Subgroups — #31884394

Order 18176.

240 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×#17985820	C₂	1	1	yes	maximal, characteristic
2	#22588219	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1450	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	—	2	1	no	—
5	C₂×#14556064	C₄	1	1	yes	characteristic
6	#17986037	C₄	1	1	yes	characteristic
7	#17985954	—	2	1	no	—
8	#17985820	—	2	1	no	—
9	#17985820	C₄	1	2	yes	complemented
10	#17985759	—	2	1	no	—
11	#431×C₇₁	—	2	1	no	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#322×C₇₁	—	2	1	no	—
14	C₈²×C₇₁	C₄	1	2	yes	complemented
15	C₂×#11765849	—	2	1	no	—
16	#14556087	—	2	1	no	—
17	#14556068	—	2	2	no	—
18	#14556068	—	4	1	no	—
19	C₂×C₇₁⋊C₁₆	—	4	1	no	—
20	#14556064	—	2	1	no	—
21	#14556064	C₈	1	2	yes	complemented
22	#14556062	—	2	1	no	—
23	#14556060	C₈	1	2	yes	complemented
24	#14556038	—	4	1	no	—
25	C₈×Dic₇₁	—	4	1	no	—
26	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
29	C₄⋊C₈×C₇₁	—	2	2	no	—
30	C₂²⋊C₈×C₇₁	—	2	2	no	—
31	C₄×C₈×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
36	#11765849	—	2	2	no	—
37	#11765849	—	4	1	no	—
38	C₂×C₇₁⋊C₈	—	4	1	no	—
39	C₄⋊Dic₇₁	—	2	2	no	—
40	C₂²⋊Dic₇₁	—	4	1	no	—
41	C₄×Dic₇₁	—	4	1	no	—
42	C₇₁⋊C₁₆	—	4	2	no	—
43	C₂²×C₄×C₇₁	—	2	1	no	—
44	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	—	2	4	no	—
46	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
55	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
57	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
58	C₄²×C₇₁	—	2	1	no	—
59	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂³×C₇₁	#104	1	1	yes	characteristic
62	C₂×C₄×C₇₁	—	2	1	no	—
63	C₂×C₄×C₇₁	—	2	1	no	—
64	C₂×C₄×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#104	1	1	yes	characteristic
71	C₂×Dic₇₁	—	4	2	no	—
72	C₈×C₇₁	—	4	4	no	—
73	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
74	C₈×C₇₁	#97	1	1	yes	characteristic
75	C₈×C₇₁	#102	1	1	yes	characteristic
76	C₈×C₇₁	#103	1	1	yes	characteristic
77	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
78	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
79	C₈×C₇₁	#106	1	1	yes	characteristic
80	C₈×C₇₁	#107	1	1	yes	characteristic
81	C₇₁⋊C₈	—	4	2	no	—
82	C₂²×C₇₁	—	4	1	no	—
83	C₂²×C₇₁	D₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
85	C₂²×C₇₁	#339	1	1	yes	characteristic
86	C₄×C₇₁	—	2	2	no	—
87	C₄×C₇₁	—	4	1	no	—
88	C₄×C₇₁	D₄⋊C₈	1	1	yes	characteristic
89	C₄×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
90	C₄×C₇₁	#336	1	1	yes	characteristic
91	C₄×C₇₁	#340	1	1	yes	characteristic
92	Dic₇₁	—	8	2	no	—
93	#7180	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	—	2	1	no	—
95	C₂×C₇₁	—	2	1	no	—
96	C₂×C₇₁	—	2	1	no	—
97	C₂×C₇₁	#1279	1	1	yes	characteristic
98	C₂×#364	—	71	1	no	—
99	#1772	—	71	1	no	—
100	#1450	D₇₁	1	1	yes	characteristic
101	C₇₁	#7180	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₃₂	—	142	1	no	—
103	#431	—	2	1	no	—
104	C₂×#108	—	71	1	no	—
105	#427	—	71	1	no	—
106	#405	—	142	1	no	—
107	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
108	#364	—	71	2	no	—
109	#364	—	142	1	no	—
110	#339	—	142	1	no	—
111	#322	—	2	1	no	—
112	C₈²	Dic₇₁	1	2	yes	complemented
113	C₂×M₁₆	—	142	1	no	—
114	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
