Finite Group Databasebeta

Subgroups — #15777316

Order 3328.

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

normal subgroups only

This group has 172 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	#13117112	C₂	1	1	yes	maximal, characteristic, complemented
2	#13117013	C₂	1	1	yes	maximal, characteristic
3	#13117009	C₂	1	1	yes	maximal, characteristic
4	#13116672	C₂	1	1	yes	maximal, characteristic
5	#13112911	C₂	1	1	yes	maximal, characteristic, complemented
6	#13112898	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#499	C₂²	1	1	yes	characteristic
9	#11718568	—	2	1	no	—
10	#11718361	C₂²	1	1	yes	characteristic
11	#11718344	C₂²	1	1	yes	characteristic
12	C₁₃×#364	C₂²	1	1	yes	characteristic
13	C₁₃×#364	C₂²	1	1	yes	characteristic
14	#11718110	—	2	1	no	—
15	#11718107	—	2	1	no	—
16	#11718105	—	2	1	no	—
17	#11718082	—	2	1	no	—
18	#11718081	—	2	1	no	—
19	#11718080	—	2	1	no	—
20	#11718079	—	2	1	no	—
21	#11718055	C₂²	1	1	yes	characteristic
22	C₄×#6090	C₂²	1	1	yes	characteristic
23	C₁₃×#125	—	2	1	no	—
24	#88158	D₄	1	1	yes	characteristic
25	#88155	—	2	1	no	—
26	#88124	—	2	1	no	—
27	C₁₃×M₃₂	—	2	1	no	—
28	C₁₃×M₃₂	—	2	1	no	—
29	C₁₃×#107	—	2	1	no	—
30	C₁₃×#106	—	2	1	no	—
31	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	#88059	D₄	1	1	yes	characteristic
33	#88058	—	2	1	no	—
34	#88057	—	2	1	no	—
35	#88052	—	2	1	no	—
36	C₂×#6090	D₄	1	1	yes	characteristic
37	C₂×#6090	D₄	1	1	yes	characteristic
38	#88045	—	2	1	no	—
39	#88038	—	4	1	no	—
40	C₁₆×D₁₃	—	4	1	no	—
41	#12464	—	13	1	no	maximal, Sylow
42	#6127	—	2	1	no	—
43	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₁₃×C₄⋊C₄	—	4	1	no	—
46	C₁₃×C₄⋊C₄	—	8	1	no	—
47	C₄²×C₁₃	#41	1	1	yes	characteristic
48	C₄⋊Dic₁₃	—	4	1	no	—
49	#6101	—	8	1	no	—
50	#6099	—	2	1	no	—
51	Dic₅₂	Q₁₆	1	1	yes	characteristic
52	D₁₀₄	Q₁₆	1	1	yes	characteristic
53	#6095	—	2	1	no	—
54	#6094	—	4	1	no	—
55	C₈×D₁₃	—	4	1	no	—
56	C₁₃×C₁₆	—	4	1	no	—
57	C₁₃×C₁₆	—	4	1	no	—
58	#6090	—	2	1	no	—
59	#6090	—	4	1	no	—
60	#6090	SD₁₆	1	1	yes	characteristic
61	#6090	SD₁₆	1	1	yes	characteristic
62	#2253	—	13	1	no	—
63	#2231	—	13	1	no	—
64	#2231	—	13	1	no	—
65	#2170	—	13	1	no	—
66	#1373	—	13	1	no	—
67	#1366	—	13	1	no	—
68	#1366	D₁₃	1	1	yes	characteristic, complemented
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	8	1	no	—
71	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
72	C₂²⋊D₁₃	—	4	1	no	—
73	C₂×Dic₁₃	—	8	1	no	—
74	D₅₂	—	2	1	no	—
75	C₄×D₁₃	—	4	1	no	—
76	Dic₂₆	—	2	1	no	—
77	C₈×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
78	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
79	C₈×C₁₃	#135	1	1	yes	characteristic
80	C₈×C₁₃	#137	1	1	yes	characteristic
81	#1055	—	4	1	no	—
82	#504	—	26	1	no	—
83	#499	—	13	1	no	—
84	#499	C₂×D₁₃	1	1	yes	characteristic
85	#443	—	13	1	no	—
86	#367	—	26	1	no	—
87	#366	—	26	1	no	—
88	#365	—	26	1	no	—
