Finite Group Databasebeta

Subgroups — #27143703

Order 12416.

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

normal subgroups only

This group has 210 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, complemented
2	#20261173	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15690753	C₂	1	1	yes	maximal, characteristic, complemented
4	#20261006	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	#15690842	—	2	2	no	—
10	#15690841	—	2	1	no	—
11	#15690838	—	2	1	no	—
12	#15690833	—	2	1	no	—
13	#15690832	C₂²	1	1	yes	characteristic
14	Q₁₆×D₉₇	—	2	1	no	—
15	C₂×D₇₇₆	C₂²	1	1	yes	characteristic
16	#15690816	C₂²	1	1	yes	characteristic
17	#15690755	—	2	1	no	—
18	#15690753	—	2	2	no	—
19	D₁₅₅₂	—	2	1	no	—
20	C₂×#13095486	C₂²	1	1	yes	characteristic
21	#15690726	—	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	Q₈×D₉₇	—	2	2	no	—
31	Q₈×D₉₇	—	4	1	no	—
32	C₂×D₃₈₈	—	2	1	no	—
33	Dic₃₈₈	D₄	1	1	yes	characteristic
34	#13095501	—	2	2	no	—
35	#13095500	—	2	1	no	—
36	#13095500	—	4	1	no	—
37	#13095498	—	2	2	no	—
38	D₇₇₆	—	2	1	no	—
39	D₇₇₆	—	2	1	no	—
40	D₇₇₆	—	4	1	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	C₂²×D₉₇	—	4	1	no	—
53	Q₈×C₉₇	—	2	2	no	—
54	Q₈×C₉₇	—	4	1	no	—
55	C₂×C₄×C₉₇	—	4	1	no	—
56	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
57	C₂²⋊D₉₇	—	2	1	no	—
58	C₂²⋊D₉₇	—	4	1	no	—
59	D₃₈₈	—	2	1	no	—
60	D₃₈₈	—	4	2	no	—
61	D₃₈₈	—	4	1	no	—
62	D₃₈₈	—	4	1	no	—
63	D₃₈₈	D₈	1	1	yes	characteristic
64	C₄×D₉₇	—	2	1	no	—
65	C₄×D₉₇	—	4	2	no	—
66	C₄×D₉₇	—	4	2	no	—
67	C₄×D₉₇	—	4	1	no	—
68	Dic₁₉₄	—	2	1	no	—
69	Dic₁₉₄	—	4	2	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	C₂×D₉₇	—	8	1	no	—
79	C₂×D₉₇	—	16	1	no	—
80	C₄×C₉₇	—	4	2	no	—
81	C₄×C₉₇	#120	1	1	yes	characteristic
82	C₄×C₉₇	C₂×D₈	1	1	yes	characteristic
83	Dic₉₇	—	2	1	no	—
84	Dic₉₇	—	4	1	no	—
85	C₂×C₉₇	—	2	1	no	—
86	C₂×C₉₇	#447	1	1	yes	characteristic
87	D₉₇	—	4	1	no	—
88	D₉₇	—	8	1	no	—
89	D₉₇	—	16	1	no	—
90	#2193	—	97	1	no	maximal, Sylow
91	C₉₇	#2193	1	1	yes	characteristic, complemented, Sylow
92	#578	—	97	1	no	—
93	#509	—	97	1	no	—
94	C₂×SD₃₂	—	97	1	no	—
95	#472	—	97	1	no	—
96	#362	D₉₇	1	1	yes	characteristic, complemented
97	#359	—	97	1	no	—
98	#350	—	97	1	no	—
99	2₋¹⁺⁴	—	194	1	no	—
100	#137	—	194	1	no	—
101	#137	—	194	2	no	—
102	AGL(1,ℤ/8)	—	194	1	no	—
103	#135	—	97	1	no	—
104	#135	—	194	1	no	—
