Finite Group Databasebeta

Subgroups — #27136208

Order 12352.

154 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 54 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×C₄⋊#11715871	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11715871	C₂	1	1	yes	maximal, characteristic
3	(C₂×#11715871)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	(C₂×#11715871)⋊C₄	C₂	1	1	yes	maximal, characteristic
5	(C₂×#11715871)⋊C₄	C₂	1	1	yes	maximal, characteristic
6	(C₂×#11715871)⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×D₁₉₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11715871	C₂²	1	1	yes	characteristic
9	C₂²×#11715871	C₂²	1	1	yes	characteristic
10	C₂²×#11715871	C₂²	1	1	yes	characteristic
11	C₂²×#11715871	C₂²	1	1	yes	characteristic
12	C₄⋊#11715871	—	2	2	no	—
13	C₄×#11715871	—	2	2	no	—
14	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₉₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₉₃	C₄	1	1	yes	characteristic, complemented
18	#15690534	C₄	1	1	yes	characteristic, complemented
19	#15690534	C₄	1	1	yes	characteristic, complemented
20	C₄⋊C₄×C₁₉₃	C₄	1	1	yes	characteristic, complemented
21	C₂×#11715871	—	2	1	no	—
22	C₂×#11715871	—	2	1	no	—
23	C₂×#11715871	—	2	1	no	—
24	C₂×#11715871	—	2	1	no	—
25	C₂×#11715871	—	2	1	no	—
26	C₂×#11715871	—	2	1	no	—
27	C₂×#11715871	—	2	1	no	—
28	C₂×#11715871	—	2	1	no	—
29	C₂×#11715871	—	2	1	no	—
30	C₂×#11715871	—	2	1	no	—
31	C₂×#11715871	D₄	1	2	yes	—
32	C₂×#11715871	Q₈	1	2	yes	—
33	C₂²×D₁₉₃	C₂³	1	1	yes	characteristic
34	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
36	C₂×Dic₁₉₃	C₂×C₄	1	1	yes	characteristic
37	C₄×D₁₉₃	—	2	1	no	—
38	C₄×D₁₉₃	—	2	1	no	—
39	C₄×D₁₉₃	—	2	1	no	—
40	C₄×D₁₉₃	—	2	1	no	—
41	C₄×D₁₉₃	—	2	1	no	—
42	C₄×D₁₉₃	—	2	1	no	—
43	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×D₁₉₃	C₂×D₄	1	1	yes	characteristic
48	C₂×D₁₉₃	C₂×Q₈	1	1	yes	characteristic
49	C₂×D₁₉₃	#41	1	1	yes	characteristic
50	C₂×D₁₉₃	#41	1	1	yes	characteristic
51	C₂×D₁₉₃	#41	1	1	yes	characteristic
52	C₂×D₁₉₃	#41	1	1	yes	characteristic
53	#11715871	—	2	2	no	—
54	#11715871	—	2	2	no	—
55	#11715871	—	4	1	no	—
56	#11715871	—	4	1	no	—
57	C₄×C₁₉₃	—	2	1	no	—
58	C₄×C₁₉₃	—	2	1	no	—
59	C₄×C₁₉₃	—	2	1	no	—
60	Dic₁₉₃	—	2	1	no	—
61	Dic₁₉₃	—	2	1	no	—
62	Dic₁₉₃	—	2	1	no	—
63	C₂×C₁₉₃	#117	1	1	yes	characteristic
64	C₂×C₁₉₃	C₄×D₄	1	1	yes	characteristic
65	C₂×C₁₉₃	C₄×Q₈	1	1	yes	characteristic
66	D₁₉₃	C₂²⋊Q₈	1	1	yes	characteristic
