Subgroups — #31888703

Order 18176.

430 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	C₄⋊C₄×C₇₁⋊C₈	C₂	1	2	yes	maximal
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985720	C₂	1	1	yes	maximal, characteristic
4	#22586181	C₂	1	2	yes	maximal
5	C₄×#106×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	—	2	2	no	—
8	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
10	#17985754	—	2	2	no	—
11	#17985752	C₄	1	4	yes	complemented
12	#17985720	C₄	1	4	yes	complemented
13	C₂×#106×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
16	#340×C₇₁	C₄	1	2	yes	—
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	—	2	1	no	—
19	C₂²×C₇₁⋊C₈	D₄	1	2	yes	—
20	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
21	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
22	C₄×C₇₁⋊C₈	—	2	1	no	—
23	C₄×C₇₁⋊C₈	—	4	4	no	—
24	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₁⋊C₈	Q₈	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×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₇₁	—	2	2	no	—
31	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
32	#106×C₇₁	C₈	1	8	yes	complemented
33	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	#95×C₇₁	—	2	2	no	—
36	C₂×C₇₁⋊C₈	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	2	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₂×C₇₁⋊C₈	—	2	4	no	—
40	C₂×C₇₁⋊C₈	—	4	2	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇₁	—	2	4	no	—
45	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
48	C₂×C₈×C₇₁	—	2	2	no	—
49	C₂×C₈×C₇₁	C₄²	1	2	yes	—
50	C₂×C₈×C₇₁	C₄²	1	2	yes	—
51	C₂×C₈×C₇₁	C₂×C₈	1	4	yes	—
52	C₄⋊C₄×C₇₁	—	2	4	no	—
53	C₄⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
54	C₄²×C₇₁	—	4	2	no	—
55	C₄²×C₇₁	—	4	2	no	—
56	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
58	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₇₁	#41	1	1	yes	characteristic
60	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
61	C₂×C₄×C₇₁	—	2	2	no	—
62	C₂×C₄×C₇₁	—	2	8	no	—
63	C₂×C₄×C₇₁	—	4	4	no	—
64	C₂×C₄×C₇₁	—	4	4	no	—
65	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
66	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₇₁	#131	1	2	yes	—
73	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
74	C₂×C₄×C₇₁	#137	1	2	yes	—
75	C₈×C₇₁	—	2	2	no	—
76	C₈×C₇₁	C₄×C₈	1	4	yes	complemented
77	C₇₁⋊C₈	—	2	4	no	—
78	C₇₁⋊C₈	—	4	2	no	—
79	C₇₁⋊C₈	—	4	4	no	—
80	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₇₁	C₈×D₄	1	2	yes	—
82	C₂²×C₇₁	#440	1	2	yes	—
83	C₂²×C₇₁	#444	1	2	yes	—
84	C₄×C₇₁	—	2	1	no	—
85	C₄×C₇₁	—	2	1	no	—
86	C₄×C₇₁	—	4	8	no	—
87	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
88	C₄×C₇₁	#405	1	1	yes	characteristic
89	C₄×C₇₁	#428	1	2	yes	—
90	C₄×C₇₁	#441	1	1	yes	characteristic
91	C₄×C₇₁	#442	1	1	yes	characteristic
92	C₄×C₇₁	C₈×Q₈	1	2	yes	—
93	#8285	—	71	1	no	maximal, Sylow
94	C₂×C₇₁	SD₁₆⋊C₈	1	4	yes	—
95	C₂×C₇₁	C₈×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₇₁	#1773	1	1	yes	characteristic
97	C₂×C₇₁	#1776	1	1	yes	characteristic
98	C₄×#106	D₇₁	1	1	yes	characteristic
99	C₈×C₄⋊C₄	—	71	2	no	—
100	C₂×#335	—	71	1	no	—
101	C₂×#322	—	71	1	no	—
102	#1279	—	71	2	no	—
103	C₇₁	#8285	1	1	yes	characteristic, complemented, Sylow
104	C₂×#106	Dic₇₁	1	2	yes	—
105	C₂×C₄⋊C₈	—	71	2	no	—
