Subgroups — C₇₁×#23247 #31844119

Order 18176.

764 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1984	C₂	1	4	yes	maximal
3	C₇₁×#1939	C₂	1	4	yes	maximal
4	C₈×C₄⋊C₄×C₇₁	C₂	1	4	yes	maximal
5	C₄×C₄⋊C₈×C₇₁	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₇₁	C₂²	1	2	yes	—
8	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈×C₇₁	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	4	yes	—
13	#391×C₇₁	C₄	1	2	yes	—
14	#389×C₇₁	C₄	1	4	yes	—
15	#384×C₇₁	C₄	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₁	C₂²	1	4	yes	—
17	C₄³×C₇₁	C₂²	1	1	yes	characteristic
18	#336×C₇₁	C₂²	1	4	yes	—
19	#336×C₇₁	C₂²	1	8	yes	—
20	C₂²×Q₈×C₇₁	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
22	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
23	C₄⋊Q₈×C₇₁	C₈	1	8	yes	complemented
24	C₄×Q₈×C₇₁	—	4	2	no	—
25	C₄×Q₈×C₇₁	—	4	2	no	—
26	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂³	1	4	yes	—
34	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
36	C₄⋊C₈×C₇₁	—	2	4	no	—
37	C₄⋊C₈×C₇₁	—	4	4	no	—
38	C₄⋊C₈×C₇₁	Q₈	1	8	yes	complemented
39	C₄×C₈×C₇₁	—	2	8	no	—
40	C₄×C₈×C₇₁	—	4	4	no	—
41	#95×C₇₁	C₂×C₄	1	8	yes	—
42	C₂×Q₈×C₇₁	—	4	2	no	—
43	C₂×Q₈×C₇₁	—	4	2	no	—
44	C₂×Q₈×C₇₁	C₂×C₈	1	8	yes	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₇₁	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₇₁	—	2	4	no	—
53	C₂×C₈×C₇₁	—	2	4	no	—
54	C₂×C₈×C₇₁	—	2	4	no	—
55	C₂×C₈×C₇₁	—	2	4	no	—
56	C₂×C₈×C₇₁	—	2	4	no	—
57	C₂×C₈×C₇₁	C₂×Q₈	1	8	yes	—
58	C₄⋊C₄×C₇₁	—	2	8	no	—
59	C₄⋊C₄×C₇₁	—	4	2	no	—
60	C₄⋊C₄×C₇₁	—	4	4	no	—
61	C₄⋊C₄×C₇₁	C₂×C₈	1	16	yes	—
62	C₄²×C₇₁	—	2	4	no	—
63	C₄²×C₇₁	—	2	4	no	—
64	C₄²×C₇₁	—	2	4	no	—
65	C₄²×C₇₁	—	2	1	no	—
66	C₄²×C₇₁	—	2	1	no	—
67	C₄²×C₇₁	—	4	1	no	—
68	C₄²×C₇₁	C₂×C₈	1	4	yes	—
69	C₄²×C₇₁	C₂×Q₈	1	4	yes	—
70	C₄²×C₇₁	#41	1	4	yes	—
71	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
72	Q₈×C₇₁	—	4	8	no	—
73	C₂×C₄×C₇₁	—	2	1	no	—
74	C₂×C₄×C₇₁	—	2	4	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	—	2	2	no	—
77	C₂×C₄×C₇₁	—	2	1	no	—
78	C₂×C₄×C₇₁	—	2	4	no	—
79	C₂×C₄×C₇₁	—	2	1	no	—
80	C₂×C₄×C₇₁	—	2	4	no	—
81	C₂×C₄×C₇₁	—	2	2	no	—
82	C₂×C₄×C₇₁	—	2	8	no	—
83	C₂×C₄×C₇₁	C₄×Q₈	1	4	yes	—
84	C₂×C₄×C₇₁	C₄×Q₈	1	4	yes	—
85	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
86	C₂×C₄×C₇₁	C₂²×C₈	1	8	yes	—
87	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₇₁	#131	1	4	yes	—
89	C₂×C₄×C₇₁	C₂²×Q₈	1	2	yes	—
90	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
91	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
93	C₈×C₇₁	—	2	8	no	—
94	C₈×C₇₁	—	4	4	no	—
95	C₂²×C₇₁	C₂×C₄×Q₈	1	2	yes	—
96	C₂²×C₇₁	#518	1	1	yes	characteristic
97	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
98	C₂²×C₇₁	C₂×#131	1	2	yes	—
99	C₂²×C₇₁	#568	1	1	yes	characteristic
100	C₄×C₇₁	—	2	4	no	—
101	C₄×C₇₁	—	2	8	no	—
102	C₄×C₇₁	—	2	2	no	—
103	C₄×C₇₁	—	4	1	no	—
104	C₄×C₇₁	—	4	4	no	—
105	C₄×C₇₁	C₈×Q₈	1	8	yes	—
106	C₄×C₇₁	#554	1	2	yes	—
107	C₄×C₇₁	#555	1	1	yes	characteristic
108	C₄×C₇₁	Q₈²	1	1	yes	characteristic
109	#23247	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂×C₇₁	#2369	1	1	yes	characteristic
111	C₂×C₇₁	C₂×C₈×Q₈	1	2	yes	—
112	C₂×C₇₁	#2962	1	2	yes	—
113	C₂×C₇₁	#2972	1	1	yes	characteristic
114	C₂×C₇₁	#2979	1	1	yes	characteristic
115	C₄×C₄⋊Q₈	C₂×C₇₁	1	1	yes	characteristic
116	#1984	C₂×C₇₁	1	4	yes	—
117	#1939	C₂×C₇₁	1	4	yes	—
118	C₈×C₄⋊C₄	C₂×C₇₁	1	4	yes	—
119	C₄×C₄⋊C₈	C₂×C₇₁	1	2	yes	—
120	C₇₁	#23247	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×C₄⋊Q₈	C₄×C₇₁	1	1	yes	characteristic
122	C₂×C₄×Q₈	C₂²×C₇₁	1	2	yes	—