67	D₁₉₃	#123	1	1	yes	characteristic
68	D₁₉₃	#125	1	1	yes	characteristic
69	D₁₉₃	#126	1	1	yes	characteristic
70	C₁₉₃	#387	1	1	yes	characteristic, complemented, Sylow
71	#387	—	193	1	no	maximal, Sylow
72	C₂×C₄⋊C₄	—	193	1	no	—
73	C₂×C₄⋊C₄	—	193	1	no	—
74	C₂×C₄²	—	193	1	no	—
75	#95	—	193	1	no	—
76	#95	—	193	1	no	—
77	#95	—	193	1	no	—
78	#95	—	193	1	no	—
79	C₂²×C₄	—	193	1	no	—
80	C₂²×C₄	—	193	1	no	—
81	C₂²×C₄	—	193	1	no	—
82	C₂²×C₄	—	193	1	no	—
83	C₂²×C₄	—	193	1	no	—
84	C₂²×C₄	—	193	1	no	—
85	C₂²×C₄	—	193	1	no	—
86	C₄⋊C₄	—	386	2	no	—
87	C₄⋊C₄	—	193	1	no	—
88	C₄⋊C₄	—	193	1	no	—
89	C₄⋊C₄	—	193	1	no	—
90	C₄⋊C₄	#11715871	1	1	yes	characteristic, complemented
91	C₄²	—	386	2	no	—
92	C₂³	—	193	1	no	—
93	C₂×C₄	—	386	1	no	—
94	C₂×C₄	—	386	1	no	—
95	C₂×C₄	—	386	1	no	—
96	C₂×C₄	—	386	1	no	—
97	C₂×C₄	—	386	1	no	—
98	C₂×C₄	—	386	1	no	—
99	C₂×C₄	—	386	1	no	—
100	C₂×C₄	—	386	1	no	—
101	C₂×C₄	—	386	1	no	—
102	C₂×C₄	—	386	1	no	—
103	C₂×C₄	—	386	1	no	—
104	C₂×C₄	—	386	1	no	—
105	C₂×C₄	—	386	1	no	—
106	C₂×C₄	—	386	1	no	—
107	C₂×C₄	—	386	1	no	—
108	C₂×C₄	—	386	1	no	—
109	C₂×C₄	—	193	2	no	—
110	C₂×C₄	—	193	2	no	—
111	C₂×C₄	—	193	1	no	—
112	C₂×C₄	—	193	1	no	—
113	C₂×C₄	—	193	1	no	—
114	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
115	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
116	C₂×C₄	C₂×#11715871	1	1	yes	characteristic
117	C₂²	—	193	1	no	—
118	C₂²	—	193	1	no	—
119	C₂²	—	193	1	no	—
120	C₂²	—	193	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	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	772	1	no	—
128	C₄	—	772	1	no	—
129	C₄	—	386	2	no	—
130	C₄	—	386	2	no	—
131	C₄	—	386	1	no	—
132	C₄	—	386	1	no	—
133	C₄	—	386	1	no	—
134	C₂	—	193	1	no	—
135	C₂	—	193	1	no	—
136	C₂	—	193	1	no	—
137	C₂	—	193	1	no	—
138	C₂	#20259978	1	1	yes	characteristic
139	C₂	D₄×#11715871	1	1	yes	characteristic
140	C₂	Q₈×#11715871	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11715871
commutator, socle	C₂²×C₁₉₃	C₂²×C₄
Fitting	C₄⋊C₄×C₁₉₃	C₄
prime radical, cosocle	C₂²×D₁₉₃	C₂³
hypercenter	C₄⋊C₄	#11715871
hypocenter	C₁₉₃	#387
LCS	G ▷ C₂²×C₁₉₃ ▷ C₁₉₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
17. C₄⋊Dic₁₉₃	C₄	semidirect
18. #15690534	C₄	semidirect
19. #15690534	C₄	semidirect
20. C₄⋊C₄×C₁₉₃	C₄	semidirect
70. C₁₉₃	#387	semidirect
90. C₄⋊C₄	#11715871	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11715871	2
C₂×C₄×#11715871	2
(C₂×#11715871)⋊C₄	2
C₄⋊C₄×D₁₉₃	2
#387	193