106	C₂×C₄×C₈	—	71	1	no	—
107	C₂×C₄×C₈	—	71	1	no	—
108	C₂×C₄×C₈	—	142	2	no	—
109	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
111	#340	Dic₇₁	1	2	yes	—
112	#336	—	142	2	no	—
113	#335	—	71	4	no	—
114	#322	—	71	4	no	—
115	C₂²×C₈	—	71	2	no	—
116	C₂²×C₈	—	142	2	no	—
117	C₂²×C₈	—	142	1	no	—
118	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
119	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
121	C₂×C₄²	—	2	2	no	—
122	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
123	#106	C₇₁⋊C₈	1	8	yes	complemented
124	C₄⋊C₈	—	71	4	no	—
125	C₄⋊C₈	—	142	2	no	—
126	C₄×C₈	—	284	4	no	—
127	C₄×C₈	—	71	2	no	—
128	C₄×C₈	—	71	2	no	—
129	C₄×C₈	—	71	2	no	—
130	C₄×C₈	—	142	1	no	—
131	C₄×C₈	C₄×D₇₁	1	2	yes	—
132	C₄×C₈	C₄×D₇₁	1	2	yes	—
133	#95	—	2	2	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
136	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
137	C₂²×C₄	#11765864	1	1	yes	characteristic
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	284	2	no	—
140	C₂×C₈	—	284	2	no	—
141	C₂×C₈	—	284	2	no	—
142	C₂×C₈	—	71	4	no	—
143	C₂×C₈	—	142	2	no	—
144	C₂×C₈	—	142	4	no	—
145	C₂×C₈	—	142	2	no	—
146	C₂×C₈	—	142	2	no	—
147	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
148	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
149	C₂×C₈	C₂×C₇₁⋊C₈	1	4	yes	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
152	C₄²	—	4	2	no	—
153	C₄²	—	4	2	no	—
154	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
155	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
156	C₄²	Q₈×D₇₁	1	1	yes	characteristic
157	C₄²	#11765866	1	1	yes	characteristic
158	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
164	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
165	C₂×C₄	#14556086	1	1	yes	characteristic
166	C₂×C₄	#14556090	1	1	yes	characteristic
167	C₂×C₄	#14556093	1	1	yes	characteristic
168	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
169	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
170	C₂×C₄	#14556135	1	2	yes	—
171	C₂×C₄	#14556155	1	2	yes	—
172	C₂×C₄	#14556157	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	—	284	2	no	—
175	C₈	—	284	4	no	—
176	C₈	—	142	4	no	—
177	C₈	C₄×C₇₁⋊C₈	1	4	yes	complemented
178	C₂²	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
179	C₂²	D₄×C₇₁⋊C₈	1	2	yes	—
180	C₂²	#17986100	1	2	yes	—
181	C₂²	#17986121	1	2	yes	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	—	4	8	no	—
185	C₄	C₂×C₄×C₇₁⋊C₈	1	1	yes	characteristic
186	C₄	#17985955	1	1	yes	characteristic
187	C₄	#17986038	1	2	yes	—
188	C₄	#17986105	1	1	yes	characteristic
189	C₄	#17986110	1	1	yes	characteristic
190	C₄	Q₈×C₇₁⋊C₈	1	2	yes	—
191	C₂	SD₁₆⋊(C₇₁⋊C₈)	1	4	yes	—
192	C₂	C₄⋊C₄×C₇₁⋊C₈	1	1	yes	characteristic
193	C₂	#22588226	1	1	yes	characteristic
194	C₂	#22588240	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄×#106×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×#106	D₇₁
hypocenter	C₇₁	#8285
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
11. #17985752	C₄	semidirect
12. #17985720	C₄	semidirect
32. #106×C₇₁	C₈	semidirect
76. C₈×C₇₁	C₄×C₈	semidirect
103. C₇₁	#8285	semidirect
123. #106	C₇₁⋊C₈	semidirect
177. C₈	C₄×C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₄×C₇₁⋊C₈	2
C₂×#17985752	2
C₂×#17985720	2
#22586181	2
C₄×#106×C₇₁	2
#8285	71