Finite Group Databasebeta

Subgroups — #27157619

Order 12416.

260 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 72 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	#20261745	C₂	1	2	yes	maximal, complemented
2	#20261714	C₂	1	2	yes	maximal
3	#20261603	C₂	1	2	yes	maximal, complemented
4	C₂×#15690818	C₂	1	1	yes	maximal, characteristic
5	#15690889	C₂²	1	4	yes	—
6	#15690879	—	2	2	no	—
7	#15690878	—	2	2	no	—
8	#15690869	—	2	2	no	—
9	#15690868	—	2	2	no	—
10	C₂×#13095511	C₂²	1	1	yes	characteristic
11	#15690818	—	2	2	no	—
12	#15690818	—	2	2	no	—
13	C₂×#13095494	C₄	1	1	yes	characteristic, complemented
14	C₂×#13095494	C₂²	1	1	yes	characteristic
15	C₂²×#11715895	C₄	1	1	yes	characteristic, complemented
16	C₂×C₈×D₉₇	C₄	1	1	yes	characteristic, complemented
17	C₂×C₈×D₉₇	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₉₇	C₄	1	1	yes	characteristic, complemented
19	C₄⋊#87604	—	2	4	no	—
20	C₂²⋊#87604	—	2	4	no	—
21	C₄×#87604	—	2	4	no	—
22	#13095511	—	2	2	no	—
23	#13095511	—	2	2	no	—
24	C₂×Dic₁₉₄	Q₈	1	1	yes	characteristic
25	C₂×D₃₈₈	D₄	1	1	yes	characteristic
26	C₂×C₂²⋊D₉₇	D₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₉₇	Q₈	1	1	yes	characteristic
28	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
30	#13095494	—	2	2	no	—
31	#13095494	—	2	2	no	—
32	C₂×#11715895	—	2	1	no	—
33	C₂×#11715895	—	2	1	no	—
34	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
35	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
36	C₈×D₉₇	—	2	1	no	—
37	C₈×D₉₇	—	2	1	no	—
38	C₈×D₉₇	D₄	1	2	yes	—
39	C₈×D₉₇	D₄	1	2	yes	—
40	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
41	M₁₆×C₉₇	—	2	2	no	—
42	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×D₉₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×D₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×#87604	—	4	4	no	—
48	C₂×C₄×C₉₇	—	2	1	no	—
49	C₂×C₄×C₉₇	—	2	1	no	—
50	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²⋊D₉₇	—	2	2	no	—
53	C₂²⋊D₉₇	—	2	2	no	—
54	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
55	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
56	D₃₈₈	—	2	2	no	—
57	C₄×D₉₇	—	2	1	no	—
58	C₄×D₉₇	—	2	1	no	—
59	C₄×D₉₇	C₂×D₄	1	2	yes	—
60	C₄×D₉₇	#41	1	2	yes	—
61	Dic₁₉₄	—	2	2	no	—
62	C₈×C₉₇	—	2	1	no	—
63	C₈×C₉₇	C₂²⋊C₄	1	2	yes	—
64	#11715895	—	2	1	no	—
65	#11715895	C₂²⋊C₄	1	1	yes	characteristic
66	#11715895	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₉₇	—	2	1	no	—
68	C₂²×C₉₇	—	4	1	no	—
69	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×D₉₇	—	2	1	no	—
71	C₂×D₉₇	—	4	1	no	—
72	C₂×D₉₇	—	4	1	no	—
73	C₂×D₉₇	#121	1	2	yes	—
74	#87604	—	8	4	no	—
75	C₄×C₉₇	—	2	1	no	—
76	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₄×C₉₇	#117	1	1	yes	characteristic
78	Dic₉₇	—	2	1	no	—
79	Dic₉₇	C₂²⋊Q₈	1	2	yes	—
80	C₂×C₉₇	—	2	1	no	—
81	C₂×C₉₇	—	4	1	no	—
82	C₂×C₉₇	#380	1	1	yes	characteristic
