Subgroups — #31912762

Order 18176.

414 conjugacy classes of subgroups.

The subgroup lattice has 218 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	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#22586417	C₂	1	2	yes	maximal
5	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
12	C₄²×Dic₇₁	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	#346×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	C₈	1	2	yes	—
21	C₂×C₄×Dic₇₁	C₈	1	2	yes	—
22	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₇₁⋊C₁₆	—	2	2	no	—
25	C₂×C₇₁⋊C₁₆	—	2	2	no	—
26	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
27	C₄×C₇₁⋊C₈	—	2	2	no	—
28	C₈×Dic₇₁	—	2	2	no	—
29	C₈×Dic₇₁	—	2	2	no	—
30	C₈×Dic₇₁	C₈	1	4	yes	—
31	C₈×Dic₇₁	C₂×C₄	1	4	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	C₂×C₁₆×C₇₁	—	2	2	no	—
36	C₂×C₁₆×C₇₁	—	2	2	no	—
37	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₇₁	D₄	1	2	yes	—
39	C₄×C₈×C₇₁	Q₈	1	2	yes	—
40	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₇₁	C₂×C₈	1	2	yes	—
42	C₂²×Dic₇₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₇₁⋊C₈	—	2	1	no	—
44	C₂×C₇₁⋊C₈	—	2	1	no	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	2	1	no	—
47	C₂×C₇₁⋊C₈	C₄²	1	2	yes	—
48	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
49	C₄×Dic₇₁	—	2	1	no	—
50	C₄×Dic₇₁	—	2	1	no	—
51	C₄×Dic₇₁	—	2	4	no	—
52	C₄×Dic₇₁	—	2	4	no	—
53	C₄×Dic₇₁	C₄²	1	2	yes	—
54	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
55	C₇₁⋊C₁₆	—	2	4	no	—
56	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
57	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
61	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₈×C₇₁	#41	1	2	yes	—
65	C₄²×C₇₁	C₄⋊C₄	1	2	yes	—
66	C₄²×C₇₁	M₁₆	1	2	yes	—
67	C₁₆×C₇₁	—	2	4	no	—
68	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	—	2	2	no	—
70	C₂×C₄×C₇₁	—	2	2	no	—
71	C₂×C₄×C₇₁	C₄⋊C₈	1	4	yes	—
72	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
75	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
78	C₂×Dic₇₁	—	2	2	no	—
79	C₂×Dic₇₁	—	2	2	no	—
80	C₂×Dic₇₁	—	2	2	no	—
81	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
82	C₂×Dic₇₁	C₄×C₈	1	2	yes	—
83	C₂×Dic₇₁	M₃₂	1	4	yes	—
84	C₂×Dic₇₁	M₃₂	1	4	yes	—
85	C₈×C₇₁	C₄⋊C₈	1	4	yes	—
86	C₈×C₇₁	C₄×D₄	1	2	yes	—
87	C₈×C₇₁	C₄×Q₈	1	2	yes	—
88	C₇₁⋊C₈	—	2	2	no	—
89	C₇₁⋊C₈	—	2	2	no	—
90	C₂²×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
91	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
92	C₂²×C₇₁	#431	1	1	yes	characteristic
93	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
94	C₂²×C₇₁	C₂×M₃₂	1	2	yes	—
95	C₄×C₇₁	—	2	4	no	—
96	C₄×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
98	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄×C₇₁	#431	1	1	yes	characteristic
100	Dic₇₁	—	2	2	no	—
101	Dic₇₁	—	2	4	no	—
102	Dic₇₁	#346	1	4	yes	complemented
103	#12653	—	71	1	no	maximal, Sylow
104	C₂×C₇₁	C₄×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₇₁	C₂×#346	1	1	yes	characteristic
106	C₂×C₇₁	C₄×M₃₂	1	1	yes	characteristic
107	C₂×C₇₁	#2150	1	2	yes	—
108	C₂×C₇₁	#2163	1	2	yes	—
109	C₂×C₄⋊C₁₆	—	71	2	no	—
110	C₂×#346	—	71	1	no	—
111	C₂×#346	D₇₁	1	1	yes	characteristic
112	C₄²×C₈	—	71	1	no	—
113	#1374	—	71	2	no	—
114	C₇₁	#12653	1	1	yes	characteristic, complemented, Sylow
115	C₂²×C₁₆	—	71	2	no	—
116	C₂²×C₁₆	C₂×D₇₁	1	2	yes	—
117	C₂×C₄×C₈	—	71	1	no	—
