Finite Group Databasebeta

Subgroups — #27157300

Order 12416.

166 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20261756	C₂	1	1	yes	maximal, characteristic
2	#20261738	C₂	1	1	yes	maximal, characteristic
3	#20261599	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261598	C₂	1	1	yes	maximal, characteristic, complemented
5	#20261597	C₂	1	1	yes	maximal, characteristic, complemented
6	#20261596	C₂	1	1	yes	maximal, characteristic, complemented
7	M₁₆×Dic₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#13095520	—	2	1	no	—
9	#15690877	—	2	1	no	—
10	C₄⋊#11715897	C₂²	1	1	yes	characteristic
11	C₄⋊#11715897	C₂²	1	1	yes	characteristic
12	C₄⋊#11715897	C₂²	1	1	yes	characteristic
13	C₄⋊#11715897	C₂²	1	1	yes	characteristic
14	C₂²⋊#11715897	—	2	1	no	—
15	#15690865	—	2	1	no	—
16	C₂×#13095495	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
18	#15690720	C₄	1	1	yes	characteristic
19	#15690720	C₄	1	1	yes	characteristic
20	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
21	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×C₉₇	C₄	1	1	yes	characteristic
23	#13095520	—	4	2	no	—
24	C₂×#11715897	—	2	1	no	—
25	C₂×#11715897	—	2	1	no	—
26	C₂×#11715897	—	2	1	no	—
27	C₂×#11715897	—	2	1	no	—
28	C₂²×Dic₉₇	D₄	1	1	yes	characteristic
29	C₂²×Dic₉₇	Q₈	1	1	yes	characteristic
30	#13095495	—	2	2	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	C₄×Dic₉₇	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
38	M₁₆×C₉₇	—	2	2	no	—
39	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	—	2	1	no	—
43	C₂×C₄×C₉₇	—	2	1	no	—
44	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₉₇	—	2	1	no	—
47	C₂×Dic₉₇	—	2	1	no	—
48	C₂×Dic₉₇	—	4	1	no	—
49	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×Dic₉₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₉₇	C₂×Q₈	1	1	yes	characteristic
53	#11715897	—	4	1	no	—
54	#11715897	—	4	1	no	—
55	#11715897	—	4	1	no	—
56	#11715897	—	4	1	no	—
57	C₈×C₉₇	—	2	1	no	—
58	C₈×C₉₇	M₁₆	1	2	yes	complemented
59	#11715895	—	2	1	no	—
60	#11715895	M₁₆	1	2	yes	complemented
61	C₂²×C₉₇	—	2	1	no	—
62	C₂²×C₉₇	—	2	1	no	—
63	C₂²×C₉₇	—	2	1	no	—
64	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	Dic₉₇	—	2	1	no	—
69	Dic₉₇	—	2	1	no	—
70	Dic₉₇	—	2	1	no	—
71	Dic₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₉₇	#137	1	1	yes	characteristic
73	C₂×C₉₇	—	4	1	no	—
74	C₂×C₉₇	#423	1	1	yes	characteristic
75	C₂×C₉₇	#428	1	1	yes	characteristic
76	C₂×C₉₇	#430	1	1	yes	characteristic
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	—	97	1	no	—
84	#334	—	97	1	no	—
85	#334	—	97	1	no	—
86	C₂×M₁₆	—	97	1	no	—
87	C₂×M₁₆	—	194	1	no	—
88	C₂×M₁₆	#87604	1	1	yes	characteristic
89	C₂×C₄²	—	97	1	no	—
90	C₄⋊C₈	—	97	1	no	—
91	C₄⋊C₈	—	97	1	no	—
92	C₄⋊C₈	—	97	1	no	—
93	C₄⋊C₈	—	97	1	no	—
94	C₄⋊C₈	—	194	1	no	—
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	—	97	1	no	—
101	C₂²×C₄	—	97	1	no	—
102	C₂²×C₄	—	97	1	no	—
103	C₂²×C₄	C₂×#87604	1	1	yes	characteristic
104	M₁₆	—	2	2	no	—
105	M₁₆	—	388	2	no	—
106	M₁₆	—	194	2	no	—
107	C₂×C₈	—	97	1	no	—
108	C₂×C₈	—	97	1	no	—
109	C₂×C₈	—	194	1	no	—
110	C₂×C₈	—	194	1	no	—
111	C₂×C₈	—	194	1	no	—
112	C₂×C₈	—	194	1	no	—
113	C₂×C₈	C₂×#87604	1	1	yes	characteristic
114	C₂×C₈	C₂×#87604	1	1	yes	characteristic
115	C₄²	—	97	1	no	—
116	C₄²	—	97	1	no	—
117	C₄²	—	97	1	no	—
118	C₄²	—	97	1	no	—
119	C₂³	C₄⋊#87604	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	97	1	no	—
123	C₂×C₄	—	97	1	no	—
124	C₂×C₄	—	97	1	no	—
125	C₂×C₄	—	97	1	no	—
126	C₂×C₄	—	388	1	no	—
127	C₂×C₄	—	194	1	no	—
128	C₂×C₄	—	194	1	no	—
129	C₂×C₄	C₄⋊#87604	1	1	yes	characteristic
130	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
131	C₈	—	2	1	no	—
132	C₈	—	97	2	no	—
133	C₈	—	388	1	no	—
134	C₈	—	388	1	no	—
135	C₈	—	388	1	no	—
136	C₈	—	388	1	no	—
137	C₈	—	194	1	no	—
138	C₈	#13095520	1	2	yes	complemented
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	97	1	no	—
145	C₄	—	97	1	no	—
146	C₄	—	194	1	no	—
147	C₄	—	194	1	no	—
148	C₄	—	194	1	no	—
149	C₄	C₂×#13095520	1	1	yes	characteristic
150	C₄	D₉₇⋊M₁₆	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#20261738	1	1	yes	characteristic
153	C₂	#20261746	1	1	yes	characteristic
154	C₂	#20261750	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	C₂×M₁₆×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₄×Dic₉₇	C₂³
hypercenter	C₂×M₁₆	#87604
hypocenter	C₉₇	#1569
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #20261599	C₂	semidirect
4. #20261598	C₂	semidirect
5. #20261597	C₂	semidirect
6. #20261596	C₂	semidirect
58. C₈×C₉₇	M₁₆	semidirect
60. #11715895	M₁₆	semidirect
78. C₉₇	#1569	semidirect
138. C₈	#13095520	semidirect

Maximal subgroups

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

subgroup	index
#20261756	2
#20261738	2
#20261599	2
#20261598	2
#20261597	2
#20261596	2
M₁₆×Dic₉₇	2
#1569	97