Subgroups — #31884366

Order 18176.

282 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	1	yes	maximal, characteristic
2	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
5	C₂×#14556062	C₄	1	2	yes	—
6	C₄×C₄⋊Dic₇₁	—	2	1	no	—
7	C₄×C₇₁⋊C₁₆	—	2	1	no	—
8	C₄×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
9	#17985762	—	2	1	no	—
10	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
11	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#335×C₇₁	—	2	1	no	—
13	#335×C₇₁	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₇₁	—	2	2	no	—
15	C₂×C₄×Dic₇₁	—	4	1	no	—
16	C₂×C₇₁⋊C₁₆	—	2	4	no	—
17	C₂×C₇₁⋊C₁₆	—	2	2	no	—
18	#14556062	—	2	2	no	—
19	#14556062	C₈	1	4	yes	complemented
20	C₇₁⋊#95	—	4	1	no	—
21	C₂²×C₈×C₇₁	—	4	1	no	—
22	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
25	C₄⋊C₈×C₇₁	—	2	2	no	—
26	C₄⋊C₈×C₇₁	—	4	1	no	—
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
31	C₂²×Dic₇₁	—	4	2	no	—
32	C₄⋊Dic₇₁	—	2	4	no	—
33	C₄⋊Dic₇₁	—	4	2	no	—
34	C₄×Dic₇₁	—	8	1	no	—
35	C₄×Dic₇₁	—	8	1	no	—
36	C₇₁⋊C₁₆	—	2	4	no	—
37	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
40	C₂×C₈×C₇₁	—	2	2	no	—
41	C₂×C₈×C₇₁	—	4	2	no	—
42	C₂×C₈×C₇₁	—	8	1	no	—
43	C₂×C₈×C₇₁	—	8	1	no	—
44	C₂×C₈×C₇₁	C₄²	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	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
50	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
53	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
54	C₂³×C₇₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₁	—	2	2	no	—
56	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
58	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#104	1	2	yes	—
62	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
63	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
64	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
66	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
67	C₂×C₄×C₇₁	Q₃₂	1	1	yes	characteristic
68	C₂×Dic₇₁	—	4	4	no	—
69	C₂×Dic₇₁	—	8	2	no	—
70	C₂×Dic₇₁	—	8	2	no	—
71	C₈×C₇₁	—	2	1	no	—
72	C₈×C₇₁	—	2	1	no	—
73	C₈×C₇₁	—	8	2	no	—
74	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
75	C₈×C₇₁	#97	1	1	yes	characteristic
76	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
77	C₂²×C₇₁	D₄⋊C₈	1	2	yes	—
78	C₂²×C₇₁	#340	1	1	yes	characteristic
79	C₂²×C₇₁	#357	1	1	yes	characteristic
80	C₂²×C₇₁	#358	1	1	yes	characteristic
81	C₂²×C₇₁	#359	1	2	yes	—
82	C₄×C₇₁	—	2	1	no	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	Q₈⋊C₈	1	2	yes	—
85	C₄×C₇₁	#336	1	1	yes	characteristic
86	C₄×C₇₁	#339	1	1	yes	characteristic
87	C₄×C₇₁	#357	1	1	yes	characteristic
88	C₄×C₇₁	#358	1	1	yes	characteristic
89	C₄×C₇₁	#366	1	1	yes	characteristic
90	C₄×C₇₁	#367	1	1	yes	characteristic
91	Dic₇₁	—	8	4	no	—
92	#7171	—	71	1	no	maximal, Sylow
93	C₂×C₇₁	#1279	1	1	yes	characteristic
94	C₂×C₇₁	#1331	1	2	yes	—
95	C₂×C₇₁	#1332	1	2	yes	—
96	C₂×C₇₁	#1380	1	1	yes	characteristic
97	C₂×C₇₁	#1382	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	71	1	no	—
99	C₄×#107	—	71	1	no	—
100	C₂×#335	D₇₁	1	1	yes	characteristic
101	C₇₁	#7171	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	142	1	no	—
103	C₂×#107	—	71	2	no	—
104	C₂×C₄⋊C₈	—	2	1	no	—
105	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	142	1	no	—
107	C₄×C₁₆	—	71	2	no	—
108	C₄×C₁₆	—	142	1	no	—
109	#340	—	142	1	no	—
110	#335	—	2	1	no	—
111	#335	Dic₇₁	1	2	yes	complemented
