Finite Group Databasebeta

← C₂×#20261092 #27152054

Subgroups — C₂×#20261092 #27152054

Order 12416.

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15690798	C₂	1	1	yes	maximal, characteristic
2	#20261092	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15690733	C₂	1	2	yes	maximal, complemented
4	C₂×#15690730	C₂	1	2	yes	maximal
5	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#125×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₁₉₄	C₂²	1	1	yes	characteristic
8	C₂²×D₃₈₈	C₂²	1	1	yes	characteristic
9	C₂²×#11715895	C₂²	1	2	yes	—
10	#15690798	—	2	2	no	—
11	#15690798	C₂²	1	4	yes	—
12	C₂×#13095488	—	2	2	no	—
13	#15690733	C₂²	1	8	yes	complemented
14	#15690730	C₂²	1	8	yes	—
15	C₄×#11715895	C₂²	1	4	yes	complemented
16	#125×C₉₇	—	2	2	no	—
17	#125×C₉₇	C₂²	1	4	yes	complemented
18	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
21	C₂³×D₉₇	—	2	1	no	—
22	C₂×Dic₁₉₄	—	2	4	no	—
23	C₂×Dic₁₉₄	—	2	1	no	—
24	C₂×Dic₁₉₄	C₂³	1	2	yes	—
25	C₂×D₃₈₈	—	2	4	no	—
26	C₂×D₃₈₈	—	2	1	no	—
27	C₂×D₃₈₈	C₂³	1	2	yes	—
28	C₂²×Dic₉₇	—	2	1	no	—
29	C₂×#11715895	—	2	4	no	—
30	C₂×#11715895	C₂³	1	4	yes	—
31	#13095488	—	2	8	no	—
32	C₂²×C₄×C₉₇	—	2	2	no	—
33	C₂²×C₄×C₉₇	D₄	1	2	yes	—
34	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₉₇	—	2	8	no	—
36	C₄⋊C₄×C₉₇	—	2	2	no	—
37	C₄⋊C₄×C₉₇	C₂³	1	4	yes	—
38	C₄²×C₉₇	D₄	1	2	yes	—
39	C₄²×C₉₇	C₂³	1	2	yes	—
40	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₉₇	—	2	2	no	—
42	C₂²×D₉₇	—	4	2	no	—
43	C₂²×D₉₇	—	4	4	no	—
44	C₂×C₄×C₉₇	—	2	4	no	—
45	C₂×C₄×C₉₇	—	2	4	no	—
46	C₂×C₄×C₉₇	—	4	4	no	—
47	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₉₇	#41	1	4	yes	—
50	C₂×C₄×C₉₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₉₇	—	2	2	no	—
52	C₂×Dic₉₇	—	4	2	no	—
53	D₃₈₈	—	2	4	no	—
54	D₃₈₈	—	4	2	no	—
55	Dic₁₉₄	—	2	4	no	—
56	Dic₁₉₄	—	4	2	no	—
57	#11715895	—	2	4	no	—
58	C₂²×C₉₇	#124	1	2	yes	—
59	C₂²×C₉₇	#135	1	4	yes	—
60	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₉₇	—	4	2	no	—
62	C₂×D₉₇	—	4	4	no	—
63	C₂×D₉₇	—	8	4	no	—
64	C₄×C₉₇	—	2	4	no	—
65	C₄×C₉₇	—	4	4	no	—
66	C₄×C₉₇	#124	1	2	yes	—
67	C₄×C₉₇	C₂×#41	1	2	yes	—
68	Dic₉₇	—	4	2	no	—
69	C₂×C₉₇	#488	1	4	yes	complemented
70	C₂×C₉₇	C₂×#124	1	1	yes	characteristic
71	C₂×C₉₇	C₂×#135	1	2	yes	—
72	D₉₇	—	8	2	no	—
73	C₂×#488	—	97	1	no	maximal, Sylow
74	C₉₇	C₂×#488	1	1	yes	characteristic, complemented, Sylow
75	C₂×#125	D₉₇	1	1	yes	characteristic, complemented
76	C₂×#124	—	97	1	no	—
77	#488	—	97	8	no	—
78	C₂×#103	—	97	2	no	—
79	C₂×#102	—	97	2	no	—
80	C₂×C₄×C₈	—	97	1	no	—
