Finite Group Databasebeta

Subgroups — #15777715

Order 3328.

238 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 75 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	#13117109	C₂	1	1	yes	maximal, characteristic
2	#13117065	C₂	1	1	yes	maximal, characteristic
3	#13117064	C₂	1	1	yes	maximal, characteristic
4	#13114445	C₂	1	1	yes	maximal, characteristic
5	#13114434	C₂	1	1	yes	maximal, characteristic
6	#13112755	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#500	C₂²	1	1	yes	characteristic
9	C₁₃×#500	C₂²	1	1	yes	characteristic
10	#11718754	—	2	1	no	—
11	#11718742	—	2	1	no	—
12	#11718742	—	2	1	no	—
13	#11718740	—	2	1	no	—
14	#11718360	C₂²	1	1	yes	characteristic
15	#11718343	C₂²	1	1	yes	characteristic
16	C₁₃×#367	—	2	2	no	—
17	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	#11718095	—	2	2	no	—
19	#11718084	—	2	2	no	—
20	#11718081	—	2	2	no	—
21	C₄×#6090	C₂²	1	1	yes	characteristic
22	C₈×#1055	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊Q₈	—	2	1	no	—
24	C₁₃×C₄⋊Q₈	—	2	1	no	—
25	C₂×Dic₅₂	—	2	1	no	—
26	C₂×Dic₅₂	—	2	1	no	—
27	#88123	—	2	1	no	—
28	#88123	—	2	1	no	—
29	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
30	C₁₃×#107	—	2	1	no	—
31	C₁₃×#107	—	2	1	no	—
32	C₁₃×#107	—	2	1	no	—
33	C₁₃×#107	—	2	1	no	—
34	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	#88075	—	4	2	no	—
36	#88075	—	4	2	no	—
37	#88058	—	2	1	no	—
38	#88058	—	2	1	no	—
39	C₂×#6090	D₄	1	2	yes	—
40	#88050	—	4	2	no	—
41	#88048	—	4	2	no	—
42	C₄×#1055	D₄	1	1	yes	characteristic
43	C₄×#1055	D₄	1	1	yes	characteristic
44	#12533	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	4	1	no	—
46	C₂×Q₈×C₁₃	—	4	1	no	—
47	C₂×Dic₂₆	—	4	1	no	—
48	C₂×Dic₂₆	—	4	1	no	—
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	—	4	1	no	—
52	C₁₃×C₄⋊C₄	—	4	1	no	—
53	C₁₃×C₄⋊C₄	—	4	1	no	—
54	C₁₃×C₄⋊C₄	—	4	1	no	—
55	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
56	C₄⋊Dic₁₃	—	4	1	no	—
57	C₄⋊Dic₁₃	—	4	1	no	—
58	C₄⋊Dic₁₃	—	4	1	no	—
59	C₄⋊Dic₁₃	—	4	1	no	—
60	C₂×#1055	D₈	1	2	yes	—
61	C₂×#1055	D₈	1	2	yes	—
62	Dic₅₂	—	4	2	no	—
63	Dic₅₂	—	4	2	no	—
64	C₁₃×C₁₆	—	2	2	no	—
65	#6090	—	2	2	no	—
66	#2254	D₁₃	1	1	yes	characteristic
67	#2252	—	13	1	no	—
68	#2241	—	13	1	no	—
69	#2241	—	13	1	no	—
70	#1710	—	13	1	no	—
71	#1707	—	13	1	no	—
72	#1312	—	13	1	no	—
73	Q₈×C₁₃	—	8	2	no	—
74	Q₈×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	—	8	1	no	—
76	C₂×C₄×C₁₃	—	8	1	no	—
77	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
80	C₂×Dic₁₃	—	8	1	no	—
81	C₂×Dic₁₃	—	8	1	no	—
82	Dic₂₆	—	8	1	no	—
83	Dic₂₆	—	8	1	no	—
84	Dic₂₆	—	8	2	no	—
85	C₈×C₁₃	#137	1	2	yes	—
86	C₈×C₁₃	#137	1	2	yes	—
87	#1055	—	2	2	no	—
88	#1055	—	2	2	no	—
89	#500	—	13	1	no	—
90	#500	C₂×D₁₃	1	1	yes	characteristic
91	#500	C₂×D₁₃	1	1	yes	characteristic
92	#498	—	26	1	no	—
93	#494	—	13	1	no	—
94	#487	—	26	1	no	—
95	#487	—	26	1	no	—
96	#487	—	26	1	no	—
97	#367	—	2	2	no	—
98	#366	—	26	2	no	—
99	#358	—	26	2	no	—
