Finite Group Databasebeta

Subgroups — #27157493

Order 12416.

284 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 86 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×#15690897	C₂	1	1	yes	maximal, characteristic
2	#20261715	C₂	1	1	yes	maximal, characteristic
3	#20261701	C₂	1	2	yes	maximal, complemented
4	#20261618	C₂	1	2	yes	maximal, complemented
5	#20260716	C₂	1	1	yes	maximal, characteristic
6	#15690897	—	2	2	no	—
7	#15690897	—	2	2	no	—
8	C₂×#13095521	C₂²	1	1	yes	characteristic
9	C₂×#13095520	C₄	1	2	yes	complemented
10	C₂×#13095519	C₂²	1	1	yes	characteristic
11	C₂²×#11715897	C₄	1	2	yes	complemented
12	#15690889	C₂²	1	2	yes	—
13	#15690868	C₄	1	4	yes	—
14	#15690864	—	2	2	no	—
15	C₈×#87604	—	2	2	no	—
16	C₂×#13095510	C₂²	1	1	yes	characteristic
17	M₁₆×D₉₇	C₂²	1	2	yes	—
18	#15690728	C₄	1	2	yes	complemented
19	#15690727	C₄	1	1	yes	characteristic, complemented
20	#100×C₉₇	C₄	1	1	yes	characteristic, complemented
21	#13095521	—	2	2	no	—
22	#13095520	—	2	4	no	—
23	#13095519	D₄	1	2	yes	—
24	#13095519	D₄	1	2	yes	—
25	C₂×#11715897	—	2	2	no	—
26	C₂×#11715897	—	2	2	no	—
27	C₂×#11715897	C₂×C₄	1	2	yes	—
28	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
29	C₂×#11715897	C₂×C₄	1	1	yes	characteristic
30	C₄⋊#87604	—	2	2	no	—
31	C₂²⋊#87604	C₂×C₄	1	4	yes	—
32	C₄×#87604	—	2	2	no	—
33	#13095510	—	2	2	no	—
34	#13095510	—	2	2	no	—
35	C₂×Dic₁₉₄	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂²⋊D₉₇	C₂×C₄	1	2	yes	—
37	C₂²×Dic₉₇	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
39	#13095495	C₂×C₄	1	2	yes	—
40	#13095494	—	2	2	no	—
41	C₈×D₉₇	—	2	2	no	—
42	C₂×D₄×C₉₇	C₂×C₄	1	1	yes	characteristic
43	M₁₆×C₉₇	C₂×C₄	1	2	yes	—
44	C₂³×C₉₇	C₄²	1	2	yes	—
45	C₂²×D₉₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×#87604	—	2	4	no	—
47	D₄×C₉₇	—	2	2	no	—
48	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²⋊D₉₇	—	2	4	no	—
50	C₂×Dic₉₇	—	2	2	no	—
51	C₂×Dic₉₇	—	2	2	no	—
52	C₂×Dic₉₇	C₄²	1	2	yes	—
53	C₂×Dic₉₇	C₂²×C₄	1	1	yes	characteristic
54	C₄×D₉₇	C₂×D₄	1	2	yes	—
55	C₄×D₉₇	#41	1	2	yes	—
56	Dic₁₉₄	—	2	2	no	—
57	#11715897	—	2	2	no	—
58	#11715897	C₂²⋊C₄	1	1	yes	characteristic
59	#11715897	C₂²⋊C₄	1	1	yes	characteristic
60	#11715897	C₂²⋊C₄	1	2	yes	—
61	C₈×C₉₇	—	2	2	no	—
62	#11715895	—	2	2	no	—
63	C₂²×C₉₇	—	2	2	no	—
64	C₂²×C₉₇	—	4	2	no	—
65	C₂²×C₉₇	C₂×C₄²	1	1	yes	characteristic
66	C₂×D₉₇	—	4	1	no	—
67	C₂×D₉₇	C₄×D₄	1	2	yes	—
68	#87604	—	4	4	no	—
69	C₄×C₉₇	C₄×D₄	1	2	yes	—
70	Dic₉₇	—	2	2	no	—
71	Dic₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	Dic₉₇	#117	1	1	yes	characteristic
73	C₂×C₉₇	—	2	1	no	—
74	C₂×C₉₇	—	4	2	no	—
75	C₂×C₉₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
76	D₉₇	—	2	2	no	—
