Finite Group Databasebeta

Subgroups — #15777298

Order 3328.

188 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 67 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	#13116745	C₂	1	1	yes	maximal, characteristic
3	#13114240	C₂	1	1	yes	maximal, characteristic
4	#13112909	C₂	1	1	yes	maximal, characteristic
5	#13112895	C₂	1	1	yes	maximal, characteristic
6	#13112894	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1676	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#500	C₂²	1	1	yes	characteristic
9	C₁₃×#477	—	2	1	no	—
10	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11718682	—	2	1	no	—
12	#11718343	C₂²	1	1	yes	characteristic
13	C₁₃×#335	C₂²	1	1	yes	characteristic
14	C₁₃×#333	—	2	1	no	—
15	#11718081	—	2	1	no	—
16	#11718081	—	2	1	no	—
17	#11718073	—	2	1	no	—
18	C₄⋊#6090	—	2	1	no	—
19	C₄⋊#6090	C₂²	1	1	yes	characteristic
20	#11718053	—	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	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
25	#88123	—	2	1	no	—
26	C₁₃×#107	—	2	1	no	—
27	C₁₃×#107	—	2	1	no	—
28	C₁₃×#107	—	4	1	no	—
29	C₁₃×C₄⋊C₈	—	2	1	no	—
30	C₁₃×C₄⋊C₈	—	2	1	no	—
31	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
32	C₁₃×#103	—	4	1	no	—
33	C₄×C₈×C₁₃	—	2	1	no	—
34	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	#88075	—	4	1	no	—
36	#88058	—	2	1	no	—
37	#88058	—	2	1	no	—
38	C₂×#6090	—	2	1	no	—
39	C₂×#6090	D₄	1	1	yes	characteristic
40	C₂×#6090	D₄	1	1	yes	characteristic
41	#88047	—	4	1	no	—
42	C₄⋊#1055	—	2	1	no	—
43	#12461	—	13	1	no	maximal, Sylow
44	C₂×Q₈×C₁₃	—	4	1	no	—
45	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
46	C₂×Dic₂₆	—	4	1	no	—
47	C₂×C₈×C₁₃	—	2	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	—	2	1	no	—
52	C₁₃×C₄⋊C₄	—	4	1	no	—
53	C₁₃×C₄⋊C₄	—	4	1	no	—
54	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
55	C₄²×C₁₃	—	2	1	no	—
56	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₄⋊Dic₁₃	—	4	1	no	—
58	C₄⋊Dic₁₃	—	4	1	no	—
59	C₂×#1055	—	4	1	no	—
60	#6090	—	2	1	no	—
61	#6090	—	4	1	no	—
62	#6090	Q₁₆	1	2	yes	—
63	#2252	—	13	1	no	—
64	#2183	—	13	1	no	—
65	#1676	—	13	1	no	—
66	#1676	D₁₃	1	1	yes	characteristic
67	#1372	—	13	1	no	—
68	#1365	—	13	1	no	—
69	#1365	—	13	1	no	—
70	Q₈×C₁₃	—	2	1	no	—
71	Q₈×C₁₃	—	2	1	no	—
72	Q₈×C₁₃	—	16	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	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	#135	1	1	yes	characteristic
79	C₂×Dic₁₃	—	8	1	no	—
80	Dic₂₆	—	16	1	no	—
81	C₈×C₁₃	—	2	1	no	—
82	C₈×C₁₃	—	4	1	no	—
83	C₈×C₁₃	—	8	1	no	—
84	C₈×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
85	C₈×C₁₃	#137	1	1	yes	characteristic
86	#1055	—	8	1	no	—
87	#500	—	13	1	no	—
88	#500	C₂×D₁₃	1	1	yes	characteristic
89	#477	—	2	1	no	—
90	#477	—	26	1	no	—
91	C₈×Q₈	C₂×D₁₃	1	1	yes	characteristic
92	#366	—	26	1	no	—
93	#366	—	26	1	no	—
94	C₄⋊C₁₆	—	13	1	no	—
95	C₄⋊C₁₆	—	26	1	no	—
96	#346	—	26	1	no	—
97	C₄×C₁₆	—	13	1	no	—
98	#335	—	13	1	no	—
99	#335	C₂×D₁₃	1	1	yes	characteristic
