Subgroups — #31888756

Order 18176.

358 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	#22587088	C₂	1	1	yes	maximal, characteristic
2	M₁₆×C₇₁⋊C₈	C₂	1	2	yes	maximal
3	C₂×#17985720	C₂	1	1	yes	maximal, characteristic
4	#22586178	C₂	1	2	yes	maximal
5	C₄×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	#17985752	C₄	1	2	yes	—
10	#17985750	C₄	1	2	yes	—
11	#17985720	—	2	2	no	—
12	#17985720	C₄	1	2	yes	—
13	#17985720	C₄	1	2	yes	—
14	C₈×C₇₁⋊C₈	—	2	2	no	—
15	C₂×#108×C₇₁	C₄	1	2	yes	—
16	C₄×M₁₆×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
18	#341×C₇₁	C₄	1	2	yes	—
19	C₂²×C₇₁⋊C₈	—	2	1	no	—
20	C₂²×C₇₁⋊C₈	Q₈	1	2	yes	—
21	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
22	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄×C₇₁⋊C₈	—	2	2	no	—
24	C₄×C₇₁⋊C₈	—	2	1	no	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
28	C₂×M₁₆×C₇₁	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	#108×C₇₁	C₈	1	8	yes	complemented
33	#97×C₇₁	—	2	2	no	—
34	C₄×C₈×C₇₁	—	2	2	no	—
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	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇₁⋊C₈	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	4	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
47	M₁₆×C₇₁	—	2	4	no	—
48	M₁₆×C₇₁	C₂×C₈	1	8	yes	—
49	C₂×C₈×C₇₁	—	2	4	no	—
50	C₂×C₈×C₇₁	—	2	2	no	—
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	C₄²	1	1	yes	characteristic
55	C₂×C₈×C₇₁	C₂×C₈	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	C₂²×C₈	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	8	no	—
74	C₈×C₇₁	C₄×C₈	1	2	yes	—
75	C₈×C₇₁	C₄×C₈	1	2	yes	—
76	C₇₁⋊C₈	—	2	4	no	—
77	C₇₁⋊C₈	—	4	2	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	C₈×Q₈	1	2	yes	—
83	C₄×C₇₁	—	2	1	no	—
84	C₄×C₇₁	—	2	1	no	—
85	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
86	C₄×C₇₁	#405	1	1	yes	characteristic
87	C₄×C₇₁	#430	1	2	yes	—
88	C₄×C₇₁	C₈×D₄	1	2	yes	—
89	C₄×C₇₁	#444	1	2	yes	—
90	#8294	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	—	2	2	no	—
92	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₇₁	#1773	1	1	yes	characteristic
94	C₂×C₇₁	#1778	1	1	yes	characteristic
95	C₄×#108	D₇₁	1	1	yes	characteristic
96	#1564	—	71	1	no	—
97	C₈×M₁₆	—	71	2	no	—
98	C₂×#322	—	71	1	no	—
99	#1278	—	71	2	no	—
100	C₇₁	#8294	1	1	yes	characteristic, complemented, Sylow
101	#431	—	71	2	no	—
102	C₂×#108	Dic₇₁	1	2	yes	—
103	C₄×M₁₆	C₂×D₇₁	1	2	yes	—
104	C₂×C₄×C₈	—	71	1	no	—
105	C₂×C₄×C₈	—	71	1	no	—
106	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
107	#341	Dic₇₁	1	2	yes	—
108	#335	—	71	2	no	—
109	#334	—	71	2	no	—
110	#322	—	71	2	no	—
111	#322	—	71	2	no	—
112	#322	—	142	2	no	—
113	C₈²	—	142	2	no	—
114	C₂×M₁₆	C₂×Dic₇₁	1	4	yes	—
115	C₂²×C₈	—	71	2	no	—
116	C₂²×C₈	—	142	1	no	—
117	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
119	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
120	#108	C₇₁⋊C₈	1	8	yes	complemented
121	C₄⋊C₈	—	71	4	no	—
122	C₂²⋊C₈	—	142	2	no	—
123	#97	—	2	2	no	—
124	C₄×C₈	—	2	2	no	—
125	C₄×C₈	—	71	2	no	—
126	C₄×C₈	—	71	2	no	—
127	C₄×C₈	—	71	2	no	—
128	C₄×C₈	—	142	2	no	—
129	C₄×C₈	—	142	1	no	—
130	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
131	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
132	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
133	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
135	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
136	C₂²×C₄	#11765866	1	1	yes	characteristic
137	M₁₆	—	2	4	no	—
138	M₁₆	C₂×C₇₁⋊C₈	1	8	yes	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	—	2	2	no	—
141	C₂×C₈	—	284	2	no	—
142	C₂×C₈	—	284	2	no	—
143	C₂×C₈	—	71	4	no	—
144	C₂×C₈	—	142	2	no	—
145	C₂×C₈	—	142	4	no	—
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	1	yes	characteristic
150	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
151	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
152	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
153	C₄²	D₄×D₇₁	1	1	yes	characteristic
154	C₄²	#11765864	1	1	yes	characteristic
155	C₂³	Q₈×Dic₇₁	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	C₄⋊C₄×D₇₁	1	1	yes	characteristic
160	C₂×C₄	#14556086	1	1	yes	characteristic
161	C₂×C₄	#14556090	1	1	yes	characteristic
162	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
163	C₂×C₄	#14556093	1	1	yes	characteristic
164	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
165	C₂×C₄	#14556135	1	2	yes	—
166	C₈	—	2	8	no	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	284	2	no	—
170	C₈	—	284	4	no	—
171	C₈	—	142	4	no	—
172	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
173	C₈	C₄×C₇₁⋊C₈	1	2	yes	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
177	C₂²	Q₈×C₇₁⋊C₈	1	2	yes	—
178	C₄	—	2	1	no	—
179	C₄	—	2	1	no	—
180	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
181	C₄	#17985955	1	1	yes	characteristic
182	C₄	#17986046	1	2	yes	—
183	C₄	D₄×C₇₁⋊C₈	1	2	yes	—
184	C₄	#17986120	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	C₄⋊C₄×C₇₁⋊C₈	1	1	yes	characteristic
187	C₂	#22588227	1	1	yes	characteristic
188	C₂	#22588248	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	C₄×Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#108	D₇₁
hypocenter	C₇₁	#8294
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
32. #108×C₇₁	C₈	semidirect
100. C₇₁	#8294	semidirect
120. #108	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22587088	2
M₁₆×C₇₁⋊C₈	2
C₂×#17985720	2
#22586178	2
C₄×#108×C₇₁	2
#8294	71