Finite Group Databasebeta

Subgroups — #13116285

Order 1664.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#11718794	C₂	1	1	yes	maximal, characteristic
2	#11718790	C₂	1	1	yes	maximal, characteristic
3	#11718627	C₂	1	1	yes	maximal, characteristic
4	C₂×#88048	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#343	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718150	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718146	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#6127	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#6099	C₂²	1	1	yes	characteristic
14	C₂²×#1055	C₂²	1	1	yes	characteristic
15	#88061	—	2	1	no	—
16	#88060	—	2	2	no	—
17	#88056	—	2	1	no	—
18	#88050	—	2	1	no	—
19	#88049	—	2	1	no	—
20	#88048	—	2	2	no	—
21	C₂²×C₄×C₁₃	—	2	1	no	—
22	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₁₃	—	2	1	no	—
24	C₂²×Dic₁₃	—	2	1	no	—
25	#6127	—	2	1	no	—
26	#6127	—	2	1	no	—
27	#6127	—	2	1	no	—
28	#6127	—	2	1	no	—
29	C₂×D₅₂	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₃	—	2	1	no	—
31	C₂×Dic₂₆	D₄	1	1	yes	characteristic
32	C₁₃×M₁₆	—	2	1	no	—
33	C₁₃×M₁₆	—	2	1	no	—
34	C₁₃×M₁₆	—	2	1	no	—
35	C₁₃×M₁₆	—	2	1	no	—
36	C₂×C₈×C₁₃	—	2	1	no	—
37	C₂×C₈×C₁₃	—	2	1	no	—
38	C₁₃×C₄⋊C₄	—	2	1	no	—
39	C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₄⋊Dic₁₃	—	2	1	no	—
41	C₄⋊Dic₁₃	—	2	1	no	—
42	#6099	—	2	1	no	—
43	#6099	—	2	1	no	—
44	C₂×#1055	—	2	1	no	—
45	C₂×#1055	—	2	1	no	—
46	C₂×#1055	—	2	1	no	—
47	C₂×#1055	D₄	1	2	yes	—
48	#2079	—	13	1	no	maximal, Sylow
49	C₂³×C₁₃	#41	1	1	yes	characteristic
50	C₂²×D₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	4	1	no	—
52	C₂×C₄×C₁₃	—	8	1	no	—
53	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₂²⋊D₁₃	—	4	1	no	—
60	C₂²⋊D₁₃	—	4	1	no	—
61	C₂×Dic₁₃	—	2	1	no	—
62	C₂×Dic₁₃	—	4	1	no	—
63	C₂×Dic₁₃	—	8	1	no	—
64	D₅₂	—	2	1	no	—
65	D₅₂	—	2	1	no	—
66	C₄×D₁₃	—	4	1	no	—
67	C₄×D₁₃	—	4	1	no	—
68	Dic₂₆	—	2	1	no	—
69	Dic₂₆	—	2	1	no	—
70	C₈×C₁₃	—	4	1	no	—
71	C₈×C₁₃	—	4	1	no	—
72	#1055	—	2	2	no	—
73	#1055	—	4	1	no	—
74	C₂×#106	—	13	1	no	—
75	#417	—	13	1	no	—
76	#417	—	13	1	no	—
77	#408	—	13	1	no	—
78	#343	—	13	1	no	—
79	#343	D₁₃	1	1	yes	characteristic, complemented
80	#341	—	13	1	no	—
81	C₂²×C₁₃	—	2	1	no	—
82	C₂²×C₁₃	—	2	1	no	—
83	C₂²×C₁₃	#121	1	1	yes	characteristic
84	C₂²×C₁₃	#126	1	1	yes	characteristic
85	C₂²×C₁₃	#126	1	1	yes	characteristic
86	C₂×D₁₃	—	4	1	no	—
87	C₂×D₁₃	—	4	1	no	—
88	C₂×D₁₃	—	4	1	no	—
89	C₄×C₁₃	—	8	1	no	—
90	C₄×C₁₃	#123	1	1	yes	characteristic
91	C₄×C₁₃	#123	1	1	yes	characteristic
92	C₄×C₁₃	#123	1	1	yes	characteristic
93	C₄×C₁₃	#124	1	1	yes	characteristic
94	Dic₁₃	—	4	1	no	—
95	Dic₁₃	—	8	1	no	—
96	C₂×#41	—	13	1	no	—
97	C₂×M₁₆	—	13	1	no	—
98	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
99	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
100	C₂²×C₈	—	13	1	no	—
101	C₂×C₄⋊C₄	—	13	1	no	—
102	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
103	#106	—	26	2	no	—
104	#103	—	26	1	no	—
