Finite Group Databasebeta

Subgroups — #27135394

Order 12352.

198 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 45 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×D₄×D₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₅₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15690540	C₂	1	1	yes	maximal, characteristic, complemented
4	#20259874	C₂	1	1	yes	maximal, characteristic, complemented
5	#20259819	C₂	1	1	yes	maximal, characteristic, complemented
6	#20259811	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₉₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₉₃	—	2	1	no	—
9	C₂×D₇₇₂	—	2	1	no	—
10	C₂×D₇₇₂	C₂²	1	1	yes	characteristic
11	C₂×D₇₇₂	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₁₉₃	—	2	1	no	—
13	D₄×D₁₉₃	—	2	2	no	—
14	D₄×D₁₉₃	—	2	2	no	—
15	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
16	D₁₅₄₄	—	2	2	no	—
17	#15690540	—	2	2	no	—
18	C₂×#13095393	C₂²	1	1	yes	characteristic, complemented
19	#15690532	—	2	1	no	—
20	C₂×D₄×C₁₉₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₉₃	C₂²	1	1	yes	characteristic, complemented
22	C₄⋊C₄×C₁₉₃	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₁₉₃	—	2	4	no	—
24	C₂²×D₁₉₃	—	2	1	no	—
25	C₂²×D₁₉₃	—	2	1	no	—
26	C₂²×D₁₉₃	—	4	2	no	—
27	C₂²×D₁₉₃	—	4	2	no	—
28	C₂²×D₁₉₃	D₄	1	1	yes	characteristic
29	D₇₇₂	—	2	1	no	—
30	D₇₇₂	—	2	1	no	—
31	D₇₇₂	—	2	1	no	—
32	D₇₇₂	—	4	2	no	—
33	D₇₇₂	D₄	1	2	yes	complemented
34	C₂²⋊D₁₉₃	—	2	4	no	—
35	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
36	C₄×D₁₉₃	—	2	1	no	—
37	C₄×D₁₉₃	—	2	1	no	—
38	#13095393	—	2	1	no	—
39	C₂³×C₁₉₃	—	2	1	no	—
40	D₄×C₁₉₃	—	2	1	no	—
41	D₄×C₁₉₃	D₄	1	2	yes	complemented
42	C₂×C₄×C₁₉₃	—	2	1	no	—
43	C₂×C₄×C₁₉₃	C₂³	1	1	yes	characteristic
44	C₈×C₁₉₃	—	2	1	no	—
45	C₂²×C₁₉₃	—	2	2	no	—
46	C₂²×C₁₉₃	—	4	1	no	—
47	C₂²×C₁₉₃	—	4	1	no	—
48	C₂²×C₁₉₃	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₉₃	—	2	1	no	—
50	C₂×D₁₉₃	—	2	1	no	—
51	C₂×D₁₉₃	—	2	2	no	—
52	C₂×D₁₉₃	—	4	8	no	—
53	C₂×D₁₉₃	—	4	1	no	—
54	C₂×D₁₉₃	—	4	1	no	—
55	C₂×D₁₉₃	—	4	1	no	—
56	C₂×D₁₉₃	—	4	1	no	—
57	C₂×D₁₉₃	—	4	1	no	—
58	C₂×D₁₉₃	D₈	1	2	yes	complemented
59	C₄×C₁₉₃	—	4	1	no	—
60	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
61	C₄×C₁₉₃	C₂×D₄	1	1	yes	characteristic
62	Dic₁₉₃	—	2	1	no	—
63	C₂×C₁₉₃	—	4	2	no	—
64	C₂×C₁₉₃	#120	1	1	yes	characteristic
65	C₂×C₁₉₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₁₉₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	D₁₉₃	—	2	2	no	—
68	D₁₉₃	—	4	2	no	—
69	D₁₉₃	—	8	1	no	—
70	C₁₉₃	#447	1	1	yes	characteristic, complemented, Sylow
71	#447	—	193	1	no	maximal, Sylow
72	C₂²×D₄	—	193	1	no	—
73	C₂×D₈	—	193	1	no	—
