Finite Group Databasebeta

Subgroups — #27143686

Order 12416.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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	#20261535	C₂	1	1	yes	maximal, characteristic
2	#20261176	C₂	1	2	yes	maximal, complemented
3	#20260544	C₂	1	2	yes	maximal, complemented
4	#20260517	C₂	1	1	yes	maximal, characteristic, complemented
5	#470×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13095511	—	2	1	no	—
7	#15690843	—	2	2	no	—
8	#15690838	—	2	2	no	—
9	#15690832	—	2	1	no	—
10	#15690832	C₂²	1	2	yes	—
11	#15690756	—	2	2	no	—
12	#15690753	—	2	2	no	—
13	#15690750	C₂²	1	2	yes	complemented
14	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic, complemented
15	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
16	C₂²⋊Q₈×C₉₇	—	2	1	no	—
17	#106×C₉₇	C₂²	1	1	yes	characteristic, complemented
18	#103×C₉₇	—	2	1	no	—
19	#13095511	—	2	2	no	—
20	#13095511	—	2	2	no	—
21	#13095511	—	4	1	no	—
22	#13095511	—	4	1	no	—
23	C₂×Dic₁₉₄	—	2	1	no	—
24	C₂×D₃₈₈	—	2	1	no	—
25	C₂×C₂²⋊D₉₇	—	4	1	no	—
26	C₂×C₄×D₉₇	—	4	1	no	—
27	Dic₃₈₈	—	2	1	no	—
28	Dic₃₈₈	D₄	1	2	yes	—
29	#13095500	—	2	2	no	—
30	#13095500	—	2	2	no	—
31	D₇₇₆	—	2	1	no	—
32	D₇₇₆	D₄	1	2	yes	—
33	#13095486	—	2	2	no	—
34	C₂×Q₈×C₉₇	—	2	1	no	—
35	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
36	Q₁₆×C₉₇	—	2	2	no	—
37	M₁₆×C₉₇	—	2	2	no	—
38	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
39	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₉₇	—	2	1	no	—
41	C₄⋊C₄×C₉₇	—	4	1	no	—
42	C₂²⋊C₄×C₉₇	—	4	1	no	—
43	C₂³×C₉₇	D₈	1	1	yes	characteristic
44	C₂²×D₉₇	—	4	1	no	—
45	Q₈×C₉₇	—	4	1	no	—
46	Q₈×C₉₇	—	4	1	no	—
47	C₂×C₄×C₉₇	—	2	1	no	—
48	C₂×C₄×C₉₇	—	2	1	no	—
49	C₂×C₄×C₉₇	—	4	1	no	—
50	C₂×C₄×C₉₇	—	4	1	no	—
51	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
52	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₉₇	—	4	2	no	—
54	C₂²⋊D₉₇	—	4	2	no	—
55	C₂×Dic₉₇	—	4	1	no	—
56	D₃₈₈	—	2	2	no	—
57	D₃₈₈	—	4	1	no	—
58	C₄×D₉₇	—	4	2	no	—
59	C₄×D₉₇	—	4	2	no	—
60	Dic₁₉₄	—	2	2	no	—
61	Dic₁₉₄	—	4	1	no	—
62	C₈×C₉₇	—	2	1	no	—
63	C₈×C₉₇	C₂×D₄	1	2	yes	—
64	C₂²×C₉₇	—	2	1	no	—
65	C₂²×C₉₇	—	4	1	no	—
66	C₂²×C₉₇	C₂×D₈	1	1	yes	characteristic
67	C₂×D₉₇	—	4	2	no	—
68	C₂×D₉₇	—	8	2	no	—
69	C₄×C₉₇	—	2	1	no	—
70	C₄×C₉₇	—	8	1	no	—
71	C₄×C₉₇	—	8	1	no	—
72	C₄×C₉₇	#120	1	1	yes	characteristic
73	C₄×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
74	Dic₉₇	—	4	2	no	—
75	C₂×C₉₇	—	2	1	no	—
76	C₂×C₉₇	—	4	1	no	—
77	C₂×C₉₇	#447	1	1	yes	characteristic
78	D₉₇	—	8	2	no	—
79	#2191	—	97	1	no	maximal, Sylow
80	C₉₇	#2191	1	1	yes	characteristic, complemented, Sylow
81	#575	—	97	1	no	—