105	C₂×Q₁₆	—	194	1	no	—
106	C₂×Q₁₆	C₂×D₉₇	1	1	yes	characteristic
107	C₂×D₈	—	97	1	no	—
108	#131	—	97	1	no	—
109	SD₃₂	—	194	1	no	—
110	SD₃₂	—	194	2	no	—
111	D₁₆	—	194	1	no	—
112	M₃₂	C₂×D₉₇	1	1	yes	characteristic
113	C₂×C₁₆	—	97	1	no	—
114	#108	C₂×D₉₇	1	1	yes	characteristic
115	#100	—	194	1	no	—
116	#41	—	97	1	no	—
117	#41	—	388	2	no	—
118	#41	—	388	1	no	—
119	#41	—	194	2	no	—
120	#41	—	194	1	no	—
121	C₂×Q₈	—	2	1	no	—
122	C₂×Q₈	—	388	1	no	—
123	C₂×Q₈	—	194	2	no	—
124	C₂×D₄	—	194	1	no	—
125	Q₁₆	—	2	1	no	—
126	Q₁₆	—	97	1	no	—
127	Q₁₆	—	194	2	no	—
128	Q₁₆	D₃₈₈	1	2	yes	—
129	SD₁₆	—	388	1	no	—
130	SD₁₆	—	194	2	no	—
131	SD₁₆	—	194	1	no	—
132	D₈	—	97	1	no	—
133	D₈	—	388	1	no	—
134	D₈	—	194	1	no	—
135	D₈	—	194	1	no	—
136	M₁₆	—	2	1	no	—
137	M₁₆	—	97	1	no	—
138	M₁₆	—	194	1	no	—
139	C₂×C₈	—	194	1	no	—
140	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
141	C₁₆	—	2	1	no	—
142	C₁₆	—	194	1	no	—
143	C₂³	—	388	1	no	—
144	Q₈	—	2	2	no	—
145	Q₈	—	4	1	no	—
146	Q₈	—	97	1	no	—
147	Q₈	—	388	2	no	—
148	Q₈	—	194	1	no	—
149	D₄	—	97	1	no	—
150	D₄	—	388	2	no	—
151	D₄	—	388	1	no	—
152	D₄	—	388	1	no	—
153	D₄	—	388	1	no	—
154	D₄	—	194	1	no	—
155	D₄	—	194	1	no	—
156	C₂×C₄	—	4	1	no	—
157	C₂×C₄	—	388	2	no	—
158	C₂×C₄	—	388	2	no	—
159	C₂×C₄	—	388	1	no	—
160	C₂×C₄	—	194	1	no	—
161	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
162	C₈	—	4	1	no	—
163	C₈	—	194	1	no	—
164	C₈	D₄×D₉₇	1	1	yes	characteristic
165	C₈	C₂×D₃₈₈	1	1	yes	characteristic
166	C₂²	—	776	1	no	—
167	C₂²	—	1552	1	no	—
168	C₂²	—	388	1	no	—
169	C₂²	—	194	1	no	—
170	C₂²	#15690837	1	1	yes	characteristic
171	C₄	—	4	2	no	—
172	C₄	—	388	1	no	—
173	C₄	—	194	1	no	—
174	C₄	#15690780	1	1	yes	characteristic
175	C₄	D₈×D₉₇	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	776	1	no	—
178	C₂	—	1552	1	no	—
179	C₂	—	388	1	no	—
180	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
hypercenter	#362	D₉₇
hypocenter	C₉₇	#2193
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
1. #20261548	C₂	semidirect
2. #20261173	C₂	semidirect
3. C₂×#15690753	C₂	semidirect
4. #20261006	C₂	semidirect
5. #20260541	C₂	semidirect
6. #20260520	C₂	semidirect
7. #362×C₉₇	C₂	semidirect
91. C₉₇	#2193	semidirect
96. #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
#20261173	2
C₂×#15690753	2
#20261006	2
#20260541	2
#20260520	2
#362×C₉₇	2
#2193	97