83	D₉₇	—	2	2	no	—
84	D₉₇	—	4	1	no	—
85	#1812	—	97	1	no	maximal, Sylow
86	C₉₇	#1812	1	1	yes	characteristic, complemented, Sylow
87	C₂×#131	—	97	1	no	—
88	#427	—	97	2	no	—
89	#410	—	97	2	no	—
90	H(3,ℤ/4)	—	97	2	no	—
91	C₂×#41	—	97	1	no	—
92	#131	—	194	2	no	—
93	#131	—	194	2	no	—
94	C₂×M₁₆	—	97	1	no	—
95	C₂×M₁₆	—	97	1	no	—
96	C₂×M₁₆	#87604	1	1	yes	characteristic, complemented
97	C₂²×C₈	—	97	1	no	—
98	C₂²×C₈	—	97	1	no	—
99	C₂²×C₈	—	97	1	no	—
100	#117	—	97	4	no	—
101	#107	—	194	2	no	—
102	#106	—	194	2	no	—
103	#99	—	194	2	no	—
104	#99	—	194	2	no	—
105	#41	—	194	2	no	—
106	#41	—	194	2	no	—
107	C₂×Q₈	—	97	1	no	—
108	C₂×D₄	—	97	1	no	—
109	C₂×D₄	—	97	1	no	—
110	C₂×D₄	—	97	1	no	—
111	C₂²×C₄	—	97	1	no	—
112	C₂²×C₄	—	97	1	no	—
113	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
114	M₁₆	—	2	2	no	—
115	M₁₆	—	194	2	no	—
116	M₁₆	—	194	2	no	—
117	C₂×C₈	—	97	1	no	—
118	C₂×C₈	—	97	2	no	—
119	C₂×C₈	—	97	2	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	C₂×#87604	1	1	yes	characteristic
126	C₂×C₈	C₂×#87604	1	1	yes	characteristic
127	C₄⋊C₄	—	194	4	no	—
128	C₂²⋊C₄	—	194	4	no	—
129	C₄²	—	194	4	no	—
130	C₂³	—	97	1	no	—
131	C₂³	—	97	1	no	—
132	C₂³	C₄⋊#87604	1	1	yes	characteristic
133	Q₈	—	194	2	no	—
134	D₄	—	194	2	no	—
135	D₄	—	194	2	no	—
136	D₄	—	194	2	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	97	1	no	—
140	C₂×C₄	—	97	2	no	—
141	C₂×C₄	—	97	2	no	—
142	C₂×C₄	—	97	1	no	—
143	C₂×C₄	—	388	4	no	—
144	C₂×C₄	—	194	1	no	—
145	C₂×C₄	—	194	1	no	—
146	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
147	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	—	97	1	no	—
150	C₈	—	97	1	no	—
151	C₈	—	194	1	no	—
152	C₈	C₂²⋊#87604	1	2	yes	—
153	C₂²	—	2	1	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	97	2	no	—
156	C₂²	—	388	1	no	—
157	C₂²	—	388	1	no	—
158	C₂²	—	194	1	no	—
159	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	776	4	no	—
162	C₄	—	97	2	no	—
163	C₄	—	194	1	no	—
164	C₄	C₂×C₂²⋊#87604	1	1	yes	characteristic
165	C₄	#15690889	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	4	1	no	—
168	C₂	—	388	1	no	—
169	C₂	—	194	2	no	—
170	C₂	#20261677	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15690889
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₂×M₁₆×C₉₇	C₄
socle	C₂×C₉₇	#380
prime radical	C₂×#13095511	C₂²
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	C₂×M₁₆	#87604
hypocenter	C₉₇	#1812
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20261745	C₂	semidirect
3. #20261603	C₂	semidirect
13. C₂×#13095494	C₄	semidirect
15. C₂²×#11715895	C₄	semidirect
16. C₂×C₈×D₉₇	C₄	semidirect
18. C₂×M₁₆×C₉₇	C₄	semidirect
86. C₉₇	#1812	semidirect
96. C₂×M₁₆	#87604	semidirect

Maximal subgroups

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

subgroup	index
#20261745	2
#20261714	2
#20261603	2
C₂×#15690818	2
#1812	97