Finite Group Databasebeta

Subgroups — #37549963

Order 23168.

220 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×#19153890	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19153890	C₂	1	1	yes	maximal, characteristic, complemented
3	#26042177	C₂	1	4	yes	maximal, complemented
4	C₂×#19153823	C₂	1	1	yes	maximal, characteristic
5	#19153890	—	2	1	no	—
6	#19153890	—	2	1	no	—
7	#19153890	—	2	1	no	—
8	#19153890	C₂²	1	2	yes	complemented
9	C₂×#15668452	—	2	1	no	—
10	C₂×#15668452	C₂²	1	2	yes	complemented
11	#19153823	—	2	1	no	—
12	#19153823	—	2	1	no	—
13	#19153823	C₄	1	2	yes	—
14	#19153823	C₂²	1	2	yes	—
15	C₂×#15668433	—	2	1	no	—
16	C₂×#15668433	C₄	1	1	yes	characteristic
17	C₂²×#13089678	—	2	1	no	—
18	C₂²×#13089678	C₂²	1	1	yes	characteristic
19	C₂×#41×C₁₈₁	C₄	1	1	yes	characteristic
20	#15668452	—	2	2	no	—
21	#15668452	—	2	2	no	—
22	#15668433	—	2	2	no	—
23	#15668433	—	2	2	no	—
24	#15668433	—	2	1	no	—
25	#15668433	C₂×C₄	1	2	yes	—
26	C₂×#13089678	—	2	2	no	—
27	C₂×#13089678	—	2	2	no	—
28	C₂×#13089678	—	2	1	no	—
29	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
30	C₂×#13089678	D₄	1	1	yes	characteristic
31	C₂×#13089678	D₄	1	1	yes	characteristic
32	C₂×#13089678	D₄	1	1	yes	characteristic
33	C₂×#13089678	D₄	1	1	yes	characteristic
34	C₂×#13089678	C₂³	1	1	yes	characteristic
35	#41×C₁₈₁	—	2	1	no	—
36	#41×C₁₈₁	—	2	1	no	—
37	#41×C₁₈₁	C₈	1	2	yes	—
38	#41×C₁₈₁	C₂×C₄	1	2	yes	—
39	C₂×Q₈×C₁₈₁	C₈	1	1	yes	characteristic
40	C₂×D₄×C₁₈₁	—	2	1	no	—
41	C₂×D₄×C₁₈₁	C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₈₁	—	2	1	no	—
43	C₂²×C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
44	#13089678	—	2	2	no	—
45	#13089678	C₂²⋊C₄	1	1	yes	characteristic
46	#13089678	C₂²⋊C₄	1	1	yes	characteristic
47	#13089678	C₂×D₄	1	1	yes	characteristic
48	#13089678	C₂×D₄	1	1	yes	characteristic
49	C₂³×C₁₈₁	—	2	1	no	—
50	C₂³×C₁₈₁	M₁₆	1	1	yes	characteristic
51	Q₈×C₁₈₁	—	2	1	no	—
52	Q₈×C₁₈₁	C₂×C₈	1	2	yes	—
53	D₄×C₁₈₁	—	2	2	no	—
54	D₄×C₁₈₁	—	2	2	no	—
55	D₄×C₁₈₁	—	2	1	no	—
56	D₄×C₁₈₁	C₂×C₈	1	2	yes	—
57	C₂×C₄×C₁₈₁	—	2	1	no	—
58	C₂×C₄×C₁₈₁	—	2	2	no	—
59	C₂×C₄×C₁₈₁	—	2	2	no	—
60	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₈₁	M₁₆	1	1	yes	characteristic
65	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₁₈₁	—	2	2	no	—
67	C₂²×C₁₈₁	—	4	1	no	—
68	C₂²×C₁₈₁	—	4	2	no	—
69	C₂²×C₁₈₁	C₂²⋊C₈	1	1	yes	characteristic
70	C₂²×C₁₈₁	C₂²⋊C₈	1	1	yes	characteristic
71	C₂²×C₁₈₁	C₂×M₁₆	1	1	yes	characteristic
72	C₄×C₁₈₁	—	2	2	no	—
73	C₄×C₁₈₁	C₂²⋊C₈	1	1	yes	characteristic
74	C₄×C₁₈₁	C₂²⋊C₈	1	1	yes	characteristic
75	C₄×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₄×C₁₈₁	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₁₈₁	—	2	1	no	—
78	C₂×C₁₈₁	—	2	1	no	—
79	C₂×C₁₈₁	—	2	1	no	—
80	C₂×C₁₈₁	—	4	2	no	—
81	C₂×C₁₈₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
