Finite Group Databasebeta

Subgroups — #15777665

Order 3328.

246 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117118	C₂	1	1	yes	maximal, characteristic
2	#13117116	C₂	1	1	yes	maximal, characteristic
3	#13117109	C₂	1	1	yes	maximal, characteristic
4	#13114453	C₂	1	1	yes	maximal, characteristic
5	#13114445	C₂	1	1	yes	maximal, characteristic
6	#13112752	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2252	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	#11718753	—	2	1	no	—
12	#11718742	—	2	1	no	—
13	#11718739	—	2	1	no	—
14	#11718343	C₂²	1	1	yes	characteristic
15	#11718343	C₂²	1	1	yes	characteristic
16	C₁₃×#366	—	2	2	no	—
17	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	#11718097	—	2	2	no	—
19	#11718082	—	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×#6107	—	4	2	no	—
26	#88123	—	2	1	no	—
27	#88123	—	2	1	no	—
28	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
29	C₁₃×#107	—	2	1	no	—
30	C₁₃×#107	—	2	1	no	—
31	C₁₃×#107	—	2	1	no	—
32	C₁₃×#107	—	2	1	no	—
33	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	#88075	—	4	2	no	—
35	#88058	—	2	1	no	—
36	#88058	—	2	1	no	—
37	#88058	—	2	1	no	—
38	#88058	—	2	1	no	—
39	C₂×#6090	D₄	1	2	yes	—
40	#88050	—	4	2	no	—
41	#88047	—	4	2	no	—
42	C₄×#1055	D₄	1	1	yes	characteristic
43	C₄×#1055	D₄	1	1	yes	characteristic
44	#12524	—	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	#6107	—	8	4	no	—
57	C₄⋊Dic₁₃	—	4	1	no	—
58	C₄⋊Dic₁₃	—	4	1	no	—
59	C₄⋊Dic₁₃	—	4	1	no	—
60	C₄⋊Dic₁₃	—	4	1	no	—
61	C₂×#1055	D₈	1	2	yes	—
62	C₂×#1055	D₈	1	2	yes	—
63	C₁₃×C₁₆	Q₁₆	1	4	yes	complemented
64	#6090	Q₁₆	1	4	yes	complemented
65	#2254	—	13	1	no	—
66	#2254	—	13	1	no	—
67	#2252	—	13	1	no	—
68	#2252	D₁₃	1	1	yes	characteristic
69	#1713	—	13	1	no	—
70	#1710	—	13	1	no	—
71	#1311	—	13	1	no	—
72	Q₈×C₁₃	—	8	2	no	—
73	Q₈×C₁₃	—	8	2	no	—
74	C₂×C₄×C₁₃	—	8	1	no	—
75	C₂×C₄×C₁₃	—	8	1	no	—
76	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
79	C₂×Dic₁₃	—	8	1	no	—
80	C₂×Dic₁₃	—	8	1	no	—
81	Dic₂₆	—	8	2	no	—
82	Dic₂₆	—	8	2	no	—
83	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
84	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
85	#1055	—	2	2	no	—
86	#1055	—	2	2	no	—
87	#500	—	13	1	no	—
88	#500	—	13	1	no	—
89	#500	—	26	1	no	—
90	#500	C₂×D₁₃	1	1	yes	characteristic
91	#500	C₂×D₁₃	1	1	yes	characteristic
92	#494	—	26	1	no	—
93	#487	—	26	1	no	—
94	#487	—	26	1	no	—
95	#367	—	26	2	no	—
96	#367	—	26	2	no	—
97	#366	—	2	2	no	—
98	#366	—	26	2	no	—
99	C₄×C₁₆	—	13	1	no	—
100	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
101	C₈²	—	13	1	no	—
102	C₂²×C₁₃	#493	1	1	yes	characteristic
103	C₄×C₁₃	—	16	1	no	—
104	C₄×C₁₃	—	16	1	no	—
105	C₄×C₁₃	#494	1	1	yes	characteristic
