Subgroups — #31888616

Order 18176.

390 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	#22587088	C₂	1	1	yes	maximal, characteristic
3	#22586633	C₂	1	2	yes	maximal
4	#22586172	C₂	1	2	yes	maximal
5	C₇₁×#1564	C₂	1	1	yes	maximal, characteristic
6	#17986052	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
9	C₂×C₈×Dic₇₁	C₄	1	2	yes	—
10	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
11	#17985752	C₄	1	2	yes	complemented
12	#17985750	C₄	1	2	yes	complemented
13	#17985721	—	2	2	no	—
14	#17985720	—	2	2	no	—
15	#431×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#335×C₇₁	C₄	1	2	yes	complemented
18	#334×C₇₁	C₄	1	2	yes	complemented
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₇₁	—	2	1	no	—
21	C₂×C₄×Dic₇₁	—	2	1	no	—
22	C₂×C₄×Dic₇₁	C₂×C₄	1	2	yes	—
23	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
24	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₄×C₇₁⋊C₈	—	2	2	no	—
26	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
27	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
28	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
29	C₈×Dic₇₁	—	2	4	no	—
30	C₈×Dic₇₁	—	2	4	no	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₇₁	C₂×C₄	1	4	yes	—
35	C₂²⋊C₈×C₇₁	—	2	2	no	—
36	C₄×C₈×C₇₁	—	2	2	no	—
37	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₇₁	—	2	2	no	—
42	C₂×C₇₁⋊C₈	—	2	2	no	—
43	C₂×C₇₁⋊C₈	—	2	2	no	—
44	C₂×C₇₁⋊C₈	—	2	4	no	—
45	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₇₁	—	2	4	no	—
48	C₄×Dic₇₁	—	2	4	no	—
49	C₄×Dic₇₁	—	2	2	no	—
50	C₄×Dic₇₁	M₁₆	1	4	yes	—
51	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×C₈×C₇₁	—	2	2	no	—
55	C₂×C₈×C₇₁	—	2	4	no	—
56	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
57	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
60	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
61	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	C₂³×C₇₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₇₁	—	2	2	no	—
67	C₂×C₄×C₇₁	—	2	2	no	—
68	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
76	C₂×Dic₇₁	—	2	4	no	—
77	C₂×Dic₇₁	—	2	4	no	—
78	C₂×Dic₇₁	—	2	4	no	—
79	C₈×C₇₁	—	2	1	no	—
80	C₈×C₇₁	—	2	1	no	—
81	C₈×C₇₁	—	4	4	no	—
82	C₈×C₇₁	#97	1	2	yes	—
83	C₈×C₇₁	#97	1	2	yes	—
84	C₇₁⋊C₈	—	2	2	no	—
85	C₇₁⋊C₈	—	2	2	no	—
86	C₇₁⋊C₈	—	4	4	no	—
87	C₂²×C₇₁	—	2	1	no	—
88	C₂²×C₇₁	—	2	1	no	—
89	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
90	C₂²×C₇₁	#432	1	2	yes	—
91	C₄×C₇₁	—	2	1	no	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
94	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
95	C₄×C₇₁	#423	1	2	yes	—
96	C₄×C₇₁	#443	1	2	yes	—
97	C₄×C₇₁	#443	1	2	yes	—
98	Dic₇₁	—	2	8	no	—
99	#8267	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	—	2	2	no	—
101	C₂×C₇₁	#1767	1	1	yes	characteristic
102	C₂×C₇₁	#1772	1	1	yes	characteristic
103	C₂×C₇₁	#1772	1	1	yes	characteristic
104	C₄²×C₈	—	71	1	no	—
105	#1564	—	71	1	no	—
106	#1564	D₇₁	1	1	yes	characteristic
107	#1452	—	71	2	no	—
108	#1276	—	71	2	no	—
109	C₇₁	#8267	1	1	yes	characteristic, complemented, Sylow
110	#431	—	71	2	no	—
111	#431	C₂×D₇₁	1	2	yes	—
112	C₂×C₄×C₈	—	71	1	no	—
113	C₂×C₄×C₈	—	71	2	no	—
114	C₂×C₄×C₈	—	71	2	no	—
115	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
116	C₄³	—	71	1	no	—
117	#335	—	71	2	no	—
118	#335	Dic₇₁	1	2	yes	complemented
119	#334	—	71	2	no	—
120	#334	Dic₇₁	1	2	yes	complemented
