Finite Group Databasebeta

Subgroups — #27136256

Order 12352.

154 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 48 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×#15690565	C₂	1	1	yes	maximal, characteristic, complemented
2	#20259978	C₂	1	1	yes	maximal, characteristic, complemented
3	#20259964	C₂	1	2	yes	maximal, complemented
4	#20259963	C₂	1	2	yes	maximal, complemented
5	#41×D₁₉₃	C₂	1	1	yes	maximal, characteristic
6	#15690565	—	2	1	no	—
7	#15690565	C₂²	1	2	yes	complemented
8	C₂×#13095400	—	2	1	no	—
9	C₄⋊#11715871	—	2	1	no	—
10	C₂²⋊#11715871	—	2	1	no	—
11	C₄×#11715871	C₂²	1	2	yes	complemented
12	#15690556	C₂²	1	1	yes	characteristic
13	#15690555	—	2	1	no	—
14	#15690555	C₄	1	1	yes	characteristic, complemented
15	#15690554	—	2	1	no	—
16	#15690554	C₂²	1	1	yes	characteristic
17	Q₈×D₁₉₃	C₄	1	1	yes	characteristic, complemented
18	D₄×D₁₉₃	—	2	1	no	—
19	D₄×D₁₉₃	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×D₁₉₃	—	2	1	no	—
21	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
22	#41×C₁₉₃	C₄	1	1	yes	characteristic, complemented
23	C₂×#11715871	—	2	2	no	—
24	#13095400	—	2	2	no	—
25	C₂²×D₁₉₃	—	2	1	no	—
26	C₂²×D₁₉₃	D₄	1	1	yes	characteristic
27	Dic₃₈₆	—	2	1	no	—
28	Dic₃₈₆	C₂×C₄	1	1	yes	characteristic
29	D₇₇₂	—	2	1	no	—
30	D₇₇₂	C₂×C₄	1	1	yes	characteristic
31	C₂²⋊D₁₉₃	—	2	1	no	—
32	C₂²⋊D₁₉₃	—	2	1	no	—
33	C₂²⋊D₁₉₃	—	2	1	no	—
34	C₂×Dic₁₉₃	—	2	1	no	—
35	C₂×Dic₁₉₃	D₄	1	1	yes	characteristic
36	C₄×D₁₉₃	—	2	1	no	—
37	C₄×D₁₉₃	—	2	1	no	—
38	C₄×D₁₉₃	—	2	1	no	—
39	C₄×D₁₉₃	C₂×C₄	1	1	yes	characteristic
40	C₄×D₁₉₃	D₄	1	1	yes	characteristic
41	C₄×D₁₉₃	D₄	1	1	yes	characteristic
42	C₄×D₁₉₃	C₂³	1	1	yes	characteristic
43	Q₈×C₁₉₃	C₂×C₄	1	1	yes	characteristic
44	D₄×C₁₉₃	—	2	1	no	—
45	D₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₃	—	2	1	no	—
47	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₁₉₃	—	2	1	no	—
49	C₂²×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₁₉₃	—	2	1	no	—
51	C₂×D₁₉₃	—	4	1	no	—
52	C₂×D₁₉₃	—	4	2	no	—
53	C₂×D₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×D₁₉₃	C₂×D₄	1	1	yes	characteristic
55	#11715871	—	4	2	no	—
56	C₄×C₁₉₃	—	2	1	no	—
57	C₄×C₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
59	Dic₁₉₃	—	2	1	no	—
60	Dic₁₉₃	C₂²⋊C₄	1	1	yes	characteristic
61	Dic₁₉₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₉₃	—	2	1	no	—
63	C₂×C₁₉₃	—	4	1	no	—
64	C₂×C₁₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	D₁₉₃	—	2	1	no	—
66	D₁₉₃	—	2	1	no	—
67	D₁₉₃	—	4	1	no	—