100	#358	—	26	2	no	—
101	C₄×C₁₆	—	13	1	no	—
102	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
103	C₈²	—	13	1	no	—
104	C₂²×C₁₃	#493	1	1	yes	characteristic
105	C₄×C₁₃	—	16	1	no	—
106	C₄×C₁₃	—	16	1	no	—
107	C₄×C₁₃	#497	1	1	yes	characteristic
108	C₄×C₁₃	#497	1	1	yes	characteristic
109	C₄×C₁₃	#508	1	2	yes	—
110	C₄×C₁₃	#510	1	2	yes	—
111	Dic₁₃	—	16	1	no	—
112	Dic₁₃	—	16	1	no	—
113	C₂×Q₁₆	—	26	1	no	—
114	C₂×Q₁₆	—	26	1	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	2	1	no	—
117	C₄⋊Q₈	—	26	1	no	—
118	C₄⋊Q₈	—	26	1	no	—
119	C₂×C₁₆	—	13	2	no	—
120	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
121	#107	—	2	1	no	—
122	#107	—	2	1	no	—
123	#107	—	2	1	no	—
124	#107	—	2	1	no	—
125	#107	—	26	1	no	—
126	#107	—	26	1	no	—
127	#106	—	52	2	no	—
128	#103	—	52	2	no	—
129	#103	—	52	2	no	—
130	#103	—	52	2	no	—
131	C₄×C₈	—	13	1	no	—
132	C₄×C₈	—	13	1	no	—
133	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
134	C₂×C₁₃	#1722	1	1	yes	characteristic
135	C₂×C₁₃	#2249	1	1	yes	characteristic
136	C₂×C₁₃	#2251	1	1	yes	characteristic
137	C₂×Q₈	—	4	1	no	—
138	C₂×Q₈	—	4	1	no	—
139	C₂×Q₈	—	52	1	no	—
140	C₂×Q₈	—	52	1	no	—
141	Q₁₆	—	52	2	no	—
142	Q₁₆	—	52	2	no	—
143	C₂×C₈	—	13	2	no	—
144	C₂×C₈	—	13	2	no	—
145	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
146	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	4	1	no	—
149	C₄⋊C₄	—	4	1	no	—
150	C₄⋊C₄	—	4	1	no	—
151	C₄⋊C₄	—	52	1	no	—
152	C₄⋊C₄	—	52	1	no	—
153	C₄⋊C₄	—	52	1	no	—
154	C₄⋊C₄	—	52	1	no	—
155	C₄²	D₄×D₁₃	1	1	yes	characteristic
156	C₁₆	—	2	2	no	—
157	C₁₆	—	26	2	no	—
158	C₁₃	#12533	1	1	yes	characteristic, complemented, Sylow
159	Q₈	—	8	2	no	—
160	Q₈	—	8	2	no	—
161	Q₈	—	104	1	no	—
162	Q₈	—	104	2	no	—
163	Q₈	—	104	1	no	—
164	C₂×C₄	—	8	1	no	—
165	C₂×C₄	—	8	1	no	—
166	C₂×C₄	—	104	1	no	—
167	C₂×C₄	—	104	1	no	—
168	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
169	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
170	C₂×C₄	#88194	1	1	yes	characteristic
171	C₈	—	26	2	no	—
172	C₈	—	26	2	no	—
173	C₈	#88172	1	2	yes	—
174	C₈	#88196	1	2	yes	—
175	C₂²	#11718812	1	1	yes	characteristic
176	C₄	—	16	1	no	—
177	C₄	—	16	1	no	—
178	C₄	—	208	1	no	—
179	C₄	—	208	1	no	—
180	C₄	#11718576	1	2	yes	—
181	C₄	#11718577	1	2	yes	—
182	C₄	#11718747	1	1	yes	characteristic
183	C₄	#11718851	1	1	yes	characteristic
184	C₂	#13114506	1	1	yes	characteristic
185	C₂	#13117096	1	1	yes	characteristic
186	C₂	#13117106	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2254	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#2254	D₁₃
hypocenter	C₁₃	#12533
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

Complements

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

N	H ≅ G/N	type
158. C₁₃	#12533	semidirect

Maximal subgroups

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

subgroup	index
#13117109	2
#13117065	2
#13117064	2
#13114445	2
#13114434	2
#13112755	2
C₁₃×#2254	2
#12533	13