Subgroups — #31912765

Order 18176.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	2	yes	maximal
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#22586418	C₂	1	2	yes	maximal
5	C₇₁×#1726	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
8	C₄⋊(C₇₁⋊C₁₆)	C₄	1	8	yes	complemented
9	#17985775	—	2	2	no	—
10	C₄×C₇₁⋊C₁₆	—	2	2	no	—
11	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
12	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
13	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#376×C₇₁	C₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	C₂×C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
23	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
31	#97×C₇₁	—	2	2	no	—
32	#97×C₇₁	—	2	2	no	—
33	#97×C₇₁	C₈	1	4	yes	—
34	C₄×C₈×C₇₁	—	2	2	no	—
35	C₄×C₈×C₇₁	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₇₁	D₄	1	2	yes	—
37	C₄×C₈×C₇₁	Q₈	1	2	yes	—
38	#95×C₇₁	C₈	1	4	yes	—
39	C₇₁⋊C₁₆	—	2	4	no	—
40	C₇₁⋊C₁₆	—	2	4	no	—
41	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	—	2	1	no	—
47	C₂×C₈×C₇₁	—	2	1	no	—
48	C₂×C₈×C₇₁	—	2	1	no	—
49	C₂×C₈×C₇₁	C₄²	1	2	yes	—
50	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
53	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
54	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₇₁	#41	1	2	yes	—
58	C₄²×C₇₁	—	2	1	no	—
59	C₄²×C₇₁	—	2	1	no	—
60	C₄²×C₇₁	C₄²	1	2	yes	—
61	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
62	C₄²×C₇₁	C₂×C₈	1	2	yes	—
63	C₄²×C₇₁	M₁₆	1	2	yes	—
64	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	4	no	—
66	C₂×C₄×C₇₁	—	2	4	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	2	4	no	—
69	C₂×C₄×C₇₁	C₄×C₈	1	4	yes	—
70	C₂×C₄×C₇₁	C₄⋊C₈	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	C₈×C₇₁	—	2	2	no	—
78	C₈×C₇₁	—	2	2	no	—
79	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
80	C₈×C₇₁	C₄×D₄	1	2	yes	—
81	C₈×C₇₁	C₄×Q₈	1	2	yes	—
82	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₇₁	#431	1	1	yes	characteristic
85	C₂²×C₇₁	#504	1	4	yes	—
86	C₄×C₇₁	—	2	4	no	—
87	C₄×C₇₁	—	2	4	no	—
88	C₄×C₇₁	—	4	2	no	—
89	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
91	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₄×C₇₁	#431	1	1	yes	characteristic
93	#12654	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₇₁	#2108	1	2	yes	—
96	C₂×C₇₁	#2151	1	2	yes	—
97	C₂×C₇₁	#2164	1	2	yes	—
98	C₂×C₄⋊C₁₆	—	71	2	no	—
99	C₂×#346	—	71	1	no	—
100	C₂×C₄×C₁₆	—	71	1	no	—
101	#1726	D₇₁	1	1	yes	characteristic
102	#1374	—	71	2	no	—
103	C₇₁	#12654	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₁₆	—	71	2	no	—
105	C₂²×C₁₆	—	71	2	no	—
106	C₂×#97	Dic₇₁	1	1	yes	characteristic
107	C₂×#97	Dic₇₁	1	1	yes	characteristic
108	C₂×#97	Dic₇₁	1	1	yes	characteristic
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
112	#376	Dic₇₁	1	1	yes	characteristic
113	C₄⋊C₁₆	—	71	8	no	—
114	#346	—	142	2	no	—
115	C₄×C₁₆	—	142	2	no	—
116	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
121	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
123	C₂×C₁₆	—	71	4	no	—
124	C₂×C₁₆	—	71	4	no	—
125	C₂×C₁₆	—	142	2	no	—
126	C₂×C₁₆	—	142	2	no	—
127	C₂×C₁₆	—	142	2	no	—
128	C₂×C₁₆	—	142	2	no	—
129	#97	—	2	2	no	—
130	#97	—	2	2	no	—
131	#97	C₇₁⋊C₈	1	4	yes	—
132	C₄×C₈	—	2	2	no	—
133	C₄×C₈	Dic₁₄₂	1	2	yes	—
134	C₄×C₈	C₄×D₇₁	1	4	yes	—
135	C₄×C₈	D₂₈₄	1	2	yes	—
136	#95	C₇₁⋊C₈	1	4	yes	—
137	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
138	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
139	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	1	no	—
145	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
146	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
147	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
148	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
149	C₂×C₈	#11765849	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
151	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
153	C₂×C₈	#11765865	1	2	yes	—
154	C₄²	—	2	1	no	—
155	C₄²	—	2	1	no	—
156	C₄²	C₄×Dic₇₁	1	2	yes	—
157	C₄²	C₄⋊Dic₇₁	1	2	yes	—
158	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
159	C₄²	#11765849	1	2	yes	—
160	C₁₆	—	142	4	no	—
161	C₁₆	—	142	4	no	—
162	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	C₄×C₇₁⋊C₈	1	4	yes	—
168	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
169	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
171	C₂×C₄	#14556087	1	1	yes	characteristic
172	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
173	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
174	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	—
178	C₈	C₄×D₂₈₄	1	2	yes	—
179	C₈	C₄×Dic₁₄₂	1	2	yes	—
180	C₂²	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
181	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
182	C₂²	#17986052	1	1	yes	characteristic
183	C₂²	#17986416	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	2	4	no	—
186	C₄	—	4	2	no	—
187	C₄	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
188	C₄	C₄×#11765849	1	1	yes	characteristic
189	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
190	C₄	#17986052	1	1	yes	characteristic
191	C₂	C₄×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
192	C₂	#22589937	1	2	yes	—
193	C₂	#22590099	1	2	yes	—
194	C₂	#22590147	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₇₁×#1726	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1726	D₇₁
hypocenter	C₇₁	#12654
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
8. C₄⋊(C₇₁⋊C₁₆)	C₄	semidirect
103. C₇₁	#12654	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
C₂×#17985775	2
C₂×C₄×C₇₁⋊C₁₆	2
#22586418	2
C₇₁×#1726	2
#12654	71