118	C₂×C₄×C₈	—	71	1	no	—
119	C₂×C₄×C₈	—	71	1	no	—
120	C₂×C₄×C₈	—	71	1	no	—
121	C₂×C₄×C₈	—	71	1	no	—
122	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
123	C₄³	—	71	1	no	—
124	C₄⋊C₁₆	—	71	8	no	—
125	#346	—	2	2	no	—
126	#346	—	142	2	no	—
127	C₂²×C₈	—	71	1	no	—
128	C₂²×C₈	—	71	1	no	—
129	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
131	C₂×C₄²	—	71	1	no	—
132	C₂×C₄²	—	71	2	no	—
133	C₂×C₄²	—	71	2	no	—
134	C₂×C₄²	—	71	1	no	—
135	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	71	4	no	—
139	C₂×C₁₆	—	142	2	no	—
140	C₂×C₁₆	—	142	2	no	—
141	C₂×C₁₆	C₄×D₇₁	1	4	yes	—
142	C₄×C₈	—	71	4	no	—
143	C₄×C₈	—	71	4	no	—
144	C₄×C₈	—	142	2	no	—
145	C₄×C₈	—	142	2	no	—
146	C₄×C₈	—	142	2	no	—
147	C₄×C₈	Dic₁₄₂	1	2	yes	—
148	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
149	C₂²×C₄	—	71	1	no	—
150	C₂²×C₄	—	71	2	no	—
151	C₂²×C₄	—	71	1	no	—
152	C₂²×C₄	C₈×D₇₁	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
154	C₂×C₈	—	71	2	no	—
155	C₂×C₈	—	71	2	no	—
156	C₂×C₈	—	142	1	no	—
157	C₂×C₈	—	142	1	no	—
158	C₂×C₈	—	142	1	no	—
159	C₂×C₈	—	142	1	no	—
160	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
161	C₂×C₈	C₈×D₇₁	1	2	yes	—
162	C₂×C₈	#11765848	1	2	yes	—
163	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
164	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
165	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
166	C₂×C₈	#11765865	1	2	yes	—
167	C₄²	—	71	2	no	—
168	C₄²	—	71	2	no	—
169	C₄²	—	142	4	no	—
170	C₄²	—	142	4	no	—
171	C₄²	—	142	1	no	—
172	C₄²	—	142	1	no	—
173	C₄²	Dic₇₁⋊C₄	1	2	yes	—
174	C₄²	#11765848	1	2	yes	—
175	C₁₆	—	2	4	no	—
176	C₁₆	—	142	4	no	—
177	C₂³	C₂×C₈×D₇₁	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	71	4	no	—
181	C₂×C₄	—	71	2	no	—
182	C₂×C₄	—	71	4	no	—
183	C₂×C₄	—	71	2	no	—
184	C₂×C₄	—	142	2	no	—
185	C₂×C₄	—	142	2	no	—
186	C₂×C₄	—	142	2	no	—
187	C₂×C₄	Dic₇₁⋊C₈	1	4	yes	—
188	C₂×C₄	C₄²×D₇₁	1	1	yes	characteristic
189	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
190	C₂×C₄	#14556085	1	1	yes	characteristic
191	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
192	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
193	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
194	C₈	—	142	2	no	—
195	C₈	—	142	2	no	—
196	C₈	Dic₇₁⋊C₈	1	4	yes	—
197	C₈	C₄×C₂²⋊D₇₁	1	2	yes	—
198	C₈	C₄×Dic₁₄₂	1	2	yes	—
199	C₂²	C₄×C₈×D₇₁	1	1	yes	characteristic
200	C₂²	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
201	C₂²	#17986050	1	1	yes	characteristic
202	C₂²	M₃₂×D₇₁	1	2	yes	—
203	C₂²	M₃₂×D₇₁	1	2	yes	—
204	C₄	—	2	4	no	—
205	C₄	—	71	4	no	—
206	C₄	—	142	4	no	—
207	C₄	—	142	2	no	—
208	C₄	C₄×Dic₇₁⋊C₄	1	1	yes	characteristic
209	C₄	C₄×#11765848	1	1	yes	characteristic
210	C₄	C₂×Dic₇₁⋊C₈	1	1	yes	characteristic
211	C₄	#17986050	1	1	yes	characteristic
212	C₂	C₄×Dic₇₁⋊C₈	1	1	yes	characteristic
213	C₂	#346×D₇₁	1	1	yes	characteristic
214	C₂	#22589933	1	1	yes	characteristic
215	C₂	#22590095	1	2	yes	—
216	C₂	#22590141	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄×C₈
Fitting	C₂×#346×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²×C₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	#12653
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. Dic₇₁⋊C₁₆	C₄	semidirect
102. Dic₇₁	#346	semidirect
114. C₇₁	#12653	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
C₄×C₈×Dic₇₁	2
#22586417	2
C₂×#346×C₇₁	2
#12653	71