Finite Group Databasebeta

Subgroups — #15783860

Order 3328.

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

normal subgroups only

This group has 176 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	#13116924	C₂	1	1	yes	maximal, characteristic
2	D₁₃×#366	C₂	1	1	yes	maximal, characteristic
3	#13113038	C₂	1	1	yes	maximal, characteristic, complemented
4	#13113037	C₂	1	1	yes	maximal, characteristic, complemented
5	#13113017	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112973	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1416	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
9	#11718849	—	2	1	no	—
10	C₂×C₁₆×D₁₃	C₂²	1	1	yes	characteristic
11	D₁₃×#107	—	2	1	no	—
12	C₂×C₁₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₁₃×#366	C₂²	1	1	yes	characteristic
14	#11718156	—	2	1	no	—
15	#11718096	C₂²	1	1	yes	characteristic
16	#11718096	C₂²	1	1	yes	characteristic
17	C₂×#88034	C₂²	1	1	yes	characteristic
18	#11718081	—	2	1	no	—
19	C₂×C₁₃×Q₁₆	—	2	1	no	—
20	#88200	—	4	1	no	—
21	C₂×C₈×D₁₃	D₄	1	1	yes	characteristic
22	C₄⋊C₄×D₁₃	—	4	1	no	—
23	C₁₃×Q₃₂	—	2	1	no	—
24	C₁₃×Q₃₂	—	2	1	no	—
25	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
26	C₁₃×#107	—	2	1	no	—
27	#88075	—	4	1	no	—
28	#88058	—	2	1	no	—
29	#88058	—	2	1	no	—
30	C₂×#6090	D₄	1	1	yes	characteristic
31	#88047	—	4	1	no	—
32	C₁₆×D₁₃	—	2	1	no	—
33	C₁₆×D₁₃	—	2	1	no	—
34	C₁₃×C₃₂	—	2	1	no	—
35	#88034	—	2	1	no	—
36	#13714	—	13	1	no	maximal, Sylow
37	C₂×Q₈×C₁₃	—	4	1	no	—
38	C₂×C₄×D₁₃	—	8	1	no	—
39	C₂×C₄×D₁₃	D₈	1	1	yes	characteristic
40	C₁₃×Q₁₆	—	4	1	no	—
41	C₁₃×Q₁₆	—	4	1	no	—
42	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	—	4	1	no	—
44	#6103	—	8	1	no	—
45	C₄⋊Dic₁₃	—	4	1	no	—
46	C₄⋊Dic₁₃	—	4	1	no	—
47	#6101	—	8	1	no	—
48	#6101	—	8	1	no	—
49	C₂×#1055	D₈	1	1	yes	characteristic
50	C₈×D₁₃	—	2	1	no	—
51	C₈×D₁₃	—	2	1	no	—
52	C₁₃×C₁₆	#41	1	1	yes	characteristic
53	C₁₃×C₁₆	#41	1	1	yes	characteristic
54	#6090	—	2	1	no	—
55	#2216	—	13	1	no	—
56	C₂×#366	—	13	1	no	—
57	#1424	—	13	1	no	—
58	#1424	—	13	1	no	—
59	#1416	—	13	1	no	—
60	#1416	D₁₃	1	1	yes	characteristic, complemented
61	C₂²⋊C₃₂	—	13	1	no	—
62	C₂²×D₁₃	D₁₆	1	1	yes	characteristic
63	Q₈×C₁₃	—	8	1	no	—
64	Q₈×C₁₃	—	8	1	no	—
65	C₂×C₄×C₁₃	—	8	1	no	—
66	C₂×C₄×C₁₃	—	8	1	no	—
67	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
68	C₂×Dic₁₃	—	8	1	no	—
69	C₂×Dic₁₃	—	8	1	no	—
70	C₂×Dic₁₃	D₁₆	1	1	yes	characteristic
71	C₄×D₁₃	—	2	1	no	—
72	C₄×D₁₃	—	2	1	no	—
73	C₄×D₁₃	—	16	2	no	—
74	C₈×C₁₃	#123	1	1	yes	characteristic
75	C₈×C₁₃	#137	1	1	yes	characteristic
76	#1055	—	2	1	no	—
77	C₂×Q₃₂	C₂×D₁₃	1	1	yes	characteristic
78	C₂²×C₁₆	—	13	1	no	—
79	#467	—	26	1	no	—
80	C₂×#107	—	26	1	no	—
81	C₂×C₃₂	—	13	1	no	—
82	C₂×C₃₂	C₂×D₁₃	1	1	yes	characteristic
83	#366	—	13	1	no	—
84	#366	—	13	1	no	—
85	#366	—	26	1	no	—
86	#366	C₂×D₁₃	1	1	yes	characteristic
87	#358	—	26	1	no	—
