Subgroups — #31891098

Order 18176.

310 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22587088	C₂	1	2	yes	maximal
2	#22586185	C₂	1	2	yes	maximal
3	#22586169	C₂	1	2	yes	maximal
4	C₇₁×#1726	C₂	1	1	yes	maximal, characteristic
5	#17986052	C₂²	1	4	yes	—
6	#17985752	—	2	2	no	—
7	#17985750	—	2	2	no	—
8	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
9	C₂×#97×C₇₁	C₄	1	1	yes	characteristic
10	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#376×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
17	C₄×C₇₁⋊C₈	—	2	4	no	—
18	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
25	#97×C₇₁	—	2	2	no	—
26	#97×C₇₁	—	2	2	no	—
27	#97×C₇₁	—	2	2	no	—
28	C₄×C₈×C₇₁	—	2	2	no	—
29	C₄×C₈×C₇₁	C₈	1	4	yes	complemented
30	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
34	#95×C₇₁	—	2	2	no	—
35	C₂×C₇₁⋊C₈	—	4	4	no	—
36	C₂²×C₄×C₇₁	—	2	1	no	—
37	C₂²×C₄×C₇₁	—	2	1	no	—
38	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₇₁	#41	1	2	yes	—
40	C₂×C₈×C₇₁	—	2	1	no	—
41	C₂×C₈×C₇₁	—	2	1	no	—
42	C₂×C₈×C₇₁	—	2	1	no	—
43	C₂×C₈×C₇₁	—	2	1	no	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₄⋊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	2	yes	—
51	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
52	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
53	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
54	C₄²×C₇₁	—	2	1	no	—
55	C₄²×C₇₁	—	2	1	no	—
56	C₄²×C₇₁	C₂×C₈	1	2	yes	—
57	C₄²×C₇₁	M₁₆	1	2	yes	—
58	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₇₁	#41	1	2	yes	—
61	C₂³×C₇₁	#117	1	1	yes	characteristic
62	C₂×C₄×C₇₁	—	2	2	no	—
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	4	2	no	—
65	C₂×C₄×C₇₁	—	4	2	no	—
66	C₂×C₄×C₇₁	—	4	2	no	—
67	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#124	1	2	yes	—
70	C₂×C₄×C₇₁	#125	1	2	yes	—
71	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
75	C₈×C₇₁	—	2	1	no	—
76	C₈×C₇₁	—	2	1	no	—
77	C₈×C₇₁	—	2	2	no	—
78	C₈×C₇₁	—	2	2	no	—
79	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
80	C₈×C₇₁	C₄⋊C₈	1	2	yes	—
81	C₇₁⋊C₈	—	8	4	no	—
82	C₂²×C₇₁	—	2	2	no	—
83	C₂²×C₇₁	#382	1	1	yes	characteristic
84	C₂²×C₇₁	#431	1	1	yes	characteristic
85	C₂²×C₇₁	#432	1	1	yes	characteristic
86	C₄×C₇₁	—	2	2	no	—
87	C₄×C₇₁	—	2	2	no	—
88	C₄×C₇₁	—	4	2	no	—
89	C₄×C₇₁	—	4	2	no	—
90	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₄×C₇₁	#423	1	1	yes	characteristic
92	C₄×C₇₁	#431	1	1	yes	characteristic
93	C₄×C₇₁	#432	1	1	yes	characteristic
94	#8780	—	71	1	no	maximal, Sylow
95	C₂×C₇₁	—	2	1	no	—
96	C₂×C₇₁	—	2	1	no	—
97	C₂×C₇₁	#1831	1	1	yes	characteristic
98	C₂×C₇₁	#1833	1	2	yes	—
99	#1726	D₇₁	1	1	yes	characteristic
100	#1564	—	71	2	no	—
101	#1281	—	71	2	no	—
102	#1274	—	71	2	no	—
103	C₇₁	#8780	1	1	yes	characteristic, complemented, Sylow
104	#431	—	71	4	no	—
105	C₂×#97	Dic₇₁	1	1	yes	characteristic
106	C₂×#97	Dic₇₁	1	1	yes	characteristic
107	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
108	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
109	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
110	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
111	#376	C₂×D₇₁	1	1	yes	characteristic
112	#335	—	142	2	no	—
113	#334	—	142	2	no	—
114	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
115	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
116	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
118	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
119	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
120	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
121	C₄⋊C₈	—	142	4	no	—
122	C₂²⋊C₈	—	142	4	no	—
123	#97	—	2	2	no	—
124	#97	—	2	2	no	—
125	#97	—	2	2	no	—
126	C₄×C₈	—	2	2	no	—
127	C₄×C₈	—	142	4	no	—
128	C₄×C₈	C₇₁⋊C₈	1	4	yes	complemented
129	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
130	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
131	C₄×C₈	D₂₈₄	1	1	yes	characteristic
132	C₄×C₈	D₂₈₄	1	1	yes	characteristic
133	#95	—	2	2	no	—
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
137	C₂²×C₄	#11765865	1	2	yes	—
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	284	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	C₂×C₇₁⋊C₈	1	2	yes	—
150	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
151	C₂×C₈	#11765849	1	2	yes	—
152	C₂×C₈	#11765849	1	2	yes	—
153	C₄²	—	2	1	no	—
154	C₄²	—	2	1	no	—
155	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
156	C₄²	#11765849	1	2	yes	—
157	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
158	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
159	C₄²	#11765865	1	2	yes	—
160	C₂³	#14556087	1	1	yes	characteristic
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	4	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
167	C₂×C₄	#14556087	1	1	yes	characteristic
168	C₂×C₄	#14556116	1	2	yes	—
169	C₂×C₄	#14556120	1	2	yes	—
170	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
171	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
172	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
173	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
174	C₈	—	2	2	no	—
175	C₈	—	2	2	no	—
176	C₈	—	2	1	no	—
177	C₈	—	2	1	no	—
178	C₈	—	568	4	no	—
179	C₈	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
180	C₈	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
181	C₂²	—	2	2	no	—
182	C₂²	#17985865	1	1	yes	characteristic
183	C₂²	#17986052	1	1	yes	characteristic
184	C₂²	#17986056	1	1	yes	characteristic
185	C₄	—	2	2	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
190	C₄	#17986023	1	1	yes	characteristic
191	C₄	#17986052	1	1	yes	characteristic
192	C₄	#17986056	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	#22588457	1	1	yes	characteristic
196	C₂	#22588464	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556087
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1726	C₂
socle	C₂²×C₇₁	#382
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1726	D₇₁
hypocenter	C₇₁	#8780
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1726

Complements

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

N	H ≅ G/N	type
29. C₄×C₈×C₇₁	C₈	semidirect
103. C₇₁	#8780	semidirect
128. C₄×C₈	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
#22586185	2
#22586169	2
C₇₁×#1726	2
#8780	71