Subgroups — #31884316

Order 18176.

230 conjugacy classes of subgroups.

The subgroup lattice has 194 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	C₂×#17985720	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2150	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	—	2	1	no	—
5	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇₁⋊C₈	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	#17985720	—	2	1	no	—
11	C₂×M₃₂×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	C₂²×C₇₁⋊C₈	—	2	1	no	—
16	C₂²×C₇₁⋊C₈	C₈	1	2	yes	—
17	#14556068	—	4	1	no	—
18	#14556068	—	4	1	no	—
19	C₂×C₇₁⋊C₁₆	—	2	2	no	—
20	C₄×C₇₁⋊C₈	—	2	1	no	—
21	C₄×C₇₁⋊C₈	—	2	1	no	—
22	C₄×C₇₁⋊C₈	—	2	1	no	—
23	C₄×C₇₁⋊C₈	C₈	1	2	yes	—
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	M₃₂×C₇₁	—	4	1	no	—
28	M₃₂×C₇₁	—	4	1	no	—
29	C₂×C₁₆×C₇₁	—	2	2	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	C₂×C₇₁⋊C₈	—	4	2	no	—
37	C₂×C₇₁⋊C₈	—	4	2	no	—
38	C₇₁⋊C₁₆	—	4	2	no	—
39	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
49	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
51	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
52	C₄²×C₇₁	C₄²	1	1	yes	characteristic
53	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
55	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
56	C₁₆×C₇₁	—	4	2	no	—
57	C₂³×C₇₁	C₄⋊C₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	—	2	2	no	—
59	C₂×C₄×C₇₁	—	2	2	no	—
60	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
61	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
70	C₈×C₇₁	—	2	1	no	—
71	C₈×C₇₁	—	2	1	no	—
72	C₈×C₇₁	#102	1	1	yes	characteristic
73	C₈×C₇₁	#102	1	1	yes	characteristic
74	C₈×C₇₁	#103	1	1	yes	characteristic
75	C₈×C₇₁	#103	1	1	yes	characteristic
76	C₇₁⋊C₈	—	4	2	no	—
77	C₇₁⋊C₈	—	4	2	no	—
78	C₂²×C₇₁	—	2	1	no	—
79	C₂²×C₇₁	—	2	1	no	—
80	C₂²×C₇₁	#334	1	1	yes	characteristic
81	C₂²×C₇₁	#335	1	1	yes	characteristic
82	C₂²×C₇₁	#336	1	1	yes	characteristic
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	—	2	1	no	—
85	C₄×C₇₁	#334	1	1	yes	characteristic
86	C₄×C₇₁	#335	1	1	yes	characteristic
87	C₄×C₇₁	#340	1	1	yes	characteristic
88	C₄×C₇₁	#341	1	1	yes	characteristic
89	C₄×C₇₁	#347	1	2	yes	—
90	C₄×C₇₁	#349	1	2	yes	—
91	#7152	—	71	1	no	maximal, Sylow
92	C₂×C₇₁	—	2	1	no	—
93	C₂×C₇₁	—	2	1	no	—
94	C₂×C₇₁	#1277	1	1	yes	characteristic
95	C₂×C₇₁	#1375	1	1	yes	characteristic
96	C₂×C₇₁	#1377	1	1	yes	characteristic
97	#2150	—	71	1	no	—
98	#2150	D₇₁	1	1	yes	characteristic
99	C₂×#322	—	71	1	no	—
100	C₇₁	#7152	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₃₂	—	2	1	no	—
102	C₂×M₃₂	—	142	1	no	—
103	C₂×C₄×C₈	—	71	1	no	—
104	C₂×C₄×C₈	—	71	1	no	—
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₄⋊C₁₆	—	2	1	no	—
107	C₄⋊C₁₆	—	71	2	no	—
108	C₄⋊C₁₆	—	142	1	no	—
109	C₄⋊C₁₆	Dic₇₁	1	2	yes	—
110	#322	—	142	1	no	—
111	#322	—	142	1	no	—
112	C₂²×C₈	—	71	2	no	—
113	C₂²×C₈	—	142	1	no	—
114	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
115	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
117	M₃₂	—	4	1	no	—
118	M₃₂	—	4	1	no	—
119	M₃₂	—	284	1	no	—
120	M₃₂	—	284	1	no	—
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	142	2	no	—
123	C₄×C₈	—	71	2	no	—
124	C₄×C₈	—	142	1	no	—
125	C₄×C₈	—	142	1	no	—
126	C₄×C₈	—	142	1	no	—
127	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
128	C₄×C₈	D₂₈₄	1	1	yes	characteristic
129	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
130	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
131	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
132	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
133	C₂²×C₄	#11765848	1	1	yes	characteristic
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	2	1	no	—
136	C₂×C₈	—	284	2	no	—
137	C₂×C₈	—	284	2	no	—
138	C₂×C₈	—	142	2	no	—
139	C₂×C₈	—	142	2	no	—
140	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
141	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
142	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
143	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
144	C₂×C₈	#11765850	1	1	yes	characteristic
145	C₂×C₈	#11765852	1	1	yes	characteristic
146	C₂×C₈	#11765853	1	1	yes	characteristic
147	C₂×C₈	#11765855	1	1	yes	characteristic
148	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
149	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
150	C₄²	C₈×D₇₁	1	1	yes	characteristic
151	C₄²	#11765848	1	1	yes	characteristic
152	C₁₆	—	4	2	no	—
153	C₁₆	—	284	2	no	—
154	C₂³	Dic₇₁⋊C₈	1	1	yes	characteristic
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
158	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
159	C₂×C₄	#14556038	1	1	yes	characteristic
160	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
161	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
162	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
163	C₂×C₄	#14556059	1	1	yes	characteristic
164	C₂×C₄	#14556061	1	1	yes	characteristic
165	C₂×C₄	#14556063	1	1	yes	characteristic
166	C₂×C₄	#14556063	1	1	yes	characteristic
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	284	2	no	—
170	C₈	—	284	2	no	—
171	C₈	#14556048	1	1	yes	characteristic
172	C₈	#14556049	1	1	yes	characteristic
173	C₈	#14556051	1	1	yes	characteristic
174	C₈	#14556052	1	1	yes	characteristic
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	#17985749	1	1	yes	characteristic
178	C₂²	#17985751	1	1	yes	characteristic
179	C₂²	#17985753	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	#17985749	1	1	yes	characteristic
183	C₄	#17985751	1	1	yes	characteristic
184	C₄	#17985761	1	1	yes	characteristic
185	C₄	#17985763	1	1	yes	characteristic
186	C₄	#17985776	1	2	yes	—
187	C₄	#17985780	1	2	yes	—
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#22586174	1	1	yes	characteristic
191	C₂	#22586419	1	1	yes	characteristic
192	C₂	#22586423	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₇₁⋊#95
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#2150	C₂
socle	C₂²×C₇₁	#336
prime radical	C₂³×C₇₁	C₄⋊C₈
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#2150	D₇₁
hypocenter	C₇₁	#7152
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
100. C₇₁	#7152	semidirect

Maximal subgroups

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

subgroup	index
#22590096	2
C₂×#17985720	2
C₇₁×#2150	2
#7152	71