Finite Group Databasebeta

Subgroups — #27157310

Order 12416.

160 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 62 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	#20261736	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	C₂×#13095519	—	2	1	no	—
9	#15690877	—	2	1	no	—
10	C₄⋊#11715897	C₂²	1	1	yes	characteristic
11	C₄⋊#11715897	C₂²	1	1	yes	characteristic
12	C₄⋊#11715897	C₂²	1	1	yes	characteristic
13	C₄⋊#11715897	C₂²	1	1	yes	characteristic
14	#15690867	—	2	1	no	—
15	#15690865	—	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	#13095519	—	4	2	no	—
24	C₂×#11715897	—	2	1	no	—
25	C₂×#11715897	—	2	1	no	—
26	C₂×#11715897	—	2	1	no	—
27	C₂×#11715897	—	2	1	no	—
28	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
30	C₂×C₄×D₉₇	Q₈	1	1	yes	characteristic
31	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
32	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₉₇	D₄	1	1	yes	characteristic
34	C₄×Dic₉₇	Q₈	1	1	yes	characteristic
35	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂²×D₉₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₉₇	—	2	2	no	—
47	C₄×D₉₇	—	2	1	no	—
48	C₄×D₉₇	—	2	1	no	—
49	C₄×D₉₇	—	2	1	no	—
50	C₄×D₉₇	—	2	1	no	—
51	#11715897	—	4	1	no	—
52	#11715897	—	4	1	no	—
53	#11715897	—	4	1	no	—
54	#11715897	—	4	1	no	—
55	C₈×C₉₇	—	2	1	no	—
56	C₈×C₉₇	—	2	1	no	—
57	#11715895	—	2	1	no	—
58	#11715895	—	2	1	no	—
59	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×D₉₇	—	2	1	no	—
61	C₂×D₉₇	—	2	1	no	—
62	C₂×D₉₇	#108	1	2	yes	—
63	C₄×C₉₇	—	2	1	no	—
64	C₄×C₉₇	—	2	1	no	—
65	C₄×C₉₇	#131	1	1	yes	characteristic
66	C₄×C₉₇	#131	1	1	yes	characteristic
67	Dic₉₇	—	2	1	no	—
68	Dic₉₇	—	2	1	no	—
69	Dic₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
70	Dic₉₇	#137	1	1	yes	characteristic
71	C₂×C₉₇	#424	1	1	yes	characteristic
72	C₂×C₉₇	#428	1	1	yes	characteristic
73	C₂×C₉₇	C₂×#108	1	1	yes	characteristic
74	D₉₇	—	2	2	no	—
75	#1573	—	97	1	no	maximal, Sylow
76	C₉₇	#1573	1	1	yes	characteristic, complemented, Sylow
77	#432	—	97	1	no	—
78	#431	—	97	1	no	—
79	C₂×C₄⋊C₈	—	97	1	no	—
80	#335	—	97	1	no	—
81	#335	—	97	1	no	—
82	#334	—	97	1	no	—
83	#334	—	97	1	no	—
84	C₂²×C₈	—	194	1	no	—
85	C₂×C₄²	—	97	1	no	—
86	C₄⋊C₈	—	97	1	no	—
87	C₄⋊C₈	—	97	1	no	—
88	C₄⋊C₈	—	97	1	no	—
89	C₄⋊C₈	—	97	1	no	—
90	C₄⋊C₈	—	97	1	no	—
91	C₄⋊C₈	—	194	1	no	—
92	C₄⋊C₈	#87604	1	1	yes	characteristic
93	C₂²⋊C₈	—	97	1	no	—
94	C₂²⋊C₈	—	97	1	no	—
95	C₂²⋊C₈	—	194	1	no	—
96	#97	—	194	1	no	—
97	C₄×C₈	—	97	1	no	—
98	C₄×C₈	—	97	1	no	—
99	C₂²×C₄	—	97	1	no	—
100	C₂²×C₄	—	97	1	no	—
101	C₂²×C₄	—	97	1	no	—
102	C₂×C₈	—	97	1	no	—
103	C₂×C₈	—	97	1	no	—
104	C₂×C₈	—	388	2	no	—
105	C₂×C₈	—	194	1	no	—
106	C₂×C₈	—	194	1	no	—
107	C₂×C₈	—	194	1	no	—
108	C₂×C₈	—	194	1	no	—
109	C₂×C₈	C₂×#87604	1	1	yes	characteristic
110	C₂×C₈	C₂×#87604	1	1	yes	characteristic
111	C₄²	—	97	1	no	—
112	C₄²	—	97	1	no	—
113	C₄²	—	97	1	no	—
114	C₄²	C₂×#87604	1	1	yes	characteristic
115	C₂³	—	97	1	no	—
116	C₂×C₄	—	97	1	no	—
117	C₂×C₄	—	97	1	no	—
118	C₂×C₄	—	97	1	no	—
119	C₂×C₄	—	194	1	no	—
120	C₂×C₄	—	194	1	no	—
121	C₂×C₄	—	194	2	no	—
122	C₂×C₄	—	194	1	no	—
123	C₂×C₄	—	194	1	no	—
124	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
125	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
126	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
127	C₈	—	2	1	no	—
128	C₈	—	2	1	no	—
129	C₈	—	388	1	no	—
130	C₈	—	388	1	no	—
131	C₈	—	388	1	no	—
132	C₈	—	388	1	no	—
133	C₈	—	194	1	no	—
134	C₈	—	194	1	no	—
135	C₂²	—	97	2	no	—
136	C₂²	—	194	1	no	—
137	C₂²	—	194	1	no	—
138	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	97	1	no	—
142	C₄	—	97	1	no	—
143	C₄	—	194	1	no	—
144	C₄	—	194	1	no	—
145	C₄	#15690897	1	1	yes	characteristic
146	C₄	#15690898	1	1	yes	characteristic
147	C₂	—	194	2	no	—
148	C₂	#20261742	1	1	yes	characteristic
149	C₂	#20261746	1	1	yes	characteristic
150	C₂	D₉₇⋊#108	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	#1573
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
76. C₉₇	#1573	semidirect

Maximal subgroups

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

subgroup	index
#20261757	2
#20261736	2
#20261599	2
#20261598	2
#20261597	2
#20261596	2
#20260787	2
#1573	97