Finite Group Databasebeta

Subgroups — #15777567

Order 3328.

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

normal subgroups only

This group has 200 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	#13116978	C₂	1	1	yes	maximal, characteristic
2	#13114445	C₂	1	1	yes	maximal, characteristic
3	#13113733	C₂	1	1	yes	maximal, characteristic
4	#13112899	C₂	1	1	yes	maximal, characteristic
5	#13112894	C₂	1	1	yes	maximal, characteristic
6	#13112890	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2227	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#500	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	#11718754	—	2	1	no	—
11	#11718742	—	2	1	no	—
12	#11718685	—	2	1	no	—
13	#11718343	C₂²	1	1	yes	characteristic
14	C₁₃×#366	—	2	1	no	—
15	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#358	—	2	1	no	—
17	#11718156	—	2	1	no	—
18	Q₈⋊#1055	—	2	1	no	—
19	#11718082	—	2	1	no	—
20	C₄⋊#6090	C₂²	1	1	yes	characteristic
21	#11718050	C₂²	1	1	yes	characteristic
22	C₈×#1055	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic
24	C₁₃×C₄⋊Q₈	—	2	1	no	—
25	C₄×Q₈×C₁₃	—	2	1	no	—
26	#88123	—	2	1	no	—
27	C₂×C₁₃×C₁₆	—	2	1	no	—
28	C₁₃×#107	—	2	1	no	—
29	C₁₃×#107	—	2	1	no	—
30	C₁₃×#107	D₄	1	1	yes	characteristic
31	C₁₃×#103	—	2	1	no	—
32	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	#88075	—	4	2	no	—
34	#88058	—	2	1	no	—
35	#88058	—	2	1	no	—
36	C₂×#6090	—	2	1	no	—
37	#88050	—	4	1	no	—
38	#88050	—	4	1	no	—
39	C₄⋊#1055	—	2	1	no	—
40	C₄⋊#1055	—	4	1	no	—
41	#88043	—	4	1	no	—
42	C₄×#1055	D₄	1	1	yes	characteristic
43	C₄×#1055	D₄	1	1	yes	characteristic
44	#12502	—	13	1	no	maximal, Sylow
45	C₂×Q₈×C₁₃	—	2	1	no	—
46	C₂×Q₈×C₁₃	—	4	1	no	—
47	C₂×Dic₂₆	—	4	1	no	—
48	C₁₃×Q₁₆	—	2	1	no	—
49	C₁₃×Q₁₆	—	2	1	no	—
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₁₃×C₄⋊C₄	—	4	1	no	—
54	C₁₃×C₄⋊C₄	—	4	1	no	—
55	C₁₃×C₄⋊C₄	—	4	1	no	—
56	C₄²×C₁₃	—	4	1	no	—
57	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Dic₁₃	—	4	1	no	—
59	C₄⋊Dic₁₃	—	4	1	no	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	—	4	1	no	—
62	C₂×#1055	D₈	1	1	yes	characteristic
63	C₂×#1055	D₈	1	1	yes	characteristic
64	C₁₃×C₁₆	—	4	1	no	—
65	#6090	—	4	1	no	—
66	#2227	D₁₃	1	1	yes	characteristic
67	#2225	—	13	1	no	—
68	#1710	—	13	1	no	—
69	#1591	—	13	1	no	—
70	#1367	—	13	1	no	—
71	#1365	—	13	1	no	—
72	#1364	—	13	1	no	—
73	Q₈×C₁₃	—	4	1	no	—
74	Q₈×C₁₃	—	4	1	no	—
75	Q₈×C₁₃	—	8	2	no	—
76	C₂×C₄×C₁₃	—	4	1	no	—
77	C₂×C₄×C₁₃	—	4	1	no	—
78	C₂×C₄×C₁₃	—	8	1	no	—
79	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×Dic₁₃	—	8	1	no	—
83	Dic₂₆	—	8	1	no	—
84	Dic₂₆	—	8	1	no	—
85	C₈×C₁₃	—	2	1	no	—
86	C₈×C₁₃	#137	1	1	yes	characteristic
87	C₈×C₁₃	#137	1	1	yes	characteristic
88	#1055	—	2	1	no	—
89	#1055	—	2	1	no	—
90	#1055	—	4	1	no	—
91	#1055	—	8	1	no	—
92	#500	—	13	1	no	—
93	#500	C₂×D₁₃	1	1	yes	characteristic
94	#487	—	26	1	no	—
95	#487	—	26	1	no	—
96	#446	—	26	1	no	—
97	C₄×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
