Subgroups — #31912462

Order 18176.

354 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	M₃₂×Dic₇₁	C₂	1	2	yes	maximal
2	#97×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586420	C₂	1	1	yes	maximal, characteristic
4	#22586419	C₂	1	2	yes	maximal
5	C₇₁×#1375	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556068	C₂²	1	2	yes	—
7	C₂×#14556039	C₄	1	1	yes	characteristic
8	C₂×#14556038	C₄	1	1	yes	characteristic
9	C₂×#14556038	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₁	C₄	1	1	yes	characteristic
13	#17985774	C₄	1	4	yes	complemented
14	C₁₆×Dic₇₁	C₄	1	4	yes	complemented
15	C₂×M₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×Dic₇₁	—	2	2	no	—
20	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
22	#14556068	—	2	2	no	—
23	#14556068	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
25	#14556039	C₈	1	4	yes	—
26	#14556038	—	2	1	no	—
27	#14556038	—	2	1	no	—
28	#14556038	C₈	1	4	yes	—
29	C₈×Dic₇₁	—	2	2	no	—
30	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
31	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	M₃₂×C₇₁	—	2	2	no	—
36	M₃₂×C₇₁	—	2	2	no	—
37	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
38	#97×C₇₁	—	2	2	no	—
39	C₂²×Dic₇₁	—	2	1	no	—
40	C₂²×Dic₇₁	M₁₆	1	2	yes	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
44	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
45	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
46	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
47	C₄×Dic₇₁	—	2	1	no	—
48	C₄×Dic₇₁	—	4	4	no	—
49	C₄×Dic₇₁	C₄²	1	1	yes	characteristic
50	C₄×Dic₇₁	C₄²	1	1	yes	characteristic
51	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
52	C₄×Dic₇₁	M₁₆	1	2	yes	—
53	C₇₁⋊C₁₆	—	2	4	no	—
54	C₂²×C₄×C₇₁	—	2	1	no	—
55	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₇₁	—	2	1	no	—
57	C₂×C₈×C₇₁	—	2	1	no	—
58	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₇₁	#41	1	2	yes	—
61	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
63	C₄²×C₇₁	—	2	1	no	—
64	C₄²×C₇₁	C₂×C₈	1	2	yes	—
65	C₁₆×C₇₁	—	2	4	no	—
66	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	—	4	2	no	—
69	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂×Dic₇₁	—	2	4	no	—
76	C₂×Dic₇₁	—	2	2	no	—
77	C₂×Dic₇₁	—	4	2	no	—
78	C₂×Dic₇₁	#97	1	2	yes	—
79	C₂×Dic₇₁	#97	1	2	yes	—
80	C₈×C₇₁	—	2	2	no	—
81	C₈×C₇₁	#117	1	2	yes	—
82	C₈×C₇₁	#117	1	2	yes	—
83	C₇₁⋊C₈	—	2	2	no	—
84	C₇₁⋊C₈	C₄×C₈	1	4	yes	—
85	C₂²×C₇₁	—	2	2	no	—
86	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
87	C₂²×C₇₁	#431	1	1	yes	characteristic
88	C₂²×C₇₁	#431	1	1	yes	characteristic
89	C₄×C₇₁	—	4	2	no	—
90	C₄×C₇₁	#376	1	1	yes	characteristic
91	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
92	C₄×C₇₁	#431	1	1	yes	characteristic
93	C₄×C₇₁	#431	1	1	yes	characteristic
94	Dic₇₁	—	2	2	no	—
95	Dic₇₁	—	4	2	no	—
96	Dic₇₁	#347	1	4	yes	—
97	#12594	—	71	1	no	maximal, Sylow
98	C₂×C₇₁	—	2	1	no	—
99	C₂×C₇₁	—	2	1	no	—
100	C₂×C₇₁	#1744	1	1	yes	characteristic
101	C₂×C₇₁	C₂×#347	1	2	yes	—
102	C₄×M₃₂	—	71	2	no	—
103	C₄×#97	—	71	1	no	—
104	#1375	—	71	2	no	—
105	#1375	—	71	1	no	—
106	#1375	D₇₁	1	1	yes	characteristic
107	C₇₁	#12594	1	1	yes	characteristic, complemented, Sylow
