Finite Group Databasebeta

Subgroups — #27143717

Order 12416.

206 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 47 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	#20261548	C₂	1	1	yes	maximal, characteristic
2	#20261178	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15690756	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261011	C₂	1	1	yes	maximal, characteristic, complemented
5	#20260541	C₂	1	1	yes	maximal, characteristic, complemented
6	#20260520	C₂	1	1	yes	maximal, characteristic, complemented
7	#362×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#15690858	—	2	1	no	—
9	#15690843	—	2	1	no	—
10	#15690842	—	2	2	no	—
11	#15690841	—	2	1	no	—
12	#15690833	—	2	1	no	—
13	#15690832	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₈₈	C₂²	1	1	yes	characteristic
15	Q₁₆×D₉₇	—	2	1	no	—
16	#15690816	C₂²	1	1	yes	characteristic
17	Dic₇₇₆	—	2	1	no	—
18	#15690756	—	2	2	no	—
19	#15690755	—	2	1	no	—
20	C₂×#13095486	C₂²	1	1	yes	characteristic
21	#15690728	—	2	1	no	—
22	C₂×Q₁₆×C₉₇	C₂²	1	1	yes	characteristic
23	M₃₂×C₉₇	C₂²	1	1	yes	characteristic
24	#108×C₉₇	C₂²	1	1	yes	characteristic
25	#13095512	—	2	2	no	—
26	#13095512	—	4	1	no	—
27	#13095511	—	2	1	no	—
28	#13095511	—	4	2	no	—
29	#13095511	D₄	1	1	yes	characteristic
30	C₂×Dic₁₉₄	—	2	1	no	—
31	Q₈×D₉₇	—	2	2	no	—
32	Q₈×D₉₇	—	4	1	no	—
33	Dic₃₈₈	—	2	1	no	—
34	Dic₃₈₈	—	2	1	no	—
35	Dic₃₈₈	—	4	1	no	—
36	Dic₃₈₈	D₄	1	1	yes	characteristic
37	#13095501	—	2	2	no	—
38	#13095500	—	2	1	no	—
39	#13095500	—	4	1	no	—
40	#13095498	—	2	2	no	—
41	D₇₇₆	D₄	1	1	yes	characteristic
42	#13095495	D₄	1	1	yes	characteristic
43	#13095494	—	2	1	no	—
44	C₈×D₉₇	—	2	1	no	—
45	#13095486	—	2	1	no	—
46	C₂×Q₈×C₉₇	—	2	1	no	—
47	Q₁₆×C₉₇	—	2	1	no	—
48	Q₁₆×C₉₇	D₄	1	2	yes	—
49	M₁₆×C₉₇	—	2	1	no	—
50	C₂×C₈×C₉₇	C₂³	1	1	yes	characteristic
51	C₁₆×C₉₇	—	2	1	no	—
52	Q₈×C₉₇	—	2	2	no	—
53	Q₈×C₉₇	—	4	1	no	—
54	C₂×C₄×C₉₇	—	4	1	no	—
55	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
56	C₂²⋊D₉₇	—	2	1	no	—
57	C₂²⋊D₉₇	—	4	1	no	—
58	C₂×Dic₉₇	—	4	1	no	—
59	D₃₈₈	—	2	1	no	—
60	D₃₈₈	—	4	2	no	—
61	D₃₈₈	D₈	1	1	yes	characteristic
62	C₄×D₉₇	—	2	1	no	—
63	C₄×D₉₇	—	4	2	no	—
64	C₄×D₉₇	—	4	2	no	—
65	C₄×D₉₇	—	4	1	no	—
66	Dic₁₉₄	—	2	1	no	—
67	Dic₁₉₄	—	4	2	no	—
68	Dic₁₉₄	—	4	1	no	—
69	Dic₁₉₄	—	4	1	no	—
70	Dic₁₉₄	D₈	1	1	yes	characteristic
71	C₈×C₉₇	—	4	1	no	—
72	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₉₇	C₂×D₄	1	1	yes	characteristic
74	#11715895	—	2	1	no	—
75	C₂²×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×D₉₇	—	2	1	no	—
77	C₂×D₉₇	—	4	1	no	—
78	C₄×C₉₇	—	4	2	no	—
79	C₄×C₉₇	#120	1	1	yes	characteristic
80	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
81	Dic₉₇	—	2	1	no	—
82	Dic₉₇	—	4	1	no	—
83	Dic₉₇	—	8	1	no	—
84	C₂×C₉₇	—	2	1	no	—
85	C₂×C₉₇	#447	1	1	yes	characteristic
