Finite Group Databasebeta

Subgroups — #27158861

Order 12416.

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

normal subgroups only

This group has 180 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	#20261848	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11715906	C₂	1	1	yes	maximal, characteristic
3	#20261825	C₂	1	2	yes	maximal, complemented
4	#20261824	C₂	1	2	yes	maximal, complemented
5	M₁₆×#87604	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11715906	—	2	1	no	—
7	C₄⋊#11715906	—	2	1	no	—
8	C₄⋊#11715906	C₂²	1	1	yes	characteristic
9	C₄⋊#11715906	C₂²	1	2	yes	—
10	C₄⋊#11715906	C₂²	1	1	yes	characteristic
11	C₂²⋊#11715906	—	2	1	no	—
12	C₄×#11715906	—	2	1	no	—
13	D₉₇⋊M₁₆	C₄	1	1	yes	characteristic
14	C₂×C₄×#87604	C₂²	1	1	yes	characteristic
15	#15690864	C₄	1	2	yes	—
16	C₈×#87604	C₂²	1	2	yes	—
17	M₁₆×D₉₇	C₄	1	1	yes	characteristic
18	C₂×#11715906	—	2	1	no	—
19	C₂×#11715906	—	2	1	no	—
20	C₂×#11715906	—	2	2	no	—
21	C₂×#11715906	—	4	2	no	—
22	C₂²×#87604	D₄	1	1	yes	characteristic
23	C₂²×#87604	Q₈	1	1	yes	characteristic
24	#13095521	—	2	1	no	—
25	#13095520	—	2	1	no	—
26	#13095519	C₂×C₄	1	2	yes	—
27	C₄×#87604	C₂×C₄	1	1	yes	characteristic
28	C₄×#87604	D₄	1	1	yes	characteristic
29	C₄×#87604	Q₈	1	1	yes	characteristic
30	C₄×#87604	C₂³	1	1	yes	characteristic
31	C₂×C₄×D₉₇	C₂×C₄	1	1	yes	characteristic
32	#13095495	C₈	1	1	yes	characteristic, complemented
33	#13095494	C₈	1	2	yes	complemented
34	C₈×D₉₇	C₂×C₄	1	2	yes	—
35	M₁₆×C₉₇	C₈	1	1	yes	characteristic, complemented
36	C₂²×D₉₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#87604	—	2	1	no	—
38	C₂×#87604	—	2	1	no	—
39	C₂×#87604	—	4	1	no	—
40	C₂×#87604	C₄⋊C₄	1	1	yes	characteristic
41	C₂×#87604	C₄⋊C₄	1	1	yes	characteristic
42	C₂×#87604	C₂×D₄	1	1	yes	characteristic
43	C₂×#87604	C₂×Q₈	1	1	yes	characteristic
44	#11715906	—	4	1	no	—
45	#11715906	—	4	2	no	—
46	#11715906	—	4	1	no	—
47	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₉₇	M₁₆	1	1	yes	characteristic
49	C₄×D₉₇	C₄⋊C₄	1	1	yes	characteristic
50	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
51	C₄×D₉₇	M₁₆	1	1	yes	characteristic
52	C₄×D₉₇	C₂²×C₄	1	1	yes	characteristic
53	#11715897	—	2	2	no	—
54	C₈×C₉₇	C₂×C₈	1	2	yes	complemented
55	#11715895	C₂×C₈	1	2	yes	complemented
56	C₂²×C₉₇	C₄⋊C₈	1	1	yes	characteristic
57	C₂×D₉₇	—	2	1	no	—
58	C₂×D₉₇	—	2	1	no	—
59	C₂×D₉₇	C₄⋊C₈	1	1	yes	characteristic
60	C₂×D₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	#87604	—	2	1	no	—
62	#87604	—	2	1	no	—
63	#87604	—	2	1	no	—
64	#87604	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#87604	#137	1	1	yes	characteristic
66	C₄×C₉₇	C₄⋊C₈	1	1	yes	characteristic
67	C₄×C₉₇	C₂²×C₈	1	1	yes	characteristic
68	Dic₉₇	C₄⋊C₈	1	1	yes	characteristic