77	#1757	—	97	1	no	maximal, Sylow
78	C₉₇	#1757	1	1	yes	characteristic, complemented, Sylow
79	C₂×#131	—	97	1	no	—
80	#413	—	97	1	no	—
81	#410	—	97	1	no	—
82	#405	—	97	2	no	—
83	#344	—	97	2	no	—
84	C₂×#41	—	97	1	no	—
85	#131	—	194	2	no	—
86	#131	—	194	2	no	—
87	C₂×M₁₆	—	97	1	no	—
88	C₂×M₁₆	—	97	2	no	—
89	C₂×M₁₆	—	97	2	no	—
90	C₂²×C₈	—	97	1	no	—
91	C₂²×C₈	—	97	2	no	—
92	#117	—	97	2	no	—
93	#101	—	97	2	no	—
94	#100	—	97	1	no	—
95	#100	#87604	1	1	yes	characteristic, complemented
96	#99	—	97	4	no	—
97	#97	—	194	2	no	—
98	C₄×C₈	—	194	2	no	—
99	#41	—	194	2	no	—
100	#41	—	194	2	no	—
101	C₂×Q₈	—	97	1	no	—
102	C₂×D₄	—	97	2	no	—
103	C₂×D₄	C₂×#87604	1	1	yes	characteristic
104	C₂²×C₄	—	97	1	no	—
105	C₂²×C₄	—	97	2	no	—
106	M₁₆	—	97	2	no	—
107	M₁₆	—	194	2	no	—
108	M₁₆	—	194	4	no	—
109	M₁₆	—	194	2	no	—
110	M₁₆	C₂×#87604	1	2	yes	—
111	C₂×C₈	—	97	1	no	—
112	C₂×C₈	—	97	1	no	—
113	C₂×C₈	—	97	2	no	—
114	C₂×C₈	—	97	2	no	—
115	C₂×C₈	—	97	2	no	—
116	C₂×C₈	—	194	2	no	—
117	C₂×C₈	—	194	2	no	—
118	C₂×C₈	—	194	2	no	—
119	C₄⋊C₄	—	194	2	no	—
120	C₂²⋊C₄	—	97	4	no	—
121	C₄²	—	194	2	no	—
122	C₂³	—	97	1	no	—
123	C₂³	C₄×#87604	1	2	yes	—
124	Q₈	—	194	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	194	4	no	—
127	C₂×C₄	—	97	1	no	—
128	C₂×C₄	—	97	2	no	—
129	C₂×C₄	—	97	2	no	—
130	C₂×C₄	—	97	2	no	—
131	C₂×C₄	—	194	2	no	—
132	C₂×C₄	—	194	2	no	—
133	C₂×C₄	—	194	4	no	—
134	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	97	1	no	—
137	C₈	—	97	1	no	—
138	C₈	—	97	2	no	—
139	C₈	—	194	2	no	—
140	C₈	—	194	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	2	no	—
143	C₂²	—	97	2	no	—
144	C₂²	—	388	1	no	—
145	C₂²	C₂×C₄×#87604	1	1	yes	characteristic
146	C₄	—	97	1	no	—
147	C₄	—	97	1	no	—
148	C₄	—	388	4	no	—
149	C₄	—	194	2	no	—
150	C₄	D₄×#87604	1	2	yes	—
151	C₂	—	2	1	no	—
152	C₂	—	4	2	no	—
153	C₂	—	194	2	no	—
154	C₂	C₂²⋊C₄×#87604	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×#87604
commutator	C₂²×C₉₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#87604
Fitting	#100×C₉₇	C₄
socle	C₂×C₉₇	C₄×C₂²⋊C₄
prime radical	C₂×#13095510	C₂²
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	#100	#87604
hypocenter	C₉₇	#1757
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

Complements

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

N	H ≅ G/N	type
3. #20261701	C₂	semidirect
4. #20261618	C₂	semidirect
9. C₂×#13095520	C₄	semidirect
11. C₂²×#11715897	C₄	semidirect
18. #15690728	C₄	semidirect
19. #15690727	C₄	semidirect
20. #100×C₉₇	C₄	semidirect
78. C₉₇	#1757	semidirect
95. #100	#87604	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690897	2
#20261715	2
#20261701	2
#20261618	2
#20260716	2
#1757	97