88	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
89	C₂×D₁₃	—	2	1	no	—
90	C₂×D₁₃	—	2	1	no	—
91	C₂×D₁₃	SD₆₄	1	2	yes	—
92	C₄×C₁₃	—	16	1	no	—
93	C₄×C₁₃	—	16	1	no	—
94	C₄×C₁₃	#480	1	1	yes	characteristic
95	C₄×C₁₃	#510	1	1	yes	characteristic
96	Dic₁₃	—	2	1	no	—
97	Dic₁₃	—	16	1	no	—
98	Dic₁₃	—	16	1	no	—
99	C₂×Q₁₆	—	2	1	no	—
100	C₂²×C₈	—	13	1	no	—
101	C₂²⋊Q₈	—	52	1	no	—
102	C₂×C₄⋊C₄	—	52	1	no	—
103	Q₃₂	—	2	1	no	—
104	Q₃₂	—	2	1	no	—
105	C₂×C₁₆	—	13	1	no	—
106	C₂×C₁₆	—	26	1	no	—
107	C₂×C₁₆	—	26	1	no	—
108	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
109	#107	—	2	1	no	—
110	#107	—	26	1	no	—
111	#107	—	26	1	no	—
112	#107	—	52	1	no	—
113	#103	—	52	1	no	—
114	C₃₂	—	2	1	no	—
115	C₃₂	—	26	1	no	—
116	C₂×C₁₃	#2232	1	1	yes	characteristic
117	C₂×C₁₃	C₂×SD₆₄	1	1	yes	characteristic
118	C₂×C₁₃	#2264	1	1	yes	characteristic
119	D₁₃	—	2	2	no	—
120	C₂×Q₈	—	4	1	no	—
121	C₂²×C₄	—	13	1	no	—
122	C₂²×C₄	—	104	1	no	—
123	Q₁₆	—	4	1	no	—
124	Q₁₆	—	4	1	no	—
125	C₂×C₈	—	13	1	no	—
126	C₂×C₈	—	26	1	no	—
127	C₂×C₈	—	26	1	no	—
128	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
129	C₄⋊C₄	—	4	1	no	—
130	C₄⋊C₄	—	52	1	no	—
131	C₄⋊C₄	—	52	1	no	—
132	C₄⋊C₄	—	104	1	no	—
133	C₄⋊C₄	—	104	1	no	—
134	C₂²⋊C₄	—	104	1	no	—
135	C₁₆	—	26	1	no	—
136	C₁₆	#6127	1	1	yes	characteristic
137	C₁₆	#6129	1	1	yes	characteristic
138	C₁₃	#13714	1	1	yes	characteristic, complemented, Sylow
139	C₂³	—	13	1	no	—
140	Q₈	—	8	1	no	—
141	Q₈	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	13	1	no	—
145	C₂×C₄	—	26	1	no	—
146	C₂×C₄	—	26	1	no	—
147	C₂×C₄	—	104	1	no	—
148	C₂×C₄	—	104	1	no	—
149	C₂×C₄	—	208	2	no	—
150	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
151	C₈	—	26	1	no	—
152	C₈	#88137	1	1	yes	characteristic
153	C₈	#88169	1	1	yes	characteristic
154	C₂²	—	13	2	no	—
155	C₂²	—	26	1	no	—
156	C₂²	—	26	1	no	—
157	C₂²	D₁₃×D₁₆	1	1	yes	characteristic
158	C₄	—	16	1	no	—
159	C₄	—	16	1	no	—
160	C₄	—	26	1	no	—
161	C₄	—	208	1	no	—
162	C₄	—	208	1	no	—
163	C₄	#11718435	1	1	yes	characteristic
164	C₄	#11718576	1	1	yes	characteristic
165	C₂	—	26	2	no	—
166	C₂	#13117015	1	1	yes	characteristic
167	C₂	D₁₃×SD₆₄	1	1	yes	characteristic
168	C₂	#13117155	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₃×D₁₆
commutator, cosocle	C₂×C₁₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₁₃
Fitting	C₁₃×#1416	C₂
socle	C₂²×C₁₃	C₂×D₁₆
prime radical	C₂²×D₁₃	D₁₆
hypercenter	#1416	D₁₃
hypocenter	C₁₃	#13714
LCS	G ▷ C₂×C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

Complements

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

N	H ≅ G/N	type
3. #13113038	C₂	semidirect
4. #13113037	C₂	semidirect
5. #13113017	C₂	semidirect
7. C₁₃×#1416	C₂	semidirect
60. #1416	D₁₃	semidirect
138. C₁₃	#13714	semidirect

Maximal subgroups

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

subgroup	index
#13116924	2
D₁₃×#366	2
#13113038	2
#13113037	2
#13113017	2
#13112973	2
C₁₃×#1416	2
#13714	13