Finite Group Databasebeta

Subgroups — #27158912

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	C₂×#15690906	C₂	1	2	yes	maximal, complemented
2	#20261834	C₂	1	2	yes	maximal, complemented
3	#20261833	C₂	1	2	yes	maximal, complemented
4	C₂×#15690897	C₂	1	1	yes	maximal, characteristic
5	#15690906	—	2	2	no	—
6	#15690906	C₂²	1	4	yes	—
7	C₂×#13095523	—	2	2	no	—
8	#15690897	—	2	2	no	—
9	#15690897	—	2	2	no	—
10	C₂×#13095521	C₂²	1	1	yes	characteristic
11	C₂×#13095520	C₄	1	1	yes	characteristic
12	C₂×#13095520	C₂²	1	1	yes	characteristic
13	C₂×#13095519	C₄	1	1	yes	characteristic
14	C₂²×#11715897	C₄	1	1	yes	characteristic
15	C₂²×#11715897	C₂²	1	1	yes	characteristic
16	C₂×#13095510	C₄	1	1	yes	characteristic
17	#13095523	—	2	4	no	—
18	#13095521	—	2	2	no	—
19	#13095520	—	2	2	no	—
20	#13095520	—	2	2	no	—
21	#13095519	—	2	1	no	—
22	#13095519	—	2	1	no	—
23	C₂×#11715897	—	2	1	no	—
24	C₂×#11715897	—	2	1	no	—
25	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
26	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
27	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
28	C₂×#11715897	D₄	1	2	yes	—
29	C₂×#11715897	D₄	1	2	yes	—
30	C₂×#11715897	C₂³	1	1	yes	characteristic
31	#13095510	—	2	2	no	—
32	#13095510	—	2	2	no	—
33	C₂×Dic₁₉₄	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊D₉₇	C₈	1	1	yes	characteristic
35	C₂×C₂²⋊D₉₇	C₈	1	1	yes	characteristic
36	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×Dic₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
40	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
41	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
43	D₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
45	C₂²⋊D₉₇	—	2	2	no	—
46	C₂²⋊D₉₇	—	2	2	no	—
47	C₂×Dic₉₇	—	2	1	no	—
48	C₂×Dic₉₇	—	2	1	no	—
49	C₂×Dic₉₇	C₂²⋊C₄	1	2	yes	—
50	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
51	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
52	C₂×Dic₉₇	M₁₆	1	2	yes	—
53	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₉₇	—	2	1	no	—
55	C₄×D₉₇	—	2	1	no	—
56	Dic₁₉₄	—	2	2	no	—
57	#11715897	—	2	1	no	—
58	#11715897	—	2	1	no	—
59	#11715897	C₂²⋊C₄	1	1	yes	characteristic
60	#11715897	C₂²⋊C₄	1	1	yes	characteristic
61	#11715897	C₂×D₄	1	2	yes	—
62	C₂²×C₉₇	—	2	1	no	—
63	C₂²×C₉₇	—	4	1	no	—
64	C₂²×C₉₇	—	4	1	no	—
65	C₂²×C₉₇	C₂²⋊C₈	1	2	yes	—
66	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
67	C₂×D₉₇	—	2	1	no	—
68	C₂×D₉₇	—	4	1	no	—
69	C₄×C₉₇	—	2	1	no	—
70	Dic₉₇	—	2	1	no	—
71	Dic₉₇	C₂²⋊C₈	1	2	yes	—
72	Dic₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	Dic₉₇	C₂×M₁₆	1	1	yes	characteristic
