no	—
36	C₄×D₄×C₇₁	—	4	2	no	—
37	#117×C₇₁	C₂³	1	4	yes	—
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	#108×C₇₁	D₄	1	8	yes	complemented
40	#107×C₇₁	—	2	2	no	—
41	#106×C₇₁	—	2	2	no	—
42	#104×C₇₁	—	2	16	no	—
43	#99×C₇₁	—	4	4	no	—
44	#97×C₇₁	—	2	4	no	—
45	#97×C₇₁	C₂×C₄	1	4	yes	—
46	C₄×C₈×C₇₁	—	2	4	no	—
47	#41×C₇₁	—	2	8	no	—
48	#41×C₇₁	—	2	8	no	—
49	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
50	C₂×D₄×C₇₁	—	2	4	no	—
51	C₂×D₄×C₇₁	—	4	2	no	—
52	C₂×D₄×C₇₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₇₁	—	2	1	no	—
54	C₂²×C₄×C₇₁	—	2	4	no	—
55	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
56	Q₁₆×C₇₁	—	2	4	no	—
57	Q₁₆×C₇₁	—	2	4	no	—
58	SD₁₆×C₇₁	—	2	8	no	—
59	SD₁₆×C₇₁	—	2	8	no	—
60	D₈×C₇₁	—	2	4	no	—
61	D₈×C₇₁	—	2	4	no	—
62	M₁₆×C₇₁	—	2	4	no	—
63	M₁₆×C₇₁	—	4	8	no	—
64	M₁₆×C₇₁	C₂×D₄	1	8	yes	—
65	C₂×C₈×C₇₁	—	2	4	no	—
66	C₂×C₈×C₇₁	—	4	8	no	—
67	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
68	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
69	C₂×C₈×C₇₁	#41	1	2	yes	—
70	C₂×C₈×C₇₁	#41	1	2	yes	—
71	C₄⋊C₄×C₇₁	—	2	4	no	—
72	C₄⋊C₄×C₇₁	—	4	2	no	—
73	C₂²⋊C₄×C₇₁	—	2	4	no	—
74	C₂²⋊C₄×C₇₁	—	4	4	no	—
75	C₄²×C₇₁	—	2	1	no	—
76	C₄²×C₇₁	—	2	4	no	—
77	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
78	C₂³×C₇₁	—	2	4	no	—
79	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
80	Q₈×C₇₁	—	2	8	no	—
81	D₄×C₇₁	—	2	8	no	—
82	D₄×C₇₁	—	4	8	no	—
83	D₄×C₇₁	—	8	4	no	—
84	C₂×C₄×C₇₁	—	2	2	no	—
85	C₂×C₄×C₇₁	—	2	4	no	—
86	C₂×C₄×C₇₁	—	4	4	no	—
87	C₂×C₄×C₇₁	—	4	4	no	—
88	C₂×C₄×C₇₁	—	4	4	no	—
89	C₂×C₄×C₇₁	—	8	1	no	—
90	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
92	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
93	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
94	C₈×C₇₁	—	2	8	no	—
95	C₈×C₇₁	C₄×D₄	1	4	yes	—
96	C₈×C₇₁	C₄×D₄	1	4	yes	—
97	C₂²×C₇₁	—	4	1	no	—
98	C₂²×C₇₁	—	4	4	no	—
99	C₂²×C₇₁	—	8	4	no	—
100	C₂²×C₇₁	#518	1	1	yes	characteristic
101	C₂²×C₇₁	#550	1	2	yes	—
102	C₄×C₇₁	—	4	2	no	—
103	C₄×C₇₁	—	4	4	no	—
104	C₄×C₇₁	—	8	4	no	—
105	C₄×C₇₁	C₂×C₄×D₄	1	2	yes	—
106	C₄×C₇₁	D₄²	1	1	yes	characteristic
107	C₄×C₇₁	#559	1	1	yes	characteristic
108	#23197	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂×C₇₁	—	2	1	no	—
110	C₂×C₇₁	—	2	2	no	—
111	C₂×C₇₁	—	8	4	no	—
112	C₂×C₇₁	#2366	1	1	yes	characteristic
113	#3150	C₂×C₇₁	1	1	yes	characteristic
114	#2957	C₂×C₇₁	1	4	yes	complemented
115	#1976	C₂×C₇₁	1	4	yes	complemented
116	#1884	C₂×C₇₁	1	4	yes	complemented
117	#1773	C₂×C₇₁	1	2	yes	complemented
118	C₇₁	#23197	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×#135	C₂²×C₇₁	1	4	yes	—
120	#568	C₂²×C₇₁	1	4	yes	—
121	#532	C₂²×C₇₁	1	2	yes	—
122	#497	C₄×C₇₁	1	4	yes	complemented
123	#496	C₄×C₇₁	1	4	yes	complemented
124	#444	—	2	8	no	—
125	#443	—	2	8	no	—
126	C₂×#108	C₂²×C₇₁	1	2	yes	—
127	#427	C₂²×C₇₁	1	2	yes	—
128	#421	C₂²×C₇₁	1	8	yes	—
129	#410	C₂²×C₇₁	1	4	yes	—
130	#405	C₂²×C₇₁	1	4	yes	—
131	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic
132	#339	C₂²×C₇₁	1	4	yes	—
133	C₂×#41	C₂³×C₇₁	1	4	yes	—
