Finite Group Databasebeta

Subgroups — #27158918

Order 12416.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20261854	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261854	C₂	1	1	yes	maximal, characteristic, complemented
3	#20261835	C₂	1	4	yes	maximal, complemented
4	C₂×#15690897	C₂	1	1	yes	maximal, characteristic
5	#15690907	—	2	1	no	—
6	#15690907	C₂²	1	2	yes	complemented
7	#15690906	—	2	1	no	—
8	#15690906	—	2	1	no	—
9	#15690905	—	2	1	no	—
10	#15690905	C₂²	1	2	yes	complemented
11	#15690897	—	2	1	no	—
12	#15690897	—	2	1	no	—
13	#15690897	C₄	1	2	yes	—
14	#15690897	C₂²	1	2	yes	—
15	C₂×#13095521	C₄	1	1	yes	characteristic
16	C₂×#13095520	—	2	1	no	—
17	C₂×#13095519	C₂²	1	1	yes	characteristic
18	C₂²×#11715897	—	2	1	no	—
19	C₂×#13095510	C₄	1	1	yes	characteristic
20	#13095523	—	2	2	no	—
21	#13095523	—	2	2	no	—
22	#13095521	—	2	1	no	—
23	#13095521	C₂×C₄	1	2	yes	—
24	#13095520	—	2	2	no	—
25	#13095520	—	2	2	no	—
26	#13095519	C₂×C₄	1	1	yes	characteristic
27	#13095519	D₄	1	1	yes	characteristic
28	#13095519	D₄	1	1	yes	characteristic
29	#13095519	C₂³	1	1	yes	characteristic
30	C₂×#11715897	—	2	1	no	—
31	C₂×#11715897	—	2	2	no	—
32	C₂×#11715897	—	2	2	no	—
33	C₂×#11715897	D₄	1	1	yes	characteristic
34	C₂×#11715897	D₄	1	1	yes	characteristic
35	#13095510	—	2	1	no	—
36	#13095510	—	2	1	no	—
37	#13095510	C₈	1	2	yes	—
38	#13095510	C₂×C₄	1	2	yes	—
39	C₂×Dic₁₉₄	C₈	1	1	yes	characteristic
40	C₂×C₂²⋊D₉₇	—	2	1	no	—
41	C₂²×Dic₉₇	—	2	1	no	—
42	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
44	C₂³×C₉₇	—	2	1	no	—
45	C₂²×D₉₇	M₁₆	1	1	yes	characteristic
46	D₄×C₉₇	—	2	1	no	—
47	D₄×C₉₇	C₂×C₈	1	2	yes	—
48	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
49	C₂²⋊D₉₇	—	2	2	no	—
50	C₂²⋊D₉₇	—	2	2	no	—
51	C₂×Dic₉₇	—	2	1	no	—
52	C₂×Dic₉₇	—	2	2	no	—
53	C₂×Dic₉₇	—	2	2	no	—
54	C₂×Dic₉₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₄×D₉₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
57	C₄×D₉₇	M₁₆	1	1	yes	characteristic
58	C₄×D₉₇	C₂²×C₄	1	1	yes	characteristic
59	Dic₁₉₄	—	2	1	no	—
60	Dic₁₉₄	C₂×C₈	1	2	yes	—
61	#11715897	—	2	2	no	—
62	#11715897	C₂²⋊C₄	1	1	yes	characteristic
63	#11715897	C₂²⋊C₄	1	1	yes	characteristic
64	#11715897	C₂×D₄	1	1	yes	characteristic
65	#11715897	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₉₇	—	2	2	no	—
67	C₂²×C₉₇	—	4	2	no	—
68	C₂²×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
69	C₂×D₉₇	—	4	1	no	—
70	C₂×D₉₇	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×D₉₇	C₂×M₁₆	1	1	yes	characteristic
72	C₄×C₉₇	C₂²⋊C₈	1	1	yes	characteristic
73	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
74	Dic₉₇	—	2	2	no	—
75	Dic₉₇	C₂²⋊C₈	1	1	yes	characteristic
76	Dic₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₉₇	—	2	1	no	—