106	C₄×C₁₃	#494	1	1	yes	characteristic
107	C₄×C₁₃	#509	1	2	yes	—
108	C₄×C₁₃	#510	1	2	yes	—
109	Dic₁₃	—	16	1	no	—
110	Dic₁₃	—	16	1	no	—
111	C₂×Q₁₆	—	52	2	no	—
112	C₄⋊Q₈	—	2	1	no	—
113	C₄⋊Q₈	—	2	1	no	—
114	C₄⋊Q₈	—	26	1	no	—
115	C₄⋊Q₈	—	26	1	no	—
116	C₂×C₁₆	—	13	2	no	—
117	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
118	#107	—	2	1	no	—
119	#107	—	2	1	no	—
120	#107	—	2	1	no	—
121	#107	—	2	1	no	—
122	#107	—	26	1	no	—
123	#107	—	26	1	no	—
124	#107	—	26	1	no	—
125	#107	—	26	1	no	—
126	#107	—	52	2	no	—
127	#103	—	52	2	no	—
128	#103	—	52	2	no	—
129	C₄×C₈	—	13	1	no	—
130	C₄×C₈	—	13	1	no	—
131	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
132	C₂×C₁₃	#1713	1	1	yes	characteristic
133	C₂×C₁₃	#2250	1	1	yes	characteristic
134	C₂×C₁₃	#2251	1	1	yes	characteristic
135	C₂×Q₈	—	4	1	no	—
136	C₂×Q₈	—	4	1	no	—
137	C₂×Q₈	—	52	1	no	—
138	C₂×Q₈	—	52	1	no	—
139	Q₁₆	—	104	4	no	—
140	C₂×C₈	—	13	2	no	—
141	C₂×C₈	—	13	2	no	—
142	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
143	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
144	C₄⋊C₄	—	4	1	no	—
145	C₄⋊C₄	—	4	1	no	—
146	C₄⋊C₄	—	4	1	no	—
147	C₄⋊C₄	—	4	1	no	—
148	C₄⋊C₄	—	52	1	no	—
149	C₄⋊C₄	—	52	1	no	—
150	C₄⋊C₄	—	52	1	no	—
151	C₄⋊C₄	—	52	1	no	—
152	C₄²	D₄×D₁₃	1	1	yes	characteristic
153	C₁₆	—	13	4	no	—
154	C₁₆	#6107	1	4	yes	complemented
155	C₁₃	#12524	1	1	yes	characteristic, complemented, Sylow
156	Q₈	—	8	2	no	—
157	Q₈	—	8	2	no	—
158	Q₈	—	104	2	no	—
159	Q₈	—	104	2	no	—
160	C₂×C₄	—	8	1	no	—
161	C₂×C₄	—	8	1	no	—
162	C₂×C₄	—	104	1	no	—
163	C₂×C₄	—	104	1	no	—
164	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
165	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
166	C₂×C₄	#88194	1	1	yes	characteristic
167	C₈	—	26	2	no	—
168	C₈	—	26	2	no	—
169	C₈	Q₁₆×D₁₃	1	2	yes	—
170	C₈	C₂×#6107	1	2	yes	—
171	C₂²	#11718812	1	1	yes	characteristic
172	C₄	—	16	1	no	—
173	C₄	—	16	1	no	—
174	C₄	—	208	1	no	—
175	C₄	—	208	1	no	—
176	C₄	#11718572	1	2	yes	—
177	C₄	#11718579	1	2	yes	—
178	C₄	#11718753	1	1	yes	characteristic
179	C₄	#11718842	1	1	yes	characteristic
180	C₂	#13114453	1	1	yes	characteristic
181	C₂	#13117099	1	1	yes	characteristic
182	C₂	#13117104	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2252	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#2252	D₁₃
hypocenter	C₁₃	#12524
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
63. C₁₃×C₁₆	Q₁₆	semidirect
64. #6090	Q₁₆	semidirect
154. C₁₆	#6107	semidirect
155. C₁₃	#12524	semidirect

Maximal subgroups

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

subgroup	index
#13117118	2
#13117116	2
#13117109	2
#13114453	2
#13114445	2
#13112752	2
C₁₃×#2252	2
#12524	13