Finite Group Databasebeta

Subgroups — #27141147

Order 12416.

176 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 55 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	#20261083	C₂	1	1	yes	maximal, characteristic, complemented
2	#20260930	C₂	1	1	yes	maximal, characteristic
3	D₄×#11715895	C₂	1	1	yes	maximal, characteristic, complemented
4	#20260485	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260484	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260478	C₂	1	1	yes	maximal, characteristic, complemented
7	#476×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13095496	—	2	1	no	—
9	C₂²×#11715895	—	2	1	no	—
10	#15690807	C₂²	1	1	yes	characteristic
11	C₄⋊#11715895	C₂²	1	1	yes	characteristic
12	C₄⋊#11715895	C₂²	1	1	yes	characteristic
13	#15690730	—	2	1	no	—
14	#15690730	—	2	1	no	—
15	#15690730	—	2	1	no	—
16	C₂²⋊#11715895	—	2	1	no	—
17	C₄×#11715895	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₉₇	C₂²	1	1	yes	characteristic
20	#106×C₉₇	—	2	1	no	—
21	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
22	#102×C₉₇	—	2	1	no	—
23	C₂×D₃₈₈	—	2	1	no	—
24	C₂×D₃₈₈	—	2	1	no	—
25	C₂×D₃₈₈	—	2	1	no	—
26	#13095496	—	4	2	no	—
27	C₂×#11715895	—	2	1	no	—
28	C₂×#11715895	—	2	1	no	—
29	C₂×#11715895	—	4	2	no	—
30	C₂×#11715895	D₄	1	1	yes	characteristic
31	C₂×#11715895	D₄	1	1	yes	characteristic
32	C₂×Q₈×C₉₇	—	2	1	no	—
33	C₂×D₄×C₉₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉₇	—	2	1	no	—
35	C₂×C₈×C₉₇	—	2	1	no	—
36	C₄⋊C₄×C₉₇	—	2	1	no	—
37	C₄⋊C₄×C₉₇	—	2	1	no	—
38	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₉₇	—	2	1	no	—
40	C₄²×C₉₇	C₂³	1	1	yes	characteristic
41	C₂³×C₉₇	—	2	1	no	—
42	C₂²×D₉₇	—	4	1	no	—
43	Q₈×C₉₇	—	8	1	no	—
44	D₄×C₉₇	—	2	1	no	—
45	D₄×C₉₇	SD₁₆	1	2	yes	—
46	C₂×C₄×C₉₇	—	2	1	no	—
47	C₂×C₄×C₉₇	—	4	1	no	—
48	C₂×C₄×C₉₇	—	4	1	no	—
49	C₂×C₄×C₉₇	—	4	1	no	—
50	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
53	D₃₈₈	—	4	1	no	—
54	D₃₈₈	—	4	1	no	—
55	D₃₈₈	—	4	1	no	—
56	D₃₈₈	—	4	1	no	—
57	D₃₈₈	—	8	1	no	—
58	C₈×C₉₇	—	4	1	no	—
59	#11715895	—	2	1	no	—
60	#11715895	—	2	1	no	—
61	#11715895	—	4	1	no	—
62	#11715895	—	4	1	no	—
63	C₂²×C₉₇	—	2	2	no	—
64	C₂²×C₉₇	—	4	1	no	—
65	C₂²×C₉₇	—	4	1	no	—
66	C₂²×C₉₇	#121	1	1	yes	characteristic
67	C₂×D₉₇	—	8	1	no	—
68	C₂×D₉₇	—	8	1	no	—
69	C₂×D₉₇	—	8	1	no	—
70	C₄×C₉₇	—	2	1	no	—
71	C₄×C₉₇	—	4	1	no	—
72	C₄×C₉₇	—	8	1	no	—
73	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
74	C₄×C₉₇	#135	1	1	yes	characteristic
75	C₄×C₉₇	#135	1	1	yes	characteristic
76	C₄×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×C₉₇	—	4	2	no	—
78	C₂×C₉₇	#463	1	1	yes	characteristic
79	C₂×C₉₇	#465	1	1	yes	characteristic
80	C₂×C₉₇	#472	1	1	yes	characteristic
