Subgroups — #31884364

Order 18176.

262 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#22588207	C₂	1	1	yes	maximal, characteristic
3	C₂×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₁₆	—	2	1	no	—
5	C₂×C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
6	C₂×#14556038	—	2	1	no	—
7	C₄×C₄⋊Dic₇₁	C₄	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₁₆)	C₄	1	2	yes	complemented
10	#17985753	—	2	1	no	—
11	C₂²×C₁₆×C₇₁	—	2	1	no	—
12	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#346×C₇₁	—	2	1	no	—
14	#346×C₇₁	C₄	1	2	yes	complemented
15	C₂²×C₇₁⋊C₈	—	2	1	no	—
16	C₂×C₄⋊Dic₇₁	C₈	1	2	yes	—
17	C₂×C₄×Dic₇₁	—	2	1	no	—
18	C₂×C₇₁⋊C₁₆	—	2	2	no	—
19	C₂×C₇₁⋊C₁₆	—	4	1	no	—
20	C₂×C₇₁⋊C₁₆	—	4	1	no	—
21	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
22	C₄⋊(C₇₁⋊C₈)	C₈	1	2	yes	—
23	#14556038	—	4	1	no	—
24	#14556038	—	4	1	no	—
25	C₇₁⋊#95	—	2	1	no	—
26	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₁	—	2	2	no	—
30	C₂×C₁₆×C₇₁	—	2	2	no	—
31	C₂×C₁₆×C₇₁	—	2	2	no	—
32	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
36	C₂²×Dic₇₁	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	2	no	—
38	C₂×C₇₁⋊C₈	—	4	1	no	—
39	C₂×C₇₁⋊C₈	—	4	1	no	—
40	C₄⋊Dic₇₁	—	4	2	no	—
41	C₄×Dic₇₁	—	4	1	no	—
42	C₄×Dic₇₁	—	4	1	no	—
43	C₇₁⋊C₁₆	—	4	2	no	—
44	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	M₁₆	1	1	yes	characteristic
47	C₂×C₈×C₇₁	—	2	1	no	—
48	C₂×C₈×C₇₁	—	2	1	no	—
49	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
54	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
57	C₄²×C₇₁	C₄²	1	1	yes	characteristic
58	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
59	C₄²×C₇₁	C₂×C₈	1	1	yes	characteristic
60	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
61	C₁₆×C₇₁	—	2	4	no	—
62	C₂³×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	2	2	no	—
65	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#97	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
70	C₂×C₄×C₇₁	#103	1	1	yes	characteristic
71	C₂×C₄×C₇₁	#104	1	1	yes	characteristic
72	C₂×C₄×C₇₁	#104	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
75	C₂×Dic₇₁	—	4	2	no	—
76	C₂×Dic₇₁	—	4	2	no	—
77	C₂×Dic₇₁	—	4	2	no	—
78	C₈×C₇₁	—	2	1	no	—
79	C₈×C₇₁	—	2	1	no	—
80	C₈×C₇₁	#102	1	1	yes	characteristic
81	C₈×C₇₁	#103	1	1	yes	characteristic
82	C₈×C₇₁	#106	1	1	yes	characteristic
83	C₈×C₇₁	#107	1	1	yes	characteristic
84	C₇₁⋊C₈	—	4	2	no	—
85	C₂²×C₇₁	D₄⋊C₈	1	1	yes	characteristic
86	C₂²×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
87	C₂²×C₇₁	#336	1	1	yes	characteristic
88	C₂²×C₇₁	#349	1	2	yes	—
89	C₂²×C₇₁	#350	1	2	yes	—
90	C₄×C₇₁	—	2	1	no	—
91	C₄×C₇₁	—	2	1	no	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	—	2	1	no	—
94	C₄×C₇₁	D₄⋊C₈	1	1	yes	characteristic
95	C₄×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
96	C₄×C₇₁	#339	1	1	yes	characteristic
97	C₄×C₇₁	#340	1	1	yes	characteristic
98	Dic₇₁	—	8	2	no	—
99	#7170	—	71	1	no	maximal, Sylow
100	C₂×C₇₁	#1279	1	1	yes	characteristic
101	C₂×C₇₁	#1330	1	2	yes	—
102	C₂×C₇₁	#1338	1	2	yes	—
103	C₂×C₇₁	#1377	1	1	yes	characteristic
104	C₂×C₇₁	#1379	1	1	yes	characteristic
105	C₂×C₄⋊C₁₆	—	71	1	no	—
106	C₂×#346	D₇₁	1	1	yes	characteristic
107	#1770	—	71	1	no	—
108	C₇₁	#7170	1	1	yes	characteristic, complemented, Sylow
109	C₂²×C₁₆	—	2	1	no	—
110	C₂²×C₁₆	—	142	1	no	—
111	C₂×C₄⋊C₈	—	71	1	no	—
112	C₂×#97	—	142	1	no	—
113	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	71	1	no	—
115	C₄⋊C₁₆	—	71	2	no	—
