Subgroups — #31884321

Order 18176.

242 conjugacy classes of subgroups.

The subgroup lattice has 184 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	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	C₂×#335×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₄×C₇₁⋊C₁₆	—	2	1	no	—
7	C₄×C₇₁⋊C₁₆	—	2	1	no	—
8	#17985752	—	2	1	no	—
9	#17985752	C₄	1	2	yes	—
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	—
14	C₂²×C₇₁⋊C₈	—	4	1	no	—
15	C₂×C₇₁⋊C₁₆	—	2	2	no	—
16	C₂×C₇₁⋊C₁₆	—	2	2	no	—
17	C₂×C₇₁⋊C₁₆	C₈	1	4	yes	complemented
18	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
19	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
20	C₂²×C₈×C₇₁	—	4	1	no	—
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₇₁	—	2	2	no	—
25	C₄⋊C₈×C₇₁	—	4	1	no	—
26	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₇₁⋊C₈	—	4	2	no	—
31	C₂×C₇₁⋊C₈	—	8	1	no	—
32	C₂×C₇₁⋊C₈	—	8	1	no	—
33	C₇₁⋊C₁₆	—	2	4	no	—
34	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
37	C₂×C₈×C₇₁	—	2	2	no	—
38	C₂×C₈×C₇₁	—	4	2	no	—
39	C₂×C₈×C₇₁	—	8	1	no	—
40	C₂×C₈×C₇₁	—	8	1	no	—
41	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
42	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
46	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
47	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₄²×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
50	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
51	C₂³×C₇₁	#107	1	1	yes	characteristic
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
54	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
55	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
56	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
57	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
58	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₇₁	#106	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#108	1	2	yes	—
61	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
62	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
63	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
64	C₂×C₄×C₇₁	Q₃₂	1	1	yes	characteristic
65	C₈×C₇₁	—	2	1	no	—
66	C₈×C₇₁	—	2	1	no	—
67	C₈×C₇₁	—	8	2	no	—
68	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
69	C₈×C₇₁	#97	1	1	yes	characteristic
70	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
71	C₇₁⋊C₈	—	8	2	no	—
72	C₂²×C₇₁	#335	1	2	yes	—
73	C₂²×C₇₁	#340	1	1	yes	characteristic
74	C₂²×C₇₁	#366	1	1	yes	characteristic
75	C₂²×C₇₁	#367	1	1	yes	characteristic
76	C₂²×C₇₁	#368	1	2	yes	—
77	C₄×C₇₁	—	2	1	no	—
78	C₄×C₇₁	—	2	1	no	—
79	C₄×C₇₁	#334	1	2	yes	—
80	C₄×C₇₁	#336	1	1	yes	characteristic
81	C₄×C₇₁	#341	1	1	yes	characteristic
82	C₄×C₇₁	#357	1	1	yes	characteristic
83	C₄×C₇₁	#357	1	1	yes	characteristic
84	C₄×C₇₁	#358	1	1	yes	characteristic
85	C₄×C₇₁	#358	1	1	yes	characteristic
86	#7154	—	71	1	no	maximal, Sylow
87	C₂×C₇₁	#1277	1	1	yes	characteristic
88	C₂×C₇₁	#1371	1	2	yes	—
89	C₂×C₇₁	#1372	1	2	yes	—
90	C₂×C₇₁	#1380	1	1	yes	characteristic
91	C₂×C₇₁	#1381	1	1	yes	characteristic
92	C₂×C₄×C₁₆	—	71	1	no	—
93	C₂×#335	—	71	1	no	—
94	C₂×#335	D₇₁	1	1	yes	characteristic
95	C₇₁	#7154	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₁₆	—	71	2	no	—
97	C₂×C₄⋊C₈	—	2	1	no	—
98	C₂×C₄⋊C₈	—	142	1	no	—
