Finite Group Databasebeta

Subgroups — #13116498

Order 1664.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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	#11719421	C₂	1	1	yes	maximal, characteristic
2	#11718401	C₂	1	2	yes	maximal, complemented
3	C₁₃×#353	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718137	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718065	C₂	1	2	yes	maximal, complemented
6	C₂×#6129	C₄	1	1	yes	characteristic, complemented
7	C₂×#6129	C₂²	1	1	yes	characteristic
8	C₁₃×#127	C₂²	1	1	yes	characteristic
9	#88192	—	2	2	no	—
10	#88190	C₄	1	1	yes	characteristic, complemented
11	#88190	C₂²	1	1	yes	characteristic
12	D₄×Dic₁₃	—	2	2	no	—
13	#88131	—	2	2	no	—
14	C₄²×D₁₃	C₄	1	1	yes	characteristic, complemented
15	#88123	C₄	1	1	yes	characteristic, complemented
16	C₁₃×#99	C₂²	1	2	yes	—
17	#88074	C₂²	1	2	yes	—
18	#88046	—	2	2	no	—
19	C₂×D₄×C₁₃	—	2	2	no	—
20	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₁₃	—	2	1	no	—
23	C₂×C₂²⋊D₁₃	D₄	1	2	yes	—
24	C₂²×Dic₁₃	—	2	1	no	—
25	C₂²×Dic₁₃	D₄	1	2	yes	—
26	#6129	—	4	1	no	—
27	#6129	—	4	1	no	—
28	#6129	—	4	1	no	—
29	#6129	—	4	1	no	—
30	C₂×C₄×D₁₃	—	2	1	no	—
31	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
33	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
34	C₁₃×C₂²⋊C₄	—	2	2	no	—
35	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²⋊Dic₁₃	—	2	2	no	—
37	C₂²⋊Dic₁₃	—	2	2	no	—
38	C₂²⋊Dic₁₃	—	2	2	no	—
39	C₄⋊Dic₁₃	—	2	2	no	—
40	#6101	—	4	2	no	—
41	C₄×Dic₁₃	—	2	1	no	—
42	C₄×Dic₁₃	—	4	2	no	—
43	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
44	#2126	—	13	1	no	maximal, Sylow
45	C₂³×C₁₃	—	2	1	no	—
46	C₂³×C₁₃	C₂×D₄	1	2	yes	—
47	C₂²×D₁₃	C₂²⋊C₄	1	1	yes	characteristic
48	D₄×C₁₃	—	4	1	no	—
49	D₄×C₁₃	—	4	4	no	—
50	D₄×C₁₃	—	4	1	no	—
51	C₂×C₄×C₁₃	—	2	1	no	—
52	C₂×C₄×C₁₃	—	4	2	no	—
53	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²⋊D₁₃	—	4	2	no	—
55	C₂²⋊D₁₃	—	4	2	no	—
56	C₂×Dic₁₃	—	2	2	no	—
57	C₂×Dic₁₃	—	2	2	no	—
58	C₂×Dic₁₃	—	2	1	no	—
59	C₂×Dic₁₃	—	2	1	no	—
60	C₂×Dic₁₃	—	4	1	no	—
61	C₂×Dic₁₃	—	4	1	no	—
62	C₂×Dic₁₃	—	4	2	no	—
63	C₂×Dic₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂×Dic₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₄×D₁₃	—	2	1	no	—
66	C₄×D₁₃	—	2	1	no	—
67	C₄×D₁₃	—	4	2	no	—
68	Dic₂₆	—	4	1	no	—
69	Dic₂₆	—	4	1	no	—
70	#532	—	13	1	no	—
71	#410	—	13	2	no	—
72	#354	—	13	2	no	—
73	#353	—	13	1	no	—
74	#353	D₁₃	1	1	yes	characteristic, complemented
75	C₂²×C₁₃	—	2	2	no	—
76	C₂²×C₁₃	—	4	2	no	—
77	C₂²×C₁₃	—	4	2	no	—
78	C₂²×C₁₃	—	4	1	no	—
79	C₂²×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×D₁₃	—	2	1	no	—
81	C₂×D₁₃	—	4	1	no	—
82	C₄×C₁₃	—	2	2	no	—
83	C₄×C₁₃	—	2	1	no	—
84	C₄×C₁₃	—	8	2	no	—
85	Dic₁₃	—	2	2	no	—
86	Dic₁₃	—	4	1	no	—
87	Dic₁₃	—	4	1	no	—
88	Dic₁₃	—	8	2	no	—
89	Dic₁₃	#99	1	1	yes	characteristic
90	Dic₁₃	#99	1	1	yes	characteristic
91	C₂×#41	—	13	1	no	—
92	C₂×#41	—	13	1	no	—
93	C₄⋊Q₈	—	13	1	no	—
