Finite Group Databasebeta

Subgroups — #15852245

Order 3328.

736 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 151 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	#13125580	C₂	1	3	yes	maximal, complemented
2	#13125207	C₂	1	3	yes	maximal
3	#13114497	C₂	1	3	yes	maximal, complemented
4	#13114493	C₂	1	3	yes	maximal, complemented
5	#13114448	C₂	1	1	yes	maximal, characteristic, complemented
6	#13113138	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1711	C₂	1	1	yes	maximal, characteristic, complemented
8	#11719433	C₂²	1	3	yes	—
9	D₁₃×#127	C₂²	1	1	yes	characteristic
10	C₁₃×#487	C₂²	1	3	yes	complemented
11	C₁₃×#486	C₂²	1	3	yes	—
12	#11718753	C₂²	1	3	yes	complemented
13	#11718749	C₂²	1	3	yes	—
14	#11718740	C₂²	1	3	yes	complemented
15	#11718731	C₂²	1	3	yes	—
16	#11718477	—	2	6	no	—
17	#11718450	—	2	6	no	—
18	#11718431	—	2	6	no	—
19	#11718425	—	2	6	no	—
20	#11718355	C₂²	1	3	yes	—
21	C₄×#6094	C₂²	1	3	yes	—
22	#11718342	C₂²	1	3	yes	complemented
23	C₈²×C₁₃	C₂²	1	1	yes	characteristic, complemented
24	#11718047	C₂²	1	3	yes	complemented
25	C₂×D₄×D₁₃	—	2	3	no	—
26	C₁₃×C₄⋊Q₈	C₂³	1	3	yes	—
27	C₁₃×#127	C₂³	1	1	yes	characteristic
28	#88200	—	2	3	no	—
29	Q₈×Dic₁₃	—	2	3	no	—
30	C₂×#6107	—	2	6	no	—
31	#88194	—	2	3	no	—
32	C₂×#6094	D₄	1	6	yes	—
33	#88151	—	2	3	no	—
34	#88146	—	2	3	no	—
35	#88144	—	2	3	no	—
36	#88141	—	2	3	no	—
37	#88128	C₂³	1	1	yes	characteristic
38	C₄²×D₁₃	C₂³	1	1	yes	characteristic
39	#88123	C₂³	1	3	yes	—
40	C₁₃×#103	—	2	6	no	—
41	C₁₃×#102	—	2	6	no	—
42	C₄×C₈×C₁₃	C₂³	1	3	yes	—
43	#88072	—	2	6	no	—
44	#88061	—	2	6	no	—
45	#88056	—	2	6	no	—
46	#88055	D₄	1	6	yes	—
47	#88049	—	2	6	no	—
48	#88048	—	2	6	no	—
49	C₄×#1055	C₂³	1	3	yes	—
50	#19753	—	13	1	no	maximal, Sylow
51	C₂³×D₁₃	—	4	1	no	—
52	C₂×Q₈×C₁₃	—	2	3	no	—
53	C₂×D₄×C₁₃	—	2	3	no	—
54	C₂×C₂²⋊D₁₃	—	4	3	no	—
55	D₄×D₁₃	—	4	12	no	—
56	C₂×D₅₂	—	2	3	no	—
57	C₂×C₄×D₁₃	C₂×D₄	1	3	yes	—
58	C₂×Dic₂₆	—	2	3	no	—
59	C₂×C₈×C₁₃	C₂×D₄	1	6	yes	—
60	C₁₃×C₄⋊C₄	—	2	3	no	—
61	C₁₃×C₄⋊C₄	—	2	3	no	—
62	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
63	#6107	—	4	12	no	—
64	#6103	—	4	3	no	—
65	C₄⋊Dic₁₃	—	2	3	no	—
66	C₄⋊Dic₁₃	—	2	3	no	—
67	C₄⋊Dic₁₃	—	4	3	no	—
68	#6101	—	4	3	no	—
69	C₄×Dic₁₃	—	4	3	no	—
70	C₄×Dic₁₃	C₂×D₄	1	3	yes	—
71	C₂×#1055	C₂×D₄	1	6	yes	—
72	#6094	—	4	6	no	—
73	#3185	—	13	3	no	—
74	#3138	—	13	3	no	—
75	#1720	—	13	3	no	—
76	#1720	—	13	3	no	—
77	#1711	—	13	1	no	—
78	#1711	D₁₃	1	1	yes	characteristic, complemented
79	#1455	—	13	1	no	—
80	C₂³×C₁₃	—	4	1	no	—
81	C₂²×D₁₃	—	4	1	no	—
82	C₂²×D₁₃	—	8	6	no	—
83	C₂²×D₁₃	#127	1	1	yes	characteristic
84	Q₈×C₁₃	—	4	6	no	—
85	D₄×C₁₃	—	4	6	no	—
86	C₂×C₄×C₁₃	—	4	3	no	—
87	C₂×C₄×C₁₃	C₂²×D₄	1	3	yes	—