108	C₂×M₃₂	—	71	2	no	—
109	C₂×M₃₂	C₂×D₇₁	1	2	yes	—
110	C₂×#97	—	71	1	no	—
111	C₂×#97	—	71	1	no	—
112	C₂×#97	—	71	1	no	—
113	C₂×#97	C₂×D₇₁	1	1	yes	characteristic
114	C₂×C₄×C₈	—	71	1	no	—
115	C₂×C₄×C₈	—	71	1	no	—
116	C₄³	—	71	1	no	—
117	#346	—	71	4	no	—
118	C₄×C₁₆	—	71	4	no	—
119	C₂²×C₈	—	71	1	no	—
120	C₂²×C₈	—	71	1	no	—
121	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
122	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
123	C₂×C₄²	—	71	1	no	—
124	C₂×C₄²	—	71	1	no	—
125	C₂×C₄²	—	142	2	no	—
126	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
127	M₃₂	—	2	2	no	—
128	M₃₂	—	2	2	no	—
129	M₃₂	—	142	2	no	—
130	M₃₂	—	142	2	no	—
131	C₂×C₁₆	—	71	4	no	—
132	C₂×C₁₆	C₄×D₇₁	1	4	yes	—
133	#97	—	2	2	no	—
134	#97	—	71	4	no	—
135	#97	—	71	4	no	—
136	#97	—	142	1	no	—
137	#97	—	142	1	no	—
138	C₄×C₈	—	71	2	no	—
139	C₄×C₈	—	71	2	no	—
140	C₄×C₈	—	142	2	no	—
141	C₂²×C₄	—	2	1	no	—
142	C₂²×C₄	—	71	2	no	—
143	C₂²×C₄	—	142	1	no	—
144	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	71	2	no	—
148	C₂×C₈	—	71	2	no	—
149	C₂×C₈	—	71	2	no	—
150	C₂×C₈	—	71	2	no	—
151	C₂×C₈	—	142	1	no	—
152	C₂×C₈	—	142	1	no	—
153	C₂×C₈	C₈×D₇₁	1	2	yes	—
154	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
155	C₂×C₈	#11765864	1	1	yes	characteristic
156	C₂×C₈	#11765864	1	2	yes	—
157	C₂×C₈	#11765866	1	1	yes	characteristic
158	C₄²	—	2	1	no	—
159	C₄²	—	284	4	no	—
160	C₄²	—	71	1	no	—
161	C₄²	—	71	1	no	—
162	C₄²	—	71	2	no	—
163	C₄²	—	71	2	no	—
164	C₄²	—	142	1	no	—
165	C₄²	C₈×D₇₁	1	2	yes	—
166	C₁₆	—	2	4	no	—
167	C₁₆	—	142	4	no	—
168	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	284	2	no	—
172	C₂×C₄	—	71	2	no	—
173	C₂×C₄	—	71	2	no	—
174	C₂×C₄	—	142	2	no	—
175	C₂×C₄	—	142	4	no	—
176	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
177	C₂×C₄	#14556084	1	1	yes	characteristic
178	C₂×C₄	#14556086	1	1	yes	characteristic
179	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
180	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
182	C₈	—	2	2	no	—
183	C₈	—	71	4	no	—
184	C₈	—	142	2	no	—
185	C₈	#14556084	1	2	yes	—
186	C₈	#14556086	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#97×D₇₁	1	1	yes	characteristic
189	C₂²	#17986049	1	1	yes	characteristic
190	C₂²	#17986051	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	—	284	2	no	—
193	C₄	—	71	4	no	—
194	C₄	—	142	2	no	—
195	C₄	#17985844	1	1	yes	characteristic
196	C₄	C₄×C₈×D₇₁	1	1	yes	characteristic
197	C₄	#17986049	1	1	yes	characteristic
198	C₄	#17986051	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#22588084	1	1	yes	characteristic
202	C₂	#347×D₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1375	C₂
socle	C₂²×C₇₁	C₂×#97
prime radical	C₂³×C₇₁	C₂×M₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1375	D₇₁
hypocenter	C₇₁	#12594
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #1375

Complements

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

N	H ≅ G/N	type
13. #17985774	C₄	semidirect
14. C₁₆×Dic₇₁	C₄	semidirect
107. C₇₁	#12594	semidirect

Maximal subgroups

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

subgroup	index
M₃₂×Dic₇₁	2
#97×Dic₇₁	2
#22586420	2
#22586419	2
C₇₁×#1375	2
#12594	71