Finite Group Databasebeta

Subgroups — #27143810

Order 12416.

170 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 45 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	#20261170	C₂	1	1	yes	maximal, characteristic, complemented
2	#20261110	C₂	1	1	yes	maximal, characteristic, complemented
3	#20260852	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260555	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260549	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260548	C₂	1	1	yes	maximal, characteristic, complemented
7	#509×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₇₇₆	C₂²	1	1	yes	characteristic
9	C₂×#13095496	—	2	1	no	—
10	#15690817	—	2	1	no	—
11	#15690806	—	2	1	no	—
12	#15690753	—	2	1	no	—
13	#15690751	—	2	1	no	—
14	#15690750	C₂²	1	1	yes	characteristic
15	#15690746	C₂²	1	1	yes	characteristic
16	#15690737	—	2	1	no	—
17	C₈×Dic₉₇	C₂²	1	1	yes	characteristic
18	#135×C₉₇	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₉₇	C₂²	1	1	yes	characteristic
20	SD₃₂×C₉₇	—	2	1	no	—
21	D₁₆×C₉₇	—	2	1	no	—
22	M₃₂×C₉₇	C₂²	1	1	yes	characteristic
23	C₂×D₃₈₈	—	2	1	no	—
24	C₂×C₂²⋊D₉₇	—	4	2	no	—
25	D₇₇₆	—	2	1	no	—
26	D₇₇₆	—	2	1	no	—
27	#13095496	—	4	1	no	—
28	#13095496	—	4	1	no	—
29	#13095495	—	2	1	no	—
30	#13095495	—	4	1	no	—
31	C₂×#11715895	—	4	1	no	—
32	C₂×#11715895	D₄	1	1	yes	characteristic
33	C₄×Dic₉₇	D₄	1	1	yes	characteristic
34	#13095486	—	2	1	no	—
35	#41×C₉₇	—	2	1	no	—
36	C₂×D₄×C₉₇	—	2	1	no	—
37	Q₁₆×C₉₇	D₄	1	1	yes	characteristic, complemented
38	SD₁₆×C₉₇	—	2	1	no	—
39	D₈×C₉₇	—	2	1	no	—
40	D₈×C₉₇	—	2	1	no	—
41	D₈×C₉₇	D₄	1	1	yes	characteristic, complemented
42	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
43	C₁₆×C₉₇	—	2	1	no	—
44	C₂³×C₉₇	—	4	1	no	—
45	C₂²×D₉₇	—	4	1	no	—
46	Q₈×C₉₇	—	2	1	no	—
47	D₄×C₉₇	—	2	1	no	—
48	D₄×C₉₇	—	4	1	no	—
49	D₄×C₉₇	—	4	1	no	—
50	D₄×C₉₇	—	4	1	no	—
51	C₂×C₄×C₉₇	—	4	1	no	—
52	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₉₇	—	8	4	no	—
54	C₂×Dic₉₇	—	2	1	no	—
55	D₃₈₈	—	4	1	no	—
56	D₃₈₈	—	4	1	no	—
57	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
58	C₈×C₉₇	#41	1	1	yes	characteristic
59	#11715895	—	4	1	no	—
60	#11715895	D₈	1	1	yes	characteristic
61	#11715895	D₈	1	1	yes	characteristic
62	C₂²×C₉₇	—	4	1	no	—
63	C₂²×C₉₇	—	8	2	no	—
64	C₂²×C₉₇	—	8	1	no	—
65	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×D₉₇	—	8	2	no	—
67	C₂×D₉₇	—	8	1	no	—
68	C₄×C₉₇	—	4	1	no	—
69	C₄×C₉₇	#121	1	1	yes	characteristic
70	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
71	Dic₉₇	—	2	2	no	—
72	C₂×C₉₇	—	2	1	no	—
73	C₂×C₉₇	—	8	1	no	—
74	C₂×C₉₇	—	16	1	no	—
75	C₂×C₉₇	#459	1	1	yes	characteristic
76	D₉₇	—	16	1	no	—
77	#2213	—	97	1	no	maximal, Sylow
78	C₉₇	#2213	1	1	yes	characteristic, complemented, Sylow
79	#509	—	97	1	no	—
80	#509	D₉₇	1	1	yes	characteristic, complemented
81	#493	—	97	1	no	—
82	#443	—	97	1	no	—
83	#364	—	97	1	no	—
84	#361	—	97	1	no	—
85	#361	—	97	1	no	—
86	#135	C₂×D₉₇	1	1	yes	characteristic
87	C₂×D₈	—	97	1	no	—
88	C₂×D₈	—	194	1	no	—
89	C₂×D₈	C₂×D₉₇	1	1	yes	characteristic
90	#131	—	194	1	no	—
91	#127	—	194	1	no	—
92	SD₃₂	—	2	1	no	—
93	SD₃₂	—	194	1	no	—
94	D₁₆	—	2	1	no	—
95	D₁₆	—	194	1	no	—
96	M₃₂	—	97	1	no	—
97	M₃₂	C₂×D₉₇	1	1	yes	characteristic
98	#108	—	97	1	no	—
99	#104	—	194	1	no	—
100	C₄×C₈	—	97	1	no	—
101	#41	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	388	2	no	—
104	C₂×D₄	—	194	1	no	—
105	Q₁₆	C₂²⋊D₉₇	1	1	yes	characteristic, complemented
106	SD₁₆	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	388	1	no	—
110	D₈	—	388	1	no	—
111	D₈	—	194	1	no	—
112	D₈	—	194	1	no	—
113	D₈	C₂²⋊D₉₇	1	1	yes	characteristic, complemented
114	M₁₆	—	388	1	no	—
115	M₁₆	—	194	1	no	—
116	C₂×C₈	—	97	1	no	—
117	C₂×C₈	—	388	1	no	—
118	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
119	C₄²	—	97	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	194	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	388	1	no	—
124	Q₈	—	2	1	no	—
125	D₄	—	2	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	776	4	no	—
130	D₄	—	388	1	no	—
131	D₄	—	388	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	194	1	no	—
134	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
135	C₈	—	97	1	no	—
136	C₈	—	97	1	no	—
137	C₈	—	388	1	no	—
138	C₈	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
139	C₈	#13095510	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	776	2	no	—
144	C₂²	—	776	1	no	—
145	C₂²	#15690834	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	194	2	no	—
148	C₄	#15690782	1	1	yes	characteristic
149	C₄	D₈×D₉₇	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	8	1	no	—
152	C₂	—	16	1	no	—
153	C₂	—	1552	1	no	—
154	C₂	#20260928	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260928
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#509×C₉₇	C₂
socle	C₂×C₉₇	#459
hypercenter	#509	D₉₇
hypocenter	C₉₇	#2213
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

Complements

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

N	H ≅ G/N	type
1. #20261170	C₂	semidirect
2. #20261110	C₂	semidirect
3. #20260852	C₂	semidirect
4. #20260555	C₂	semidirect
5. #20260549	C₂	semidirect
6. #20260548	C₂	semidirect
7. #509×C₉₇	C₂	semidirect
37. Q₁₆×C₉₇	D₄	semidirect
41. D₈×C₉₇	D₄	semidirect
78. C₉₇	#2213	semidirect
80. #509	D₉₇	semidirect
105. Q₁₆	C₂²⋊D₉₇	semidirect
113. D₈	C₂²⋊D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261170	2
#20261110	2
#20260852	2
#20260555	2
#20260549	2
#20260548	2
#509×C₉₇	2
#2213	97