Subgroups — #31912756

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₁⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₇₁	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#22586418	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1374	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₁	C₄	1	1	yes	characteristic
13	Dic₇₁⋊C₁₆	C₄	1	8	yes	complemented
14	C₁₆×Dic₇₁	—	2	4	no	—
15	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₁	C₈	1	2	yes	—
20	C₂×C₄×Dic₇₁	C₈	1	2	yes	—
21	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
26	C₄×C₇₁⋊C₈	—	2	2	no	—
27	C₈×Dic₇₁	—	2	2	no	—
28	C₈×Dic₇₁	C₈	1	4	yes	—
29	C₈×Dic₇₁	C₂×C₄	1	4	yes	—
30	C₈×Dic₇₁	D₄	1	2	yes	—
31	C₈×Dic₇₁	Q₈	1	2	yes	—
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	—	2	2	no	—
37	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₇₁	—	2	2	no	—
39	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
45	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
47	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
48	C₄×Dic₇₁	—	2	4	no	—
49	C₄×Dic₇₁	C₁₆	1	8	yes	complemented
50	C₄×Dic₇₁	C₄²	1	2	yes	—
51	C₄×Dic₇₁	C₄⋊C₄	1	2	yes	—
52	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
53	C₄×Dic₇₁	M₁₆	1	2	yes	—
54	C₇₁⋊C₁₆	—	2	4	no	—
55	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₇₁	—	2	1	no	—
58	C₂×C₈×C₇₁	—	2	1	no	—
59	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₈×C₇₁	#41	1	2	yes	—
64	C₄²×C₇₁	—	2	1	no	—
65	C₄²×C₇₁	—	2	1	no	—
66	C₁₆×C₇₁	—	2	4	no	—
67	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	—	2	2	no	—
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	C₂×C₁₆	1	4	yes	—
71	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×Dic₇₁	—	2	2	no	—
78	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
79	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
80	C₂×Dic₇₁	C₄⋊C₈	1	4	yes	—
81	C₂×Dic₇₁	C₂×C₁₆	1	4	yes	—
82	C₂×Dic₇₁	C₂×C₁₆	1	4	yes	—
83	C₂×Dic₇₁	M₃₂	1	4	yes	—
84	C₈×C₇₁	—	2	2	no	—
85	C₈×C₇₁	C₄×D₄	1	2	yes	—
86	C₈×C₇₁	C₄×Q₈	1	2	yes	—
87	C₇₁⋊C₈	—	2	2	no	—
88	C₇₁⋊C₈	C₄⋊C₈	1	4	yes	—
89	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
90	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²×C₇₁	#431	1	1	yes	characteristic
92	C₂²×C₇₁	C₂²×C₁₆	1	2	yes	—
93	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
94	C₄×C₇₁	—	2	4	no	—
95	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
98	C₄×C₇₁	#431	1	1	yes	characteristic
99	Dic₇₁	—	2	2	no	—
100	Dic₇₁	C₄×C₁₆	1	4	yes	complemented
101	Dic₇₁	C₄⋊C₁₆	1	8	yes	complemented
102	C₄×C₄⋊C₁₆	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
104	C₂×C₇₁	C₂×C₄×C₁₆	1	1	yes	characteristic
105	C₂×C₇₁	C₄×M₃₂	1	1	yes	characteristic
106	C₂×C₇₁	C₂×C₄⋊C₁₆	1	2	yes	—
107	C₂×C₇₁	#2162	1	2	yes	—
108	C₂×C₄⋊C₁₆	—	71	2	no	—
109	C₂×C₄×C₁₆	—	71	2	no	—
110	C₄²×C₈	—	71	1	no	—
111	#1374	—	71	1	no	—
112	#1374	D₇₁	1	1	yes	characteristic
113	C₇₁	C₄×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
114	C₂²×C₁₆	—	71	2	no	—
115	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
116	C₂×C₄×C₈	—	71	1	no	—
117	C₂×C₄×C₈	—	71	1	no	—
118	C₂×C₄×C₈	—	71	1	no	—
119	C₂×C₄×C₈	—	71	1	no	—
