Subgroups — #31888759

Order 18176.

334 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	C₄×#14556064	C₂	1	1	yes	maximal, characteristic
2	#22587088	C₂	1	1	yes	maximal, characteristic
3	#22586629	C₂	1	2	yes	maximal
4	#22586177	C₂	1	2	yes	maximal
5	C₂×#322×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×#14556064	C₄	1	2	yes	—
8	C₄×#11765849	C₂²	1	2	yes	—
9	#17985764	C₄	1	2	yes	—
10	#17985752	C₄	1	2	yes	—
11	#17985750	C₄	1	2	yes	—
12	#17985721	—	2	2	no	—
13	#17985720	—	2	2	no	—
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	#322×C₇₁	C₄	1	2	yes	—
18	#322×C₇₁	C₄	1	2	yes	—
19	C₂×#11765849	C₂×C₄	1	4	yes	—
20	#14556064	—	2	4	no	—
21	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
23	#14556039	—	2	2	no	—
24	C₄×C₇₁⋊C₈	—	2	2	no	—
25	C₄×C₇₁⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₇₁	—	2	1	no	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	Q₈	1	2	yes	—
30	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₇₁	—	2	1	no	—
32	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₇₁	D₄	1	2	yes	—
39	#11765849	—	2	4	no	—
40	#11765849	—	2	4	no	—
41	C₂×C₇₁⋊C₈	—	2	4	no	—
42	C₂×C₇₁⋊C₈	—	2	4	no	—
43	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
46	C₂×C₈×C₇₁	—	2	2	no	—
47	C₂×C₈×C₇₁	—	2	2	no	—
48	C₂×C₈×C₇₁	—	4	2	no	—
49	C₂×C₈×C₇₁	—	4	2	no	—
50	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
53	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₄⋊C₄	1	4	yes	—
55	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
56	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₇₁	#41	1	1	yes	characteristic
60	C₂³×C₇₁	C₄×Q₈	1	1	yes	characteristic
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
70	C₂×C₄×C₇₁	#131	1	2	yes	—
71	C₈×C₇₁	—	2	1	no	—
72	C₈×C₇₁	—	2	1	no	—
73	C₈×C₇₁	—	2	4	no	—
74	C₈×C₇₁	—	4	2	no	—
75	C₈×C₇₁	#97	1	2	yes	—
76	C₈×C₇₁	#97	1	2	yes	—
77	C₇₁⋊C₈	—	4	4	no	—
78	C₇₁⋊C₈	—	4	4	no	—
79	C₂²×C₇₁	—	2	1	no	—
80	C₂²×C₇₁	—	2	1	no	—
81	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₇₁	#446	1	2	yes	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	—	2	1	no	—
85	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
86	C₄×C₇₁	#405	1	1	yes	characteristic
87	C₄×C₇₁	#430	1	2	yes	—
88	C₄×C₇₁	#436	1	2	yes	—
89	C₄×C₇₁	#444	1	2	yes	—
90	#8295	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	2	no	—
92	C₂×C₇₁	#1770	1	1	yes	characteristic
93	C₂×C₇₁	#1773	1	1	yes	characteristic
94	C₂×C₇₁	#1778	1	1	yes	characteristic
95	C₄×#108	—	71	1	no	—
96	#1564	—	71	1	no	—
97	#1451	—	71	2	no	—
98	C₂×#322	D₇₁	1	1	yes	characteristic
99	#1278	—	71	2	no	—
100	C₇₁	#8295	1	1	yes	characteristic, complemented, Sylow
101	#431	—	71	2	no	—
102	C₂×#108	—	71	2	no	—
103	C₄×M₁₆	—	71	2	no	—
104	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
107	#341	—	71	2	no	—
108	#335	—	71	2	no	—
109	#334	—	71	2	no	—
110	#322	—	142	2	no	—
111	#322	—	142	2	no	—
112	#322	Dic₇₁	1	2	yes	—
113	#322	Dic₇₁	1	2	yes	—
114	C₂×M₁₆	—	71	4	no	—
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	Dic₁₄₂	1	2	yes	—
117	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
118	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
120	#108	—	142	4	no	—
121	C₄⋊C₈	—	71	4	no	—
122	C₂²⋊C₈	—	142	2	no	—
123	#97	—	142	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	142	2	no	—
126	C₄×C₈	—	142	2	no	—
127	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
128	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
129	C₄×C₈	D₂₈₄	1	2	yes	—
130	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
131	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
133	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
134	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
135	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
136	C₂²×C₄	#11765865	1	1	yes	characteristic
137	M₁₆	—	142	4	no	—
138	M₁₆	—	142	4	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	—	4	2	no	—
143	C₂×C₈	—	142	4	no	—
144	C₂×C₈	—	142	4	no	—
145	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
146	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
147	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
148	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
149	C₂×C₈	C₄⋊Dic₇₁	1	4	yes	—
150	C₂×C₈	#11765848	1	4	yes	—
151	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
152	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
153	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
154	C₄²	#11765865	1	1	yes	characteristic
155	C₂³	C₄×Dic₁₄₂	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
159	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
160	C₂×C₄	#14556087	1	1	yes	characteristic
161	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
162	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
163	C₂×C₄	C₄×Dic₁₄₂	1	1	yes	characteristic
164	C₂×C₄	C₂×#11765848	1	2	yes	—
165	C₂×C₄	#14556134	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	2	no	—
170	C₈	—	284	4	no	—
171	C₈	—	284	4	no	—
172	C₈	#14556038	1	2	yes	—
173	C₈	#14556038	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄×C₄⋊Dic₇₁	1	1	yes	characteristic
177	C₂²	#17986129	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×#14556038	1	1	yes	characteristic
181	C₄	#17985954	1	1	yes	characteristic
182	C₄	#17986048	1	2	yes	—
183	C₄	#17986078	1	2	yes	—
184	C₄	#17986122	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	#22588207	1	1	yes	characteristic
187	C₂	#22588224	1	1	yes	characteristic
188	C₂	#22588246	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#322×C₇₁	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	C₄×Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#322	D₇₁
hypocenter	C₇₁	#8295
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
100. C₇₁	#8295	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556064	2
#22587088	2
#22586629	2
#22586177	2
C₂×#322×C₇₁	2
#8295	71