Finite Group Databasebeta

Subgroups — #27158119

Order 12416.

164 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 64 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×#15690885	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15690883	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#13095514	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261630	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261630	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261623	C₂	1	1	yes	maximal, characteristic, complemented
7	#20260800	C₂	1	1	yes	maximal, characteristic
8	#15690885	—	2	2	no	—
9	#15690883	—	2	2	no	—
10	C₂×#13095514	C₂²	1	1	yes	characteristic
11	C₂×#13095514	C₂²	1	1	yes	characteristic
12	C₂×#13095514	C₂²	1	1	yes	characteristic
13	C₂×#13095514	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
16	C₄×D₃₈₈	C₄	1	1	yes	characteristic
17	#15690722	C₄	1	1	yes	characteristic
18	#15690722	C₄	1	1	yes	characteristic
19	C₄×#11715895	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₉₇	C₄	1	1	yes	characteristic
21	#13095514	—	2	1	no	—
22	#13095514	—	2	1	no	—
23	#13095514	—	2	1	no	—
24	#13095514	D₄	1	2	yes	complemented
25	C₂×D₃₈₈	C₈	1	1	yes	characteristic
26	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
29	C₂×#11715895	C₂³	1	1	yes	characteristic
30	C₈×D₉₇	—	2	1	no	—
31	C₈×D₉₇	—	2	1	no	—
32	C₈×D₉₇	—	2	1	no	—
33	C₈×D₉₇	—	2	1	no	—
34	C₄⋊Dic₉₇	C₈	1	1	yes	characteristic
35	#13095488	C₈	1	1	yes	characteristic
36	#13095488	C₈	1	1	yes	characteristic
37	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₉₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
46	C₂×Dic₉₇	C₂×C₈	1	1	yes	characteristic
47	D₃₈₈	—	2	2	no	—
48	C₄×D₉₇	—	2	1	no	—
49	C₄×D₉₇	—	2	1	no	—
50	C₄×D₉₇	—	2	1	no	—
51	C₄×D₉₇	—	2	1	no	—
52	C₈×C₉₇	—	2	1	no	—
53	C₈×C₉₇	—	2	1	no	—
54	#11715895	—	2	1	no	—
55	#11715895	C₂×D₄	1	1	yes	characteristic
56	#11715895	#41	1	1	yes	characteristic
57	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
58	C₂×D₉₇	—	2	1	no	—
59	C₂×D₉₇	—	2	1	no	—
60	C₂×D₉₇	—	2	1	no	—
61	C₂×D₉₇	—	2	1	no	—
62	C₂×D₉₇	—	2	1	no	—
63	C₂×D₉₇	M₃₂	1	2	yes	complemented
64	C₄×C₉₇	—	2	1	no	—
65	C₄×C₉₇	—	2	1	no	—
66	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
67	C₄×C₉₇	#131	1	1	yes	characteristic
68	Dic₉₇	—	2	1	no	—
69	Dic₉₇	—	2	1	no	—
70	C₂×C₉₇	C₈×D₄	1	1	yes	characteristic
71	C₂×C₉₇	C₂×M₃₂	1	1	yes	characteristic
72	C₂×C₉₇	#504	1	1	yes	characteristic
73	D₉₇	—	2	2	no	—
74	D₉₇	—	4	1	no	—
75	#2168	—	97	1	no	maximal, Sylow
