Finite Group Databasebeta

Subgroups — #27143716

Order 12416.

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

normal subgroups only

This group has 206 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	#20261552	C₂	1	1	yes	maximal, characteristic
2	#20261178	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₇₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261010	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260552	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260543	C₂	1	1	yes	maximal, characteristic, complemented
7	#350×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#15690857	—	2	1	no	—
9	#15690843	—	2	2	no	—
10	#15690843	—	2	1	no	—
11	#15690840	—	2	1	no	—
12	#15690832	—	2	1	no	—
13	#15690832	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₈₈	—	2	1	no	—
15	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
17	Dic₇₇₆	—	2	2	no	—
18	Dic₇₇₆	—	2	1	no	—
19	#15690755	—	2	1	no	—
20	#15690746	C₂²	1	1	yes	characteristic
21	#15690728	—	2	1	no	—
22	#131×C₉₇	C₂²	1	1	yes	characteristic
23	M₃₂×C₉₇	C₂²	1	1	yes	characteristic
24	C₂×C₁₆×C₉₇	C₂²	1	1	yes	characteristic
25	#13095511	—	2	2	no	—
26	#13095511	—	2	1	no	—
27	#13095510	—	4	2	no	—
28	#13095510	—	4	1	no	—
29	C₂×Dic₁₉₄	—	2	2	no	—
30	C₂×Dic₁₉₄	—	2	1	no	—
31	C₂×Dic₁₉₄	—	2	1	no	—
32	Q₈×D₉₇	—	4	1	no	—
33	Dic₃₈₈	—	2	2	no	—
34	Dic₃₈₈	—	2	1	no	—
35	Dic₃₈₈	—	2	1	no	—
36	Dic₃₈₈	—	2	1	no	—
37	Dic₃₈₈	D₄	1	1	yes	characteristic
38	Dic₃₈₈	D₄	1	2	yes	—
39	#13095501	—	4	1	no	—
40	#13095500	—	2	2	no	—
41	#13095500	—	2	1	no	—
42	#13095499	—	4	1	no	—
43	D₇₇₆	D₄	1	1	yes	characteristic
44	#13095495	—	2	1	no	—
45	#41×C₉₇	D₄	1	1	yes	characteristic
46	M₁₆×C₉₇	—	2	1	no	—
47	M₁₆×C₉₇	D₄	1	1	yes	characteristic
48	C₂×C₈×C₉₇	—	2	1	no	—
49	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
50	C₁₆×C₉₇	—	2	1	no	—
51	C₁₆×C₉₇	—	2	1	no	—
52	Q₈×C₉₇	D₈	1	1	yes	characteristic
53	D₄×C₉₇	—	2	1	no	—
54	D₄×C₉₇	D₈	1	1	yes	characteristic
55	C₂×C₄×C₉₇	—	2	1	no	—
56	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₉₇	—	4	2	no	—
58	C₂²⋊D₉₇	—	4	1	no	—
59	C₂×Dic₉₇	—	4	2	no	—
60	C₂×Dic₉₇	—	4	1	no	—
61	C₂×Dic₉₇	—	4	1	no	—
62	D₃₈₈	—	2	1	no	—
63	C₄×D₉₇	—	4	2	no	—
64	C₄×D₉₇	—	4	1	no	—
65	Dic₁₉₄	—	2	2	no	—
66	Dic₁₉₄	—	2	1	no	—
67	Dic₁₉₄	—	4	2	no	—
68	Dic₁₉₄	—	4	1	no	—
69	Dic₁₉₄	—	4	1	no	—
70	Dic₁₉₄	—	4	1	no	—
71	C₈×C₉₇	—	2	1	no	—
72	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
74	#11715895	—	4	1	no	—
75	C₂²×C₉₇	—	2	1	no	—
76	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₉₇	—	4	1	no	—
78	C₄×C₉₇	—	2	1	no	—
79	C₄×C₉₇	#120	1	1	yes	characteristic
80	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
81	Dic₉₇	—	4	1	no	—
82	Dic₉₇	—	4	2	no	—
83	Dic₉₇	—	8	1	no	—
84	C₂×C₉₇	—	2	1	no	—
85	C₂×C₉₇	—	4	1	no	—
86	C₂×C₉₇	#447	1	1	yes	characteristic
87	D₉₇	—	8	1	no	—
88	#2195	—	97	1	no	maximal, Sylow
