Subgroups — #31915488

Order 18176.

326 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17985827	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22589074	C₂	1	2	yes	maximal
4	#22586431	C₂	1	2	yes	maximal
5	C₂×#367×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14556062	C₂²	1	2	yes	—
8	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
9	#17985898	—	2	2	no	—
10	#17985827	—	2	2	no	—
11	C₁₆×Dic₇₁	—	2	2	no	—
12	#17985761	—	2	2	no	—
13	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×#107×C₇₁	C₂²	1	2	yes	—
15	#367×C₇₁	C₄	1	4	yes	complemented
16	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇₁	—	2	2	no	—
18	C₂×C₄⋊Dic₇₁	—	4	2	no	—
19	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
20	C₂×C₇₁⋊C₁₆	—	2	1	no	—
21	C₂×C₇₁⋊C₁₆	—	2	1	no	—
22	C₂×C₇₁⋊C₁₆	Q₈	1	2	yes	—
23	#14556062	—	2	2	no	—
24	#14556062	—	2	2	no	—
25	C₈×Dic₇₁	—	2	1	no	—
26	C₈×Dic₇₁	—	2	1	no	—
27	C₇₁⋊#95	—	4	2	no	—
28	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
32	C₂×C₁₆×C₇₁	Q₈	1	2	yes	—
33	#107×C₇₁	—	2	2	no	—
34	#107×C₇₁	C₂×C₄	1	4	yes	—
35	C₂²×Dic₇₁	—	4	2	no	—
36	C₂²×Dic₇₁	D₈	1	2	yes	—
37	C₂×C₇₁⋊C₈	—	2	1	no	—
38	C₂×C₇₁⋊C₈	—	2	1	no	—
39	C₂×C₇₁⋊C₈	Q₁₆	1	2	yes	—
40	C₄⋊Dic₇₁	—	4	2	no	—
41	C₄⋊Dic₇₁	—	4	2	no	—
42	C₄⋊Dic₇₁	—	8	4	no	—
43	C₄×Dic₇₁	—	2	1	no	—
44	C₄×Dic₇₁	—	2	1	no	—
45	C₇₁⋊C₁₆	—	2	2	no	—
46	C₂²×C₄×C₇₁	—	4	2	no	—
47	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
53	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
54	C₄⋊C₄×C₇₁	—	2	4	no	—
55	C₄⋊C₄×C₇₁	—	4	2	no	—
56	C₁₆×C₇₁	C₄⋊C₄	1	4	yes	complemented
57	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₇₁	—	4	4	no	—
59	C₂×C₄×C₇₁	—	8	2	no	—
60	C₂×C₄×C₇₁	—	8	2	no	—
61	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
62	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
63	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
66	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
67	C₂×Dic₇₁	—	2	2	no	—
68	C₂×Dic₇₁	—	2	2	no	—
69	C₂×Dic₇₁	—	8	2	no	—
70	C₂×Dic₇₁	—	8	2	no	—
71	C₂×Dic₇₁	—	8	2	no	—
72	C₂×Dic₇₁	SD₃₂	1	4	yes	—
73	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
75	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₈×C₇₁	#125	1	1	yes	characteristic
77	C₇₁⋊C₈	—	2	2	no	—
78	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
79	C₂²×C₇₁	#466	1	1	yes	characteristic
80	C₂²×C₇₁	#500	1	1	yes	characteristic
81	C₂²×C₇₁	C₂×SD₃₂	1	2	yes	—
82	C₂²×C₇₁	#508	1	2	yes	—
83	C₄×C₇₁	—	8	4	no	—
84	C₄×C₇₁	#389	1	1	yes	characteristic
85	C₄×C₇₁	C₄×Q₁₆	1	1	yes	characteristic
86	C₄×C₇₁	#467	1	1	yes	characteristic
87	C₄×C₇₁	#499	1	1	yes	characteristic
88	Dic₇₁	—	2	4	no	—
89	Dic₇₁	—	16	2	no	—
90	#13166	—	71	1	no	maximal, Sylow
91	C₂×C₇₁	#1942	1	1	yes	characteristic
92	C₂×C₇₁	C₄×SD₃₂	1	2	yes	—
93	C₂×C₇₁	#2217	1	2	yes	—
94	C₂×C₇₁	#2253	1	1	yes	characteristic
95	C₂×C₇₁	#2254	1	1	yes	characteristic
96	C₂×#367	—	71	1	no	—
97	C₂×#367	D₇₁	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	71	1	no	—
99	#1942	—	71	2	no	—
100	#1380	—	71	2	no	—
101	C₇₁	#13166	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₁₆	—	71	1	no	—
103	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