74	C₂×D₈	—	193	1	no	—
75	#121	—	193	1	no	—
76	#102	—	193	1	no	—
77	#102	D₁₉₃	1	1	yes	characteristic, complemented
78	C₂²⋊C₈	—	193	1	no	—
79	C₂⁴	—	386	1	no	—
80	C₂×D₄	—	386	1	no	—
81	C₂×D₄	—	386	2	no	—
82	C₂×D₄	—	386	2	no	—
83	C₂×D₄	—	386	1	no	—
84	C₂×D₄	—	193	1	no	—
85	C₂×D₄	—	193	1	no	—
86	C₂×D₄	C₂×D₁₉₃	1	1	yes	characteristic
87	C₂²×C₄	—	193	1	no	—
88	D₈	—	386	2	no	—
89	D₈	—	386	2	no	—
90	C₂×C₈	—	193	1	no	—
91	C₂×C₈	C₂×D₁₉₃	1	1	yes	characteristic, complemented
92	C₄⋊C₄	C₂×D₁₉₃	1	1	yes	characteristic, complemented
93	C₂²⋊C₄	—	386	1	no	—
94	C₂³	—	2	1	no	—
95	C₂³	—	772	2	no	—
96	C₂³	—	772	2	no	—
97	C₂³	—	386	4	no	—
98	C₂³	—	386	1	no	—
99	C₂³	—	386	1	no	—
100	C₂³	—	193	1	no	—
101	D₄	—	2	1	no	—
102	D₄	—	772	2	no	—
103	D₄	—	386	4	no	—
104	D₄	—	386	1	no	—
105	D₄	—	386	1	no	—
106	D₄	—	386	1	no	—
107	D₄	—	193	2	no	—
108	D₄	D₇₇₂	1	2	yes	complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	386	1	no	—
111	C₂×C₄	—	386	1	no	—
112	C₂×C₄	—	193	1	no	—
113	C₂×C₄	C₂²×D₁₉₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	386	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	4	1	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	772	8	no	—
120	C₂²	—	772	1	no	—
121	C₂²	—	772	1	no	—
122	C₂²	—	772	1	no	—
123	C₂²	—	772	1	no	—
124	C₂²	—	772	1	no	—
125	C₂²	—	386	2	no	—
126	C₂²	—	386	1	no	—
127	C₂²	—	386	1	no	—
128	C₂²	—	193	2	no	—
129	C₂²	D₄×D₁₉₃	1	1	yes	characteristic
130	C₄	—	4	1	no	—
131	C₄	—	386	1	no	—
132	C₄	D₄×D₁₉₃	1	1	yes	characteristic
133	C₄	C₂×D₇₇₂	1	1	yes	characteristic
134	C₂	—	4	2	no	—
135	C₂	—	772	2	no	—
136	C₂	—	1544	1	no	—
137	C₂	—	386	2	no	—
138	C₂	#20259869	1	1	yes	characteristic
139	C₂	D₈×D₁₉₃	1	1	yes	characteristic
140	C₂	#20259926	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₉₃
commutator, cosocle	C₂×C₄×C₁₉₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₉₃
Fitting	#102×C₁₉₃	C₂
socle	C₂²×C₁₉₃	C₂×D₄
hypercenter	#102	D₁₉₃
hypocenter	C₁₉₃	#447
LCS	G ▷ C₂×C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₁₉₃	C₂	semidirect
2. C₂×D₁₅₄₄	C₂	semidirect
3. C₂×#15690540	C₂	semidirect
4. #20259874	C₂	semidirect
5. #20259819	C₂	semidirect
6. #20259811	C₂	semidirect
7. #102×C₁₉₃	C₂	semidirect
11. C₂×D₇₇₂	C₂²	semidirect
18. C₂×#13095393	C₂²	semidirect
21. C₂×C₈×C₁₉₃	C₂²	semidirect
22. C₄⋊C₄×C₁₉₃	C₂²	semidirect
33. D₇₇₂	D₄	semidirect
41. D₄×C₁₉₃	D₄	semidirect
58. C₂×D₁₉₃	D₈	semidirect
70. C₁₉₃	#447	semidirect
77. #102	D₁₉₃	semidirect
91. C₂×C₈	C₂×D₁₉₃	semidirect
92. C₄⋊C₄	C₂×D₁₉₃	semidirect
108. D₄	D₇₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₁₉₃	2
C₂×D₁₅₄₄	2
C₂×#15690540	2
#20259874	2
#20259819	2
#20259811	2
#102×C₁₉₃	2
#447	193