Finite Group Databasebeta

Subgroups — #27157803

Order 12416.

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

normal subgroups only

This group has 186 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	#20261742	C₂	1	1	yes	maximal, characteristic
2	#20261700	C₂	1	1	yes	maximal, characteristic
3	#20261611	C₂	1	1	yes	maximal, characteristic
4	#20261605	C₂	1	1	yes	maximal, characteristic
5	#20261605	C₂	1	1	yes	maximal, characteristic
6	#20261600	C₂	1	1	yes	maximal, characteristic
7	#20260770	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095521	C₂²	1	1	yes	characteristic
9	C₂×#13095521	C₂²	1	1	yes	characteristic
10	C₂×#13095519	C₂²	1	1	yes	characteristic
11	C₂×C₄×#87604	C₂²	1	1	yes	characteristic
12	#15690877	—	2	1	no	—
13	C₄⋊#11715897	—	2	1	no	—
14	#15690865	—	2	1	no	—
15	C₄×#11715897	—	2	1	no	—
16	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
17	#15690768	C₂²	1	1	yes	characteristic
18	#15690768	C₂²	1	1	yes	characteristic
19	#15690743	C₄	1	1	yes	characteristic, complemented
20	#15690743	C₄	1	1	yes	characteristic, complemented
21	#15690742	C₄	1	1	yes	characteristic, complemented
22	#106×C₉₇	C₄	1	1	yes	characteristic, complemented
23	C₂²×#87604	—	2	1	no	—
24	#13095521	—	2	1	no	—
25	#13095521	—	2	1	no	—
26	#13095521	—	2	1	no	—
27	#13095521	Q₈	1	2	yes	—
28	#13095519	—	2	1	no	—
29	#13095519	—	2	1	no	—
30	C₂×#11715897	—	2	1	no	—
31	C₂×#11715897	—	2	1	no	—
32	C₂×#11715897	D₄	1	1	yes	characteristic
33	C₂×#11715897	D₄	1	1	yes	characteristic
34	C₄×#87604	—	2	1	no	—
35	C₄×#87604	—	2	1	no	—
36	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
37	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
38	C₈×D₉₇	—	2	1	no	—
39	C₈×D₉₇	—	2	1	no	—
40	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₉₇	C₂×C₄	1	1	yes	characteristic
42	#13095488	—	2	1	no	—
43	#13095488	—	2	1	no	—
44	C₄×Dic₉₇	—	2	1	no	—
45	C₄×Dic₉₇	—	2	1	no	—
46	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
47	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
48	C₄⋊C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×D₉₇	#41	1	1	yes	characteristic
50	C₂×#87604	—	2	2	no	—
51	C₂×#87604	—	4	1	no	—
52	C₂×#87604	—	4	1	no	—
53	C₂×C₄×C₉₇	—	2	1	no	—
54	C₂×C₄×C₉₇	—	2	1	no	—
55	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₉₇	—	2	1	no	—
57	C₂×Dic₉₇	—	2	1	no	—
58	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
59	C₄×D₉₇	C₂×Q₈	1	1	yes	characteristic
60	C₄×D₉₇	#41	1	1	yes	characteristic
61	C₄×D₉₇	#41	1	1	yes	characteristic
62	C₄×D₉₇	#41	1	1	yes	characteristic
63	#11715897	—	2	2	no	—
64	#11715897	—	2	1	no	—
65	#11715897	—	2	1	no	—
66	#11715897	—	4	1	no	—
67	C₈×C₉₇	—	2	1	no	—
68	#11715895	—	2	1	no	—
69	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
70	C₂×D₉₇	—	2	1	no	—
71	C₂×D₉₇	—	2	1	no	—
72	C₂×D₉₇	#125	1	1	yes	characteristic
73	C₂×D₉₇	#126	1	1	yes	characteristic
74	#87604	—	4	2	no	—
75	C₄×C₉₇	—	4	1	no	—
76	C₄×C₉₇	—	4	1	no	—
77	C₄×C₉₇	#117	1	1	yes	characteristic
78	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