89	C₉₇	#2195	1	1	yes	characteristic, complemented, Sylow
90	#578	—	97	1	no	—
91	#510	—	97	1	no	—
92	C₂×Q₃₂	—	97	1	no	—
93	#473	—	97	1	no	—
94	#362	—	97	1	no	—
95	#359	—	97	1	no	—
96	#350	D₉₇	1	1	yes	characteristic, complemented
97	2₋¹⁺⁴	—	194	1	no	—
98	#137	—	194	2	no	—
99	#137	—	194	1	no	—
100	#137	—	194	1	no	—
101	#135	—	97	1	no	—
102	#135	—	194	1	no	—
103	C₂×Q₁₆	—	97	1	no	—
104	C₂×Q₁₆	—	97	1	no	—
105	C₂×Q₁₆	—	194	1	no	—
106	#131	C₂×D₉₇	1	1	yes	characteristic
107	Q₃₂	—	194	2	no	—
108	Q₃₂	—	194	1	no	—
109	SD₃₂	—	194	1	no	—
110	M₃₂	C₂×D₉₇	1	1	yes	characteristic
111	C₂×C₁₆	C₂×D₉₇	1	1	yes	characteristic
112	#108	—	97	1	no	—
113	#101	—	194	1	no	—
114	#41	—	388	2	no	—
115	#41	—	388	1	no	—
116	#41	—	194	2	no	—
117	#41	—	194	1	no	—
118	#41	D₃₈₈	1	1	yes	characteristic
119	C₂×Q₈	—	388	1	no	—
120	C₂×Q₈	—	194	2	no	—
121	C₂×Q₈	—	194	1	no	—
122	C₂×Q₈	—	194	1	no	—
123	Q₁₆	—	97	2	no	—
124	Q₁₆	—	97	1	no	—
125	Q₁₆	—	388	1	no	—
126	Q₁₆	—	194	2	no	—
127	Q₁₆	—	194	1	no	—
128	Q₁₆	—	194	1	no	—
129	Q₁₆	—	194	1	no	—
130	SD₁₆	—	388	1	no	—
131	SD₁₆	—	194	2	no	—
132	SD₁₆	—	194	1	no	—
133	D₈	—	97	1	no	—
134	M₁₆	—	2	1	no	—
135	M₁₆	—	194	1	no	—
136	M₁₆	D₃₈₈	1	1	yes	characteristic
137	C₂×C₈	—	2	1	no	—
138	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	2	1	no	—
141	Q₈	—	388	2	no	—
142	Q₈	—	388	1	no	—
143	Q₈	—	388	1	no	—
144	Q₈	—	388	1	no	—
145	Q₈	—	194	2	no	—
146	Q₈	—	194	1	no	—
147	Q₈	D₇₇₆	1	1	yes	characteristic
148	D₄	—	2	1	no	—
149	D₄	—	388	2	no	—
150	D₄	—	388	1	no	—
151	D₄	—	194	1	no	—
152	D₄	D₇₇₆	1	1	yes	characteristic
153	C₂×C₄	—	2	1	no	—
154	C₂×C₄	—	388	2	no	—
155	C₂×C₄	—	388	1	no	—
156	C₂×C₄	—	388	2	no	—
157	C₂×C₄	—	388	1	no	—
158	C₂×C₄	—	388	1	no	—
159	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
160	C₈	—	2	1	no	—
161	C₈	—	388	1	no	—
162	C₈	D₄×D₉₇	1	1	yes	characteristic
163	C₈	D₄×D₉₇	1	1	yes	characteristic
164	C₂²	—	2	1	no	—
165	C₂²	—	388	1	no	—
166	C₂²	#15690838	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	776	1	no	—
169	C₄	—	388	2	no	—
170	C₄	—	388	1	no	—
171	C₄	#15690780	1	1	yes	characteristic
172	C₄	C₂×D₇₇₆	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	4	1	no	—
175	C₂	—	776	1	no	—
176	C₂	#20260871	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260871
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#350×C₉₇	C₂
socle	C₂×C₉₇	#447
prime radical	#20261552	C₂
hypercenter	#350	D₉₇
hypocenter	C₉₇	#2195
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #350

Complements

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

N	H ≅ G/N	type
2. #20261178	C₂	semidirect
3. C₂×Dic₇₇₆	C₂	semidirect
4. #20261010	C₂	semidirect
5. #20260552	C₂	semidirect
6. #20260543	C₂	semidirect
7. #350×C₉₇	C₂	semidirect
89. C₉₇	#2195	semidirect
96. #350	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261552	2
#20261178	2
C₂×Dic₇₇₆	2
#20261010	2
#20260552	2
#20260543	2
#350×C₉₇	2
#2195	97