123	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
124	C₂×C₄⋊C₈	C₂²×C₇₁	1	4	yes	—
125	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₇₁	1	4	yes	—
127	C₂×C₄×C₈	C₂²×C₇₁	1	4	yes	—
128	#391	C₄×C₇₁	1	2	yes	—
129	#389	C₄×C₇₁	1	4	yes	—
130	#384	C₄×C₇₁	1	1	yes	characteristic
131	C₄×C₄⋊C₄	C₂²×C₇₁	1	4	yes	—
132	C₄³	C₂²×C₇₁	1	1	yes	characteristic
133	#336	C₂²×C₇₁	1	8	yes	—
134	#336	C₂²×C₇₁	1	4	yes	—
135	C₂²×Q₈	C₂×C₄×C₇₁	1	2	yes	—
136	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
137	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
138	C₄⋊Q₈	C₈×C₇₁	1	8	yes	complemented
139	C₄×Q₈	—	4	2	no	—
140	C₄×Q₈	—	4	2	no	—
141	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
143	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
144	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	4	yes	—
145	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
147	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
148	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
149	C₂×C₄²	C₂³×C₇₁	1	4	yes	—
150	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	4	4	no	—
153	C₄⋊C₈	Q₈×C₇₁	1	8	yes	complemented
154	C₄×C₈	—	2	8	no	—
155	C₄×C₈	—	4	4	no	—
156	#95	C₂×C₄×C₇₁	1	8	yes	—
157	C₂×Q₈	—	4	2	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂×C₈×C₇₁	1	8	yes	—
160	C₂²×C₄	C₂²×C₄×C₇₁	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
162	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
163	C₂²×C₄	C₂²×C₄×C₇₁	1	4	yes	—
164	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
166	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	2	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	C₂×Q₈×C₇₁	1	8	yes	—
173	C₄⋊C₄	—	2	8	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	C₂×C₈×C₇₁	1	16	yes	—
177	C₄²	—	2	4	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	2	4	no	—
180	C₄²	—	2	1	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	4	1	no	—
183	C₄²	C₂×C₈×C₇₁	1	4	yes	—
184	C₄²	C₂×Q₈×C₇₁	1	4	yes	—
185	C₄²	#41×C₇₁	1	4	yes	—
186	C₂³	C₂³×C₄×C₇₁	1	1	yes	characteristic
187	Q₈	—	4	8	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	C₄×Q₈×C₇₁	1	4	yes	—
199	C₂×C₄	C₄×Q₈×C₇₁	1	4	yes	—
200	C₂×C₄	C₂²×C₈×C₇₁	1	2	yes	—
201	C₂×C₄	C₂²×C₈×C₇₁	1	8	yes	—
202	C₂×C₄	C₂²×C₈×C₇₁	1	4	yes	—
203	C₂×C₄	#131×C₇₁	1	4	yes	—
204	C₂×C₄	C₂²×Q₈×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	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
208	C₈	—	2	8	no	—
209	C₈	—	4	4	no	—
210	C₂²	C₂×C₄×Q₈×C₇₁	1	2	yes	—
211	C₂²	#518×C₇₁	1	1	yes	characteristic
212	C₂²	C₂³×C₈×C₇₁	1	1	yes	characteristic
213	C₂²	C₂×#131×C₇₁	1	2	yes	—
214	C₂²	#568×C₇₁	1	1	yes	characteristic
215	C₄	—	2	4	no	—
216	C₄	—	2	2	no	—
217	C₄	—	2	8	no	—
218	C₄	—	4	1	no	—
219	C₄	—	4	4	no	—
220	C₄	C₈×Q₈×C₇₁	1	8	yes	—
221	C₄	#554×C₇₁	1	2	yes	—
222	C₄	#555×C₇₁	1	1	yes	characteristic
223	C₄	Q₈²×C₇₁	1	1	yes	characteristic
224	C₂	C₇₁×#2369	1	1	yes	characteristic
225	C₂	C₂×C₈×Q₈×C₇₁	1	2	yes	—
226	C₂	C₇₁×#2962	1	2	yes	—
227	C₂	C₇₁×#2972	1	1	yes	characteristic
228	C₂	C₇₁×#2979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle, prime radical	C₂³×C₇₁	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
23. C₄⋊Q₈×C₇₁	C₈	semidirect
38. C₄⋊C₈×C₇₁	Q₈	semidirect
109. #23247	C₇₁	direct (N × H)
120. C₇₁	#23247	direct (N × H)
138. C₄⋊Q₈	C₈×C₇₁	semidirect
153. C₄⋊C₈	Q₈×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₇₁	2
C₇₁×#1984	2
C₇₁×#1939	2
C₈×C₄⋊C₄×C₇₁	2
C₄×C₄⋊C₈×C₇₁	2
#23247	71

Subgroups — C71×#23247 #31844119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23247 #31844119