Finite Group Databasebeta

Subgroups — #15777308

Order 3328.

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

normal subgroups only

This group has 194 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	#13117116	C₂	1	1	yes	maximal, characteristic
2	#13116741	C₂	1	1	yes	maximal, characteristic
3	#13114237	C₂	1	1	yes	maximal, characteristic
4	#13114231	C₂	1	1	yes	maximal, characteristic
5	#13112907	C₂	1	1	yes	maximal, characteristic
6	#13112890	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1365	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#500	C₂²	1	1	yes	characteristic
9	#11718752	—	2	1	no	—
10	#11718510	—	2	1	no	—
11	#11718343	C₂²	1	1	yes	characteristic
12	C₈×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₃×#335	C₂²	1	1	yes	characteristic
15	#11718138	—	2	1	no	—
16	#11718094	—	2	1	no	—
17	#11718093	—	2	1	no	—
18	#11718082	—	2	1	no	—
19	#11718082	—	2	1	no	—
20	#11718074	—	2	1	no	—
21	C₄×#6090	C₂²	1	1	yes	characteristic
22	#11718050	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊Q₈	—	2	1	no	—
24	C₂×Dic₅₂	—	4	1	no	—
25	#88123	—	2	1	no	—
26	C₄×Dic₂₆	D₄	1	1	yes	characteristic
27	C₂×C₁₃×C₁₆	—	2	1	no	—
28	C₁₃×#107	—	2	1	no	—
29	C₁₃×#107	—	2	1	no	—
30	C₁₃×C₄⋊C₈	—	2	1	no	—
31	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	#88058	—	2	1	no	—
33	#88058	—	2	1	no	—
34	#88056	—	4	1	no	—
35	#88054	—	2	1	no	—
36	C₈×Dic₁₃	—	2	1	no	—
37	C₂×#6090	D₄	1	1	yes	characteristic
38	C₂×#6090	D₄	1	1	yes	characteristic
39	#88050	—	4	1	no	—
40	#88047	—	4	1	no	—
41	C₄⋊#1055	—	2	1	no	—
42	C₄⋊#1055	D₄	1	1	yes	characteristic
43	#12462	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	—	4	1	no	—
45	C₂×Dic₂₆	—	4	1	no	—
46	C₂×Dic₂₆	D₈	1	1	yes	characteristic
47	C₂×C₈×C₁₃	—	4	1	no	—
48	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
50	C₁₃×C₄⋊C₄	—	4	1	no	—
51	C₁₃×C₄⋊C₄	—	4	1	no	—
52	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₄⋊Dic₁₃	—	4	1	no	—
54	C₄⋊Dic₁₃	—	4	1	no	—
55	C₄⋊Dic₁₃	D₈	1	1	yes	characteristic
56	#6101	—	2	1	no	—
57	C₄×Dic₁₃	—	2	1	no	—
58	C₂×#1055	—	2	1	no	—
59	C₂×#1055	—	4	1	no	—
60	Dic₅₂	—	8	2	no	—
61	C₁₃×C₁₆	—	4	1	no	—
62	#6090	—	2	1	no	—
63	#6090	Q₁₆	1	2	yes	complemented
64	#2254	—	13	1	no	—
65	#2183	—	13	1	no	—
66	#1676	—	13	1	no	—
67	#1674	—	13	1	no	—
68	#1371	—	13	1	no	—
69	#1365	D₁₃	1	1	yes	characteristic
70	#1364	—	13	1	no	—
71	Q₈×C₁₃	—	16	1	no	—
72	C₂×C₄×C₁₃	—	8	1	no	—
73	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
76	C₂×Dic₁₃	—	2	1	no	—
77	C₂×Dic₁₃	—	2	1	no	—
78	C₂×Dic₁₃	—	8	1	no	—
79	Dic₂₆	—	2	1	no	—
80	Dic₂₆	—	2	1	no	—
81	Dic₂₆	—	8	1	no	—
82	Dic₂₆	—	8	1	no	—
83	C₈×C₁₃	—	2	1	no	—
84	C₈×C₁₃	—	8	1	no	—
85	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₈×C₁₃	#137	1	1	yes	characteristic
87	#1055	—	4	1	no	—
88	#1055	—	8	1	no	—
89	#500	—	13	1	no	—
90	#500	C₂×D₁₃	1	1	yes	characteristic
91	#477	—	26	1	no	—
92	#462	—	26	1	no	—
93	C₈×Q₈	—	13	1	no	—
94	#367	—	26	1	no	—
95	#367	—	26	1	no	—
96	C₄⋊C₁₆	—	26	1	no	—
97	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