104	C₂×#107	—	71	2	no	—
105	C₂×#107	C₂×D₇₁	1	2	yes	—
106	C₂×C₄×C₈	—	71	1	no	—
107	#389	—	142	2	no	—
108	#367	—	142	2	no	—
109	#367	Dic₇₁	1	4	yes	complemented
110	C₄×C₁₆	—	142	2	no	—
111	#340	—	142	2	no	—
112	C₂²×C₈	—	71	1	no	—
113	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	284	2	no	—
116	C₂×C₄⋊C₄	—	142	2	no	—
117	C₂×C₄²	—	71	1	no	—
118	C₂×C₁₆	—	71	2	no	—
119	C₂×C₁₆	—	142	1	no	—
120	C₂×C₁₆	—	142	1	no	—
121	C₂×C₁₆	Dic₁₄₂	1	2	yes	—
122	C₂×C₁₆	D₂₈₄	1	2	yes	—
123	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	142	2	no	—
126	#107	—	142	2	no	—
127	#107	C₂×Dic₇₁	1	4	yes	—
128	C₄×C₈	—	142	1	no	—
129	C₄×C₈	—	142	1	no	—
130	#95	—	284	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	284	2	no	—
133	C₂²×C₄	—	71	2	no	—
134	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
135	C₂×C₈	—	71	2	no	—
136	C₂×C₈	—	142	1	no	—
137	C₂×C₈	—	142	1	no	—
138	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
139	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
140	C₂×C₈	Q₈×D₇₁	1	1	yes	characteristic
141	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
142	C₂×C₈	#11765864	1	1	yes	characteristic
143	C₂×C₈	#11765866	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	284	2	no	—
147	C₄⋊C₄	—	284	2	no	—
148	C₄⋊C₄	—	568	4	no	—
149	C₄²	—	142	1	no	—
150	C₄²	—	142	1	no	—
151	C₁₆	—	142	2	no	—
152	C₁₆	C₄⋊Dic₇₁	1	4	yes	complemented
153	C₂³	D₈×D₇₁	1	1	yes	characteristic
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	568	2	no	—
158	C₂×C₄	—	568	2	no	—
159	C₂×C₄	—	568	2	no	—
160	C₂×C₄	—	71	4	no	—
161	C₂×C₄	—	142	2	no	—
162	C₂×C₄	—	142	2	no	—
163	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
164	C₂×C₄	#14556103	1	1	yes	characteristic
165	C₂×C₄	#14556127	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
167	C₂×C₄	#14556147	1	1	yes	characteristic
168	C₂×C₄	#14556151	1	1	yes	characteristic
169	C₈	—	142	2	no	—
170	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
171	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
172	C₈	#14556108	1	1	yes	characteristic
173	C₈	#14556119	1	1	yes	characteristic
174	C₂²	D₈×Dic₇₁	1	1	yes	characteristic
175	C₂²	#17986236	1	1	yes	characteristic
176	C₂²	#17986403	1	1	yes	characteristic
177	C₂²	SD₃₂×D₇₁	1	2	yes	—
178	C₂²	#17986430	1	2	yes	—
179	C₄	—	8	4	no	—
180	C₄	—	1136	2	no	—
181	C₄	—	142	4	no	—
182	C₄	#17985898	1	1	yes	characteristic
183	C₄	Q₁₆×Dic₇₁	1	1	yes	characteristic
184	C₄	#17986241	1	1	yes	characteristic
185	C₄	#17986400	1	1	yes	characteristic
186	C₂	#22589074	1	1	yes	characteristic
187	C₂	SD₃₂×Dic₇₁	1	2	yes	—
188	C₂	#22590437	1	2	yes	—
189	C₂	#22590618	1	1	yes	characteristic
190	C₂	#22590621	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#367×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×D₈
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#367	D₇₁
hypocenter	C₇₁	#13166
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
15. #367×C₇₁	C₄	semidirect
56. C₁₆×C₇₁	C₄⋊C₄	semidirect
101. C₇₁	#13166	semidirect
109. #367	Dic₇₁	semidirect
152. C₁₆	C₄⋊Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17985827	2
C₂×C₁₆×Dic₇₁	2
#22589074	2
#22586431	2
C₂×#367×C₇₁	2
#13166	71