94	#127	C₂×D₁₃	1	1	yes	characteristic
95	#124	—	13	1	no	—
96	#124	—	13	1	no	—
97	#121	—	26	2	no	—
98	C₄×D₄	—	26	2	no	—
99	#117	—	26	2	no	—
100	C₂×C₄²	—	13	1	no	—
101	#99	—	13	2	no	—
102	#99	—	26	2	no	—
103	#99	C₂×D₁₃	1	2	yes	—
104	C₂×C₁₃	—	2	1	no	—
105	C₂×C₁₃	—	4	2	no	—
106	C₂×C₁₃	—	8	1	no	—
107	C₂×C₁₃	C₂×#99	1	1	yes	characteristic
108	D₁₃	—	4	1	no	—
109	#41	—	52	1	no	—
110	#41	—	52	1	no	—
111	#41	—	52	1	no	—
112	#41	—	52	1	no	—
113	C₂×Q₈	—	13	1	no	—
114	C₂×Q₈	—	13	1	no	—
115	C₂×D₄	—	2	2	no	—
116	C₂×D₄	—	13	2	no	—
117	C₂×D₄	—	26	1	no	—
118	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
119	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
120	C₂²×C₄	—	13	2	no	—
121	C₂²×C₄	—	13	1	no	—
122	C₂²×C₄	—	26	1	no	—
123	C₂²×C₄	—	26	1	no	—
124	C₄⋊C₄	—	26	2	no	—
125	C₄⋊C₄	—	52	2	no	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	26	2	no	—
128	C₂²⋊C₄	—	26	2	no	—
129	C₂²⋊C₄	—	26	2	no	—
130	C₄²	—	13	1	no	—
131	C₄²	—	26	1	no	—
132	C₄²	—	52	2	no	—
133	C₄²	C₄×D₁₃	1	1	yes	characteristic
134	C₁₃	#2126	1	1	yes	characteristic, complemented, Sylow
135	C₂³	—	2	1	no	—
136	C₂³	—	13	1	no	—
137	C₂³	D₄×D₁₃	1	2	yes	—
138	Q₈	—	52	1	no	—
139	Q₈	—	52	1	no	—
140	D₄	—	4	1	no	—
141	D₄	—	4	1	no	—
142	D₄	—	4	4	no	—
143	D₄	—	52	2	no	—
144	D₄	—	52	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	13	1	no	—
148	C₂×C₄	—	13	2	no	—
149	C₂×C₄	—	26	1	no	—
150	C₂×C₄	—	26	1	no	—
151	C₂×C₄	—	26	2	no	—
152	C₂×C₄	—	26	2	no	—
153	C₂×C₄	—	26	1	no	—
154	C₂×C₄	—	26	1	no	—
155	C₂×C₄	—	52	2	no	—
156	C₂×C₄	—	52	1	no	—
157	C₂×C₄	—	52	1	no	—
158	C₂×C₄	—	52	2	no	—
159	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
160	C₂²	—	2	2	no	—
161	C₂²	—	4	2	no	—
162	C₂²	—	4	2	no	—
163	C₂²	—	4	1	no	—
164	C₂²	—	26	1	no	—
165	C₂²	—	52	1	no	—
166	C₂²	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
167	C₄	—	2	2	no	—
168	C₄	—	2	1	no	—
169	C₄	—	8	2	no	—
170	C₄	—	13	1	no	—
171	C₄	—	13	1	no	—
172	C₄	—	26	2	no	—
173	C₄	—	52	1	no	—
174	C₄	—	52	1	no	—
175	C₄	—	104	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	4	2	no	—
178	C₂	—	8	1	no	—
179	C₂	—	52	1	no	—
180	C₂	D₁₃×#99	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₁₃×#99
commutator	C₂×C₄×C₁₃	C₂²×C₄
Frattini	C₂×D₄	C₂²×D₁₃
Fitting	C₁₃×#353	C₂
socle	C₂×C₁₃	C₂×#99
prime radical	#11719421	C₂
cosocle	C₂×D₄×C₁₃	C₂³
hypercenter	#353	D₁₃
hypocenter	C₁₃	#2126
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #353

Complements

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

N	H ≅ G/N	type
2. #11718401	C₂	semidirect
3. C₁₃×#353	C₂	semidirect
4. #11718137	C₂	semidirect
5. #11718065	C₂	semidirect
6. C₂×#6129	C₄	semidirect
10. #88190	C₄	semidirect
14. C₄²×D₁₃	C₄	semidirect
15. #88123	C₄	semidirect
74. #353	D₁₃	semidirect
134. C₁₃	#2126	semidirect

Maximal subgroups

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

subgroup	index
#11719421	2
#11718401	2
C₁₃×#353	2
#11718137	2
#11718065	2
#2126	13