120	C₂×C₄×C₈	—	71	1	no	—
121	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
122	C₄³	—	71	1	no	—
123	C₄⋊C₁₆	—	71	8	no	—
124	C₄×C₁₆	—	142	4	no	—
125	C₂²×C₈	—	71	1	no	—
126	C₂²×C₈	—	71	1	no	—
127	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
128	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
129	C₂×C₄²	—	71	2	no	—
130	C₂×C₄²	—	71	1	no	—
131	C₂×C₄²	—	71	1	no	—
132	C₂×C₄²	—	71	2	no	—
133	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	2	no	—
136	C₂×C₁₆	—	71	4	no	—
137	C₂×C₁₆	—	142	2	no	—
138	C₂×C₁₆	—	142	2	no	—
139	C₂×C₁₆	C₄×D₇₁	1	4	yes	—
140	C₄×C₈	—	2	2	no	—
141	C₄×C₈	—	71	4	no	—
142	C₄×C₈	—	71	4	no	—
143	C₄×C₈	—	71	2	no	—
144	C₄×C₈	—	71	2	no	—
145	C₄×C₈	—	142	2	no	—
146	C₄×C₈	—	142	2	no	—
147	C₂²×C₄	—	71	1	no	—
148	C₂²×C₄	—	71	2	no	—
149	C₂²×C₄	—	71	1	no	—
150	C₂²×C₄	C₈×D₇₁	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	71	2	no	—
155	C₂×C₈	—	71	2	no	—
156	C₂×C₈	—	71	2	no	—
157	C₂×C₈	—	71	2	no	—
158	C₂×C₈	—	142	1	no	—
159	C₂×C₈	—	142	1	no	—
160	C₂×C₈	C₈×D₇₁	1	2	yes	—
161	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
162	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
163	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
164	C₂×C₈	#11765864	1	2	yes	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	1	no	—
167	C₄²	—	71	8	no	—
168	C₄²	—	71	2	no	—
169	C₄²	—	71	2	no	—
170	C₄²	—	71	2	no	—
171	C₄²	—	71	2	no	—
172	C₄²	—	142	4	no	—
173	C₁₆	—	2	4	no	—
174	C₁₆	—	142	4	no	—
175	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	71	4	no	—
179	C₂×C₄	—	71	4	no	—
180	C₂×C₄	—	71	4	no	—
181	C₂×C₄	—	71	2	no	—
182	C₂×C₄	—	71	4	no	—
183	C₂×C₄	—	71	2	no	—
184	C₂×C₄	—	142	2	no	—
185	C₂×C₄	C₁₆×D₇₁	1	4	yes	—
186	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
187	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
188	C₂×C₄	#14556084	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
190	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	71	4	no	—
194	C₈	—	142	2	no	—
195	C₈	(C₂²⋊D₇₁)⋊C₄	1	2	yes	—
196	C₈	#14556093	1	2	yes	—
197	C₂²	C₄×C₈×D₇₁	1	1	yes	characteristic
198	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
199	C₂²	#17986049	1	1	yes	characteristic
200	C₂²	C₂×C₁₆×D₇₁	1	2	yes	—
201	C₂²	M₃₂×D₇₁	1	2	yes	—
202	C₄	—	2	4	no	—
203	C₄	—	71	8	no	—
204	C₄	—	71	4	no	—
205	C₄	—	142	2	no	—
206	C₄	#17985850	1	1	yes	characteristic
207	C₄	#17985951	1	1	yes	characteristic
208	C₄	C₄⋊C₈×D₇₁	1	1	yes	characteristic
209	C₄	#17986049	1	1	yes	characteristic
210	C₂	#22588188	1	1	yes	characteristic
211	C₂	C₄×C₁₆×D₇₁	1	1	yes	characteristic
212	C₂	#22589933	1	1	yes	characteristic
213	C₂	C₄⋊C₁₆×D₇₁	1	2	yes	—
214	C₂	#22590136	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂×C₇₁	C₂×C₄×C₁₆
Fitting	C₇₁×#1374	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1374	D₇₁
hypocenter	C₇₁	C₄×C₄⋊C₁₆
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
13. Dic₇₁⋊C₁₆	C₄	semidirect
49. C₄×Dic₇₁	C₁₆	semidirect
100. Dic₇₁	C₄×C₁₆	semidirect
101. Dic₇₁	C₄⋊C₁₆	semidirect
113. C₇₁	C₄×C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₁⋊C₁₆	2
C₂×C₁₆×Dic₇₁	2
C₄×C₈×Dic₇₁	2
#22586418	2
C₇₁×#1374	2
C₄×C₄⋊C₁₆	71