Subgroups — #31912766

Order 18176.

382 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₁⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
3	#22588026	C₂	1	1	yes	maximal, characteristic
4	#22586417	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	C₂×#14556039	C₄	1	1	yes	characteristic
8	C₂×#14556038	C₄	1	1	yes	characteristic
9	C₂×#14556038	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	#17985845	C₄	1	1	yes	characteristic
13	Dic₇₁⋊C₁₆	C₄	1	8	yes	complemented
14	#17985775	—	2	2	no	—
15	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₄×C₁₆×C₇₁	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇₁⋊C₁₆	—	2	2	no	—
23	C₂×C₇₁⋊C₁₆	—	2	2	no	—
24	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
25	#14556039	—	2	2	no	—
26	#14556038	—	2	2	no	—
27	#14556038	C₈	1	4	yes	—
28	C₈×Dic₇₁	—	2	2	no	—
29	C₈×Dic₇₁	C₂×C₄	1	4	yes	—
30	C₇₁⋊#95	C₈	1	4	yes	—
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₇₁	—	2	2	no	—
35	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₇₁	D₄	1	2	yes	—
38	C₄×C₈×C₇₁	Q₈	1	2	yes	—
39	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
40	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
41	C₂×C₇₁⋊C₈	—	2	1	no	—
42	C₂×C₇₁⋊C₈	—	2	1	no	—
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	2	1	no	—
45	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
46	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
47	C₄×Dic₇₁	—	2	1	no	—
48	C₄×Dic₇₁	—	2	1	no	—
49	C₄×Dic₇₁	C₄²	1	2	yes	—
50	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
51	C₇₁⋊C₁₆	—	2	4	no	—
52	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
56	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
57	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₇₁	#41	1	2	yes	—
61	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
62	C₄²×C₇₁	M₁₆	1	2	yes	—
63	C₁₆×C₇₁	—	2	4	no	—
64	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₇₁	—	2	4	no	—
66	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
67	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
73	C₂×Dic₇₁	—	2	4	no	—
74	C₂×Dic₇₁	—	2	2	no	—
75	C₂×Dic₇₁	—	2	4	no	—
76	C₂×Dic₇₁	C₄×C₈	1	4	yes	—
77	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
78	C₈×C₇₁	C₄×D₄	1	2	yes	—
79	C₈×C₇₁	C₄×Q₈	1	2	yes	—
80	C₇₁⋊C₈	—	2	2	no	—
81	C₇₁⋊C₈	—	2	2	no	—
82	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
83	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₇₁	#431	1	1	yes	characteristic
85	C₂²×C₇₁	#504	1	4	yes	—
86	C₄×C₇₁	—	2	4	no	—
87	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
89	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
90	C₄×C₇₁	#431	1	1	yes	characteristic
91	Dic₇₁	—	2	4	no	—
92	Dic₇₁	—	4	2	no	—
93	#12654	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
95	C₂×C₇₁	#2108	1	2	yes	—
96	C₂×C₇₁	#2151	1	2	yes	—
97	C₂×C₇₁	#2164	1	2	yes	—
98	C₂×C₄⋊C₁₆	—	71	2	no	—
99	C₂×#346	—	71	1	no	—
100	C₂×C₄×C₁₆	D₇₁	1	1	yes	characteristic
101	#1726	—	71	1	no	—
102	#1374	—	71	2	no	—
103	C₇₁	#12654	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₁₆	—	71	2	no	—
105	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
106	C₂×#97	—	71	1	no	—
107	C₂×#97	—	71	1	no	—
108	C₂×#97	—	71	1	no	—
109	C₂×C₄×C₈	—	71	1	no	—
110	C₂×C₄×C₈	—	71	1	no	—
111	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
112	#376	—	71	1	no	—
113	C₄⋊C₁₆	—	71	8	no	—
114	#346	—	142	2	no	—
115	C₄×C₁₆	—	2	2	no	—
116	C₂²×C₈	—	71	1	no	—
117	C₂²×C₈	—	71	1	no	—
118	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
120	C₂×C₄²	—	71	1	no	—
121	C₂×C₄²	—	71	1	no	—
122	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	C₂×C₁₆	—	2	2	no	—
125	C₂×C₁₆	—	71	4	no	—
126	C₂×C₁₆	—	142	2	no	—
127	C₂×C₁₆	—	142	2	no	—
128	C₂×C₁₆	C₄×D₇₁	1	4	yes	—
129	#97	—	71	4	no	—
130	#97	—	142	2	no	—
131	#97	—	142	2	no	—
132	C₄×C₈	—	71	4	no	—
133	C₄×C₈	—	142	2	no	—
134	C₄×C₈	Dic₁₄₂	1	2	yes	—
135	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
136	#95	—	71	4	no	—
137	C₂²×C₄	—	71	2	no	—
138	C₂²×C₄	—	71	2	no	—
139	C₂²×C₄	C₈×D₇₁	1	2	yes	—
140	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
141	C₂×C₈	—	71	2	no	—
142	C₂×C₈	—	71	2	no	—
143	C₂×C₈	—	142	1	no	—
144	C₂×C₈	—	142	1	no	—
145	C₂×C₈	—	142	1	no	—
146	C₂×C₈	—	142	1	no	—
147	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
148	C₂×C₈	C₈×D₇₁	1	2	yes	—
149	C₂×C₈	#11765848	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
151	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
152	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
153	C₂×C₈	#11765865	1	2	yes	—
154	C₄²	—	71	2	no	—
155	C₄²	—	71	2	no	—
156	C₄²	—	142	1	no	—
157	C₄²	—	142	1	no	—
158	C₄²	Dic₇₁⋊C₄	1	2	yes	—
159	C₄²	#11765848	1	2	yes	—
160	C₁₆	—	2	4	no	—
161	C₁₆	—	142	4	no	—
162	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	71	4	no	—
165	C₂×C₄	—	142	2	no	—
166	C₂×C₄	—	142	4	no	—
167	C₂×C₄	—	142	4	no	—
168	C₂×C₄	Dic₇₁⋊C₈	1	4	yes	—
169	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
171	C₂×C₄	#14556085	1	1	yes	characteristic
172	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
173	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
174	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
175	C₈	—	142	2	no	—
176	C₈	—	142	2	no	—
177	C₈	Dic₇₁⋊C₈	1	4	yes	—
178	C₈	C₄×C₂²⋊D₇₁	1	2	yes	—
179	C₈	C₄×Dic₁₄₂	1	2	yes	—
180	C₂²	C₄×C₈×D₇₁	1	1	yes	characteristic
181	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
182	C₂²	#17986050	1	1	yes	characteristic
183	C₂²	#17986415	1	4	yes	—
184	C₄	—	2	4	no	—
185	C₄	—	284	2	no	—
186	C₄	—	142	4	no	—
187	C₄	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
188	C₄	C₄×#11765848	1	1	yes	characteristic
189	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
190	C₄	#17986050	1	1	yes	characteristic
191	C₂	C₄×Dic₇₁⋊C₈	1	1	yes	characteristic
192	C₂	#22589935	1	2	yes	—
193	C₂	#22590098	1	2	yes	—
194	C₂	#22590145	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₂×C₄×C₁₆×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×C₄×C₁₆	D₇₁
hypocenter	C₇₁	#12654
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
13. Dic₇₁⋊C₁₆	C₄	semidirect
103. C₇₁	#12654	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₇₁⋊C₁₆	2
C₂×#17985775	2
#22588026	2
#22586417	2
C₂×C₄×C₁₆×C₇₁	2
#12654	71