98	#367	—	26	1	no	—
99	#366	—	2	1	no	—
100	C₄⋊C₁₆	—	13	1	no	—
101	C₄⋊C₁₆	C₂×D₁₃	1	1	yes	characteristic
102	#358	—	2	1	no	—
103	#358	—	26	1	no	—
104	#335	—	13	1	no	—
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	—	8	1	no	—
110	C₄×C₁₃	—	8	1	no	—
111	C₄×C₁₃	—	16	1	no	—
112	C₄×C₁₃	#470	1	1	yes	characteristic
113	C₄×C₁₃	#473	1	1	yes	characteristic
114	C₄×C₁₃	#508	1	1	yes	characteristic
115	C₄×C₁₃	#510	1	1	yes	characteristic
116	Dic₁₃	—	16	1	no	—
117	C₂×Q₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
118	C₄⋊Q₈	—	2	1	no	—
119	C₄⋊Q₈	—	26	1	no	—
120	C₄×Q₈	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	26	1	no	—
123	#107	—	2	1	no	—
124	#107	—	2	1	no	—
125	#107	—	26	1	no	—
126	#107	—	26	1	no	—
127	#107	C₂²⋊D₁₃	1	1	yes	characteristic
128	C₄⋊C₈	—	26	1	no	—
129	C₄⋊C₈	—	52	1	no	—
130	#103	—	2	1	no	—
131	#103	—	52	1	no	—
132	#103	—	52	1	no	—
133	#103	—	52	2	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	#2211	1	1	yes	characteristic
140	C₂×C₁₃	#2212	1	1	yes	characteristic
141	C₂×Q₈	—	2	1	no	—
142	C₂×Q₈	—	4	1	no	—
143	C₂×Q₈	—	52	1	no	—
144	Q₁₆	—	2	1	no	—
145	Q₁₆	—	2	1	no	—
146	C₂×C₈	—	13	1	no	—
147	C₂×C₈	—	13	1	no	—
148	C₂×C₈	—	26	1	no	—
149	C₂×C₈	—	52	1	no	—
150	C₂×C₈	#6129	1	1	yes	characteristic
151	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₄⋊C₄	—	4	1	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄⋊C₄	—	52	1	no	—
157	C₄⋊C₄	—	52	1	no	—
158	C₄²	—	4	1	no	—
159	C₄²	D₄×D₁₃	1	1	yes	characteristic
160	C₁₆	—	4	1	no	—
161	C₁₆	—	52	1	no	—
162	C₁₃	#12502	1	1	yes	characteristic, complemented, Sylow
163	Q₈	—	4	1	no	—
164	Q₈	—	4	1	no	—
165	Q₈	—	8	2	no	—
166	Q₈	—	104	1	no	—
167	Q₈	—	104	1	no	—
168	C₂×C₄	—	4	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	—	8	1	no	—
171	C₂×C₄	—	104	1	no	—
172	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
173	C₂×C₄	#88165	1	1	yes	characteristic
174	C₂×C₄	#88193	1	1	yes	characteristic
175	C₈	—	2	1	no	—
176	C₈	—	26	1	no	—
177	C₈	—	26	1	no	—
178	C₈	—	52	1	no	—
179	C₈	—	104	1	no	—
180	C₈	#88196	1	1	yes	characteristic
181	C₈	#88205	1	1	yes	characteristic
182	C₂²	#11718811	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	8	1	no	—
185	C₄	—	8	1	no	—
186	C₄	—	16	1	no	—
187	C₄	—	208	1	no	—
188	C₄	#11718579	1	1	yes	characteristic
189	C₄	#11718580	1	1	yes	characteristic
190	C₄	#11718713	1	1	yes	characteristic
191	C₄	#11718865	1	1	yes	characteristic
192	C₂	#13114323	1	1	yes	characteristic
193	C₂	#13116900	1	1	yes	characteristic
194	C₂	#13116903	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718811
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2227	C₂
socle, prime radical	C₂²×C₁₃	#459
hypercenter	#2227	D₁₃
hypocenter	C₁₃	#12502
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
162. C₁₃	#12502	semidirect

Maximal subgroups

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

subgroup	index
#13116978	2
#13114445	2
#13113733	2
#13112899	2
#13112894	2
#13112890	2
C₁₃×#2227	2
#12502	13