76	C₉₇	#2168	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₃₂	—	97	1	no	—
78	C₂²×C₁₆	—	97	1	no	—
79	C₈×D₄	—	97	1	no	—
80	C₄⋊C₁₆	—	97	1	no	—
81	C₂²⋊C₁₆	—	97	1	no	—
82	C₂²⋊C₁₆	—	97	1	no	—
83	#346	—	97	1	no	—
84	C₂²×C₈	—	97	1	no	—
85	C₂²×C₈	—	97	1	no	—
86	C₄×D₄	—	97	1	no	—
87	M₃₂	—	194	2	no	—
88	C₂×C₁₆	—	97	1	no	—
89	C₂×C₁₆	—	97	1	no	—
90	C₂×C₁₆	—	97	1	no	—
91	C₂×C₁₆	—	97	1	no	—
92	C₂×C₁₆	—	194	2	no	—
93	C₄⋊C₈	#87604	1	1	yes	characteristic
94	C₂²⋊C₈	—	97	1	no	—
95	C₂²⋊C₈	—	97	1	no	—
96	C₄×C₈	—	97	1	no	—
97	C₂×D₄	—	97	1	no	—
98	C₂²×C₄	—	97	1	no	—
99	C₂²×C₄	—	97	1	no	—
100	C₂×C₈	—	97	1	no	—
101	C₂×C₈	—	97	1	no	—
102	C₂×C₈	—	194	1	no	—
103	C₂×C₈	—	194	1	no	—
104	C₂×C₈	—	194	1	no	—
105	C₂×C₈	—	194	1	no	—
106	C₂×C₈	C₂×#87604	1	1	yes	characteristic
107	C₂×C₈	C₂×#87604	1	1	yes	characteristic
108	C₄⋊C₄	—	97	1	no	—
109	C₂²⋊C₄	—	97	1	no	—
110	C₂²⋊C₄	—	97	1	no	—
111	C₄²	C₂×#87604	1	1	yes	characteristic
112	C₁₆	—	97	2	no	—
113	C₁₆	—	194	1	no	—
114	C₁₆	—	194	1	no	—
115	C₁₆	—	194	1	no	—
116	C₂³	—	97	1	no	—
117	C₂³	—	97	1	no	—
118	D₄	—	194	2	no	—
119	C₂×C₄	—	97	1	no	—
120	C₂×C₄	—	97	1	no	—
121	C₂×C₄	—	194	1	no	—
122	C₂×C₄	—	194	1	no	—
123	C₂×C₄	—	194	1	no	—
124	C₂×C₄	—	194	1	no	—
125	C₂×C₄	#13095519	1	1	yes	characteristic
126	C₂×C₄	#13095519	1	1	yes	characteristic
127	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	97	1	no	—
131	C₈	—	97	1	no	—
132	C₈	—	194	1	no	—
133	C₂²	—	97	2	no	—
134	C₂²	—	194	1	no	—
135	C₂²	—	194	1	no	—
136	C₂²	—	194	1	no	—
137	C₂²	—	194	1	no	—
138	C₂²	—	194	1	no	—
139	C₂²	C₂×#13095519	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	194	1	no	—
143	C₄	—	194	1	no	—
144	C₄	D₄×#87604	1	1	yes	characteristic
145	C₄	#15690898	1	1	yes	characteristic
146	C₂	—	388	1	no	—
147	C₂	—	194	2	no	—
148	C₂	#20261763	1	1	yes	characteristic
149	C₂	#20261791	1	1	yes	characteristic
150	C₂	#20261794	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87604
commutator, socle	C₂²×C₉₇	C₂²×C₈
Fitting	C₄⋊C₈×C₉₇	C₄
prime radical	C₂×D₃₈₈	C₈
cosocle	C₂×#11715895	C₂³
hypercenter	C₄⋊C₈	#87604
hypocenter	C₉₇	#2168
LCS	G ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15690885	C₂	semidirect
2. C₂×#15690883	C₂	semidirect
3. C₄⋊#13095514	C₂	semidirect
4. #20261630	C₂	semidirect
5. #20261630	C₂	semidirect
6. #20261623	C₂	semidirect
24. #13095514	D₄	semidirect
63. C₂×D₉₇	M₃₂	semidirect
76. C₉₇	#2168	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15690885	2
C₂×#15690883	2
C₄⋊#13095514	2
#20261630	2
#20261623	2
#20260800	2
#2168	97