105	#103	—	26	1	no	—
106	#102	—	26	1	no	—
107	#102	—	26	1	no	—
108	C₂²⋊C₈	—	26	2	no	—
109	C₂×C₁₃	—	2	1	no	—
110	C₂×C₁₃	—	2	1	no	—
111	C₂×C₁₃	#397	1	1	yes	characteristic
112	C₂×C₁₃	#471	1	1	yes	characteristic
113	C₂×C₁₃	#471	1	1	yes	characteristic
114	D₁₃	—	8	1	no	—
115	#41	—	26	1	no	—
116	#41	—	26	1	no	—
117	#41	—	26	1	no	—
118	#41	—	26	1	no	—
119	C₂×Q₈	—	13	1	no	—
120	C₂×D₄	—	13	1	no	—
121	C₂×D₄	—	26	1	no	—
122	C₂²×C₄	—	2	1	no	—
123	C₂²×C₄	—	26	1	no	—
124	C₂²×C₄	—	26	1	no	—
125	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
126	M₁₆	—	2	1	no	—
127	M₁₆	—	2	1	no	—
128	M₁₆	—	2	1	no	—
129	M₁₆	—	2	1	no	—
130	M₁₆	—	26	1	no	—
131	M₁₆	—	26	1	no	—
132	C₂×C₈	—	2	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	13	2	no	—
135	C₂×C₈	—	26	1	no	—
136	C₂×C₈	—	26	1	no	—
137	C₂×C₈	—	26	1	no	—
138	C₄⋊C₄	—	2	1	no	—
139	C₄⋊C₄	—	2	1	no	—
140	C₄⋊C₄	—	26	1	no	—
141	C₄⋊C₄	—	26	1	no	—
142	C₁₃	#2079	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	26	1	no	—
144	C₂³	#6127	1	1	yes	characteristic
145	Q₈	—	26	1	no	—
146	Q₈	—	26	1	no	—
147	D₄	—	26	1	no	—
148	D₄	—	26	1	no	—
149	D₄	—	52	1	no	—
150	D₄	—	52	1	no	—
151	C₂×C₄	—	4	1	no	—
152	C₂×C₄	—	8	1	no	—
153	C₂×C₄	—	26	1	no	—
154	C₂×C₄	—	52	1	no	—
155	C₂×C₄	—	52	1	no	—
156	C₂×C₄	—	52	1	no	—
157	C₂×C₄	—	104	1	no	—
158	C₂×C₄	#6127	1	1	yes	characteristic
159	C₂×C₄	#6127	1	1	yes	characteristic
160	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
161	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
162	C₂×C₄	#6129	1	1	yes	characteristic
163	C₂×C₄	#6131	1	1	yes	characteristic
164	C₈	—	4	1	no	—
165	C₈	—	4	1	no	—
166	C₈	—	26	2	no	—
167	C₈	—	52	1	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	52	1	no	—
171	C₂²	—	52	1	no	—
172	C₂²	—	52	1	no	—
173	C₂²	#88130	1	1	yes	characteristic
174	C₂²	#88138	1	1	yes	characteristic
175	C₂²	#88152	1	1	yes	characteristic
176	C₄	—	8	1	no	—
177	C₄	—	52	1	no	—
178	C₄	—	104	1	no	—
179	C₄	#88137	1	1	yes	characteristic
180	C₄	#88139	1	1	yes	characteristic
181	C₄	#88148	1	1	yes	characteristic
182	C₄	#88148	1	1	yes	characteristic
183	C₂	—	2	1	no	—
184	C₂	—	2	1	no	—
185	C₂	—	104	1	no	—
186	C₂	#11718336	1	1	yes	characteristic
187	C₂	#11718409	1	1	yes	characteristic
188	C₂	#11718415	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#88138
commutator, cosocle	C₂²×C₄×C₁₃	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#343	C₂
socle	C₂²×C₁₃	#121
prime radical	C₂×#6127	C₂²
hypercenter	#343	D₁₃
hypocenter	C₁₃	#2079
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

Complements

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

N	H ≅ G/N	type
4. C₂×#88048	C₂	semidirect
5. C₁₃×#343	C₂	semidirect
6. #11718150	C₂	semidirect
7. #11718146	C₂	semidirect
79. #343	D₁₃	semidirect
142. C₁₃	#2079	semidirect

Maximal subgroups

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

subgroup	index
#11718794	2
#11718790	2
#11718627	2
C₂×#88048	2
C₁₃×#343	2
#11718150	2
#11718146	2
#2079	13