Subgroups — #31915924

Order 18176.

494 conjugacy classes of subgroups.

The subgroup lattice has 214 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	C₂×#17985803	C₂	1	4	yes	maximal
3	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14556062	C₂²	1	2	yes	—
7	C₄×C₄⋊Dic₇₁	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₁₆)	—	2	2	no	—
9	#17985803	—	2	8	no	—
10	#17985762	—	2	1	no	—
11	C₂²×Q₁₆×C₇₁	—	2	1	no	—
12	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₇₁	—	2	1	no	—
14	#494×C₇₁	—	2	2	no	—
15	#494×C₇₁	C₄	1	4	yes	complemented
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₄×Dic₇₁	—	4	1	no	—
19	C₂×C₇₁⋊C₁₆	—	2	4	no	—
20	C₂×C₇₁⋊C₁₆	—	2	2	no	—
21	#14556062	—	2	2	no	—
22	#14556062	D₄	1	4	yes	—
23	C₇₁⋊#95	—	4	1	no	—
24	C₂²×Q₈×C₇₁	—	2	2	no	—
25	C₂×Q₁₆×C₇₁	—	2	4	no	—
26	C₂×Q₁₆×C₇₁	—	2	4	no	—
27	C₂×Q₁₆×C₇₁	—	2	2	no	—
28	C₂×Q₁₆×C₇₁	—	2	4	no	—
29	C₂×Q₁₆×C₇₁	—	4	4	no	—
30	C₂×Q₁₆×C₇₁	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₇₁	—	2	4	no	—
34	C₄⋊Q₈×C₇₁	—	4	2	no	—
35	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
36	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₇₁	D₄	1	2	yes	—
40	C₂²×Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	2	4	no	—
42	C₄⋊Dic₇₁	—	4	2	no	—
43	C₄×Dic₇₁	—	8	2	no	—
44	C₇₁⋊C₁₆	—	4	2	no	—
45	C₂×Q₈×C₇₁	—	2	4	no	—
46	C₂×Q₈×C₇₁	—	4	2	no	—
47	C₂×Q₈×C₇₁	—	4	4	no	—
48	C₂×Q₈×C₇₁	—	4	4	no	—
49	C₂²×C₄×C₇₁	—	4	2	no	—
50	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₇₁	—	2	8	no	—
54	Q₁₆×C₇₁	—	4	8	no	—
55	Q₁₆×C₇₁	—	4	2	no	—
56	Q₁₆×C₇₁	—	4	2	no	—
57	C₂×C₈×C₇₁	—	2	2	no	—
58	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₇₁	#41	1	2	yes	—
63	C₄⋊C₄×C₇₁	—	4	4	no	—
64	C₄⋊C₄×C₇₁	—	4	4	no	—
65	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₇₁	SD₁₆	1	2	yes	—
67	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
68	Q₈×C₇₁	—	4	4	no	—
69	Q₈×C₇₁	—	4	4	no	—
70	Q₈×C₇₁	—	8	4	no	—
71	C₂×C₄×C₇₁	—	2	2	no	—
72	C₂×C₄×C₇₁	—	4	4	no	—
73	C₂×C₄×C₇₁	—	8	4	no	—
74	C₂×C₄×C₇₁	#102	1	2	yes	—
75	C₂×C₄×C₇₁	#102	1	2	yes	—
76	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
77	C₂×C₄×C₇₁	Q₃₂	1	2	yes	—
78	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₁	#121	1	2	yes	—
80	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
82	C₂×Dic₇₁	—	4	4	no	—
83	C₂×Dic₇₁	—	8	2	no	—
84	C₂×Dic₇₁	—	8	2	no	—
85	C₈×C₇₁	—	2	2	no	—
86	C₈×C₇₁	C₄×D₄	1	2	yes	—
87	C₈×C₇₁	#124	1	1	yes	characteristic
88	C₈×C₇₁	#127	1	1	yes	characteristic
89	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
90	C₂²×C₇₁	#459	1	2	yes	—
91	C₂²×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
92	C₂²×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
93	C₂²×C₇₁	#510	1	2	yes	—
94	C₄×C₇₁	—	2	2	no	—
95	C₄×C₇₁	—	8	4	no	—
96	C₄×C₇₁	#358	1	4	yes	complemented
97	C₄×C₇₁	#390	1	1	yes	characteristic
98	C₄×C₇₁	#418	1	1	yes	characteristic
99	C₄×C₇₁	#460	1	2	yes	—
100	Dic₇₁	—	8	4	no	—
101	#13253	—	71	1	no	maximal, Sylow
102	C₂×C₇₁	#1982	1	1	yes	characteristic
103	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
104	C₂×C₇₁	#2141	1	1	yes	characteristic
105	C₂×C₇₁	#2208	1	2	yes	—
106	C₂×C₇₁	#2209	1	2	yes	—
107	C₂×#494	D₇₁	1	1	yes	characteristic
108	C₂×C₄⋊C₁₆	—	71	1	no	—
109	C₂×#358	—	71	4	no	—
110	C₄×#107	—	71	1	no	—
111	C₇₁	#13253	1	1	yes	characteristic, complemented, Sylow
112	C₂²×Q₁₆	—	2	1	no	—
113	C₂²×Q₁₆	C₂×D₇₁	1	2	yes	—
114	C₂×C₄⋊Q₈	—	2	1	no	—
115	C₂²×C₁₆	—	71	2	no	—
