Finite Group Databasebeta

Subgroups — #22589741

Order 9088.

196 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986048	C₂	1	1	yes	maximal, characteristic
2	#17986047	C₂	1	1	yes	maximal, characteristic
3	#17986047	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556047	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556047	C₂	1	1	yes	maximal, characteristic
6	C₇₁⋊#338	C₂	1	1	yes	maximal, characteristic
7	C₂×#101×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765849	C₂²	1	1	yes	characteristic
9	C₂×#11765849	C₂²	1	1	yes	characteristic
10	C₂×#11765849	C₂²	1	1	yes	characteristic
11	C₂×#11765849	C₂²	1	1	yes	characteristic
12	#14556064	—	2	2	no	—
13	#14556063	—	2	2	no	—
14	#14556063	—	2	2	no	—
15	#14556047	—	2	2	no	—
16	#14556047	—	2	2	no	—
17	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	#101×C₇₁	—	2	2	no	—
21	#11765849	—	2	1	no	—
22	#11765849	—	2	1	no	—
23	#11765849	—	2	1	no	—
24	#11765849	—	2	1	no	—
25	#11765849	—	2	1	no	—
26	#11765849	—	2	1	no	—
27	#11765849	D₄	1	2	yes	—
28	#11765849	D₄	1	2	yes	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	—	2	1	no	—
31	C₂×C₇₁⋊C₈	—	2	1	no	—
32	C₂×C₇₁⋊C₈	—	2	1	no	—
33	C₂×Q₈×C₇₁	—	2	1	no	—
34	C₂×Q₈×C₇₁	—	2	1	no	—
35	C₂×Q₈×C₇₁	—	2	1	no	—
36	C₂×Q₈×C₇₁	—	2	1	no	—
37	C₂×Q₈×C₇₁	—	2	1	no	—
38	C₂×Q₈×C₇₁	—	2	1	no	—
39	C₂²×C₄×C₇₁	Q₈	1	2	yes	—
40	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
41	M₁₆×C₇₁	—	2	1	no	—
42	M₁₆×C₇₁	—	2	1	no	—
43	M₁₆×C₇₁	—	2	1	no	—
44	M₁₆×C₇₁	D₄	1	2	yes	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	—	2	1	no	—
47	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
48	Q₈×C₇₁	—	4	1	no	—
49	Q₈×C₇₁	—	4	1	no	—
50	Q₈×C₇₁	—	4	1	no	—
51	Q₈×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	—	2	2	no	—
53	C₂×C₄×C₇₁	—	2	2	no	—
54	C₂×C₄×C₇₁	—	2	2	no	—
55	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
59	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
60	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₈×C₇₁	—	2	2	no	—
62	C₈×C₇₁	—	4	1	no	—
63	C₇₁⋊C₈	—	2	2	no	—
64	C₇₁⋊C₈	—	2	2	no	—
65	C₇₁⋊C₈	—	4	1	no	—
66	C₇₁⋊C₈	—	4	1	no	—
67	C₂²×C₇₁	—	4	1	no	—
68	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₇₁	—	4	2	no	—
72	C₄×C₇₁	#121	1	1	yes	characteristic
73	C₄×C₇₁	#121	1	1	yes	characteristic
74	C₄×C₇₁	#121	1	1	yes	characteristic
75	C₄×C₇₁	#126	1	1	yes	characteristic
76	C₂×C₇₁	—	2	1	no	—
77	C₂×C₇₁	—	2	1	no	—
78	C₂×C₇₁	—	2	1	no	—
79	C₂×C₇₁	#396	1	1	yes	characteristic
80	#2070	—	71	1	no	maximal, Sylow
81	C₇₁	#2070	1	1	yes	characteristic, complemented, Sylow
82	#430	—	71	1	no	—
83	#430	—	71	1	no	—
84	#430	—	71	1	no	—
85	C₂×#101	—	71	1	no	—
86	C₂×#101	—	71	1	no	—
87	C₂×#101	D₇₁	1	1	yes	characteristic
88	#338	—	71	1	no	—
89	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
90	C₂×M₁₆	—	71	1	no	—
91	C₂×M₁₆	—	71	1	no	—
92	C₂×M₁₆	—	71	1	no	—
93	C₂×M₁₆	—	71	1	no	—
94	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
95	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
96	#108	—	142	2	no	—
97	#108	—	142	2	no	—
98	#108	—	142	2	no	—
99	#101	—	2	2	no	—
100	#101	—	142	2	no	—
101	#101	—	142	2	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂²×C₄	Dic₁₄₂	1	2	yes	—
109	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	1	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	71	2	no	—
114	M₁₆	—	71	2	no	—
115	M₁₆	—	142	1	no	—
116	M₁₆	—	142	1	no	—
117	M₁₆	—	142	1	no	—
118	M₁₆	—	142	1	no	—
119	M₁₆	—	142	1	no	—
120	M₁₆	—	142	1	no	—
121	M₁₆	C₂²⋊D₇₁	1	2	yes	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	142	1	no	—
125	C₂×C₈	—	142	1	no	—
126	C₂×C₈	—	142	1	no	—
127	C₂×C₈	—	142	1	no	—
128	C₂³	C₂×Dic₁₄₂	1	1	yes	characteristic
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
137	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
138	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
139	C₂×C₄	#11765864	1	1	yes	characteristic
140	C₂×C₄	#11765864	1	1	yes	characteristic
141	C₂×C₄	#11765865	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	284	1	no	—
145	C₈	—	284	1	no	—
146	C₈	—	142	2	no	—
147	C₈	—	142	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#14556105	1	1	yes	characteristic
150	C₂²	#14556105	1	1	yes	characteristic
151	C₂²	#14556107	1	1	yes	characteristic
152	C₄	—	4	2	no	—
153	C₄	#14556099	1	1	yes	characteristic
154	C₄	#14556100	1	1	yes	characteristic
155	C₄	#14556101	1	1	yes	characteristic
156	C₄	#14556121	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#17985922	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#17985922
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#101×C₇₁	C₂
socle	C₂×C₇₁	#396
hypercenter	C₂×#101	D₇₁
hypocenter	C₇₁	#2070
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

Complements

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

N	H ≅ G/N	type
81. C₇₁	#2070	semidirect