134	#135	—	2	4	no	—
135	#135	—	2	4	no	—
136	#135	—	2	4	no	—
137	#135	—	2	4	no	—
138	C₂×Q₁₆	C₂×C₄×C₇₁	1	4	yes	—
139	C₂×SD₁₆	C₂×C₄×C₇₁	1	8	yes	—
140	C₂×D₈	C₂×C₄×C₇₁	1	4	yes	—
141	#131	—	2	16	no	—
142	C₂×M₁₆	—	4	4	no	—
143	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
144	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
145	C₄⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
146	#127	C₂×C₄×C₇₁	1	2	yes	—
147	#124	C₂×C₄×C₇₁	1	4	yes	—
148	#121	—	4	2	no	—
149	C₄×D₄	—	4	2	no	—
150	#117	C₂³×C₇₁	1	4	yes	—
151	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
152	#108	D₄×C₇₁	1	8	yes	complemented
153	#107	—	2	2	no	—
154	#106	—	2	2	no	—
155	#104	—	2	16	no	—
156	#99	—	4	4	no	—
157	#97	—	2	4	no	—
158	#97	C₂×C₄×C₇₁	1	4	yes	—
159	C₄×C₈	—	2	4	no	—
160	#41	—	2	8	no	—
161	#41	—	2	8	no	—
162	C₂×Q₈	C₂²×C₄×C₇₁	1	4	yes	—
163	C₂×D₄	—	2	4	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	C₂²×C₄×C₇₁	1	4	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
169	Q₁₆	—	2	4	no	—
170	Q₁₆	—	2	4	no	—
171	SD₁₆	—	2	8	no	—
172	SD₁₆	—	2	8	no	—
173	D₈	—	2	4	no	—
174	D₈	—	2	4	no	—
175	M₁₆	—	2	4	no	—
176	M₁₆	—	4	8	no	—
177	M₁₆	C₂×D₄×C₇₁	1	8	yes	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	8	no	—
180	C₂×C₈	C₂²×C₄×C₇₁	1	4	yes	—
181	C₂×C₈	C₂×D₄×C₇₁	1	4	yes	—
182	C₂×C₈	#41×C₇₁	1	2	yes	—
183	C₂×C₈	#41×C₇₁	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	—	2	4	no	—
187	C₂²⋊C₄	—	4	4	no	—
188	C₄²	—	2	1	no	—
189	C₄²	—	2	4	no	—
190	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	8	1	no	—
203	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
204	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
205	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
206	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
207	C₈	—	2	8	no	—
208	C₈	C₄×D₄×C₇₁	1	4	yes	—
209	C₈	C₄×D₄×C₇₁	1	4	yes	—
210	C₂²	—	4	1	no	—
211	C₂²	—	4	4	no	—
212	C₂²	—	8	4	no	—
213	C₂²	#518×C₇₁	1	1	yes	characteristic
214	C₂²	#550×C₇₁	1	2	yes	—
215	C₄	—	4	2	no	—
216	C₄	—	4	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₂×C₄×D₄×C₇₁	1	2	yes	—
219	C₄	D₄²×C₇₁	1	1	yes	characteristic
220	C₄	#559×C₇₁	1	1	yes	characteristic
221	C₂	—	2	1	no	—
222	C₂	—	2	2	no	—
223	C₂	—	8	4	no	—
224	C₂	C₇₁×#2366	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle	C₂×C₇₁	#2366
prime radical	C₇₁×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#2957	C₂	semidirect
3. C₇₁×#1976	C₂	semidirect
4. C₇₁×#1884	C₂	semidirect
5. C₇₁×#1773	C₂	semidirect
9. #497×C₇₁	C₄	semidirect
10. #496×C₇₁	C₄	semidirect
39. #108×C₇₁	D₄	semidirect
108. #23197	C₇₁	direct (N × H)
114. #2957	C₂×C₇₁	semidirect
115. #1976	C₂×C₇₁	semidirect
116. #1884	C₂×C₇₁	semidirect
117. #1773	C₂×C₇₁	semidirect
118. C₇₁	#23197	direct (N × H)
122. #497	C₄×C₇₁	semidirect
123. #496	C₄×C₇₁	semidirect
152. #108	D₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#3150	2
C₇₁×#2957	2
C₇₁×#1976	2
C₇₁×#1884	2
C₇₁×#1773	2
#23197	71

Subgroups — C71×#23197 #31844069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23197 #31844069