Finite Group Databasebeta

Subgroups — #13115548

Order 1664.

222 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 75 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	#11718790	C₂	1	1	yes	maximal, characteristic
2	#11718663	C₂	1	1	yes	maximal, characteristic
3	#11718660	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88045	C₂	1	1	yes	maximal, characteristic
5	C₂×#88043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#339	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718150	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	C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#6099	C₂²	1	1	yes	characteristic
13	C₂²×#1055	C₂²	1	1	yes	characteristic
14	C₂²×#1055	C₂²	1	1	yes	characteristic
15	#88060	—	2	2	no	—
16	#88050	C₄	1	2	yes	complemented
17	#88049	C₄	1	2	yes	complemented
18	#88045	—	2	2	no	—
19	C₄⋊#1055	—	2	2	no	—
20	#88043	—	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	#6127	—	2	1	no	—
25	#6127	—	2	1	no	—
26	#6127	—	2	1	no	—
27	#6127	—	2	1	no	—
28	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₁₃	—	2	1	no	—
30	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
31	C₁₃×M₁₆	—	2	1	no	—
32	C₁₃×M₁₆	—	2	1	no	—
33	C₂×C₈×C₁₃	—	2	1	no	—
34	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₁₃×C₂²⋊C₄	—	2	1	no	—
36	C₄²×C₁₃	—	2	1	no	—
37	C₄²×C₁₃	—	2	1	no	—
38	C₄²×C₁₃	D₄	1	2	yes	—
39	#6099	—	2	1	no	—
40	#6099	—	2	1	no	—
41	C₂×#1055	—	2	1	no	—
42	C₂×#1055	—	2	1	no	—
43	C₂×#1055	—	2	1	no	—
44	C₂×#1055	—	2	1	no	—
45	C₂×#1055	—	2	1	no	—
46	C₂×#1055	C₂×C₄	1	2	yes	—
47	C₂×#1055	D₄	1	2	yes	—
48	#1938	—	13	1	no	maximal, Sylow
49	C₂³×C₁₃	#41	1	1	yes	characteristic
50	C₂²×D₁₃	—	2	1	no	—
51	C₂×C₄×C₁₃	—	2	2	no	—
52	C₂×C₄×C₁₃	—	2	2	no	—
53	C₂×C₄×C₁₃	—	4	2	no	—
54	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
58	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
59	C₂×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₂²⋊D₁₃	—	4	1	no	—
61	C₂²⋊D₁₃	—	4	1	no	—
62	C₂×Dic₁₃	—	2	1	no	—
63	D₅₂	—	2	1	no	—
64	D₅₂	—	2	1	no	—
65	C₄×D₁₃	—	4	1	no	—
66	C₄×D₁₃	—	4	1	no	—
67	Dic₂₆	—	2	1	no	—
68	Dic₂₆	—	2	1	no	—
69	C₈×C₁₃	—	4	1	no	—
70	#1055	—	2	2	no	—
71	#1055	—	2	2	no	—
72	#1055	—	4	1	no	—
73	#424	—	13	1	no	—
74	C₂×#104	—	13	1	no	—
75	#416	—	13	1	no	—
76	#408	—	13	1	no	—
77	C₂×#97	—	13	1	no	—
78	#341	—	13	1	no	—
79	#339	D₁₃	1	1	yes	characteristic, complemented
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	C₄×D₄	1	1	yes	characteristic
82	C₂²×C₁₃	#123	1	1	yes	characteristic
83	C₂²×C₁₃	#126	1	1	yes	characteristic
84	C₂×D₁₃	—	4	1	no	—
85	C₂×D₁₃	—	4	2	no	—
86	C₄×C₁₃	—	4	2	no	—
87	C₄×C₁₃	—	4	2	no	—
88	C₄×C₁₃	#117	1	1	yes	characteristic
89	C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
90	C₄×C₁₃	#121	1	1	yes	characteristic
91	C₄×C₁₃	#124	1	1	yes	characteristic
92	Dic₁₃	—	4	1	no	—
93	C₂×#41	—	13	1	no	—
94	C₂×M₁₆	—	13	1	no	—
95	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