100	#333	—	2	1	no	—
101	#333	—	26	1	no	—
102	C₂²×C₁₃	#466	1	1	yes	characteristic
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	—	4	1	no	—
105	C₄×C₁₃	—	4	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	#510	1	1	yes	characteristic
110	C₄×C₁₃	#510	1	1	yes	characteristic
111	Dic₁₃	—	16	1	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	26	1	no	—
114	C₄×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
115	C₂×C₁₆	—	13	1	no	—
116	C₂×C₁₆	—	13	1	no	—
117	C₂×C₁₆	—	26	1	no	—
118	#107	—	2	1	no	—
119	#107	—	2	1	no	—
120	#107	—	4	1	no	—
121	#107	—	26	1	no	—
122	#107	—	26	1	no	—
123	#107	—	52	1	no	—
124	C₄⋊C₈	—	2	1	no	—
125	C₄⋊C₈	—	2	1	no	—
126	C₄⋊C₈	—	26	1	no	—
127	C₄⋊C₈	C₂²⋊D₁₃	1	1	yes	characteristic
128	#103	—	4	1	no	—
129	#103	—	52	1	no	—
130	C₄×C₈	—	2	1	no	—
131	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	C₂×C₁₃	#1674	1	1	yes	characteristic
133	C₂×C₁₃	#2219	1	1	yes	characteristic
134	C₂×C₁₃	#2221	1	1	yes	characteristic
135	C₂×Q₈	—	4	1	no	—
136	C₂×Q₈	—	52	1	no	—
137	C₂×Q₈	#6104	1	1	yes	characteristic
138	C₂×C₈	—	2	1	no	—
139	C₂×C₈	—	4	1	no	—
140	C₂×C₈	—	52	1	no	—
141	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
142	C₂×C₈	#6129	1	1	yes	characteristic
143	C₄⋊C₄	—	2	1	no	—
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	52	1	no	—
147	C₄⋊C₄	—	52	1	no	—
148	C₄⋊C₄	#6104	1	1	yes	characteristic
149	C₄²	—	2	1	no	—
150	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
151	C₁₆	—	13	2	no	—
152	C₁₆	—	26	1	no	—
153	C₁₆	—	52	1	no	—
154	C₁₃	#12461	1	1	yes	characteristic, complemented, Sylow
155	Q₈	—	2	1	no	—
156	Q₈	—	2	1	no	—
157	Q₈	—	16	1	no	—
158	Q₈	—	208	1	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	104	1	no	—
163	C₂×C₄	C₂×#6104	1	1	yes	characteristic
164	C₂×C₄	#88193	1	1	yes	characteristic
165	C₂×C₄	#88204	1	1	yes	characteristic
166	C₈	—	2	1	no	—
167	C₈	—	4	1	no	—
168	C₈	—	8	1	no	—
169	C₈	—	104	1	no	—
170	C₈	Q₁₆×D₁₃	1	1	yes	characteristic
171	C₈	#88172	1	1	yes	characteristic
172	C₂²	#11718699	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	4	1	no	—
175	C₄	—	4	1	no	—
176	C₄	—	16	1	no	—
177	C₄	—	208	1	no	—
178	C₄	#11718840	1	1	yes	characteristic
179	C₄	#11718843	1	1	yes	characteristic
180	C₄	#11718856	1	1	yes	characteristic
181	C₄	#11718865	1	1	yes	characteristic
182	C₂	#13114228	1	1	yes	characteristic
183	C₂	#13116942	1	1	yes	characteristic
184	C₂	#13116954	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718699
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1676	C₂
socle, prime radical	C₂²×C₁₃	#466
hypercenter	#1676	D₁₃
hypocenter	C₁₃	#12461
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈ ◁ #1676

Complements

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

N	H ≅ G/N	type
154. C₁₃	#12461	semidirect

Maximal subgroups

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

subgroup	index
#13117109	2
#13116745	2
#13114240	2
#13112909	2
#13112895	2
#13112894	2
C₁₃×#1676	2
#12461	13