82	C₁₈₁	#2117	1	1	yes	characteristic, complemented, Sylow
83	#2117	—	181	1	no	maximal, Sylow
84	C₂×#131	—	181	1	no	—
85	C₂×M₃₂	—	181	1	no	—
86	C₂×M₃₂	—	181	1	no	—
87	#350	—	181	4	no	—
88	C₂×#41	#10624382	1	1	yes	characteristic
89	#131	—	362	1	no	—
90	#131	—	362	1	no	—
91	#131	—	181	2	no	—
92	#131	—	181	2	no	—
93	C₂×M₁₆	—	362	1	no	—
94	C₂×M₁₆	—	181	1	no	—
95	C₂²×C₈	—	362	1	no	—
96	C₂²×C₈	—	181	1	no	—
97	M₃₂	—	362	1	no	—
98	M₃₂	—	362	1	no	—
99	M₃₂	—	362	1	no	—
100	M₃₂	—	181	2	no	—
101	C₂×C₁₆	—	362	1	no	—
102	C₂×C₁₆	—	181	2	no	—
103	#41	—	2	1	no	—
104	#41	—	2	1	no	—
105	#41	#13089685	1	2	yes	—
106	#41	C₂×#10624382	1	2	yes	—
107	C₂×Q₈	#13089685	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	#13089685	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	C₂×#10624382	1	1	yes	characteristic
112	M₁₆	—	362	1	no	—
113	M₁₆	—	362	2	no	—
114	M₁₆	—	362	2	no	—
115	M₁₆	—	181	2	no	—
116	C₂×C₈	—	362	1	no	—
117	C₂×C₈	—	362	2	no	—
118	C₂×C₈	—	362	2	no	—
119	C₂×C₈	—	181	1	no	—
120	C₂×C₈	—	181	1	no	—
121	C₂×C₈	—	181	1	no	—
122	C₂×C₈	—	181	1	no	—
123	C₂×C₈	—	181	1	no	—
124	C₂×C₈	—	181	1	no	—
125	C₁₆	—	362	2	no	—
126	C₁₆	—	362	2	no	—
127	C₂³	—	2	1	no	—
128	C₂³	#15668458	1	1	yes	characteristic
129	Q₈	—	2	1	no	—
130	Q₈	C₂×#13089685	1	2	yes	—
131	D₄	—	2	2	no	—
132	D₄	—	2	2	no	—
133	D₄	—	2	1	no	—
134	D₄	C₂×#13089685	1	2	yes	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₂²⋊#10624382	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#10624382	1	1	yes	characteristic
140	C₂×C₄	C₂×#13089685	1	1	yes	characteristic
141	C₂×C₄	C₂×#13089685	1	1	yes	characteristic
142	C₂×C₄	#15668458	1	1	yes	characteristic
143	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
144	C₈	—	362	2	no	—
145	C₈	—	181	1	no	—
146	C₈	—	181	1	no	—
147	C₈	—	181	1	no	—
148	C₈	—	181	1	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂²⋊#13089685	1	1	yes	characteristic
153	C₂²	C₂²⋊#13089685	1	1	yes	characteristic
154	C₂²	C₂×#15668458	1	1	yes	characteristic
155	C₄	—	2	2	no	—
156	C₄	C₂²⋊#13089685	1	1	yes	characteristic
157	C₄	C₂²⋊#13089685	1	1	yes	characteristic
158	C₄	C₂×C₂²⋊#10624382	1	1	yes	characteristic
159	C₄	C₂²×#13089685	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	—	2	1	no	—
163	C₂	—	4	2	no	—
164	C₂	C₂×C₂²⋊#13089685	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₂²⋊#10624382
commutator	C₄×C₁₈₁	C₂²×C₈
Frattini	C₂×C₄	C₂²×#10624382
Fitting, prime radical	C₂×#41×C₁₈₁	C₄
socle	C₂×C₁₈₁	C₂×C₂²⋊C₈
cosocle	C₂×#13089678	C₂³
hypercenter	C₂×#41	#10624382
hypocenter	C₁₈₁	#2117
LCS	G ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#19153890	C₂	semidirect
2. C₂×#19153890	C₂	semidirect
3. #26042177	C₂	semidirect
8. #19153890	C₂²	semidirect
10. C₂×#15668452	C₂²	semidirect
82. C₁₈₁	#2117	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19153890	2
#26042177	2
C₂×#19153823	2
#2117	181