Subgroups — #31906497

Order 18176.

250 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985775	C₂	1	1	yes	maximal, characteristic
2	#22586629	C₂	1	1	yes	maximal, characteristic
3	#22586262	C₂	1	2	yes	maximal, complemented
4	#22586260	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₄	1	2	yes	—
7	#17986052	C₄	1	1	yes	characteristic
8	C₄×#11765849	C₄	1	1	yes	characteristic
9	#17985775	C₄	1	2	yes	—
10	#17985775	C₂²	1	1	yes	characteristic
11	#17985775	C₂²	1	1	yes	characteristic
12	#17985721	C₄	1	2	yes	—
13	#17985720	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₇₁	C₄	1	2	yes	—
16	C₂²⋊C₁₆×C₇₁	C₄	1	2	yes	—
17	#346×C₇₁	C₂²	1	2	yes	—
18	C₂×#11765849	—	2	1	no	—
19	C₂×C₇₁⋊C₁₆	—	2	4	no	—
20	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
23	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
24	#14556039	—	2	1	no	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₇₁	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
35	#11765849	—	4	2	no	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	2	no	—
38	C₇₁⋊C₁₆	—	2	2	no	—
39	C₇₁⋊C₁₆	M₁₆	1	4	yes	complemented
40	C₂²×C₄×C₇₁	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₇₁	—	2	2	no	—
44	C₂×C₈×C₇₁	—	2	2	no	—
45	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₇₁	C₄²	1	1	yes	characteristic
50	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
51	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
52	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₇₁	—	2	4	no	—
54	C₂³×C₇₁	#97	1	1	yes	characteristic
55	C₂×C₄×C₇₁	—	2	2	no	—
56	C₂×C₄×C₇₁	—	2	1	no	—
57	C₂×C₄×C₇₁	—	2	1	no	—
58	C₂×C₄×C₇₁	—	4	1	no	—
59	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
61	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
62	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₇₁	—	2	1	no	—
66	C₈×C₇₁	—	2	1	no	—
67	C₈×C₇₁	C₂×M₁₆	1	2	yes	—
68	C₈×C₇₁	#131	1	2	yes	—
69	C₇₁⋊C₈	—	4	2	no	—
70	C₇₁⋊C₈	—	4	2	no	—
71	C₂²×C₇₁	—	2	1	no	—
72	C₂²×C₇₁	—	2	1	no	—
73	C₂²×C₇₁	#347	1	2	yes	—
74	C₂²×C₇₁	C₂×#97	1	1	yes	characteristic
75	C₄×C₇₁	—	2	1	no	—
76	C₄×C₇₁	—	2	1	no	—
77	C₄×C₇₁	—	2	1	no	—
78	C₄×C₇₁	—	2	1	no	—
79	C₄×C₇₁	—	2	1	no	—
80	C₄×C₇₁	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₇₁	#405	1	1	yes	characteristic
82	#11413	—	71	1	no	maximal, Sylow
83	C₂×C₇₁	—	2	2	no	—
84	C₂×C₇₁	#1451	1	1	yes	characteristic
85	C₂×C₇₁	C₂×#347	1	1	yes	characteristic
86	C₂×C₇₁	#2110	1	1	yes	characteristic
87	#2163	D₇₁	1	1	yes	characteristic
88	C₂×#346	—	71	1	no	—
89	#1451	—	71	1	no	—
90	#1313	—	71	2	no	—
91	#1313	—	71	2	no	—
92	C₇₁	#11413	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	71	2	no	—
94	#431	—	71	1	no	—
95	C₄×M₁₆	—	71	1	no	—
96	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
97	C₄⋊C₁₆	Dic₇₁	1	2	yes	—
98	C₂²⋊C₁₆	Dic₇₁	1	2	yes	—
99	#346	—	71	2	no	—
100	#346	—	71	1	no	—
101	#346	—	71	1	no	—
102	#346	C₂×D₇₁	1	2	yes	—
103	#322	—	71	2	no	—
104	#322	—	71	2	no	—
105	C₂×M₁₆	—	142	1	no	—
106	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
109	C₂×C₁₆	—	71	2	no	—
110	C₂×C₁₆	—	71	2	no	—
111	C₂×C₁₆	—	142	4	no	—
112	C₂×C₁₆	C₂×Dic₇₁	1	4	yes	—
113	C₄⋊C₈	—	142	1	no	—
114	C₂²⋊C₈	—	142	1	no	—
115	#97	—	142	1	no	—
116	C₄×C₈	—	71	2	no	—
117	C₄×C₈	—	71	2	no	—
118	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
119	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
122	C₂²×C₄	C₄×Dic₇₁	1	1	yes	characteristic
123	C₂²×C₄	#11765848	1	1	yes	characteristic
124	C₂²×C₄	#11765848	1	1	yes	characteristic
125	M₁₆	—	284	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	142	2	no	—
129	C₂×C₈	—	142	2	no	—
130	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₇₁	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
134	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
135	C₄²	#11765848	1	1	yes	characteristic
136	C₄²	#11765848	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₁₆	—	71	4	no	—
140	C₁₆	—	142	2	no	—
141	C₂³	#14556038	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#14556038	1	1	yes	characteristic
147	C₂×C₄	#14556038	1	1	yes	characteristic
148	C₂×C₄	#14556038	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
150	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
151	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	284	2	no	—
155	C₈	—	284	2	no	—
156	C₈	M₁₆×D₇₁	1	2	yes	—
157	C₈	#14556135	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#17985776	1	2	yes	—
161	C₂²	C₂×#14556038	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	M₁₆×Dic₇₁	1	1	yes	characteristic
168	C₄	#17985955	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#22586630	1	1	yes	characteristic
171	C₂	C₂×#17985776	1	1	yes	characteristic
172	C₂	#22589944	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#2163	C₂
socle	C₂²×C₇₁	C₂×#97
prime radical	C₂³×C₇₁	#97
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#2163	D₇₁
hypocenter	C₇₁	#11413
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #22586262	C₂	semidirect
4. #22586260	C₂	semidirect
39. C₇₁⋊C₁₆	M₁₆	semidirect
92. C₇₁	#11413	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985775	2
#22586629	2
#22586262	2
#22586260	2
C₇₁×#2163	2
#11413	71