Subgroups — C₇₁×#7920 #31828792

Order 18176.

614 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2920	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1281	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₇₁	C₄	1	2	yes	—
5	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
6	#432×C₇₁	—	2	2	no	—
7	#432×C₇₁	C₄	1	4	yes	—
8	#431×C₇₁	—	2	2	no	—
9	#431×C₇₁	C₂²	1	4	yes	—
10	#423×C₇₁	—	2	2	no	—
11	#407×C₇₁	C₄	1	4	yes	—
12	C₄×M₁₆×C₇₁	—	2	2	no	—
13	C₄×M₁₆×C₇₁	—	2	2	no	—
14	C₄×M₁₆×C₇₁	—	2	2	no	—
15	C₂×#97×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₄	1	2	yes	—
17	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
19	C₂×M₁₆×C₇₁	—	2	2	no	—
20	C₂×M₁₆×C₇₁	—	2	2	no	—
21	C₂×M₁₆×C₇₁	—	2	4	no	—
22	C₂×M₁₆×C₇₁	—	2	8	no	—
23	C₂²×C₈×C₇₁	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₇₁	D₄	1	2	yes	—
28	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	D₄	1	2	yes	—
30	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₇₁	—	2	4	no	—
34	C₄⋊C₈×C₇₁	—	2	4	no	—
35	C₂²⋊C₈×C₇₁	—	2	4	no	—
36	C₂²⋊C₈×C₇₁	C₂×C₄	1	8	yes	—
37	#97×C₇₁	—	2	4	no	—
38	#97×C₇₁	—	2	4	no	—
39	C₄×C₈×C₇₁	—	2	4	no	—
40	C₄×C₈×C₇₁	—	2	4	no	—
41	C₂⁴×C₇₁	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	—	2	4	no	—
43	C₂²×C₄×C₇₁	—	4	4	no	—
44	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
46	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	M₁₆×C₇₁	—	2	8	no	—
57	M₁₆×C₇₁	—	2	8	no	—
58	M₁₆×C₇₁	—	4	8	no	—
59	C₂×C₈×C₇₁	—	2	4	no	—
60	C₂×C₈×C₇₁	—	2	4	no	—
61	C₂×C₈×C₇₁	—	2	8	no	—
62	C₂×C₈×C₇₁	M₁₆	1	8	yes	complemented
63	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
67	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
68	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₇₁	C₂×D₄	1	2	yes	—
72	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
73	C₂³×C₇₁	—	2	2	no	—
74	C₂³×C₇₁	—	4	2	no	—
75	C₂³×C₇₁	#95	1	1	yes	characteristic
76	C₂³×C₇₁	#95	1	1	yes	characteristic
77	C₂³×C₇₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₁	—	2	2	no	—
79	C₂×C₄×C₇₁	—	2	4	no	—
80	C₂×C₄×C₇₁	—	2	4	no	—
81	C₂×C₄×C₇₁	—	2	4	no	—
82	C₂×C₄×C₇₁	—	2	2	no	—
83	C₂×C₄×C₇₁	—	2	2	no	—
84	C₂×C₄×C₇₁	—	2	2	no	—
85	C₂×C₄×C₇₁	—	2	2	no	—
86	C₂×C₄×C₇₁	—	4	2	no	—
87	C₂×C₄×C₇₁	—	4	2	no	—
88	C₂×C₄×C₇₁	—	4	2	no	—
89	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
90	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
91	C₂×C₄×C₇₁	#95	1	2	yes	—
92	C₂×C₄×C₇₁	#95	1	2	yes	—
93	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
98	C₈×C₇₁	—	2	8	no	—
99	C₈×C₇₁	—	4	8	no	—
100	C₂²×C₇₁	—	2	2	no	—
101	C₂²×C₇₁	—	2	1	no	—
102	C₂²×C₇₁	—	2	1	no	—
103	C₂²×C₇₁	—	2	2	no	—
104	C₂²×C₇₁	—	2	2	no	—
105	C₂²×C₇₁	—	4	4	no	—
106	C₂²×C₇₁	C₂×#95	1	1	yes	characteristic
107	C₂²×C₇₁	C₄×M₁₆	1	2	yes	—
108	C₄×C₇₁	—	2	1	no	—
109	C₄×C₇₁	—	2	1	no	—
110	C₄×C₇₁	—	2	2	no	—
111	C₄×C₇₁	—	2	2	no	—
112	C₄×C₇₁	—	2	2	no	—
113	C₄×C₇₁	#344	1	4	yes	—
114	C₄×C₇₁	#407	1	2	yes	—
115	C₄×C₇₁	#423	1	2	yes	—
116	#7920	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	C₂×C₇₁	—	2	2	no	—
118	C₂×C₇₁	—	4	1	no	—
119	C₂×C₇₁	—	4	1	no	—
120	C₂×C₇₁	#1728	1	1	yes	characteristic
121	C₂×C₇₁	C₂×#344	1	2	yes	—
122	#2920	C₂×C₇₁	1	2	yes	—
123	C₂×C₄×M₁₆	C₂×C₇₁	1	1	yes	characteristic
124	#1281	C₂×C₇₁	1	4	yes	complemented
125	C₇₁	#7920	1	1	yes	characteristic, direct product factor, Sylow
126	C₂²×M₁₆	C₄×C₇₁	1	2	yes	—
127	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
128	#432	—	2	2	no	—
129	#432	C₄×C₇₁	1	4	yes	—
130	#431	—	2	2	no	—
131	#431	C₂²×C₇₁	1	4	yes	—
132	#423	—	2	2	no	—
133	#407	C₄×C₇₁	1	4	yes	—
134	C₄×M₁₆	—	2	2	no	—
135	C₄×M₁₆	—	2	2	no	—
