Subgroups — #31888693

Order 18176.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588209	C₂	1	2	yes	maximal
2	C₂×#17985750	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×C₇₁⋊C₈	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×#14556039	—	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	#17985750	C₄	1	4	yes	complemented
12	C₈×C₇₁⋊C₈	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	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
22	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	4	yes	—
23	#14556039	—	4	4	no	—
24	C₄×C₇₁⋊C₈	—	2	1	no	—
25	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₇₁⋊C₈	Q₈	1	2	yes	—
28	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₇₁	—	2	2	no	—
32	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
33	#106×C₇₁	—	2	4	no	—
34	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
36	#95×C₇₁	—	2	2	no	—
37	C₂×C₇₁⋊C₈	—	2	4	no	—
38	C₂×C₇₁⋊C₈	—	2	2	no	—
39	C₂×C₇₁⋊C₈	—	2	2	no	—
40	C₂×C₇₁⋊C₈	—	2	4	no	—
41	C₂×C₇₁⋊C₈	—	4	2	no	—
42	C₂×C₇₁⋊C₈	—	4	2	no	—
43	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇₁⋊C₈	C₄⋊C₄	1	2	yes	—
45	C₂×C₇₁⋊C₈	SD₁₆	1	4	yes	—
46	C₂²×C₄×C₇₁	—	2	4	no	—
47	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
50	C₂×C₈×C₇₁	—	2	2	no	—
51	C₂×C₈×C₇₁	C₄²	1	2	yes	—
52	C₂×C₈×C₇₁	C₄²	1	2	yes	—
53	C₂×C₈×C₇₁	M₁₆	1	4	yes	—
54	C₄⋊C₄×C₇₁	—	2	4	no	—
55	C₄⋊C₄×C₇₁	—	2	4	no	—
56	C₄²×C₇₁	—	4	2	no	—
57	C₄²×C₇₁	—	4	2	no	—
58	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
60	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₇₁	#41	1	1	yes	characteristic
62	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₇₁	—	2	2	no	—
64	C₂×C₄×C₇₁	—	4	4	no	—
65	C₂×C₄×C₇₁	—	4	4	no	—
66	C₂×C₄×C₇₁	—	4	4	no	—
67	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
74	C₂×C₄×C₇₁	#131	1	2	yes	—
75	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₇₁	#135	1	2	yes	—
77	C₈×C₇₁	—	2	2	no	—
78	C₈×C₇₁	#97	1	4	yes	complemented
79	C₇₁⋊C₈	—	2	4	no	—
80	C₇₁⋊C₈	—	2	2	no	—
81	C₇₁⋊C₈	—	4	4	no	—
82	C₇₁⋊C₈	#106	1	4	yes	complemented
83	C₂²×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₇₁	#436	1	2	yes	—
85	C₂²×C₇₁	C₄×SD₁₆	1	2	yes	—
86	C₂²×C₇₁	#443	1	2	yes	—
87	C₄×C₇₁	—	2	1	no	—
88	C₄×C₇₁	—	2	1	no	—
89	C₄×C₇₁	—	8	4	no	—
90	C₄×C₇₁	C₂×#97	1	1	yes	characteristic
91	C₄×C₇₁	#405	1	1	yes	characteristic
92	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
93	C₄×C₇₁	#427	1	1	yes	characteristic
94	C₄×C₇₁	C₄×SD₁₆	1	2	yes	—
95	C₄×C₇₁	#446	1	2	yes	—
96	#8283	—	71	1	no	maximal, Sylow
97	C₂×C₇₁	#1590	1	4	yes	—
98	C₂×C₇₁	#1770	1	1	yes	characteristic
99	C₂×C₇₁	#1772	1	1	yes	characteristic
100	C₂×C₇₁	C₄×#106	1	1	yes	characteristic
101	C₄×#106	D₇₁	1	1	yes	characteristic
102	#1770	—	71	2	no	—
103	C₂×#334	—	71	1	no	—
104	C₂×C₈²	—	71	1	no	—
105	#1279	—	71	2	no	—
106	C₇₁	#8283	1	1	yes	characteristic, complemented, Sylow
107	C₂×#106	Dic₇₁	1	2	yes	—
108	C₂×C₄⋊C₈	—	71	2	no	—
109	C₂×#97	—	142	2	no	—
110	C₂×C₄×C₈	—	71	1	no	—
111	C₂×C₄×C₈	—	71	1	no	—
112	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
