Subgroups — #31912572

Order 18176.

382 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589961	C₂	1	1	yes	maximal, characteristic
2	#22589944	C₂	1	2	yes	maximal, complemented
3	#22588140	C₂	1	1	yes	maximal, characteristic
4	#22586428	C₂	1	2	yes	maximal, complemented
5	C₂×#504×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	#17985983	C₄	1	1	yes	characteristic
8	#17985973	C₄	1	1	yes	characteristic
9	#17985954	C₂²	1	2	yes	—
10	#17985781	C₄	1	4	yes	complemented
11	#17985776	—	2	4	no	—
12	#17985770	C₄	1	2	yes	—
13	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
14	#504×C₇₁	—	2	4	no	—
15	C₂×M₃₂×C₇₁	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₇₁	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#11765849	C₂×C₄	1	2	yes	—
20	#14556087	C₂×C₄	1	2	yes	—
21	#14556068	—	2	2	no	—
22	#14556068	C₂×C₄	1	4	yes	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	#14556047	—	2	1	no	—
25	#14556045	—	2	1	no	—
26	#14556043	—	2	2	no	—
27	#14556038	—	2	2	no	—
28	C₈×Dic₇₁	—	2	2	no	—
29	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
30	#131×C₇₁	—	2	2	no	—
31	#131×C₇₁	—	2	2	no	—
32	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₇₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
36	M₃₂×C₇₁	—	2	4	no	—
37	M₃₂×C₇₁	D₄	1	4	yes	—
38	C₂×C₁₆×C₇₁	—	2	2	no	—
39	C₂×C₁₆×C₇₁	—	2	4	no	—
40	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
43	#11765849	—	2	2	no	—
44	#11765849	—	2	2	no	—
45	C₂×C₇₁⋊C₈	—	2	2	no	—
46	C₄⋊Dic₇₁	—	2	2	no	—
47	C₂²⋊Dic₇₁	—	2	2	no	—
48	C₄×Dic₇₁	—	2	2	no	—
49	C₇₁⋊C₁₆	—	2	4	no	—
50	#41×C₇₁	—	2	2	no	—
51	#41×C₇₁	—	2	2	no	—
52	C₂×Q₈×C₇₁	M₁₆	1	1	yes	characteristic
53	C₂×D₄×C₇₁	M₁₆	1	2	yes	—
54	C₂×D₄×C₇₁	M₁₆	1	1	yes	characteristic
55	C₂²×C₄×C₇₁	C₄²	1	2	yes	—
56	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	M₁₆×C₇₁	—	2	2	no	—
58	M₁₆×C₇₁	—	2	4	no	—
59	C₂×C₈×C₇₁	—	2	2	no	—
60	C₂×C₈×C₇₁	—	2	2	no	—
61	C₂×C₈×C₇₁	C₄²	1	2	yes	—
62	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₇₁	#41	1	2	yes	—
66	C₁₆×C₇₁	—	2	1	no	—
67	C₁₆×C₇₁	—	2	1	no	—
68	C₁₆×C₇₁	C₂²⋊C₄	1	4	yes	—
69	C₂³×C₇₁	#97	1	2	yes	—
70	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
71	Q₈×C₇₁	—	2	2	no	—
72	D₄×C₇₁	—	2	4	no	—
73	D₄×C₇₁	—	2	2	no	—
74	C₂×C₄×C₇₁	—	2	2	no	—
75	C₂×C₄×C₇₁	—	2	2	no	—
76	C₂×C₄×C₇₁	#97	1	2	yes	—
77	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
78	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#131	1	2	yes	—
81	C₂×Dic₇₁	—	4	2	no	—
82	C₈×C₇₁	—	2	2	no	—
83	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₈×C₇₁	#117	1	1	yes	characteristic
85	C₈×C₇₁	C₄×D₄	1	2	yes	—
86	C₇₁⋊C₈	—	4	2	no	—
87	C₂²×C₇₁	—	2	2	no	—
88	C₂²×C₇₁	—	4	1	no	—
89	C₂²×C₇₁	—	4	2	no	—
90	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
91	C₂²×C₇₁	#435	1	2	yes	—
92	C₄×C₇₁	—	2	2	no	—
93	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₇₁	#405	1	1	yes	characteristic
95	C₄×C₇₁	#435	1	2	yes	—
96	Dic₇₁	—	8	2	no	—
97	#12617	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	—	2	2	no	—
99	C₂×C₇₁	—	2	1	no	—
100	C₂×C₇₁	—	4	2	no	—
101	C₂×C₇₁	#1752	1	1	yes	characteristic
102	C₂×#504	D₇₁	1	1	yes	characteristic
103	#2115	—	71	1	no	—
104	#2110	—	71	2	no	—
105	#1757	—	71	1	no	—
106	#1379	—	71	2	no	—
107	C₇₁	#12617	1	1	yes	characteristic, complemented, Sylow
108	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
109	#504	—	2	4	no	—
110	C₂×M₃₂	—	71	2	no	—
111	C₂×M₃₂	Dic₇₁	1	2	yes	complemented
112	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