115	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
116	#117	—	142	1	no	—
117	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
118	M₃₂	—	284	1	no	—
119	M₃₂	—	142	2	no	—
120	C₂×C₁₆	—	284	1	no	—
121	#108	—	71	2	no	—
122	#108	—	142	1	no	—
123	#107	—	142	1	no	—
124	#106	—	71	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₂²⋊C₈	—	2	2	no	—
127	#97	—	284	1	no	—
128	C₄×C₈	—	2	2	no	—
129	C₄×C₈	—	284	1	no	—
130	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
131	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
132	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
133	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
136	M₁₆	—	284	1	no	—
137	M₁₆	—	142	2	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	—	284	1	no	—
140	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
141	C₂×C₈	(C₂×D₇₁)⋊C₄	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	C₈×D₇₁	1	1	yes	characteristic
145	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
146	C₂×C₈	#11765848	1	1	yes	characteristic
147	C₂×C₈	#11765848	1	1	yes	characteristic
148	C₂×C₈	D₅₆₈	1	1	yes	characteristic
149	C₂×C₈	#11765854	1	1	yes	characteristic
150	C₂×C₈	#11765854	1	1	yes	characteristic
151	C₂×C₈	Dic₂₈₄	1	1	yes	characteristic
152	C₄⋊C₄	—	142	2	no	—
153	C₂²⋊C₄	—	284	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	—	284	1	no	—
156	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
157	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
158	C₁₆	—	284	2	no	—
159	C₂³	#14556056	1	1	yes	characteristic
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	2	no	—
163	C₂×C₄	—	284	2	no	—
164	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
165	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
166	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
167	C₂×C₄	#14556050	1	1	yes	characteristic
168	C₂×C₄	#14556053	1	1	yes	characteristic
169	C₂×C₄	#14556056	1	1	yes	characteristic
170	C₈	—	4	4	no	—
171	C₈	—	284	2	no	—
172	C₈	C₈×Dic₇₁	1	1	yes	characteristic
173	C₈	#14556038	1	1	yes	characteristic
174	C₈	#14556050	1	1	yes	characteristic
175	C₈	#14556053	1	1	yes	characteristic
176	C₈	Dic₇₁⋊C₈	1	1	yes	characteristic
177	C₈	Dic₇₁⋊C₈	1	1	yes	characteristic
178	C₈	#14556060	1	1	yes	characteristic
179	C₈	#14556062	1	1	yes	characteristic
180	C₂²	—	4	1	no	—
181	C₂²	#17985728	1	1	yes	characteristic
182	C₂²	#17985731	1	1	yes	characteristic
183	C₂²	#17985759	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	4	1	no	—
186	C₄	—	568	2	no	—
187	C₄	#17985728	1	1	yes	characteristic
188	C₄	#17985731	1	1	yes	characteristic
189	C₄	#17985753	1	1	yes	characteristic
190	C₄	#17985762	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	—	2	1	no	—
194	C₂	#22586180	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556060
commutator	C₈×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₇₁×#1450	C₂
socle	C₂×C₇₁	#1279
prime radical	C₂³×C₇₁	#104
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	#1450	D₇₁
hypocenter	C₇₁	#7180
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
9. #17985820	C₄	semidirect
14. C₈²×C₇₁	C₄	semidirect
21. #14556064	C₈	semidirect
23. #14556060	C₈	semidirect
101. C₇₁	#7180	semidirect
112. C₈²	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985820	2
#22588219	2
C₇₁×#1450	2
#7180	71