Finite Group Databasebeta

Subgroups — #27140502

Order 12416.

166 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20260791	C₂	1	1	yes	maximal, characteristic
2	#20260757	C₂	1	1	yes	maximal, characteristic
3	C₄×#13095494	C₂	1	1	yes	maximal, characteristic
4	#20260487	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260486	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260485	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13095494	—	2	1	no	—
9	C₂×#13095494	C₄	1	1	yes	characteristic
10	C₂×#13095494	C₄	1	1	yes	characteristic
11	C₄²×D₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊#11715895	C₂²	1	1	yes	characteristic
13	C₄⋊#11715895	C₂²	1	1	yes	characteristic
14	#15690739	—	2	1	no	—
15	#15690722	—	2	1	no	—
16	#15690720	—	2	1	no	—
17	#15690720	C₄	1	1	yes	characteristic
18	#15690720	C₄	1	1	yes	characteristic
19	C₄×#11715895	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₉₇	D₄	1	1	yes	characteristic
25	C₂×C₄×D₉₇	Q₈	1	1	yes	characteristic
26	#13095494	—	2	2	no	—
27	#13095494	—	2	2	no	—
28	#13095494	—	4	2	no	—
29	C₂×#11715895	—	2	1	no	—
30	C₂×#11715895	—	2	1	no	—
31	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
32	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₉₇	D₄	1	1	yes	characteristic
35	C₄×Dic₉₇	Q₈	1	1	yes	characteristic
36	C₂×C₈×C₉₇	—	2	1	no	—
37	C₂×C₈×C₉₇	—	2	1	no	—
38	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₉₇	C₂³	1	1	yes	characteristic
41	C₂²×D₉₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
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	C₄×D₉₇	—	4	1	no	—
53	C₈×C₉₇	—	2	1	no	—
54	C₈×C₉₇	—	4	1	no	—
55	C₈×C₉₇	—	4	1	no	—
56	C₈×C₉₇	M₁₆	1	2	yes	complemented
57	#11715895	—	2	1	no	—
58	#11715895	—	4	1	no	—
59	#11715895	—	4	1	no	—
60	#11715895	M₁₆	1	2	yes	complemented
61	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×D₉₇	—	2	1	no	—
63	C₂×D₉₇	—	2	1	no	—
64	C₂×D₉₇	—	2	1	no	—
65	C₄×C₉₇	—	2	1	no	—
66	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
68	C₄×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₉₇	#137	1	1	yes	characteristic
70	Dic₉₇	—	2	1	no	—
71	Dic₉₇	—	2	1	no	—
72	Dic₉₇	—	2	1	no	—
73	C₂×C₉₇	#423	1	1	yes	characteristic
74	C₂×C₉₇	#428	1	1	yes	characteristic
75	C₂×C₉₇	#430	1	1	yes	characteristic
76	D₉₇	—	4	1	no	—
77	#1569	—	97	1	no	maximal, Sylow
78	C₉₇	#1569	1	1	yes	characteristic, complemented, Sylow
79	#432	—	97	1	no	—
80	#423	—	97	1	no	—
81	C₄×M₁₆	—	97	1	no	—
82	#335	—	97	1	no	—
83	#335	D₉₇	1	1	yes	characteristic, complemented
84	#334	—	97	1	no	—
85	#334	—	97	1	no	—
86	C₂×M₁₆	—	97	1	no	—
87	C₂×M₁₆	—	97	1	no	—
88	C₂×M₁₆	—	194	1	no	—
89	C₂×C₄²	—	97	1	no	—
90	C₄⋊C₈	—	97	1	no	—
91	C₄⋊C₈	—	97	1	no	—
92	C₄⋊C₈	—	194	1	no	—
93	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
94	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
95	C₂²⋊C₈	—	194	1	no	—
96	#97	—	97	1	no	—
97	#97	—	97	1	no	—
98	#97	—	194	1	no	—
99	C₄×C₈	—	97	1	no	—
100	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
101	C₂²×C₄	—	97	1	no	—
102	C₂²×C₄	—	97	1	no	—
103	C₂²×C₄	—	97	1	no	—
104	M₁₆	—	388	2	no	—
105	M₁₆	—	194	2	no	—
106	M₁₆	—	194	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	97	1	no	—
110	C₂×C₈	—	97	1	no	—
111	C₂×C₈	—	194	1	no	—
112	C₂×C₈	—	194	1	no	—
113	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
114	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
115	C₄²	—	97	1	no	—
116	C₄²	—	97	1	no	—
117	C₄²	—	97	1	no	—
118	C₄²	C₂²×D₉₇	1	1	yes	characteristic
119	C₂³	—	97	1	no	—
120	C₂×C₄	—	97	1	no	—
121	C₂×C₄	—	97	1	no	—
122	C₂×C₄	—	97	1	no	—
123	C₂×C₄	—	388	1	no	—
124	C₂×C₄	—	194	1	no	—
125	C₂×C₄	—	194	1	no	—
126	C₂×C₄	—	194	1	no	—
127	C₂×C₄	—	194	1	no	—
128	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
129	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
130	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	4	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	97	2	no	—
135	C₈	—	388	1	no	—
136	C₈	—	388	1	no	—
137	C₈	—	194	1	no	—
138	C₈	#13095494	1	2	yes	complemented
139	C₂²	—	194	1	no	—
140	C₂²	—	194	1	no	—
141	C₂²	—	194	1	no	—
142	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	194	1	no	—
145	C₄	—	194	1	no	—
146	C₄	—	194	1	no	—
147	C₄	M₁₆×D₉₇	1	1	yes	characteristic
148	C₄	C₂×#13095494	1	1	yes	characteristic
149	C₄	#15690834	1	1	yes	characteristic
150	C₄	#15690842	1	1	yes	characteristic
151	C₂	—	388	1	no	—
152	C₂	#20260757	1	1	yes	characteristic
153	C₂	#20260776	1	1	yes	characteristic
154	C₂	#20260782	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting	#335×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂²×D₉₇	C₄⋊C₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	#335	D₉₇
hypocenter	C₉₇	#1569
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
4. #20260487	C₂	semidirect
5. #20260486	C₂	semidirect
6. #20260485	C₂	semidirect
7. #335×C₉₇	C₂	semidirect
56. C₈×C₉₇	M₁₆	semidirect
60. #11715895	M₁₆	semidirect
78. C₉₇	#1569	semidirect
83. #335	D₉₇	semidirect
138. C₈	#13095494	semidirect

Maximal subgroups

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

subgroup	index
#20260791	2
#20260757	2
C₄×#13095494	2
#20260487	2
#20260486	2
#20260485	2
#335×C₉₇	2
#1569	97