Finite Group Databasebeta

Subgroups — #15783538

Order 3328.

182 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 67 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	#13116722	C₂	1	1	yes	maximal, characteristic
2	#13116707	C₂	1	1	yes	maximal, characteristic, complemented
3	#13113042	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113029	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113023	C₂	1	1	yes	maximal, characteristic
6	#13112971	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2264	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#508	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
10	#11718857	—	2	1	no	—
11	Q₁₆×Dic₁₃	—	2	1	no	—
12	C₁₃×Q₆₄	C₄	1	2	yes	complemented
13	C₁₃×SD₆₄	C₄	1	2	yes	complemented
14	C₁₃×M₆₄	C₂²	1	1	yes	characteristic
15	#11718159	—	2	1	no	—
16	#11718156	—	2	1	no	—
17	#11718098	C₂²	1	1	yes	characteristic
18	#11718097	C₂²	1	1	yes	characteristic
19	#11718094	C₂²	1	1	yes	characteristic
20	#11718091	C₂²	1	1	yes	characteristic
21	C₁₃×#135	—	2	1	no	—
22	C₂×C₁₃×Q₁₆	—	2	1	no	—
23	#88202	—	4	1	no	—
24	Q₈×Dic₁₃	—	4	1	no	—
25	C₁₃×Q₃₂	—	2	1	no	—
26	C₁₃×Q₃₂	C₂×C₄	1	2	yes	—
27	C₁₃×Q₃₂	C₂×C₄	1	1	yes	characteristic
28	C₁₃×SD₃₂	—	2	1	no	—
29	C₁₃×D₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
31	#88077	—	4	1	no	—
32	#88075	—	4	1	no	—
33	#88059	—	2	1	no	—
34	#88058	—	2	1	no	—
35	C₈×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂×#6090	D₄	1	1	yes	characteristic
37	C₁₃×C₃₂	C₂×C₄	1	2	yes	—
38	#88034	—	2	1	no	—
39	#13657	—	13	1	no	maximal, Sylow
40	C₁₃×#41	—	4	1	no	—
41	C₂×Q₈×C₁₃	—	4	1	no	—
42	C₁₃×Q₁₆	—	2	1	no	—
43	C₁₃×Q₁₆	—	2	2	no	—
44	C₁₃×Q₁₆	—	4	1	no	—
45	C₁₃×SD₁₆	—	4	1	no	—
46	C₁₃×D₈	—	2	1	no	—
47	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₁₃	—	4	1	no	—
49	C₄⋊Dic₁₃	—	8	1	no	—
50	C₄×Dic₁₃	—	8	1	no	—
51	C₄×Dic₁₃	D₈	1	1	yes	characteristic
52	#6099	—	4	1	no	—
53	#6099	—	8	1	no	—
54	C₂×#1055	—	8	1	no	—
55	C₂×#1055	D₈	1	1	yes	characteristic
56	C₁₃×C₁₆	C₂²×C₄	1	1	yes	characteristic
57	C₁₃×C₁₆	#41	1	1	yes	characteristic
58	#6090	—	2	1	no	—
59	#2264	D₁₃	1	1	yes	characteristic
60	#2179	—	13	1	no	—
61	#2176	—	13	1	no	—
62	#1426	—	13	1	no	—
63	#1420	—	13	1	no	—
64	#1418	—	13	1	no	—
65	#1398	—	13	1	no	—
66	Q₈×C₁₃	—	4	1	no	—
67	Q₈×C₁₃	—	4	2	no	—
68	Q₈×C₁₃	—	8	1	no	—
69	D₄×C₁₃	—	4	1	no	—
70	D₄×C₁₃	—	8	1	no	—
71	C₂×C₄×C₁₃	—	8	1	no	—
72	C₂×C₄×C₁₃	—	8	1	no	—
73	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
74	C₂×Dic₁₃	—	2	1	no	—
75	C₂×Dic₁₃	—	8	1	no	—
76	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
77	C₈×C₁₃	#135	1	1	yes	characteristic
78	#1055	—	2	1	no	—
79	#1055	—	8	1	no	—
80	#508	C₂×D₁₃	1	1	yes	characteristic
81	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
82	#443	—	26	1	no	—
83	C₄×Q₁₆	—	26	1	no	—
84	Q₆₄	Dic₁₃	1	2	yes	complemented
85	SD₆₄	Dic₁₃	1	2	yes	complemented
86	M₆₄	—	13	1	no	—
87	M₆₄	C₂×D₁₃	1	1	yes	characteristic
88	#368	—	13	1	no	—
