Finite Group Databasebeta

Subgroups — #27157308

Order 12416.

150 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 60 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	#20261757	C₂	1	1	yes	maximal, characteristic
2	#20261754	C₂	1	1	yes	maximal, characteristic
3	#20261599	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261598	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261597	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261596	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260787	C₂	1	1	yes	maximal, characteristic
8	C₄⋊#11715897	C₂²	1	1	yes	characteristic
9	C₄⋊#11715897	C₂²	1	1	yes	characteristic
10	C₄⋊#11715897	C₂²	1	1	yes	characteristic
11	C₄⋊#11715897	C₂²	1	1	yes	characteristic
12	#15690867	—	2	1	no	—
13	#15690867	—	2	1	no	—
14	#15690865	—	2	1	no	—
15	C₄×#11715897	—	2	1	no	—
16	C₄²×D₉₇	C₂²	1	1	yes	characteristic
17	C₄⋊#11715895	C₄	1	1	yes	characteristic
18	#15690722	C₄	1	1	yes	characteristic
19	#15690722	C₄	1	1	yes	characteristic
20	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
21	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₉₇	C₄	1	1	yes	characteristic
23	C₂×#11715897	—	2	1	no	—
24	C₂×#11715897	—	2	1	no	—
25	C₂×#11715897	—	2	1	no	—
26	C₂×#11715897	—	2	1	no	—
27	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
29	C₂×C₄×D₉₇	Q₈	1	1	yes	characteristic
30	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
31	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₉₇	D₄	1	1	yes	characteristic
33	C₄×Dic₉₇	Q₈	1	1	yes	characteristic
34	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×D₉₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
45	C₄×D₉₇	—	2	1	no	—
46	C₄×D₉₇	—	2	1	no	—
47	C₄×D₉₇	—	2	1	no	—
48	C₄×D₉₇	—	2	1	no	—
49	C₄×D₉₇	—	4	1	no	—
50	#11715897	—	4	1	no	—
51	#11715897	—	4	1	no	—
52	#11715897	—	4	1	no	—
53	#11715897	—	4	1	no	—
54	C₈×C₉₇	—	2	1	no	—
55	C₈×C₉₇	—	2	1	no	—
56	#11715895	—	2	1	no	—
57	#11715895	—	2	1	no	—
58	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₉₇	—	2	1	no	—
60	C₂×D₉₇	—	2	1	no	—
61	C₂×D₉₇	—	2	1	no	—
62	C₄×C₉₇	—	2	1	no	—
63	C₄×C₉₇	—	2	1	no	—
64	C₄×C₉₇	#131	1	1	yes	characteristic
65	C₄×C₉₇	#131	1	1	yes	characteristic
66	Dic₉₇	—	2	1	no	—
67	Dic₉₇	—	2	1	no	—
68	Dic₉₇	#135	1	1	yes	characteristic
69	Dic₉₇	#135	1	1	yes	characteristic
70	C₂×C₉₇	#424	1	1	yes	characteristic
71	C₂×C₉₇	#427	1	1	yes	characteristic
72	C₂×C₉₇	#430	1	1	yes	characteristic
73	D₉₇	—	4	1	no	—
74	#1572	—	97	1	no	maximal, Sylow
75	C₉₇	#1572	1	1	yes	characteristic, complemented, Sylow
76	#432	—	97	1	no	—
77	#431	—	97	1	no	—
78	#431	—	97	1	no	—
79	#335	—	97	1	no	—
80	#335	—	97	1	no	—
81	#334	—	97	1	no	—
82	#334	—	97	1	no	—
83	C₂×C₄²	—	97	1	no	—
84	C₄⋊C₈	—	97	1	no	—
85	C₄⋊C₈	—	97	1	no	—
86	C₄⋊C₈	—	97	1	no	—
87	C₄⋊C₈	—	97	1	no	—
88	C₄⋊C₈	—	97	1	no	—
89	C₄⋊C₈	#87604	1	1	yes	characteristic
90	C₂²⋊C₈	—	97	1	no	—
91	C₂²⋊C₈	—	97	1	no	—
92	C₂²⋊C₈	—	194	1	no	—
93	C₂²⋊C₈	—	194	1	no	—
94	#97	—	194	1	no	—
95	C₄×C₈	—	97	1	no	—
96	C₄×C₈	—	97	1	no	—
97	C₄×C₈	—	194	1	no	—
98	C₂²×C₄	—	97	1	no	—
99	C₂²×C₄	—	97	1	no	—
100	C₂²×C₄	—	97	1	no	—
101	C₂×C₈	—	97	1	no	—
102	C₂×C₈	—	97	1	no	—
103	C₂×C₈	—	194	1	no	—
104	C₂×C₈	—	194	1	no	—
105	C₂×C₈	—	194	1	no	—
106	C₂×C₈	—	194	1	no	—
107	C₂×C₈	C₂×#87604	1	1	yes	characteristic
108	C₂×C₈	C₂×#87604	1	1	yes	characteristic
109	C₄²	—	97	1	no	—
110	C₄²	—	97	1	no	—
111	C₄²	—	97	1	no	—
112	C₄²	C₂×#87604	1	1	yes	characteristic
113	C₂³	—	97	1	no	—
114	C₂×C₄	—	97	1	no	—
115	C₂×C₄	—	97	1	no	—
116	C₂×C₄	—	97	1	no	—
117	C₂×C₄	—	388	1	no	—
118	C₂×C₄	—	194	1	no	—
119	C₂×C₄	—	194	1	no	—
120	C₂×C₄	—	194	1	no	—
121	C₂×C₄	—	194	1	no	—
122	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
123	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
124	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	388	1	no	—
128	C₈	—	388	1	no	—
129	C₈	—	388	1	no	—
130	C₈	—	388	1	no	—
131	C₈	—	194	1	no	—
132	C₈	—	194	1	no	—
133	C₂²	—	194	1	no	—
134	C₂²	—	194	1	no	—
135	C₂²	—	194	1	no	—
136	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	97	1	no	—
140	C₄	—	97	1	no	—
141	C₄	—	194	1	no	—
142	C₄	—	194	1	no	—
143	C₄	#15690897	1	1	yes	characteristic
144	C₄	#15690898	1	1	yes	characteristic
145	C₂	—	388	1	no	—
146	C₂	#20261742	1	1	yes	characteristic
147	C₂	#20261745	1	1	yes	characteristic
148	C₂	#20261750	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₄⋊C₈×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂²×D₉₇	C₄⋊C₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	C₄⋊C₈	#87604
hypocenter	C₉₇	#1572
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #20261599	C₂	semidirect
4. #20261598	C₂	semidirect
5. #20261597	C₂	semidirect
6. #20261596	C₂	semidirect
75. C₉₇	#1572	semidirect

Maximal subgroups

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

subgroup	index
#20261757	2
#20261754	2
#20261599	2
#20261598	2
#20261597	2
#20261596	2
#20260787	2
#1572	97