Finite Group Databasebeta

Subgroups — #15777552

Order 3328.

212 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 63 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	#13116989	C₂	1	1	yes	maximal, characteristic
2	#13116985	C₂	1	1	yes	maximal, characteristic
3	#13114453	C₂	1	1	yes	maximal, characteristic
4	#13113732	C₂	1	1	yes	maximal, characteristic
5	#13112899	C₂	1	1	yes	maximal, characteristic
6	#13112895	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1365	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#500	C₂²	1	1	yes	characteristic
9	#11718753	—	2	1	no	—
10	#11718739	—	2	1	no	—
11	#11718492	—	2	1	no	—
12	C₄×Dic₅₂	C₂²	1	1	yes	characteristic
13	#11718343	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₁₃×#335	C₂²	1	1	yes	characteristic
16	#11718096	—	2	1	no	—
17	#11718095	—	2	1	no	—
18	#11718084	—	2	1	no	—
19	#11718081	—	2	1	no	—
20	#11718076	—	2	1	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	C₂×#6107	—	4	2	no	—
25	C₂×Dic₅₂	D₄	1	1	yes	characteristic
26	#88123	—	2	1	no	—
27	C₄×Dic₂₆	—	2	1	no	—
28	C₂×C₁₃×C₁₆	—	2	1	no	—
29	C₁₃×#107	—	2	1	no	—
30	C₁₃×#107	—	2	1	no	—
31	C₁₃×C₄⋊C₈	—	2	1	no	—
32	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	#88058	—	2	1	no	—
34	#88058	—	2	1	no	—
35	#88058	D₄	1	1	yes	characteristic
36	#88056	—	2	1	no	—
37	#88055	—	4	1	no	—
38	#88054	—	4	1	no	—
39	C₂×#6090	—	2	1	no	—
40	#88047	—	4	1	no	—
41	#88047	—	4	1	no	—
42	C₄×#1055	D₄	1	1	yes	characteristic
43	C₄×#1055	D₄	1	1	yes	characteristic
44	#12500	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	4	1	no	—
46	C₂×Dic₂₆	—	2	1	no	—
47	C₂×Dic₂₆	—	4	1	no	—
48	C₂×C₈×C₁₃	—	4	1	no	—
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
51	C₁₃×C₄⋊C₄	—	4	1	no	—
52	C₁₃×C₄⋊C₄	—	4	1	no	—
53	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
54	#6107	—	8	4	no	—
55	C₄⋊Dic₁₃	—	2	1	no	—
56	C₄⋊Dic₁₃	—	4	1	no	—
57	C₄⋊Dic₁₃	—	4	1	no	—
58	#6101	—	4	1	no	—
59	C₄×Dic₁₃	—	4	1	no	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	D₈	1	1	yes	characteristic
62	C₂×#1055	D₈	1	1	yes	characteristic
63	Dic₅₂	—	2	1	no	—
64	Dic₅₂	—	2	1	no	—
65	C₁₃×C₁₆	—	4	1	no	—
66	#6090	—	4	1	no	—
67	#2227	—	13	1	no	—
68	#2227	—	13	1	no	—
69	#1713	—	13	1	no	—
70	#1591	—	13	1	no	—
71	#1367	—	13	1	no	—
72	#1365	—	13	1	no	—
73	#1365	D₁₃	1	1	yes	characteristic
74	Q₈×C₁₃	—	8	2	no	—
75	C₂×C₄×C₁₃	—	8	1	no	—
76	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	C₂×Dic₁₃	—	4	1	no	—
80	C₂×Dic₁₃	—	4	1	no	—
81	C₂×Dic₁₃	—	8	1	no	—
82	Dic₂₆	—	4	1	no	—
83	Dic₂₆	—	4	1	no	—
84	Dic₂₆	—	8	2	no	—
85	C₈×C₁₃	—	2	1	no	—
86	C₈×C₁₃	—	8	1	no	—
87	C₈×C₁₃	#137	1	1	yes	characteristic
88	C₈×C₁₃	#137	1	1	yes	characteristic
89	#1055	—	2	2	no	—
90	#1055	—	2	1	no	—
91	#1055	Q₃₂	1	2	yes	—
92	#500	—	13	1	no	—
93	#500	—	26	1	no	—
94	#500	C₂×D₁₃	1	1	yes	characteristic
95	#494	—	26	1	no	—
96	#446	—	26	1	no	—
97	C₄×Q₁₆	—	13	1	no	—
98	#366	—	26	1	no	—
99	#366	—	26	1	no	—
100	C₄⋊C₁₆	—	13	1	no	—
101	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
102	#358	—	26	1	no	—
103	#358	—	26	1	no	—
104	#335	C₂×D₁₃	1	1	yes	characteristic