113	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
114	#340	Dic₇₁	1	2	yes	—
115	#336	—	142	2	no	—
116	#334	—	71	4	no	—
117	C₈²	—	71	4	no	—
118	C₂²×C₈	—	71	1	no	—
119	C₂²×C₈	—	71	1	no	—
120	C₂²×C₈	—	142	2	no	—
121	C₂²×C₈	—	142	1	no	—
122	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
125	C₂×C₄²	—	2	2	no	—
126	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
127	#106	—	2	4	no	—
128	C₄⋊C₈	—	71	4	no	—
129	C₄⋊C₈	—	142	2	no	—
130	#97	—	284	4	no	—
131	C₄×C₈	—	71	2	no	—
132	C₄×C₈	—	71	2	no	—
133	C₄×C₈	—	71	2	no	—
134	C₄×C₈	—	142	1	no	—
135	C₄×C₈	C₄×D₇₁	1	2	yes	—
136	C₄×C₈	C₄×D₇₁	1	2	yes	—
137	#95	—	2	2	no	—
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
140	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
141	C₂²×C₄	#11765864	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	284	2	no	—
144	C₂×C₈	—	284	2	no	—
145	C₂×C₈	—	71	2	no	—
146	C₂×C₈	—	71	4	no	—
147	C₂×C₈	—	71	2	no	—
148	C₂×C₈	—	142	4	no	—
149	C₂×C₈	—	142	4	no	—
150	C₂×C₈	—	142	2	no	—
151	C₂×C₈	—	142	2	no	—
152	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
153	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
154	C₂×C₈	#11765849	1	4	yes	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄²	—	4	2	no	—
158	C₄²	—	4	2	no	—
159	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
161	C₄²	Q₈×D₇₁	1	1	yes	characteristic
162	C₄²	#11765866	1	1	yes	characteristic
163	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
169	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
170	C₂×C₄	#14556086	1	1	yes	characteristic
171	C₂×C₄	#14556090	1	1	yes	characteristic
172	C₂×C₄	#14556093	1	1	yes	characteristic
173	C₂×C₄	Q₈×Dic₇₁	1	1	yes	characteristic
174	C₂×C₄	C₂×#11765849	1	2	yes	—
175	C₂×C₄	#14556135	1	2	yes	—
176	C₂×C₄	SD₁₆×D₇₁	1	2	yes	—
177	C₂×C₄	#14556149	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	284	4	no	—
180	C₈	—	71	4	no	—
181	C₈	—	142	4	no	—
182	C₈	—	142	2	no	—
183	C₈	#14556039	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
185	C₂²	#17986077	1	2	yes	—
186	C₂²	SD₁₆×Dic₇₁	1	2	yes	—
187	C₂²	#17986116	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	8	4	no	—
191	C₄	C₂×#14556039	1	1	yes	characteristic
192	C₄	#17985955	1	1	yes	characteristic
193	C₄	#106×D₇₁	1	1	yes	characteristic
194	C₄	#17986034	1	1	yes	characteristic
195	C₄	#17986086	1	2	yes	—
196	C₄	#17986130	1	2	yes	—
197	C₂	#22587231	1	4	yes	—
198	C₂	#22588209	1	1	yes	characteristic
199	C₂	#22588221	1	1	yes	characteristic
200	C₂	#106×Dic₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂×C₄²
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	#8283
LCS	G ▷ C₂×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. #17985750	C₄	semidirect
12. C₈×C₇₁⋊C₈	C₄	semidirect
78. C₈×C₇₁	#97	semidirect
82. C₇₁⋊C₈	#106	semidirect
106. C₇₁	#8283	semidirect
183. C₈	#14556039	semidirect

Maximal subgroups

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

subgroup	index
#22588209	2
C₂×#17985750	2
C₂×C₈×C₇₁⋊C₈	2
#22586181	2
C₄×#106×C₇₁	2
#8283	71