116	C₄⋊C₁₆	—	142	1	no	—
117	#346	—	2	1	no	—
118	#346	Dic₇₁	1	2	yes	complemented
119	#336	—	142	1	no	—
120	C₂²×C₈	—	142	1	no	—
121	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
123	C₂×C₄⋊C₄	—	71	2	no	—
124	C₂×C₄²	—	142	1	no	—
125	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	2	2	no	—
129	C₂×C₁₆	—	284	1	no	—
130	C₂×C₁₆	—	284	1	no	—
131	C₂×C₁₆	—	142	2	no	—
132	C₄⋊C₈	—	71	2	no	—
133	C₄⋊C₈	—	142	1	no	—
134	#97	—	284	1	no	—
135	#97	—	284	1	no	—
136	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
137	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
138	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
140	#95	—	142	1	no	—
141	C₂²×C₄	—	142	2	no	—
142	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
143	C₂²×C₄	C₈×D₇₁	1	1	yes	characteristic
144	C₂²×C₄	#11765848	1	1	yes	characteristic
145	C₂×C₈	—	2	1	no	—
146	C₂×C₈	—	2	1	no	—
147	C₂×C₈	—	284	1	no	—
148	C₂×C₈	—	284	1	no	—
149	C₂×C₈	—	142	2	no	—
150	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
152	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
153	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
154	C₂×C₈	D₅₆₈	1	1	yes	characteristic
155	C₂×C₈	#11765854	1	1	yes	characteristic
156	C₂×C₈	#11765854	1	1	yes	characteristic
157	C₂×C₈	Dic₂₈₄	1	1	yes	characteristic
158	C₄⋊C₄	—	284	2	no	—
159	C₄²	—	284	1	no	—
160	C₄²	—	284	1	no	—
161	C₄²	C₄×Dic₇₁	1	1	yes	characteristic
162	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
163	C₄²	C₈×D₇₁	1	1	yes	characteristic
164	C₄²	#11765848	1	1	yes	characteristic
165	C₁₆	—	2	4	no	—
166	C₁₆	—	284	2	no	—
167	C₂³	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	284	2	no	—
171	C₂×C₄	—	284	2	no	—
172	C₂×C₄	—	284	2	no	—
173	C₂×C₄	C₇₁⋊#95	1	1	yes	characteristic
174	C₂×C₄	C₈×Dic₇₁	1	1	yes	characteristic
175	C₂×C₄	#14556038	1	1	yes	characteristic
176	C₂×C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
177	C₂×C₄	#14556050	1	1	yes	characteristic
178	C₂×C₄	#14556053	1	1	yes	characteristic
179	C₂×C₄	#14556056	1	1	yes	characteristic
180	C₂×C₄	#14556056	1	1	yes	characteristic
181	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
182	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
183	C₈	—	2	1	no	—
184	C₈	—	2	1	no	—
185	C₈	—	284	2	no	—
186	C₈	#14556050	1	1	yes	characteristic
187	C₈	#14556053	1	1	yes	characteristic
188	C₈	#14556060	1	1	yes	characteristic
189	C₈	#14556062	1	1	yes	characteristic
190	C₂²	#17985728	1	1	yes	characteristic
191	C₂²	#17985731	1	1	yes	characteristic
192	C₂²	#17985753	1	1	yes	characteristic
193	C₂²	#17985780	1	2	yes	—
194	C₂²	#17985784	1	2	yes	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	2	1	no	—
199	C₄	—	568	2	no	—
200	C₄	#17985728	1	1	yes	characteristic
201	C₄	#17985731	1	1	yes	characteristic
202	C₄	#17985759	1	1	yes	characteristic
203	C₄	#17985762	1	1	yes	characteristic
204	C₂	#22586180	1	1	yes	characteristic
205	C₂	#22586299	1	2	yes	—
206	C₂	#22586323	1	2	yes	—
207	C₂	#22586423	1	1	yes	characteristic
208	C₂	#22586430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₇₁
Fitting	C₂×#346×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₇₁	C₂²
hypercenter	C₂×#346	D₇₁
hypocenter	C₇₁	#7170
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
9. C₄⋊(C₇₁⋊C₁₆)	C₄	semidirect
14. #346×C₇₁	C₄	semidirect
108. C₇₁	#7170	semidirect
118. #346	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₇₁⋊C₁₆)	2
#22588207	2
C₂×#346×C₇₁	2
#7170	71