Subgroups — #31884320

Order 18176.

218 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590096	C₂	1	1	yes	maximal, characteristic
2	#22586623	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	—	2	1	no	—
5	#17986052	C₄	1	1	yes	characteristic
6	#17986052	C₄	1	1	yes	characteristic
7	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	—
9	#17985720	—	2	1	no	—
10	C₈×C₇₁⋊C₈	—	2	1	no	—
11	C₂²×C₁₆×C₇₁	—	2	1	no	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₁₆×C₇₁	—	2	1	no	—
14	C₄⋊C₁₆×C₇₁	C₄	1	2	yes	—
15	#14556068	—	4	1	no	—
16	#14556068	—	4	1	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
19	C₄⋊(C₇₁⋊C₈)	C₈	1	2	yes	—
20	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
21	C₂²⋊(C₇₁⋊C₈)	C₈	1	2	yes	—
22	C₄×C₇₁⋊C₈	—	2	1	no	—
23	C₄×C₇₁⋊C₈	—	2	1	no	—
24	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
26	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₁	—	2	2	no	—
28	C₂×C₁₆×C₇₁	—	4	1	no	—
29	C₂×C₁₆×C₇₁	—	4	1	no	—
30	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
34	C₂×C₇₁⋊C₈	—	2	2	no	—
35	C₂×C₇₁⋊C₈	—	2	2	no	—
36	C₇₁⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
47	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
48	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
50	C₄²×C₇₁	C₄²	1	1	yes	characteristic
51	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
54	C₁₆×C₇₁	—	4	2	no	—
55	C₂³×C₇₁	C₄⋊C₈	1	1	yes	characteristic
56	C₂×C₄×C₇₁	—	2	2	no	—
57	C₂×C₄×C₇₁	—	4	1	no	—
58	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
59	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
68	C₈×C₇₁	—	2	1	no	—
69	C₈×C₇₁	—	2	1	no	—
70	C₈×C₇₁	#102	1	1	yes	characteristic
71	C₈×C₇₁	#102	1	1	yes	characteristic
72	C₈×C₇₁	#103	1	1	yes	characteristic
73	C₈×C₇₁	#103	1	1	yes	characteristic
74	C₇₁⋊C₈	—	4	2	no	—
75	C₇₁⋊C₈	—	4	2	no	—
76	C₂²×C₇₁	—	2	1	no	—
77	C₂²×C₇₁	—	2	1	no	—
78	C₂²×C₇₁	#334	1	1	yes	characteristic
79	C₂²×C₇₁	#335	1	1	yes	characteristic
80	C₂²×C₇₁	#336	1	1	yes	characteristic
81	C₄×C₇₁	—	2	1	no	—
82	C₄×C₇₁	—	2	1	no	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	#334	1	1	yes	characteristic
85	C₄×C₇₁	#335	1	1	yes	characteristic
86	C₄×C₇₁	#340	1	1	yes	characteristic
87	C₄×C₇₁	#341	1	1	yes	characteristic
88	C₄×C₇₁	#350	1	2	yes	—
89	#7153	—	71	1	no	maximal, Sylow
90	C₂×C₇₁	—	2	1	no	—
91	C₂×C₇₁	—	2	1	no	—
92	C₂×C₇₁	#1277	1	1	yes	characteristic
93	C₂×C₇₁	#1378	1	1	yes	characteristic
94	C₂×C₇₁	#1379	1	1	yes	characteristic
95	#2150	—	71	1	no	—
96	C₂×C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
97	#1450	—	71	1	no	—
98	C₇₁	#7153	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₃₂	—	142	1	no	—
100	C₂²×C₁₆	—	2	1	no	—
101	#431	—	71	1	no	—
102	#431	—	71	1	no	—
103	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
104	C₄⋊C₁₆	—	2	1	no	—
105	C₄⋊C₁₆	—	71	2	no	—
106	C₄⋊C₁₆	—	142	1	no	—
107	C₄⋊C₁₆	Dic₇₁	1	2	yes	—
108	#322	—	142	1	no	—
109	C₈²	—	142	1	no	—
110	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
111	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
112	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
113	M₃₂	—	284	1	no	—
114	M₃₂	—	284	1	no	—
115	C₂×C₁₆	—	2	2	no	—
116	C₂×C₁₆	—	4	1	no	—
117	C₂×C₁₆	—	4	1	no	—
118	C₂×C₁₆	—	142	2	no	—
119	C₄⋊C₈	—	71	2	no	—
120	C₄⋊C₈	—	142	1	no	—
121	C₂²⋊C₈	—	71	2	no	—
122	C₂²⋊C₈	—	142	1	no	—
123	C₄×C₈	—	142	1	no	—
124	C₄×C₈	—	142	1	no	—
125	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
126	C₄×C₈	D₂₈₄	1	1	yes	characteristic
127	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
128	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
129	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
130	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
131	C₂²×C₄	#11765848	1	1	yes	characteristic
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	142	2	no	—
135	C₂×C₈	—	142	2	no	—
136	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
137	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
138	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
140	C₂×C₈	#11765850	1	1	yes	characteristic
141	C₂×C₈	#11765852	1	1	yes	characteristic
142	C₂×C₈	#11765853	1	1	yes	characteristic
143	C₂×C₈	#11765855	1	1	yes	characteristic
144	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
145	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
146	C₄²	C₈×D₇₁	1	1	yes	characteristic
147	C₄²	#11765848	1	1	yes	characteristic
148	C₁₆	—	4	2	no	—
149	C₁₆	—	284	2	no	—
150	C₂³	Dic₇₁⋊C₈	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
154	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
155	C₂×C₄	#14556038	1	1	yes	characteristic
156	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
157	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
158	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
159	C₂×C₄	#14556059	1	1	yes	characteristic
160	C₂×C₄	#14556061	1	1	yes	characteristic
161	C₂×C₄	#14556063	1	1	yes	characteristic
162	C₂×C₄	#14556063	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	284	2	no	—
166	C₈	—	284	2	no	—
167	C₈	#14556048	1	1	yes	characteristic
168	C₈	#14556049	1	1	yes	characteristic
169	C₈	#14556051	1	1	yes	characteristic
170	C₈	#14556052	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#17985749	1	1	yes	characteristic
174	C₂²	#17985751	1	1	yes	characteristic
175	C₂²	#17985753	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#17985749	1	1	yes	characteristic
180	C₄	#17985751	1	1	yes	characteristic
181	C₄	#17985761	1	1	yes	characteristic
182	C₄	#17985763	1	1	yes	characteristic
183	C₄	#17985782	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#22586174	1	1	yes	characteristic
187	C₂	#22586427	1	1	yes	characteristic
188	C₂	#22586428	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×C₄⋊C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#7153
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
98. C₇₁	#7153	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
#22586623	2
C₂×C₄⋊C₁₆×C₇₁	2
#7153	71