88	C₂²⋊D₁₃	—	8	6	no	—
89	C₂×Dic₁₃	—	4	3	no	—
90	C₂×Dic₁₃	#127	1	3	yes	—
91	D₅₂	—	4	6	no	—
92	C₄×D₁₃	—	2	6	no	—
93	Dic₂₆	—	4	6	no	—
94	C₈×C₁₃	—	2	6	no	—
95	#1055	—	2	6	no	—
96	#533	—	13	3	no	—
97	C₂×#127	—	13	1	no	—
98	#498	—	13	3	no	—
99	#494	—	13	3	no	—
100	#490	—	26	6	no	—
101	#487	—	13	3	no	—
102	#487	C₂×D₁₃	1	3	yes	complemented
103	#486	—	13	3	no	—
104	#486	—	13	3	no	—
105	#486	—	13	3	no	—
106	#486	C₂×D₁₃	1	3	yes	—
107	#470	—	26	6	no	—
108	#441	—	26	6	no	—
109	#418	—	26	6	no	—
110	C₄×M₁₆	—	13	3	no	—
111	#322	—	13	3	no	—
112	C₈²	C₂×D₁₃	1	1	yes	characteristic, complemented
113	C₂²×C₁₃	—	8	3	no	—
114	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
115	C₂×D₁₃	—	2	3	no	—
116	C₂×D₁₃	—	8	3	no	—
117	C₂×D₁₃	—	16	4	no	—
118	C₄×C₁₃	—	8	3	no	—
119	C₄×C₁₃	#575	1	6	yes	—
120	Dic₁₃	—	2	3	no	—
121	Dic₁₃	—	8	3	no	—
122	C₂²×D₄	—	26	3	no	—
123	C₂×Q₁₆	—	26	6	no	—
124	C₂×M₁₆	—	13	6	no	—
125	C₄⋊Q₈	—	13	3	no	—
126	C₄⋊Q₈	C₂²×D₁₃	1	3	yes	—
127	#127	—	13	1	no	—
128	#127	—	26	3	no	—
129	#127	C₂²×D₁₃	1	1	yes	characteristic
130	#125	—	26	3	no	—
131	C₂²⋊Q₈	—	26	3	no	—
132	C₂²⋊Q₈	—	26	3	no	—
133	C₄×Q₈	—	26	3	no	—
134	C₄×Q₈	—	26	3	no	—
135	#117	—	26	3	no	—
136	C₂×C₄²	—	13	1	no	—
137	#106	—	26	6	no	—
138	#103	—	2	6	no	—
139	#103	—	26	6	no	—
140	#102	—	2	6	no	—
141	#102	—	26	6	no	—
142	#102	—	26	6	no	—
143	#102	—	26	6	no	—
144	#97	—	13	6	no	—
145	C₄×C₈	—	13	3	no	—
146	C₄×C₈	C₂²×D₁₃	1	3	yes	—
147	C₂×C₁₃	—	16	1	no	—
148	C₂×C₁₃	#3153	1	3	yes	—
149	D₁₃	—	4	1	no	—
150	D₁₃	—	16	1	no	—
151	C₂⁴	—	52	1	no	—
152	C₂×Q₈	—	2	3	no	—
153	C₂×Q₈	—	26	3	no	—
154	C₂×D₄	—	2	3	no	—
155	C₂×D₄	—	26	3	no	—
156	C₂×D₄	—	52	3	no	—
157	C₂×D₄	—	52	12	no	—
158	C₂²×C₄	—	13	3	no	—
159	Q₁₆	—	52	12	no	—
160	M₁₆	—	52	6	no	—
161	C₂×C₈	—	13	6	no	—
162	C₂×C₈	D₄×D₁₃	1	6	yes	—
163	C₄⋊C₄	—	2	3	no	—
164	C₄⋊C₄	—	2	3	no	—
165	C₄⋊C₄	—	26	3	no	—
166	C₄⋊C₄	—	26	3	no	—
167	C₄⋊C₄	—	52	3	no	—
168	C₄⋊C₄	—	52	3	no	—
169	C₂²⋊C₄	—	52	3	no	—
170	C₄²	—	13	3	no	—
171	C₄²	—	52	3	no	—
172	C₄²	C₂³×D₁₃	1	1	yes	characteristic
173	C₁₃	#19753	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	4	1	no	—
175	C₂³	—	13	1	no	—
176	C₂³	—	52	1	no	—
177	C₂³	—	104	6	no	—
178	Q₈	—	4	6	no	—
179	Q₈	—	52	6	no	—
180	D₄	—	4	6	no	—
181	D₄	—	52	6	no	—
182	D₄	—	104	6	no	—
183	C₂×C₄	—	4	3	no	—
184	C₂×C₄	—	13	3	no	—
185	C₂×C₄	—	26	6	no	—
186	C₂×C₄	—	52	3	no	—
187	C₂×C₄	C₂×D₄×D₁₃	1	3	yes	—
188	C₈	—	2	6	no	—
189	C₈	—	26	6	no	—
190	C₂²	—	8	3	no	—
191	C₂²	—	26	3	no	—
192	C₂²	—	104	3	no	—
193	C₂²	—	208	4	no	—
194	C₂²	D₁₃×#127	1	1	yes	characteristic
195	C₄	—	8	3	no	—
196	C₄	—	26	3	no	—
197	C₄	—	104	3	no	—
198	C₄	#11719473	1	6	yes	—
199	C₂	—	16	1	no	—
200	C₂	—	52	1	no	—
201	C₂	—	208	1	no	—
202	C₂	#13125297	1	3	yes	—