89	#365	—	26	1	no	—
90	#364	—	13	1	no	—
91	#364	C₂×D₁₃	1	1	yes	characteristic
92	#364	C₂×D₁₃	1	1	yes	characteristic
93	#360	—	26	1	no	—
94	#360	—	26	1	no	—
95	#350	—	26	1	no	—
96	C₄×C₁₆	—	13	1	no	—
97	C₂²×C₁₃	#472	1	1	yes	characteristic
98	C₂×D₁₃	—	4	1	no	—
99	C₄×C₁₃	—	4	1	no	—
100	C₄×C₁₃	—	16	1	no	—
101	C₄×C₁₃	#461	1	1	yes	characteristic
102	C₄×C₁₃	#467	1	1	yes	characteristic
103	Dic₁₃	—	4	1	no	—
104	Dic₁₃	—	16	1	no	—
105	#135	—	13	1	no	—
106	#131	—	26	1	no	—
107	#125	—	2	1	no	—
108	#125	—	26	1	no	—
109	M₃₂	—	2	1	no	—
110	M₃₂	—	2	1	no	—
111	M₃₂	—	26	1	no	—
112	M₃₂	—	52	1	no	—
113	C₂×C₁₆	—	13	1	no	—
114	C₂×C₁₆	—	13	1	no	—
115	C₂×C₁₆	—	52	1	no	—
116	#108	—	13	1	no	—
117	#107	—	2	1	no	—
118	#107	—	26	1	no	—
119	#106	—	2	1	no	—
120	#106	—	26	1	no	—
121	#104	—	26	1	no	—
122	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₁₃	—	2	1	no	—
124	C₂×C₁₃	#1675	1	1	yes	characteristic
125	D₁₃	—	8	1	no	—
126	#41	—	26	1	no	—
127	Q₁₆	—	13	1	no	—
128	SD₁₆	—	26	1	no	—
129	D₈	—	13	1	no	—
130	M₁₆	—	26	1	no	—
131	M₁₆	—	52	1	no	—
132	C₂×C₈	—	52	1	no	—
133	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
134	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
135	C₄⋊C₄	—	4	1	no	—
136	C₄⋊C₄	—	8	1	no	—
137	C₄⋊C₄	—	52	1	no	—
138	C₄⋊C₄	—	104	1	no	—
139	C₄²	#6127	1	1	yes	characteristic
140	C₁₆	—	4	1	no	—
141	C₁₆	—	4	1	no	—
142	C₁₆	—	13	1	no	—
143	C₁₆	—	13	1	no	—
144	C₁₆	—	26	1	no	—
145	C₁₆	—	52	1	no	—
146	C₁₃	#12464	1	1	yes	characteristic, complemented, Sylow
147	Q₈	—	26	1	no	—
148	D₄	—	26	1	no	—
149	D₄	—	52	1	no	—
150	C₂×C₄	—	2	1	no	—
151	C₂×C₄	—	8	1	no	—
152	C₂×C₄	—	52	1	no	—
153	C₂×C₄	—	104	1	no	—
154	C₂×C₄	#88138	1	1	yes	characteristic
155	C₈	—	52	1	no	—
156	C₈	SD₁₆×D₁₃	1	1	yes	characteristic
157	C₈	#88170	1	1	yes	characteristic
158	C₈	Q₁₆×D₁₃	1	1	yes	characteristic
159	C₈	#88172	1	1	yes	characteristic
160	C₂²	—	52	1	no	—
161	C₂²	#11718410	1	1	yes	characteristic
162	C₄	—	4	1	no	—
163	C₄	—	16	1	no	—
164	C₄	—	52	1	no	—
165	C₄	—	208	1	no	—
166	C₄	#11718474	1	1	yes	characteristic
167	C₄	#11718479	1	1	yes	characteristic
168	C₂	—	2	1	no	—
169	C₂	—	104	1	no	—
170	C₂	#13114234	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#13114234
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1366	C₂
socle	C₂×C₁₃	#1675
prime radical	#88158	D₄
hypercenter	#1366	D₁₃
hypocenter	C₁₃	#12464
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1366

Complements

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

N	H ≅ G/N	type
1. #13117112	C₂	semidirect
5. #13112911	C₂	semidirect
6. #13112898	C₂	semidirect
7. C₁₃×#1366	C₂	semidirect
68. #1366	D₁₃	semidirect
146. C₁₃	#12464	semidirect

Maximal subgroups

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

subgroup	index
#13117112	2
#13117013	2
#13117009	2
#13116672	2
#13112911	2
#13112898	2
C₁₃×#1366	2
#12464	13