121	#322	—	142	2	no	—
122	#322	—	142	2	no	—
123	C₂²×C₈	—	71	2	no	—
124	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
125	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
126	C₂×C₄²	—	71	2	no	—
127	C₂×C₄²	—	142	1	no	—
128	C₂×C₄²	—	142	1	no	—
129	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
130	C₄⋊C₈	—	71	4	no	—
131	C₄⋊C₈	C₂×Dic₇₁	1	4	yes	—
132	C₂²⋊C₈	—	2	2	no	—
133	C₂²⋊C₈	—	142	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	—	71	1	no	—
136	C₄×C₈	—	71	1	no	—
137	C₄×C₈	—	71	2	no	—
138	C₄×C₈	—	142	4	no	—
139	C₄×C₈	—	142	4	no	—
140	C₄×C₈	—	142	2	no	—
141	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
142	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
143	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
144	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
145	C₂²×C₄	—	142	2	no	—
146	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
147	C₂²×C₄	#11765864	1	1	yes	characteristic
148	C₂²×C₄	#11765866	1	1	yes	characteristic
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	71	2	no	—
152	C₂×C₈	—	71	2	no	—
153	C₂×C₈	—	142	4	no	—
154	C₂×C₈	—	142	2	no	—
155	C₂×C₈	—	142	2	no	—
156	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
157	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
158	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
159	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
160	C₂×C₈	#11765848	1	4	yes	—
161	C₄²	—	71	4	no	—
162	C₄²	—	142	4	no	—
163	C₄²	—	142	4	no	—
164	C₄²	—	142	2	no	—
165	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
166	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
167	C₄²	D₄×D₇₁	1	1	yes	characteristic
168	C₄²	Q₈×D₇₁	1	1	yes	characteristic
169	C₂³	#14556086	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	142	4	no	—
173	C₂×C₄	—	142	4	no	—
174	C₂×C₄	—	142	4	no	—
175	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
176	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
177	C₂×C₄	#14556090	1	1	yes	characteristic
178	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
179	C₂×C₄	#14556093	1	1	yes	characteristic
180	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
181	C₂×C₄	M₁₆×D₇₁	1	2	yes	—
182	C₂×C₄	C₂×#11765848	1	2	yes	—
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	4	4	no	—
186	C₈	—	284	4	no	—
187	C₈	—	142	2	no	—
188	C₈	—	142	2	no	—
189	C₈	#14556038	1	2	yes	—
190	C₈	#14556038	1	2	yes	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
194	C₂²	#17986055	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	142	8	no	—
198	C₄	C₂×#14556038	1	1	yes	characteristic
199	C₄	M₁₆×Dic₇₁	1	1	yes	characteristic
200	C₄	#17986021	1	2	yes	—
201	C₄	#17986115	1	2	yes	—
202	C₄	#17986116	1	2	yes	—
203	C₂	—	2	2	no	—
204	C₂	#22588195	1	1	yes	characteristic
205	C₂	#22588221	1	1	yes	characteristic
206	C₂	#22588222	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1564	C₂
socle	C₂²×C₇₁	C₄×C₄⋊C₄
prime radical	C₂³×C₇₁	#117
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1564	D₇₁
hypocenter	C₇₁	#8267
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
11. #17985752	C₄	semidirect
12. #17985750	C₄	semidirect
17. #335×C₇₁	C₄	semidirect
18. #334×C₇₁	C₄	semidirect
109. C₇₁	#8267	semidirect
118. #335	Dic₇₁	semidirect
120. #334	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₈×Dic₇₁	2
#22587088	2
#22586633	2
#22586172	2
C₇₁×#1564	2
#8267	71