79	Dic₉₇	—	4	1	no	—
80	Dic₉₇	—	4	1	no	—
81	Dic₉₇	C₂²⋊Q₈	1	1	yes	characteristic
82	Dic₉₇	#123	1	1	yes	characteristic
83	C₂×C₉₇	#387	1	1	yes	characteristic
84	C₂×C₉₇	#443	1	1	yes	characteristic
85	C₂×C₉₇	#444	1	1	yes	characteristic
86	D₉₇	—	2	1	no	—
87	D₉₇	—	2	1	no	—
88	#1922	—	97	1	no	maximal, Sylow
89	C₉₇	#1922	1	1	yes	characteristic, complemented, Sylow
90	#427	—	97	1	no	—
91	#424	—	97	1	no	—
92	#405	—	97	1	no	—
93	#341	—	97	1	no	—
94	#339	—	97	1	no	—
95	#339	—	97	1	no	—
96	#336	—	97	1	no	—
97	C₂×M₁₆	—	97	1	no	—
98	C₂×M₁₆	—	97	1	no	—
99	C₂²×C₈	—	97	1	no	—
100	C₂²×C₈	—	97	1	no	—
101	#117	—	97	1	no	—
102	#117	—	97	1	no	—
103	C₂×C₄²	—	97	1	no	—
104	#107	—	97	1	no	—
105	#107	—	97	1	no	—
106	#106	—	97	1	no	—
107	#106	#87604	1	1	yes	characteristic, complemented
108	C₄⋊C₈	—	194	1	no	—
109	C₄⋊C₈	—	194	1	no	—
110	#97	—	194	1	no	—
111	C₄×C₈	—	194	1	no	—
112	C₂²×C₄	—	97	1	no	—
113	C₂²×C₄	—	194	1	no	—
114	M₁₆	—	97	2	no	—
115	M₁₆	—	194	1	no	—
116	M₁₆	—	194	1	no	—
117	M₁₆	—	194	1	no	—
118	C₂×C₈	—	97	1	no	—
119	C₂×C₈	—	97	1	no	—
120	C₂×C₈	—	97	1	no	—
121	C₂×C₈	—	194	1	no	—
122	C₂×C₈	—	194	1	no	—
123	C₂×C₈	—	194	1	no	—
124	C₂×C₈	—	194	1	no	—
125	C₂×C₈	—	194	1	no	—
126	C₂×C₈	—	194	1	no	—
127	C₂×C₈	C₂×#87604	1	1	yes	characteristic
128	C₄⋊C₄	—	97	1	no	—
129	C₄⋊C₄	—	97	1	no	—
130	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic
131	C₄⋊C₄	C₂×#87604	1	1	yes	characteristic
132	C₂²⋊C₄	—	194	1	no	—
133	C₂²⋊C₄	—	194	1	no	—
134	C₄²	—	194	1	no	—
135	C₄²	—	194	1	no	—
136	C₄²	—	194	1	no	—
137	C₄²	—	194	1	no	—
138	C₂³	—	97	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	97	1	no	—
142	C₂×C₄	—	97	1	no	—
143	C₂×C₄	—	97	1	no	—
144	C₂×C₄	—	97	1	no	—
145	C₂×C₄	—	97	1	no	—
146	C₂×C₄	—	388	1	no	—
147	C₂×C₄	—	388	1	no	—
148	C₂×C₄	—	194	2	no	—
149	C₂×C₄	—	194	1	no	—
150	C₂×C₄	—	194	1	no	—
151	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	388	1	no	—
154	C₈	—	194	1	no	—
155	C₈	—	194	1	no	—
156	C₈	—	194	2	no	—
157	C₈	—	194	1	no	—
158	C₂²	—	97	1	no	—
159	C₂²	—	97	1	no	—
160	C₂²	—	194	1	no	—
161	C₂²	—	194	1	no	—
162	C₂²	D₄×#87604	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	97	1	no	—
166	C₄	—	97	1	no	—
167	C₄	—	388	2	no	—
168	C₄	—	388	1	no	—
169	C₄	—	388	1	no	—
170	C₄	#15690889	1	1	yes	characteristic
171	C₄	Q₈×#87604	1	1	yes	characteristic
172	C₂	—	194	1	no	—
173	C₂	—	194	1	no	—
174	C₂	#20261684	1	1	yes	characteristic
175	C₂	#20261778	1	1	yes	characteristic
176	C₂	#20261780	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	#106×C₉₇	C₄
socle	C₂²×C₉₇	C₄×D₄
prime radical	C₂²×D₉₇	#41
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	#106	#87604
hypocenter	C₉₇	#1922
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
19. #15690743	C₄	semidirect
20. #15690743	C₄	semidirect
21. #15690742	C₄	semidirect
22. #106×C₉₇	C₄	semidirect
89. C₉₇	#1922	semidirect
107. #106	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261742	2
#20261700	2
#20261611	2
#20261605	2
#20261600	2
#20260770	2
#1922	97