113	C₂²×C₁₆	Dic₇₁	1	2	yes	complemented
114	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
115	#413	—	71	1	no	—
116	#410	—	71	1	no	—
117	#405	—	71	2	no	—
118	#349	—	71	4	no	—
119	#347	—	142	4	no	—
120	#344	—	71	2	no	—
121	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
122	#131	—	2	2	no	—
123	#131	—	2	2	no	—
124	C₂×M₁₆	—	71	2	no	—
125	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
126	C₂×M₁₆	C₄×D₇₁	1	2	yes	—
127	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
128	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
129	#117	—	71	2	no	—
130	M₃₂	—	2	4	no	—
131	M₃₂	—	71	4	no	—
132	M₃₂	—	142	2	no	—
133	M₃₂	C₂²⋊D₇₁	1	4	yes	—
134	C₂×C₁₆	—	2	2	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	—	142	2	no	—
137	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
138	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
139	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
140	#101	—	142	1	no	—
141	#100	—	142	1	no	—
142	#99	—	142	2	no	—
143	#97	—	142	2	no	—
144	C₄×C₈	—	142	2	no	—
145	#41	—	2	2	no	—
146	#41	—	2	2	no	—
147	C₂×Q₈	#11765848	1	1	yes	characteristic
148	C₂×D₄	#11765848	1	2	yes	—
149	C₂×D₄	#11765848	1	1	yes	characteristic
150	C₂²×C₄	C₄×Dic₇₁	1	2	yes	—
151	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
152	M₁₆	—	2	2	no	—
153	M₁₆	—	2	4	no	—
154	M₁₆	—	142	2	no	—
155	M₁₆	—	142	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	142	2	no	—
159	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
160	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
161	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
162	C₂×C₈	C₂×C₂²⋊D₇₁	1	2	yes	—
163	C₂×C₈	#11765865	1	2	yes	—
164	C₄⋊C₄	—	142	2	no	—
165	C₂²⋊C₄	—	142	2	no	—
166	C₄²	—	142	2	no	—
167	C₁₆	—	2	1	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	—	142	4	no	—
170	C₁₆	C₂²⋊Dic₇₁	1	4	yes	—
171	C₂³	#14556038	1	2	yes	—
172	C₂³	C₂×#11765848	1	1	yes	characteristic
173	Q₈	—	2	2	no	—
174	D₄	—	2	4	no	—
175	D₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	284	2	no	—
179	C₂×C₄	#14556038	1	2	yes	—
180	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
181	C₂×C₄	C₄×C₂²⋊D₇₁	1	2	yes	—
182	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
183	C₂×C₄	#14556134	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	284	2	no	—
186	C₈	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
187	C₈	#14556087	1	1	yes	characteristic
188	C₈	C₄×C₂²⋊D₇₁	1	2	yes	—
189	C₂²	—	2	2	no	—
190	C₂²	—	4	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	C₂×#14556038	1	1	yes	characteristic
193	C₂²	#17986070	1	2	yes	—
194	C₄	—	2	2	no	—
195	C₄	—	568	2	no	—
196	C₄	C₄×C₂²⋊Dic₇₁	1	1	yes	characteristic
197	C₄	#17985954	1	1	yes	characteristic
198	C₄	#17986070	1	2	yes	—
199	C₂	—	2	2	no	—
200	C₂	—	2	1	no	—
201	C₂	—	4	2	no	—
202	C₂	#22588116	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556087
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#504×C₇₁	C₂
socle	C₂×C₇₁	#1752
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#504	D₇₁
hypocenter	C₇₁	#12617
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
2. #22589944	C₂	semidirect
4. #22586428	C₂	semidirect
10. #17985781	C₄	semidirect
15. C₂×M₃₂×C₇₁	C₄	semidirect
17. C₂²×C₁₆×C₇₁	C₄	semidirect
107. C₇₁	#12617	semidirect
111. C₂×M₃₂	Dic₇₁	semidirect
113. C₂²×C₁₆	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22589961	2
#22589944	2
#22588140	2
#22586428	2
C₂×#504×C₇₁	2
#12617	71