86	D₉₇	—	4	1	no	—
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	D₉₇	1	1	yes	characteristic, complemented
95	#359	—	97	1	no	—
96	#350	—	97	1	no	—
97	2₋¹⁺⁴	—	194	1	no	—
98	#137	—	194	1	no	—
99	#137	—	194	1	no	—
100	#137	—	194	2	no	—
101	#135	—	97	1	no	—
102	#135	—	194	1	no	—
103	C₂×Q₁₆	—	97	1	no	—
104	C₂×Q₁₆	—	194	1	no	—
105	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic
106	#131	—	97	1	no	—
107	Q₃₂	—	194	1	no	—
108	Q₃₂	—	194	2	no	—
109	SD₃₂	—	194	1	no	—
110	M₃₂	C₂×D₉₇	1	1	yes	characteristic
111	C₂×C₁₆	—	97	1	no	—
112	#108	C₂×D₉₇	1	1	yes	characteristic
113	#101	—	194	1	no	—
114	#41	—	97	1	no	—
115	#41	—	388	2	no	—
116	#41	—	388	1	no	—
117	#41	—	194	2	no	—
118	#41	—	194	1	no	—
119	C₂×Q₈	—	2	1	no	—
120	C₂×Q₈	—	388	1	no	—
121	C₂×Q₈	—	194	2	no	—
122	C₂×Q₈	—	194	1	no	—
123	Q₁₆	—	2	1	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	D₃₈₈	1	2	yes	—
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	—	97	1	no	—
136	M₁₆	—	194	1	no	—
137	C₂×C₈	—	194	1	no	—
138	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
139	C₁₆	—	2	1	no	—
140	C₁₆	—	194	1	no	—
141	Q₈	—	2	2	no	—
142	Q₈	—	4	1	no	—
143	Q₈	—	97	1	no	—
144	Q₈	—	388	2	no	—
145	Q₈	—	388	1	no	—
146	Q₈	—	388	1	no	—
147	Q₈	—	194	1	no	—
148	D₄	—	97	1	no	—
149	D₄	—	388	2	no	—
150	D₄	—	388	1	no	—
151	D₄	—	194	1	no	—
152	D₄	—	194	1	no	—
153	C₂×C₄	—	4	1	no	—
154	C₂×C₄	—	388	2	no	—
155	C₂×C₄	—	388	2	no	—
156	C₂×C₄	—	388	1	no	—
157	C₂×C₄	—	388	1	no	—
158	C₂×C₄	—	194	1	no	—
159	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
160	C₈	—	4	1	no	—
161	C₈	—	194	1	no	—
162	C₈	D₄×D₉₇	1	1	yes	characteristic
163	C₈	C₂×D₃₈₈	1	1	yes	characteristic
164	C₂²	—	388	1	no	—
165	C₂²	—	194	1	no	—
166	C₂²	#15690837	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	776	1	no	—
169	C₄	—	388	1	no	—
170	C₄	—	194	1	no	—
171	C₄	#15690780	1	1	yes	characteristic
172	C₄	D₈×D₉₇	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	776	1	no	—
175	C₂	—	388	1	no	—
176	C₂	#20260870	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20260870
commutator, cosocle	C₂×C₈×C₉₇	C₂³
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	#362×C₉₇	C₂
socle	C₂×C₉₇	#447
prime radical	#20261548	C₂
hypercenter	#362	D₉₇
hypocenter	C₉₇	#2195
LCS	G ▷ C₂×C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #362

Complements

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

N	H ≅ G/N	type
2. #20261178	C₂	semidirect
3. C₂×#15690756	C₂	semidirect
4. #20261011	C₂	semidirect
5. #20260541	C₂	semidirect
6. #20260520	C₂	semidirect
7. #362×C₉₇	C₂	semidirect
89. C₉₇	#2195	semidirect
94. #362	D₉₇	semidirect

Maximal subgroups

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

subgroup	index
#20261548	2
#20261178	2
C₂×#15690756	2
#20261011	2
#20260541	2
#20260520	2
#362×C₉₇	2
#2195	97