136	C₄×M₁₆	—	2	2	no	—
137	C₂×#97	C₂²×C₇₁	1	2	yes	—
138	C₂×C₄×C₈	C₄×C₇₁	1	2	yes	—
139	C₂³×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
140	C₂³×C₄	C₂×C₄×C₇₁	1	2	yes	—
141	C₂×M₁₆	—	2	2	no	—
142	C₂×M₁₆	—	2	2	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	—	2	8	no	—
145	C₂²×C₈	C₂×C₄×C₇₁	1	4	yes	—
146	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
147	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
148	C₂×C₄²	D₄×C₇₁	1	2	yes	—
149	C₂×C₄²	D₄×C₇₁	1	2	yes	—
150	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
151	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
152	C₂×C₄²	Q₈×C₇₁	1	1	yes	characteristic
153	C₂×C₄²	Q₈×C₇₁	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	C₄⋊C₈	—	2	4	no	—
157	C₂²⋊C₈	—	2	4	no	—
158	C₂²⋊C₈	C₂×C₄×C₇₁	1	8	yes	—
159	#97	—	2	4	no	—
160	#97	—	2	4	no	—
161	C₄×C₈	—	2	4	no	—
162	C₄×C₈	—	2	4	no	—
163	C₂⁴	C₄²×C₇₁	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₄²×C₇₁	1	1	yes	characteristic
167	C₂²×C₄	C₄²×C₇₁	1	2	yes	—
168	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
169	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
170	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
171	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊C₄×C₇₁	1	2	yes	—
175	C₂²×C₄	C₄⋊C₄×C₇₁	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
177	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
178	M₁₆	—	2	8	no	—
179	M₁₆	—	2	8	no	—
180	M₁₆	—	4	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	M₁₆×C₇₁	1	8	yes	complemented
185	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
186	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
187	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
188	C₄²	C₄⋊C₄×C₇₁	1	2	yes	—
189	C₄²	C₄⋊C₄×C₇₁	1	1	yes	characteristic
190	C₄²	C₄⋊C₄×C₇₁	1	2	yes	—
191	C₄²	C₄⋊C₄×C₇₁	1	1	yes	characteristic
192	C₄²	C₂×D₄×C₇₁	1	2	yes	—
193	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
194	C₄²	C₂×Q₈×C₇₁	1	1	yes	characteristic
195	C₂³	—	2	2	no	—
196	C₂³	—	4	2	no	—
197	C₂³	#95×C₇₁	1	1	yes	characteristic
198	C₂³	#95×C₇₁	1	1	yes	characteristic
199	C₂³	C₂×C₄²×C₇₁	1	1	yes	characteristic
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	#95×C₇₁	1	1	yes	characteristic
212	C₂×C₄	#95×C₇₁	1	1	yes	characteristic
213	C₂×C₄	#95×C₇₁	1	2	yes	—
214	C₂×C₄	#95×C₇₁	1	2	yes	—
215	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
217	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	2	yes	—
218	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
219	C₂×C₄	C₂×M₁₆×C₇₁	1	4	yes	—
220	C₈	—	2	8	no	—
221	C₈	—	4	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	4	4	no	—
228	C₂²	C₂×#95×C₇₁	1	1	yes	characteristic
229	C₂²	C₄×M₁₆×C₇₁	1	2	yes	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	#344×C₇₁	1	4	yes	—
236	C₄	#407×C₇₁	1	2	yes	—
237	C₄	#423×C₇₁	1	2	yes	—
238	C₂	—	2	2	no	—
239	C₂	—	4	1	no	—
240	C₂	—	4	1	no	—
241	C₂	C₇₁×#1728	1	1	yes	characteristic
242	C₂	C₂×#344×C₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂×C₄²×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
socle	C₂²×C₇₁	C₂×#95
prime radical	C₂⁴×C₇₁	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂²×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₁×#1281	C₂	semidirect
62. C₂×C₈×C₇₁	M₁₆	semidirect
116. #7920	C₇₁	direct (N × H)
124. #1281	C₂×C₇₁	semidirect
125. C₇₁	#7920	direct (N × H)
184. C₂×C₈	M₁₆×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#2920	2
C₂×C₄×M₁₆×C₇₁	2
C₇₁×#1281	2
#7920	71

Subgroups — C71×#7920 #31828792

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7920 #31828792