96	C₂²×C₈	—	13	1	no	—
97	C₂²×C₈	—	13	1	no	—
98	#117	C₂×D₁₃	1	1	yes	characteristic
99	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
100	C₄⋊C₈	—	26	2	no	—
101	#104	—	26	2	no	—
102	#103	—	13	2	no	—
103	#102	—	13	2	no	—
104	C₂²⋊C₈	—	26	2	no	—
105	#97	—	26	2	no	—
106	C₂×C₁₃	—	2	1	no	—
107	C₂×C₁₃	—	2	1	no	—
108	C₂×C₁₃	#388	1	1	yes	characteristic
109	C₂×C₁₃	#444	1	2	yes	—
110	D₁₃	—	8	1	no	—
111	#41	—	26	1	no	—
112	#41	—	26	1	no	—
113	#41	—	26	1	no	—
114	#41	—	26	1	no	—
115	C₂×Q₈	—	13	1	no	—
116	C₂×D₄	—	13	1	no	—
117	C₂×D₄	—	26	1	no	—
118	C₂²×C₄	—	2	1	no	—
119	C₂²×C₄	—	26	1	no	—
120	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
121	M₁₆	—	2	1	no	—
122	M₁₆	—	2	1	no	—
123	M₁₆	—	26	1	no	—
124	M₁₆	—	26	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	13	2	no	—
127	C₂×C₈	—	13	2	no	—
128	C₂×C₈	—	26	1	no	—
129	C₂×C₈	—	26	1	no	—
130	C₂×C₈	—	26	1	no	—
131	C₂×C₈	—	26	1	no	—
132	C₂×C₈	—	26	1	no	—
133	C₄⋊C₄	C₄×D₁₃	1	2	yes	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	D₅₂	1	2	yes	—
138	C₁₃	#1938	1	1	yes	characteristic, complemented, Sylow
139	C₂³	—	26	1	no	—
140	C₂³	#6129	1	1	yes	characteristic
141	Q₈	—	26	1	no	—
142	Q₈	—	26	1	no	—
143	D₄	—	26	1	no	—
144	D₄	—	26	1	no	—
145	D₄	—	52	1	no	—
146	D₄	—	52	1	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	—	26	1	no	—
151	C₂×C₄	—	52	1	no	—
152	C₂×C₄	—	52	1	no	—
153	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
155	C₂×C₄	#6127	1	1	yes	characteristic
156	C₂×C₄	D₄×D₁₃	1	1	yes	characteristic
157	C₂×C₄	#6129	1	1	yes	characteristic
158	C₂×C₄	#6131	1	1	yes	characteristic
159	C₈	—	4	1	no	—
160	C₈	—	26	2	no	—
161	C₈	—	26	2	no	—
162	C₈	—	52	1	no	—
163	C₂²	—	2	2	no	—
164	C₂²	—	52	2	no	—
165	C₂²	—	52	1	no	—
166	C₂²	#88135	1	1	yes	characteristic
167	C₂²	#88140	1	1	yes	characteristic
168	C₂²	#88152	1	1	yes	characteristic
169	C₄	—	4	2	no	—
170	C₄	—	4	2	no	—
171	C₄	—	52	1	no	—
172	C₄	C₄×D₅₂	1	1	yes	characteristic
173	C₄	#88139	1	1	yes	characteristic
174	C₄	#88146	1	1	yes	characteristic
175	C₄	#88149	1	1	yes	characteristic
176	C₂	—	2	1	no	—
177	C₂	—	2	1	no	—
178	C₂	—	104	1	no	—
179	C₂	#11718331	1	1	yes	characteristic
180	C₂	#11718484	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃
Fitting	C₁₃×#339	C₂
socle	C₂²×C₁₃	C₄×D₄
prime radical	C₂×#6127	C₂²
cosocle	C₂²×C₄×C₁₃	C₂³
hypercenter	#339	D₁₃
hypocenter	C₁₃	#1938
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #339

Complements

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

N	H ≅ G/N	type
3. #11718660	C₂	semidirect
5. C₂×#88043	C₂	semidirect
6. C₁₃×#339	C₂	semidirect
7. #11718150	C₂	semidirect
16. #88050	C₄	semidirect
17. #88049	C₄	semidirect
79. #339	D₁₃	semidirect
138. C₁₃	#1938	semidirect

Maximal subgroups

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

subgroup	index
#11718790	2
#11718663	2
#11718660	2
C₂×#88045	2
C₂×#88043	2
C₁₃×#339	2
#11718150	2
#1938	13