69	Dic₉₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₉₇	—	2	1	no	—
71	C₂×C₉₇	C₂×C₄⋊C₈	1	1	yes	characteristic
72	D₉₇	—	2	1	no	—
73	D₉₇	#428	1	1	yes	characteristic
74	D₉₇	#430	1	1	yes	characteristic
75	#1565	—	97	1	no	maximal, Sylow
76	C₉₇	#1565	1	1	yes	characteristic, complemented, Sylow
77	#431	—	97	1	no	—
78	C₂×C₄⋊C₈	—	97	1	no	—
79	C₄×M₁₆	—	97	1	no	—
80	#335	—	97	2	no	—
81	#334	—	97	2	no	—
82	C₂×M₁₆	—	97	1	no	—
83	C₂×M₁₆	—	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	2	no	—
88	C₄⋊C₈	—	97	1	no	—
89	C₄⋊C₈	—	194	1	no	—
90	C₂²⋊C₈	—	194	1	no	—
91	#97	—	97	2	no	—
92	C₄×C₈	—	97	2	no	—
93	C₄×C₈	—	194	1	no	—
94	C₂²×C₄	—	97	1	no	—
95	C₂²×C₄	—	97	1	no	—
96	C₂²×C₄	—	97	1	no	—
97	M₁₆	—	97	1	no	—
98	M₁₆	—	97	2	no	—
99	M₁₆	—	194	1	no	—
100	M₁₆	—	194	1	no	—
101	M₁₆	#11715906	1	1	yes	characteristic, complemented
102	C₂×C₈	—	97	2	no	—
103	C₂×C₈	—	97	2	no	—
104	C₂×C₈	—	388	2	no	—
105	C₂×C₈	—	194	1	no	—
106	C₂×C₈	—	194	1	no	—
107	C₂×C₈	—	194	2	no	—
108	C₄²	—	97	1	no	—
109	C₄²	—	97	1	no	—
110	C₄²	—	97	1	no	—
111	C₄²	—	97	1	no	—
112	C₂³	—	97	1	no	—
113	C₂×C₄	—	97	1	no	—
114	C₂×C₄	—	97	1	no	—
115	C₂×C₄	—	97	1	no	—
116	C₂×C₄	—	97	1	no	—
117	C₂×C₄	—	97	1	no	—
118	C₂×C₄	—	97	1	no	—
119	C₂×C₄	—	97	1	no	—
120	C₂×C₄	—	97	1	no	—
121	C₂×C₄	—	97	1	no	—
122	C₂×C₄	—	388	1	no	—
123	C₂×C₄	—	194	1	no	—
124	C₂×C₄	—	194	1	no	—
125	C₂×C₄	C₂×#11715906	1	1	yes	characteristic
126	C₈	—	97	2	no	—
127	C₈	—	388	2	no	—
128	C₈	—	388	1	no	—
129	C₈	—	388	1	no	—
130	C₈	—	194	2	no	—
131	C₈	C₂×#11715906	1	2	yes	complemented
132	C₂²	—	97	1	no	—
133	C₂²	—	97	1	no	—
134	C₂²	—	194	1	no	—
135	C₂²	—	194	1	no	—
136	C₂²	C₄⋊#11715906	1	1	yes	characteristic
137	C₄	—	97	1	no	—
138	C₄	—	97	1	no	—
139	C₄	—	97	1	no	—
140	C₄	—	97	1	no	—
141	C₄	—	194	1	no	—
142	C₄	—	194	1	no	—
143	C₄	—	194	1	no	—
144	C₄	C₄⋊#11715906	1	1	yes	characteristic
145	C₄	C₂²×#11715906	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	97	1	no	—
148	C₂	—	97	1	no	—
149	C₂	—	194	1	no	—
150	C₂	C₂×C₄⋊#11715906	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₄⋊#11715906
commutator	C₄×C₉₇	C₂²×C₈
Frattini	C₄	C₂²×#11715906
Fitting	M₁₆×C₉₇	C₈
socle	C₂×C₉₇	C₂×C₄⋊C₈
prime radical	C₂²×D₉₇	C₄⋊C₄
cosocle	C₄×#87604	C₂³
hypercenter	M₁₆	#11715906
hypocenter	C₉₇	#1565
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
3. #20261825	C₂	semidirect
4. #20261824	C₂	semidirect
32. #13095495	C₈	semidirect
33. #13095494	C₈	semidirect
35. M₁₆×C₉₇	C₈	semidirect
54. C₈×C₉₇	C₂×C₈	semidirect
55. #11715895	C₂×C₈	semidirect
76. C₉₇	#1565	semidirect
101. M₁₆	#11715906	semidirect
131. C₈	C₂×#11715906	semidirect

Maximal subgroups

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

subgroup	index
#20261848	2
C₂×C₄⋊#11715906	2
#20261825	2
#20261824	2
M₁₆×#87604	2
#1565	97