116	#494	—	2	2	no	—
117	#494	Dic₇₁	1	4	yes	complemented
118	C₂×#107	—	71	2	no	—
119	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
120	C₄×C₄⋊C₄	—	142	1	no	—
121	C₄⋊C₁₆	—	142	2	no	—
122	#358	—	142	8	no	—
123	#340	—	142	1	no	—
124	C₂²×Q₈	—	2	2	no	—
125	C₂×Q₁₆	—	2	4	no	—
126	C₂×Q₁₆	—	2	4	no	—
127	C₂×Q₁₆	—	2	2	no	—
128	C₂×Q₁₆	—	2	4	no	—
129	C₂×Q₁₆	—	4	4	no	—
130	C₂×Q₁₆	C₂×Dic₇₁	1	4	yes	—
131	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
132	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	4	no	—
134	C₄⋊Q₈	—	4	2	no	—
135	C₂×C₄⋊C₄	—	4	1	no	—
136	C₂×C₄⋊C₄	—	4	1	no	—
137	C₂×C₄⋊C₄	—	142	2	no	—
138	C₂×C₄²	—	284	1	no	—
139	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
140	C₂×C₁₆	—	142	4	no	—
141	C₂×C₁₆	—	142	2	no	—
142	#107	—	71	4	no	—
143	#107	—	142	2	no	—
144	C₄×C₈	C₂×Dic₇₁	1	2	yes	—
145	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
146	#95	—	284	1	no	—
147	C₂×Q₈	—	2	4	no	—
148	C₂×Q₈	—	4	2	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂×Q₈	—	4	4	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	—	284	2	no	—
153	C₂²×C₄	#11765850	1	1	yes	characteristic
154	C₂²×C₄	#11765850	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
156	Q₁₆	—	2	8	no	—
157	Q₁₆	—	4	8	no	—
158	Q₁₆	—	4	2	no	—
159	Q₁₆	—	4	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
162	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
163	C₂×C₈	D₄×D₇₁	1	2	yes	—
164	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
165	C₂×C₈	#11765864	1	2	yes	—
166	C₄⋊C₄	—	4	4	no	—
167	C₄⋊C₄	—	4	4	no	—
168	C₄⋊C₄	—	284	2	no	—
169	C₄⋊C₄	—	142	4	no	—
170	C₄²	—	568	2	no	—
171	C₄²	C₂²⋊Dic₇₁	1	2	yes	—
172	C₄²	#11765853	1	2	yes	—
173	C₁₆	—	284	2	no	—
174	C₂³	C₂×#11765850	1	1	yes	characteristic
175	Q₈	—	4	4	no	—
176	Q₈	—	4	4	no	—
177	Q₈	—	8	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	8	4	no	—
181	C₂×C₄	—	284	4	no	—
182	C₂×C₄	—	568	2	no	—
183	C₂×C₄	—	568	2	no	—
184	C₂×C₄	#14556049	1	2	yes	—
185	C₂×C₄	#14556049	1	2	yes	—
186	C₂×C₄	#14556071	1	2	yes	—
187	C₂×C₄	#14556074	1	2	yes	—
188	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
189	C₂×C₄	#14556099	1	2	yes	—
190	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
191	C₂×C₄	#14556154	1	2	yes	—
192	C₈	—	2	2	no	—
193	C₈	D₄×Dic₇₁	1	2	yes	—
194	C₈	#14556115	1	1	yes	characteristic
195	C₈	#14556124	1	1	yes	characteristic
196	C₂²	C₂×#14556049	1	1	yes	characteristic
197	C₂²	#17986191	1	2	yes	—
198	C₂²	C₂×#14556071	1	1	yes	characteristic
199	C₂²	C₂×#14556074	1	1	yes	characteristic
200	C₂²	#17986440	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	8	4	no	—
203	C₄	—	568	4	no	—
204	C₄	#17985803	1	4	yes	complemented
205	C₄	#17985902	1	1	yes	characteristic
206	C₄	#17986004	1	1	yes	characteristic
207	C₄	#17986196	1	2	yes	—
208	C₂	#22589268	1	1	yes	characteristic
209	C₂	C₂×#17985803	1	1	yes	characteristic
210	C₂	#22590060	1	1	yes	characteristic
211	C₂	#22590382	1	2	yes	—
212	C₂	#22590387	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#494×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#494	D₇₁
hypocenter	C₇₁	#13253
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
15. #494×C₇₁	C₄	semidirect
96. C₄×C₇₁	#358	semidirect
111. C₇₁	#13253	semidirect
117. #494	Dic₇₁	semidirect
204. C₄	#17985803	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
C₂×#17985803	2
C₄×#14556062	2
C₂×#494×C₇₁	2
#13253	71