68	C₁₉₃	#421	1	1	yes	characteristic, complemented, Sylow
69	#421	—	193	1	no	maximal, Sylow
70	C₂×#41	—	193	1	no	—
71	C₂×M₁₆	—	193	1	no	—
72	#117	—	193	1	no	—
73	#104	—	193	2	no	—
74	#104	—	193	2	no	—
75	#41	—	386	1	no	—
76	#41	—	386	1	no	—
77	#41	—	193	1	no	—
78	#41	—	193	1	no	—
79	#41	—	193	1	no	—
80	#41	#11715871	1	1	yes	characteristic, complemented
81	C₂×Q₈	—	193	1	no	—
82	C₂×D₄	—	386	1	no	—
83	C₂×D₄	—	193	1	no	—
84	C₂²×C₄	—	386	1	no	—
85	C₂²×C₄	—	193	1	no	—
86	M₁₆	—	386	1	no	—
87	M₁₆	—	193	2	no	—
88	C₂×C₈	—	386	1	no	—
89	C₄⋊C₄	—	386	1	no	—
90	C₂²⋊C₄	—	386	1	no	—
91	C₄²	—	193	2	no	—
92	C₂³	—	386	1	no	—
93	C₂³	—	193	1	no	—
94	Q₈	—	386	1	no	—
95	Q₈	—	193	1	no	—
96	Q₈	C₂×#11715871	1	1	yes	characteristic
97	D₄	—	2	1	no	—
98	D₄	—	386	1	no	—
99	D₄	—	386	1	no	—
100	D₄	—	386	1	no	—
101	D₄	—	386	1	no	—
102	D₄	—	193	1	no	—
103	D₄	C₂×#11715871	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	386	1	no	—
106	C₂×C₄	—	386	1	no	—
107	C₂×C₄	—	386	2	no	—
108	C₂×C₄	—	386	1	no	—
109	C₂×C₄	—	386	1	no	—
110	C₂×C₄	—	193	1	no	—
111	C₂×C₄	—	193	1	no	—
112	C₂×C₄	—	193	1	no	—
113	C₂×C₄	—	193	1	no	—
114	C₂×C₄	—	193	1	no	—
115	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
116	C₈	—	386	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	772	1	no	—
119	C₂²	—	772	2	no	—
120	C₂²	—	386	1	no	—
121	C₂²	—	193	1	no	—
122	C₂²	—	193	1	no	—
123	C₂²	C₂²⋊#11715871	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	772	2	no	—
126	C₄	—	386	1	no	—
127	C₄	—	193	1	no	—
128	C₄	—	193	1	no	—
129	C₄	C₂²⋊#11715871	1	1	yes	characteristic
130	C₄	C₂²×#11715871	1	1	yes	characteristic
131	C₂	—	2	1	no	—
132	C₂	—	4	1	no	—
133	C₂	—	772	1	no	—
134	C₂	—	386	1	no	—
135	C₂	—	386	1	no	—
136	C₂	C₂×C₂²⋊#11715871	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11715871
commutator	C₄×C₁₉₃	C₂²×C₄
Frattini	C₄	C₂²×#11715871
Fitting	#41×C₁₉₃	C₄
socle	C₂×C₁₉₃	C₂×C₂²⋊C₄
prime radical	#41×D₁₉₃	C₂
cosocle	C₄×D₁₉₃	C₂³
hypercenter	#41	#11715871
hypocenter	C₁₉₃	#421
LCS	G ▷ C₄×C₁₉₃ ▷ C₂×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₂×#15690565	C₂	semidirect
2. #20259978	C₂	semidirect
3. #20259964	C₂	semidirect
4. #20259963	C₂	semidirect
7. #15690565	C₂²	semidirect
11. C₄×#11715871	C₂²	semidirect
14. #15690555	C₄	semidirect
17. Q₈×D₁₉₃	C₄	semidirect
19. D₄×D₁₉₃	C₄	semidirect
22. #41×C₁₉₃	C₄	semidirect
68. C₁₉₃	#421	semidirect
80. #41	#11715871	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690565	2
#20259978	2
#20259964	2
#20259963	2
#41×D₁₉₃	2
#421	193