105	Q₈⋊C₈	—	26	1	no	—
106	C₈²	—	13	1	no	—
107	C₂²×C₁₃	#459	1	1	yes	characteristic
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	16	1	no	—
110	C₄×C₁₃	#470	1	1	yes	characteristic
111	C₄×C₁₃	#473	1	1	yes	characteristic
112	C₄×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
113	C₄×C₁₃	#510	1	1	yes	characteristic
114	Dic₁₃	—	8	1	no	—
115	Dic₁₃	—	8	1	no	—
116	Dic₁₃	—	16	1	no	—
117	C₂×Q₁₆	—	13	1	no	—
118	C₂×Q₁₆	—	52	2	no	—
119	C₄⋊Q₈	—	2	1	no	—
120	C₄⋊Q₈	—	26	1	no	—
121	C₄×Q₈	—	26	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	26	1	no	—
124	#107	—	2	1	no	—
125	#107	—	2	1	no	—
126	#107	—	13	1	no	—
127	#107	—	26	1	no	—
128	#107	—	26	1	no	—
129	#107	—	52	1	no	—
130	#107	—	52	1	no	—
131	C₄⋊C₈	—	2	1	no	—
132	C₄⋊C₈	—	52	1	no	—
133	#103	—	26	1	no	—
134	#97	—	52	1	no	—
135	C₄×C₈	—	13	1	no	—
136	C₄×C₈	—	13	1	no	—
137	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
138	C₂×C₁₃	#1689	1	1	yes	characteristic
139	C₂×C₁₃	#2209	1	1	yes	characteristic
140	C₂×C₁₃	#2213	1	1	yes	characteristic
141	C₂×Q₈	—	4	1	no	—
142	C₂×Q₈	—	26	1	no	—
143	C₂×Q₈	—	52	1	no	—
144	Q₁₆	—	26	1	no	—
145	Q₁₆	—	26	1	no	—
146	Q₁₆	—	104	4	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	13	1	no	—
149	C₂×C₈	—	13	1	no	—
150	C₂×C₈	—	26	1	no	—
151	C₂×C₈	#6127	1	1	yes	characteristic
152	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	26	1	no	—
156	C₄⋊C₄	—	52	1	no	—
157	C₄⋊C₄	—	52	1	no	—
158	C₄⋊C₄	—	52	1	no	—
159	C₄²	—	52	1	no	—
160	C₄²	D₄×D₁₃	1	1	yes	characteristic
161	C₁₆	—	4	1	no	—
162	C₁₆	—	52	1	no	—
163	C₁₃	#12500	1	1	yes	characteristic, complemented, Sylow
164	Q₈	—	8	2	no	—
165	Q₈	—	52	1	no	—
166	Q₈	—	52	1	no	—
167	Q₈	—	104	2	no	—
168	C₂×C₄	—	8	1	no	—
169	C₂×C₄	—	52	1	no	—
170	C₂×C₄	—	52	1	no	—
171	C₂×C₄	—	104	1	no	—
172	C₂×C₄	#88138	1	1	yes	characteristic
173	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
174	C₂×C₄	#88168	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	8	1	no	—
177	C₈	—	13	2	no	—
178	C₈	—	26	2	no	—
179	C₈	—	26	1	no	—
180	C₈	#88169	1	1	yes	characteristic
181	C₈	#88172	1	1	yes	characteristic
182	C₂²	#11718447	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	16	1	no	—
185	C₄	—	104	1	no	—
186	C₄	—	104	1	no	—
187	C₄	—	208	1	no	—
188	C₄	#11718416	1	1	yes	characteristic
189	C₄	#11718477	1	1	yes	characteristic
190	C₄	#11718576	1	1	yes	characteristic
191	C₄	D₁₃×Q₃₂	1	1	yes	characteristic
192	C₂	#13114324	1	1	yes	characteristic
193	C₂	#13116885	1	1	yes	characteristic
194	C₂	#13116911	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718447
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1365	C₂
socle, prime radical	C₂²×C₁₃	#459
hypercenter	#1365	D₁₃
hypocenter	C₁₃	#12500
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1365

Complements

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

N	H ≅ G/N	type
163. C₁₃	#12500	semidirect

Maximal subgroups

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

subgroup	index
#13116989	2
#13116985	2
#13114453	2
#13113732	2
#13112899	2
#13112895	2
C₁₃×#1365	2
#12500	13