98	#346	—	26	1	no	—
99	C₄×C₁₆	—	13	1	no	—
100	#335	—	13	1	no	—
101	#335	C₂×D₁₃	1	1	yes	characteristic
102	#333	—	26	1	no	—
103	#332	—	26	1	no	—
104	C₂²×C₁₃	#466	1	1	yes	characteristic
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	16	1	no	—
107	C₄×C₁₃	#462	1	1	yes	characteristic
108	C₄×C₁₃	#473	1	1	yes	characteristic
109	C₄×C₁₃	#509	1	1	yes	characteristic
110	C₄×C₁₃	#510	1	1	yes	characteristic
111	Dic₁₃	—	4	1	no	—
112	Dic₁₃	—	4	1	no	—
113	Dic₁₃	—	16	1	no	—
114	C₂×Q₁₆	—	52	1	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	C₄⋊Q₈	—	26	1	no	—
117	C₄×Q₈	—	13	1	no	—
118	C₂×C₁₆	—	2	1	no	—
119	C₂×C₁₆	—	13	1	no	—
120	C₂×C₁₆	—	13	1	no	—
121	#107	—	2	1	no	—
122	#107	—	2	1	no	—
123	#107	—	26	1	no	—
124	#107	—	26	1	no	—
125	#107	—	52	1	no	—
126	C₄⋊C₈	—	2	1	no	—
127	C₄⋊C₈	—	13	1	no	—
128	C₄⋊C₈	—	26	1	no	—
129	C₄⋊C₈	—	26	1	no	—
130	#103	—	52	1	no	—
131	#103	—	52	1	no	—
132	C₄×C₈	—	26	1	no	—
133	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
134	C₂×C₁₃	#1674	1	1	yes	characteristic
135	C₂×C₁₃	#2220	1	1	yes	characteristic
136	C₂×C₁₃	#2223	1	1	yes	characteristic
137	C₂×Q₈	—	4	1	no	—
138	C₂×Q₈	—	13	1	no	—
139	C₂×Q₈	—	52	1	no	—
140	Q₁₆	—	104	2	no	—
141	C₂×C₈	—	4	1	no	—
142	C₂×C₈	—	26	1	no	—
143	C₂×C₈	—	52	1	no	—
144	C₂×C₈	#6127	1	1	yes	characteristic
145	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	13	1	no	—
149	C₄⋊C₄	—	26	1	no	—
150	C₄⋊C₄	—	52	1	no	—
151	C₄⋊C₄	—	52	1	no	—
152	C₄²	—	26	1	no	—
153	C₄²	D₄×D₁₃	1	1	yes	characteristic
154	C₁₆	—	4	1	no	—
155	C₁₆	—	13	2	no	—
156	C₁₆	—	26	1	no	—
157	C₁₃	#12462	1	1	yes	characteristic, complemented, Sylow
158	Q₈	—	16	1	no	—
159	Q₈	—	26	1	no	—
160	Q₈	—	26	1	no	—
161	Q₈	—	104	1	no	—
162	Q₈	—	104	1	no	—
163	C₂×C₄	—	8	1	no	—
164	C₂×C₄	—	26	1	no	—
165	C₂×C₄	—	26	1	no	—
166	C₂×C₄	—	104	1	no	—
167	C₂×C₄	#88138	1	1	yes	characteristic
168	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
169	C₂×C₄	#88170	1	1	yes	characteristic
170	C₈	—	2	1	no	—
171	C₈	—	8	1	no	—
172	C₈	—	52	1	no	—
173	C₈	—	104	1	no	—
174	C₈	#88169	1	1	yes	characteristic
175	C₈	Q₁₆×D₁₃	1	1	yes	characteristic
176	C₂²	#11718436	1	1	yes	characteristic
177	C₄	—	2	1	no	—
178	C₄	—	16	1	no	—
179	C₄	—	52	1	no	—
180	C₄	—	52	1	no	—
181	C₄	—	208	1	no	—
182	C₄	#11718416	1	1	yes	characteristic
183	C₄	#11718454	1	1	yes	characteristic
184	C₄	#11718575	1	1	yes	characteristic
185	C₄	#11718579	1	1	yes	characteristic
186	C₂	#13114229	1	1	yes	characteristic
187	C₂	#13116952	1	1	yes	characteristic
188	C₂	#13116965	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718436
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1365	C₂
socle, prime radical	C₂²×C₁₃	#466
hypercenter	#1365	D₁₃
hypocenter	C₁₃	#12462
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
63. #6090	Q₁₆	semidirect
157. C₁₃	#12462	semidirect

Maximal subgroups

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

subgroup	index
#13117116	2
#13116741	2
#13114237	2
#13114231	2
#13112907	2
#13112890	2
C₁₃×#1365	2
#12462	13