74	C₂×C₉₇	—	2	1	no	—
75	C₂×C₉₇	—	2	2	no	—
76	C₂×C₉₇	—	4	1	no	—
77	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	D₉₇	—	4	1	no	—
79	#2115	—	97	1	no	maximal, Sylow
80	C₉₇	#2115	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	97	1	no	—
82	C₂×M₃₂	—	97	2	no	—
83	#349	—	97	2	no	—
84	#349	—	97	2	no	—
85	C₂×#41	—	97	1	no	—
86	#131	—	194	2	no	—
87	#131	—	194	2	no	—
88	C₂×M₁₆	—	97	1	no	—
89	C₂×M₁₆	—	97	1	no	—
90	C₂×M₁₆	—	97	1	no	—
91	C₂²×C₈	—	97	1	no	—
92	C₂²×C₈	—	97	1	no	—
93	C₂²×C₈	—	97	1	no	—
94	M₃₂	—	97	4	no	—
95	M₃₂	—	194	2	no	—
96	C₂×C₁₆	—	194	2	no	—
97	#41	—	194	2	no	—
98	#41	—	194	2	no	—
99	C₂×Q₈	—	97	1	no	—
100	C₂×D₄	—	97	1	no	—
101	C₂×D₄	—	97	1	no	—
102	C₂×D₄	#11715906	1	1	yes	characteristic
103	C₂²×C₄	—	97	1	no	—
104	C₂²×C₄	—	97	1	no	—
105	C₂²×C₄	—	97	1	no	—
106	M₁₆	—	194	2	no	—
107	M₁₆	—	194	2	no	—
108	M₁₆	—	194	2	no	—
109	C₂×C₈	—	97	1	no	—
110	C₂×C₈	—	97	2	no	—
111	C₂×C₈	—	97	2	no	—
112	C₂×C₈	—	97	1	no	—
113	C₂×C₈	—	97	1	no	—
114	C₂×C₈	—	97	1	no	—
115	C₂×C₈	—	194	1	no	—
116	C₂×C₈	—	194	1	no	—
117	C₂×C₈	—	194	1	no	—
118	C₂×C₈	—	194	1	no	—
119	C₁₆	—	194	4	no	—
120	C₂³	—	97	1	no	—
121	C₂³	C₂×#11715906	1	1	yes	characteristic
122	C₂³	C₂×#11715906	1	1	yes	characteristic
123	Q₈	—	194	2	no	—
124	D₄	—	2	2	no	—
125	D₄	—	194	2	no	—
126	D₄	—	194	2	no	—
127	C₂×C₄	—	97	2	no	—
128	C₂×C₄	—	97	2	no	—
129	C₂×C₄	—	97	1	no	—
130	C₂×C₄	—	97	1	no	—
131	C₂×C₄	—	97	1	no	—
132	C₂×C₄	—	194	1	no	—
133	C₂×C₄	—	194	1	no	—
134	C₂×C₄	—	194	1	no	—
135	C₂×C₄	—	194	1	no	—
136	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
137	C₈	—	97	2	no	—
138	C₈	—	97	1	no	—
139	C₈	—	97	1	no	—
140	C₈	—	194	1	no	—
141	C₈	—	194	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	388	1	no	—
146	C₂²	—	194	1	no	—
147	C₂²	C₂²⋊#11715906	1	2	yes	—
148	C₂²	C₂²×#11715906	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	97	1	no	—
151	C₄	—	97	1	no	—
152	C₄	—	97	2	no	—
153	C₄	—	194	1	no	—
154	C₂	—	2	1	no	—
155	C₂	—	2	2	no	—
156	C₂	—	4	1	no	—
157	C₂	—	388	1	no	—
158	C₂	C₂×C₂²⋊#11715906	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11715906
commutator	C₂²×C₉₇	C₂²×C₈
Frattini	C₂²	C₂²×#11715906
Fitting	C₂×D₄×C₉₇	C₈
socle	C₂×C₉₇	C₂×C₂²⋊C₈
prime radical	C₂×#13095510	C₄
cosocle	C₂×#11715897	C₂³
hypercenter	C₂×D₄	#11715906
hypocenter	C₉₇	#2115
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15690906	C₂	semidirect
2. #20261834	C₂	semidirect
3. #20261833	C₂	semidirect
80. C₉₇	#2115	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690906	2
#20261834	2
#20261833	2
C₂×#15690897	2
#2115	97