99	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
100	C₄×C₁₆	—	142	1	no	—
101	C₄×C₁₆	—	142	1	no	—
102	#335	—	2	1	no	—
103	#335	—	71	2	no	—
104	#335	—	142	1	no	—
105	#335	Dic₇₁	1	2	yes	—
106	C₂²×C₈	—	4	1	no	—
107	C₂²×C₈	—	284	1	no	—
108	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
109	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
110	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
111	C₂×C₁₆	—	71	4	no	—
112	C₂×C₁₆	—	142	2	no	—
113	C₂×C₁₆	—	142	2	no	—
114	C₄⋊C₈	—	2	2	no	—
115	C₄⋊C₈	—	4	1	no	—
116	C₄⋊C₈	—	284	1	no	—
117	C₄⋊C₈	—	142	2	no	—
118	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
119	C₄×C₈	D₂₈₄	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
121	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
122	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
123	C₂²×C₄	#11765850	1	1	yes	characteristic
124	C₂²×C₄	#11765855	1	1	yes	characteristic
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	8	1	no	—
128	C₂×C₈	—	8	1	no	—
129	C₂×C₈	—	284	2	no	—
130	C₂×C₈	—	568	1	no	—
131	C₂×C₈	—	568	1	no	—
132	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
133	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
134	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
135	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
136	C₂×C₈	C₈×D₇₁	1	2	yes	—
137	C₂×C₈	#11765848	1	2	yes	—
138	C₄²	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
139	C₄²	C₂²⋊Dic₇₁	1	1	yes	characteristic
140	C₄²	#11765852	1	1	yes	characteristic
141	C₄²	#11765853	1	1	yes	characteristic
142	C₁₆	—	142	4	no	—
143	C₂³	#14556061	1	1	yes	characteristic
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
146	C₂×C₄	#14556048	1	1	yes	characteristic
147	C₂×C₄	#14556049	1	1	yes	characteristic
148	C₂×C₄	#14556051	1	1	yes	characteristic
149	C₂×C₄	#14556052	1	1	yes	characteristic
150	C₂×C₄	Dic₇₁⋊C₈	1	2	yes	—
151	C₂×C₄	#14556059	1	1	yes	characteristic
152	C₂×C₄	#14556063	1	2	yes	—
153	C₂×C₄	#14556069	1	1	yes	characteristic
154	C₂×C₄	#14556071	1	1	yes	characteristic
155	C₂×C₄	#14556072	1	1	yes	characteristic
156	C₂×C₄	#14556074	1	1	yes	characteristic
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	8	2	no	—
160	C₈	—	568	2	no	—
161	C₈	C₈×Dic₇₁	1	1	yes	characteristic
162	C₈	#14556038	1	1	yes	characteristic
163	C₈	(C₂×D₇₁)⋊C₈	1	2	yes	—
164	C₂²	#17985751	1	2	yes	—
165	C₂²	#17985761	1	1	yes	characteristic
166	C₂²	#17985824	1	1	yes	characteristic
167	C₂²	#17985826	1	1	yes	characteristic
168	C₂²	#17985828	1	2	yes	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	#17985749	1	2	yes	—
172	C₄	#17985753	1	1	yes	characteristic
173	C₄	#17985763	1	1	yes	characteristic
174	C₄	#17985799	1	1	yes	characteristic
175	C₄	#17985800	1	1	yes	characteristic
176	C₄	#17985802	1	1	yes	characteristic
177	C₄	#17985803	1	1	yes	characteristic
178	C₂	#22586174	1	1	yes	characteristic
179	C₂	#22586411	1	2	yes	—
180	C₂	#22586413	1	2	yes	—
181	C₂	#22586431	1	1	yes	characteristic
182	C₂	#22586433	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	#107
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	#7154
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
17. C₂×C₇₁⋊C₁₆	C₈	semidirect
95. C₇₁	#7154	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
C₂×#17985752	2
C₂×#335×C₇₁	2
#7154	71