89	#367	—	13	1	no	—
90	#359	—	26	1	no	—
91	#358	—	26	1	no	—
92	#346	—	13	1	no	—
93	C₂²×C₁₃	—	8	1	no	—
94	C₂²×C₁₃	#509	1	1	yes	characteristic
95	C₄×C₁₃	—	8	1	no	—
96	C₄×C₁₃	—	8	2	no	—
97	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
98	C₄×C₁₃	#509	1	1	yes	characteristic
99	Dic₁₃	—	4	1	no	—
100	Dic₁₃	—	16	1	no	—
101	#135	—	2	1	no	—
102	C₂×Q₁₆	—	2	1	no	—
103	#131	—	52	1	no	—
104	C₄×Q₈	—	52	1	no	—
105	Q₃₂	—	2	1	no	—
106	Q₃₂	C₂×Dic₁₃	1	2	yes	—
107	Q₃₂	C₂×Dic₁₃	1	1	yes	characteristic
108	SD₃₂	—	2	1	no	—
109	D₁₆	C₂×Dic₁₃	1	1	yes	characteristic
110	C₂×C₁₆	—	13	1	no	—
111	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
112	#108	—	26	1	no	—
113	#107	—	26	1	no	—
114	#104	—	52	1	no	—
115	#103	—	52	1	no	—
116	C₄×C₈	—	13	1	no	—
117	C₃₂	—	26	1	no	—
118	C₃₂	C₂×Dic₁₃	1	2	yes	—
119	C₂×C₁₃	—	2	1	no	—
120	C₂×C₁₃	—	16	1	no	—
121	C₂×C₁₃	#2177	1	1	yes	characteristic
122	#41	—	4	1	no	—
123	C₂×Q₈	—	4	1	no	—
124	Q₁₆	—	2	1	no	—
125	Q₁₆	—	2	2	no	—
126	Q₁₆	—	4	1	no	—
127	SD₁₆	—	4	1	no	—
128	D₈	—	2	1	no	—
129	M₁₆	—	52	1	no	—
130	M₁₆	—	104	1	no	—
131	C₂×C₈	—	13	1	no	—
132	C₂×C₈	—	104	1	no	—
133	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
134	C₄⋊C₄	—	52	1	no	—
135	C₄⋊C₄	—	104	1	no	—
136	C₄²	—	13	1	no	—
137	C₄²	—	104	1	no	—
138	C₁₆	—	26	1	no	—
139	C₁₆	#6129	1	1	yes	characteristic
140	C₁₆	C₂²×Dic₁₃	1	1	yes	characteristic
141	C₁₃	#13657	1	1	yes	characteristic, complemented, Sylow
142	Q₈	—	4	1	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	8	1	no	—
145	D₄	—	4	1	no	—
146	D₄	—	8	1	no	—
147	C₂×C₄	—	8	1	no	—
148	C₂×C₄	—	8	1	no	—
149	C₂×C₄	—	26	1	no	—
150	C₂×C₄	—	104	1	no	—
151	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
152	C₈	—	26	1	no	—
153	C₈	—	104	1	no	—
154	C₈	#88166	1	1	yes	characteristic
155	C₈	D₄×Dic₁₃	1	1	yes	characteristic
156	C₂²	—	8	1	no	—
157	C₂²	#11718572	1	1	yes	characteristic
158	C₄	—	8	1	no	—
159	C₄	—	8	2	no	—
160	C₄	—	52	1	no	—
161	C₄	—	208	1	no	—
162	C₄	#11718572	1	1	yes	characteristic
163	C₄	D₈×Dic₁₃	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	16	1	no	—
166	C₂	#13116712	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13116712
commutator	C₁₃×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₁₃×#2264	C₂
socle	C₂×C₁₃	#2177
prime radical	C₁₃×#508	C₂²
cosocle	C₂×C₁₃×C₁₆	C₂³
hypercenter	#2264	D₁₃
hypocenter	C₁₃	#13657
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2264

Complements

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

N	H ≅ G/N	type
2. #13116707	C₂	semidirect
3. #13113042	C₂	semidirect
4. #13113029	C₂	semidirect
6. #13112971	C₂	semidirect
12. C₁₃×Q₆₄	C₄	semidirect
13. C₁₃×SD₆₄	C₄	semidirect
84. Q₆₄	Dic₁₃	semidirect
85. SD₆₄	Dic₁₃	semidirect
141. C₁₃	#13657	semidirect

Maximal subgroups

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

subgroup	index
#13116722	2
#13116707	2
#13113042	2
#13113029	2
#13113023	2
#13112971	2
C₁₃×#2264	2
#13657	13