78	C₂×C₉₇	—	4	2	no	—
79	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
80	D₉₇	—	2	1	no	—
81	D₉₇	—	2	1	no	—
82	#2117	—	97	1	no	maximal, Sylow
83	C₉₇	#2117	1	1	yes	characteristic, complemented, Sylow
84	C₂×#131	—	97	1	no	—
85	C₂×M₃₂	—	97	1	no	—
86	C₂×M₃₂	—	97	1	no	—
87	#350	—	97	4	no	—
88	C₂×#41	—	97	1	no	—
89	#131	—	97	2	no	—
90	#131	—	97	2	no	—
91	#131	—	194	1	no	—
92	#131	—	194	1	no	—
93	C₂×M₁₆	—	97	1	no	—
94	C₂×M₁₆	—	194	1	no	—
95	C₂²×C₈	—	97	1	no	—
96	C₂²×C₈	—	194	1	no	—
97	M₃₂	—	97	2	no	—
98	M₃₂	—	194	1	no	—
99	M₃₂	—	194	1	no	—
100	M₃₂	—	194	1	no	—
101	C₂×C₁₆	—	97	2	no	—
102	C₂×C₁₆	—	194	1	no	—
103	#41	—	97	2	no	—
104	#41	—	97	2	no	—
105	#41	—	194	1	no	—
106	#41	—	194	1	no	—
107	C₂×Q₈	—	97	1	no	—
108	C₂×D₄	—	194	1	no	—
109	C₂×D₄	#11715906	1	1	yes	characteristic
110	C₂²×C₄	—	97	1	no	—
111	C₂²×C₄	—	194	1	no	—
112	M₁₆	—	97	2	no	—
113	M₁₆	—	194	2	no	—
114	M₁₆	—	194	2	no	—
115	M₁₆	—	194	1	no	—
116	C₂×C₈	—	97	1	no	—
117	C₂×C₈	—	97	1	no	—
118	C₂×C₈	—	97	1	no	—
119	C₂×C₈	—	97	1	no	—
120	C₂×C₈	—	97	1	no	—
121	C₂×C₈	—	97	1	no	—
122	C₂×C₈	—	194	2	no	—
123	C₂×C₈	—	194	2	no	—
124	C₂×C₈	—	194	1	no	—
125	C₁₆	—	194	2	no	—
126	C₁₆	—	194	2	no	—
127	C₂³	—	2	1	no	—
128	C₂³	—	97	1	no	—
129	Q₈	—	97	2	no	—
130	Q₈	—	194	1	no	—
131	D₄	—	2	1	no	—
132	D₄	—	194	2	no	—
133	D₄	—	194	2	no	—
134	D₄	C₂×#11715906	1	2	yes	—
135	C₂×C₄	—	97	1	no	—
136	C₂×C₄	—	97	1	no	—
137	C₂×C₄	—	97	1	no	—
138	C₂×C₄	—	97	1	no	—
139	C₂×C₄	—	97	1	no	—
140	C₂×C₄	—	194	2	no	—
141	C₂×C₄	—	194	2	no	—
142	C₂×C₄	—	194	1	no	—
143	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
144	C₈	—	97	1	no	—
145	C₈	—	97	1	no	—
146	C₈	—	97	1	no	—
147	C₈	—	97	1	no	—
148	C₈	—	194	2	no	—
149	C₂²	—	2	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	97	1	no	—
152	C₂²	—	97	1	no	—
153	C₂²	—	388	1	no	—
154	C₂²	C₂²⋊#11715906	1	1	yes	characteristic
155	C₄	—	97	1	no	—
156	C₄	—	97	1	no	—
157	C₄	—	194	2	no	—
158	C₄	C₂²⋊#11715906	1	1	yes	characteristic
159	C₄	C₂²×#11715906	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	4	2	no	—
162	C₂	—	194	1	no	—
163	C₂	—	194	1	no	—
164	C₂	C₂×C₂²⋊#11715906	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11715906
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₄	C₂²×#11715906
Fitting	C₂×D₄×C₉₇	C₈
socle	C₂×C₉₇	C₂×C₂²⋊C₈
prime radical	C₂×#13095510	C₄
cosocle	#13095519	C₂³
hypercenter	C₂×D₄	#11715906
hypocenter	C₉₇	#2117
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #20261854	C₂	semidirect
2. #20261854	C₂	semidirect
3. #20261835	C₂	semidirect
6. #15690907	C₂²	semidirect
10. #15690905	C₂²	semidirect
83. C₉₇	#2117	semidirect

Maximal subgroups

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

subgroup	index
#20261854	2
#20261835	2
C₂×#15690897	2
#2117	97