	subgroup	G/H
center	C₂²	D₁₃×#127
commutator, cosocle	C₄²×C₁₃	C₂⁴
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1711	C₂
socle	C₂²×C₁₃	C₂×#127
prime radical	D₁₃×#127	C₂²
hypercenter	#1711	D₁₃
hypocenter	C₁₃	#19753
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1711

Complements

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

N	H ≅ G/N	type
1. #13125580	C₂	semidirect
3. #13114497	C₂	semidirect
4. #13114493	C₂	semidirect
5. #13114448	C₂	semidirect
6. #13113138	C₂	semidirect
7. C₁₃×#1711	C₂	semidirect
10. C₁₃×#487	C₂²	semidirect
12. #11718753	C₂²	semidirect
14. #11718740	C₂²	semidirect
22. #11718342	C₂²	semidirect
23. C₈²×C₁₃	C₂²	semidirect
24. #11718047	C₂²	semidirect
78. #1711	D₁₃	semidirect
102. #487	C₂×D₁₃	semidirect
112. C₈²	C₂×D₁₃	semidirect
173. C₁₃	#19753	semidirect

Maximal subgroups

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

subgroup	index
#13125580	2
#13125207	2
#13114497	2
#13114493	2
#13114448	2
#13113138	2
C₁₃×#1711	2
#19753	13