81	D₉₇	—	16	1	no	—
82	#1677	—	97	1	no	maximal, Sylow
83	C₉₇	#1677	1	1	yes	characteristic, complemented, Sylow
84	#486	—	97	1	no	—
85	#476	D₉₇	1	1	yes	characteristic, complemented
86	#459	—	97	1	no	—
87	C₈×D₄	—	97	1	no	—
88	#334	—	97	1	no	—
89	#333	—	97	1	no	—
90	#331	—	97	1	no	—
91	C₂×D₈	—	194	1	no	—
92	C₂²×C₈	—	194	1	no	—
93	C₄⋊Q₈	C₂×D₉₇	1	1	yes	characteristic
94	#127	—	97	1	no	—
95	C₄×D₄	C₂×D₉₇	1	1	yes	characteristic
96	#106	—	2	1	no	—
97	C₄⋊C₈	—	97	1	no	—
98	C₄⋊C₈	—	97	1	no	—
99	C₄⋊C₈	C₂×D₉₇	1	1	yes	characteristic
100	#102	—	2	1	no	—
101	#102	—	194	1	no	—
102	#102	—	194	1	no	—
103	#102	—	194	1	no	—
104	C₂²⋊C₈	—	194	1	no	—
105	C₄×C₈	—	97	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	194	1	no	—
108	C₂×D₄	—	194	1	no	—
109	C₂×D₄	—	194	1	no	—
110	C₂×D₄	C₂²⋊D₉₇	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	388	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	97	1	no	—
115	C₂×C₈	—	97	1	no	—
116	C₂×C₈	—	388	2	no	—
117	C₂×C₈	—	194	1	no	—
118	C₂×C₈	—	194	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	C₂²⋊D₉₇	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₉₇	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	388	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	776	1	no	—
129	D₄	—	388	1	no	—
130	D₄	—	388	1	no	—
131	D₄	—	388	1	no	—
132	D₄	—	388	1	no	—
133	D₄	#13095498	1	2	yes	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
140	C₂×C₄	#13095510	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	388	1	no	—
143	C₈	—	388	1	no	—
144	C₈	—	194	1	no	—
145	C₈	—	194	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	776	1	no	—
150	C₂²	—	776	1	no	—
151	C₂²	—	776	1	no	—
152	C₂²	#15690782	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	1	no	—
156	C₄	C₂×#13095498	1	1	yes	characteristic
157	C₄	#15690830	1	1	yes	characteristic
158	C₄	#15690831	1	1	yes	characteristic
159	C₄	#15690837	1	1	yes	characteristic
160	C₂	—	4	2	no	—
161	C₂	—	1552	1	no	—
162	C₂	#20260952	1	1	yes	characteristic
163	C₂	#20260966	1	1	yes	characteristic
164	C₂	#20261003	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15690782
commutator, cosocle	C₄²×C₉₇	C₂³
Frattini	C₄²	C₂²×D₉₇
Fitting	#476×C₉₇	C₂
socle	C₂²×C₉₇	#121
prime radical	#20260930	C₂
hypercenter	#476	D₉₇
hypocenter	C₉₇	#1677
LCS	G ▷ C₄²×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #20261083	C₂	semidirect
3. D₄×#11715895	C₂	semidirect
4. #20260485	C₂	semidirect
5. #20260484	C₂	semidirect
6. #20260478	C₂	semidirect
7. #476×C₉₇	C₂	semidirect
83. C₉₇	#1677	semidirect
85. #476	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261083	2
#20260930	2
D₄×#11715895	2
#20260485	2
#20260484	2
#20260478	2
#476×C₉₇	2
#1677	97