82	#510	—	97	2	no	—
83	#470	D₉₇	1	1	yes	characteristic, complemented
84	#360	—	97	2	no	—
85	#349	—	97	1	no	—
86	C₂×#41	—	194	1	no	—
87	#137	—	194	2	no	—
88	AGL(1,ℤ/8)	—	194	2	no	—
89	#135	—	97	2	no	—
90	#135	—	194	1	no	—
91	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic, complemented
92	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
93	C₂²⋊Q₈	—	2	1	no	—
94	Q₃₂	—	194	2	no	—
95	SD₃₂	—	194	2	no	—
96	M₃₂	—	97	2	no	—
97	#106	C₂×D₉₇	1	1	yes	characteristic, complemented
98	#103	—	2	1	no	—
99	#41	—	388	1	no	—
100	#41	—	388	1	no	—
101	#41	—	194	2	no	—
102	#41	—	194	2	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	194	1	no	—
105	C₂×D₄	—	388	1	no	—
106	C₂×D₄	—	194	1	no	—
107	C₂²×C₄	—	388	1	no	—
108	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	97	2	no	—
111	Q₁₆	—	194	1	no	—
112	SD₁₆	—	194	2	no	—
113	SD₁₆	—	194	2	no	—
114	D₈	—	97	2	no	—
115	D₈	—	194	1	no	—
116	M₁₆	—	2	2	no	—
117	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₂²⋊C₄	—	4	1	no	—
122	C₁₆	—	194	2	no	—
123	C₂³	—	388	1	no	—
124	C₂³	#13095496	1	1	yes	characteristic
125	Q₈	—	4	1	no	—
126	Q₈	—	4	1	no	—
127	Q₈	—	388	1	no	—
128	Q₈	—	194	2	no	—
129	D₄	—	388	1	no	—
130	D₄	—	388	2	no	—
131	D₄	—	388	2	no	—
132	D₄	—	194	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	388	2	no	—
138	C₂×C₄	—	388	1	no	—
139	C₂×C₄	—	388	2	no	—
140	C₂×C₄	#13095496	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	D₄×D₉₇	1	2	yes	—
144	C₂²	—	2	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	776	2	no	—
147	C₂²	—	388	2	no	—
148	C₂²	C₂×#13095496	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	8	1	no	—
151	C₄	—	8	1	no	—
152	C₄	—	388	2	no	—
153	C₄	#15690778	1	1	yes	characteristic
154	C₄	#15690835	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	4	1	no	—
157	C₂	—	776	2	no	—
158	C₂	#20260868	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260868
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#470×C₉₇	C₂
socle	C₂×C₉₇	#447
prime radical	#20261535	C₂
hypercenter	#470	D₉₇
hypocenter	C₉₇	#2191
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #470

Complements

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

N	H ≅ G/N	type
2. #20261176	C₂	semidirect
3. #20260544	C₂	semidirect
4. #20260517	C₂	semidirect
5. #470×C₉₇	C₂	semidirect
13. #15690750	C₂²	semidirect
14. C₂×Q₁₆×C₉₇	C₂²	semidirect
17. #106×C₉₇	C₂²	semidirect
80. C₉₇	#2191	semidirect
83. #470	D₉₇	semidirect
91. C₂×Q₁₆	C₂×D₉₇	semidirect
97. #106	C₂×D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261535	2
#20261176	2
#20260544	2
#20260517	2
#470×C₉₇	2
#2191	97