81	C₂²×Q₈	—	97	1	no	—
82	C₂²×D₄	—	97	1	no	—
83	C₂²×C₈	—	97	2	no	—
84	#125	—	2	2	no	—
85	#125	C₂×D₉₇	1	4	yes	complemented
86	#124	—	97	4	no	—
87	#124	—	194	2	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
90	C₂×C₂²⋊C₄	—	194	2	no	—
91	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
92	#103	—	97	8	no	—
93	#102	—	97	8	no	—
94	C₄×C₈	—	97	4	no	—
95	C₂⁴	—	194	1	no	—
96	C₂×Q₈	—	97	2	no	—
97	C₂×Q₈	—	194	4	no	—
98	C₂×Q₈	—	194	1	no	—
99	C₂×D₄	—	97	2	no	—
100	C₂×D₄	—	194	1	no	—
101	C₂×D₄	—	194	4	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	194	1	no	—
104	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
106	C₂×C₈	—	97	4	no	—
107	C₂×C₈	—	194	4	no	—
108	C₄⋊C₄	—	2	8	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	C₂²×D₉₇	1	4	yes	—
111	C₂²⋊C₄	—	194	8	no	—
112	C₄²	C₂²⋊D₉₇	1	2	yes	—
113	C₄²	C₂²×D₉₇	1	2	yes	—
114	C₂³	—	388	2	no	—
115	C₂³	—	388	4	no	—
116	C₂³	—	194	2	no	—
117	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
118	Q₈	—	388	2	no	—
119	Q₈	—	194	4	no	—
120	D₄	—	388	2	no	—
121	D₄	—	194	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	388	2	no	—
126	C₂×C₄	—	194	2	no	—
127	C₂×C₄	C₂×C₂²⋊D₉₇	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
129	C₂×C₄	#13095512	1	4	yes	—
130	C₂×C₄	C₂³×D₉₇	1	1	yes	characteristic
131	C₈	—	194	4	no	—
132	C₂²	—	776	4	no	—
133	C₂²	—	388	2	no	—
134	C₂²	—	388	4	no	—
135	C₂²	#15690799	1	2	yes	—
136	C₂²	#15690830	1	4	yes	—
137	C₂²	C₂²×C₂²⋊D₉₇	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	4	4	no	—
140	C₄	—	388	2	no	—
141	C₄	#15690799	1	2	yes	—
142	C₄	C₂×#13095512	1	2	yes	—
143	C₂	—	776	2	no	—
144	C₂	#20261092	1	4	yes	direct product factor
145	C₂	C₂×#15690799	1	1	yes	characteristic
146	C₂	C₂×#15690830	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator, cosocle	C₂×C₄×C₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₉₇
Fitting	C₂×#125×C₉₇	C₂
socle	C₂³×C₉₇	C₂×D₄
prime radical	C₂²×D₃₈₈	C₂²
hypercenter	C₂×#125	D₉₇
hypocenter	C₉₇	C₂×#488
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
2. #20261092	C₂	direct (N × H)
3. C₂×#15690733	C₂	semidirect
5. C₂×C₄×#11715895	C₂	semidirect
6. C₂×#125×C₉₇	C₂	semidirect
13. #15690733	C₂²	semidirect
15. C₄×#11715895	C₂²	semidirect
17. #125×C₉₇	C₂²	semidirect
69. C₂×C₉₇	#488	semidirect
74. C₉₇	C₂×#488	semidirect
75. C₂×#125	D₉₇	semidirect
85. #125	C₂×D₉₇	semidirect
144. C₂	#20261092	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15690798	2
#20261092	2
C₂×#15690733	2
C₂×#15690730	2
C₂×C₄×#11715895	2
C₂×#125×C₉₇	2
C₂×#488	97