112	C₂²×C₈	—	4	1	no	—
113	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
114	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	142	2	no	—
116	C₂×C₄²	—	284	1	no	—
117	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
118	C₂×C₁₆	—	142	4	no	—
119	C₂×C₁₆	—	142	2	no	—
120	#107	—	71	4	no	—
121	#107	—	142	2	no	—
122	C₄⋊C₈	—	2	2	no	—
123	C₄⋊C₈	—	4	1	no	—
124	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
125	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
128	#95	—	284	1	no	—
129	C₂²×C₄	—	284	2	no	—
130	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	#11765850	1	1	yes	characteristic
132	C₂²×C₄	#11765852	1	1	yes	characteristic
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	8	1	no	—
136	C₂×C₈	—	8	1	no	—
137	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
138	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
139	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
141	C₂×C₈	C₈×D₇₁	1	2	yes	—
142	C₂×C₈	#11765848	1	2	yes	—
143	C₄⋊C₄	—	284	2	no	—
144	C₄⋊C₄	—	142	4	no	—
145	C₄²	—	568	1	no	—
146	C₄²	—	568	1	no	—
147	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
148	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
149	C₄²	#11765853	1	1	yes	characteristic
150	C₄²	#11765855	1	1	yes	characteristic
151	C₁₆	—	142	4	no	—
152	C₂³	#14556048	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	284	4	no	—
155	C₂×C₄	—	568	2	no	—
156	C₂×C₄	—	568	2	no	—
157	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
158	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
159	C₂×C₄	#14556049	1	1	yes	characteristic
160	C₂×C₄	#14556051	1	1	yes	characteristic
161	C₂×C₄	#14556052	1	1	yes	characteristic
162	C₂×C₄	#14556054	1	2	yes	—
163	C₂×C₄	#14556059	1	1	yes	characteristic
164	C₂×C₄	#14556061	1	1	yes	characteristic
165	C₂×C₄	#14556069	1	1	yes	characteristic
166	C₂×C₄	#14556071	1	1	yes	characteristic
167	C₂×C₄	#14556072	1	1	yes	characteristic
168	C₂×C₄	#14556074	1	1	yes	characteristic
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	—	8	2	no	—
172	C₈	C₈×Dic₇₁	1	1	yes	characteristic
173	C₈	#14556038	1	1	yes	characteristic
174	C₈	Dic₇₁⋊C₈	1	2	yes	—
175	C₂²	#17985726	1	2	yes	—
176	C₂²	#17985761	1	1	yes	characteristic
177	C₂²	#17985799	1	1	yes	characteristic
178	C₂²	#17985802	1	1	yes	characteristic
179	C₂²	#17985805	1	2	yes	—
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	568	4	no	—
183	C₄	#17985729	1	2	yes	—
184	C₄	#17985753	1	1	yes	characteristic
185	C₄	#17985760	1	1	yes	characteristic
186	C₄	#17985800	1	1	yes	characteristic
187	C₄	#17985803	1	1	yes	characteristic
188	C₄	#17985824	1	1	yes	characteristic
189	C₄	#17985826	1	1	yes	characteristic
190	C₂	#22586179	1	1	yes	characteristic
191	C₂	#22586300	1	2	yes	—
192	C₂	#22586303	1	2	yes	—
193	C₂	#22586431	1	1	yes	characteristic
194	C₂	#22586437	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11765850
commutator	C₈×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×#335×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	#102
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	#7171
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
8. C₄×C₇₁⋊C₁₆	C₄	semidirect
13. #335×C₇₁	C₄	semidirect
19. #14556062	C₈	semidirect
101. C₇₁	#7171	semidirect
111. #335	Dic₇₁	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
C₄×#14556062	2
C₂×#335×C₇₁	2
#7171	71