Finite Group Databasebeta

Subgroups — #15777716

Order 3328.

238 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 83 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	#13117111	C₂	1	1	yes	maximal, characteristic
3	#13117064	C₂	1	1	yes	maximal, characteristic
4	#13117064	C₂	1	1	yes	maximal, characteristic
5	#13114446	C₂	1	1	yes	maximal, characteristic
6	#13114433	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1312	C₂	1	1	yes	maximal, characteristic
8	#11718360	C₂²	1	1	yes	characteristic
9	#11718343	C₂²	1	1	yes	characteristic
10	#11718343	C₂²	1	1	yes	characteristic
11	#11718343	C₂²	1	1	yes	characteristic
12	#11718342	—	2	1	no	—
13	#11718342	—	2	1	no	—
14	#11718342	—	2	1	no	—
15	#11718341	—	2	1	no	—
16	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₈²×C₁₃	C₂²	1	1	yes	characteristic
19	#11718097	—	2	2	no	—
20	#11718096	—	2	2	no	—
21	#11718095	—	2	2	no	—
22	#11718095	—	2	2	no	—
23	C₂×Dic₅₂	—	2	1	no	—
24	C₂×Dic₅₂	—	2	1	no	—
25	#88123	—	2	1	no	—
26	#88123	—	2	1	no	—
27	#88123	—	2	1	no	—
28	#88123	—	2	1	no	—
29	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
30	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
31	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	#88058	—	2	1	no	—
35	#88058	—	2	1	no	—
36	#88058	—	2	1	no	—
37	#88058	—	2	1	no	—
38	#88058	—	2	1	no	—
39	#88058	—	2	1	no	—
40	#88057	—	4	2	no	—
41	#88056	—	4	2	no	—
42	#88056	—	4	2	no	—
43	#88056	—	4	2	no	—
44	#12533	—	13	1	no	maximal, Sylow
45	C₂×Dic₂₆	—	4	1	no	—
46	C₂×Dic₂₆	—	4	1	no	—
47	C₂×Dic₂₆	—	4	1	no	—
48	C₂×Dic₂₆	—	4	1	no	—
49	C₂×C₈×C₁₃	D₈	1	2	yes	—
50	C₂×C₈×C₁₃	D₈	1	2	yes	—
51	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Dic₁₃	—	4	1	no	—
55	C₄⋊Dic₁₃	—	4	1	no	—
56	C₄⋊Dic₁₃	—	4	1	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	C₄⋊Dic₁₃	—	4	1	no	—
62	Dic₅₂	—	4	2	no	—
63	Dic₅₂	—	4	2	no	—
64	C₁₃×C₁₆	—	2	2	no	—
65	C₁₃×C₁₆	—	2	2	no	—
66	#2254	—	13	1	no	—
67	#2252	—	13	1	no	—
68	#2241	—	13	1	no	—
69	#2241	—	13	1	no	—
70	#1710	—	13	1	no	—
71	#1707	—	13	1	no	—
72	#1312	D₁₃	1	1	yes	characteristic
73	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
76	C₂×Dic₁₃	—	8	1	no	—
77	C₂×Dic₁₃	—	8	1	no	—
78	C₂×Dic₁₃	—	8	1	no	—
79	C₂×Dic₁₃	—	8	1	no	—
80	Dic₂₆	—	8	1	no	—
81	Dic₂₆	—	8	1	no	—
82	Dic₂₆	—	8	2	no	—
83	Dic₂₆	—	8	2	no	—
84	Dic₂₆	—	8	2	no	—
85	C₈×C₁₃	—	2	2	no	—
86	C₈×C₁₃	—	2	2	no	—
87	C₈×C₁₃	#137	1	2	yes	—
88	C₈×C₁₃	#137	1	2	yes	—
89	#500	—	13	1	no	—
90	#500	—	13	1	no	—
91	#500	—	13	1	no	—
92	#498	—	26	1	no	—
93	#494	—	13	1	no	—
94	#487	—	26	1	no	—
95	#487	—	26	1	no	—
96	#487	—	26	1	no	—
97	#367	—	26	2	no	—
98	#366	—	26	2	no	—
99	#358	—	26	2	no	—
100	#358	—	26	2	no	—
101	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
102	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
103	C₈²	C₂×D₁₃	1	1	yes	characteristic
104	C₂²×C₁₃	#493	1	1	yes	characteristic
105	C₄×C₁₃	#497	1	1	yes	characteristic
106	C₄×C₁₃	#497	1	1	yes	characteristic
107	C₄×C₁₃	#508	1	2	yes	—
108	C₄×C₁₃	#510	1	2	yes	—
109	Dic₁₃	—	16	1	no	—
110	Dic₁₃	—	16	1	no	—
111	Dic₁₃	—	16	1	no	—
112	Dic₁₃	—	16	1	no	—
113	C₂×Q₁₆	—	26	1	no	—
114	C₂×Q₁₆	—	26	1	no	—
115	C₄⋊Q₈	—	26	1	no	—
116	C₄⋊Q₈	—	26	1	no	—
117	C₄⋊Q₈	—	26	1	no	—
118	C₄⋊Q₈	—	26	1	no	—
119	C₂×C₁₆	D₅₂	1	2	yes	—
120	C₂×C₁₆	D₅₂	1	2	yes	—
121	#107	—	26	1	no	—
122	#107	—	26	1	no	—
123	#107	—	26	1	no	—
124	#107	—	26	1	no	—
125	#107	—	26	1	no	—
126	#107	—	26	1	no	—
127	#106	—	52	2	no	—
128	#103	—	52	2	no	—
129	#103	—	52	2	no	—
130	#103	—	52	2	no	—
131	C₄×C₈	D₅₂	1	1	yes	characteristic
132	C₄×C₈	D₅₂	1	1	yes	characteristic
133	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
134	C₂×C₁₃	#1722	1	1	yes	characteristic
135	C₂×C₁₃	#2249	1	1	yes	characteristic
136	C₂×C₁₃	#2251	1	1	yes	characteristic
137	C₂×Q₈	—	52	1	no	—
138	C₂×Q₈	—	52	1	no	—
139	C₂×Q₈	—	52	1	no	—
140	C₂×Q₈	—	52	1	no	—
141	Q₁₆	—	52	2	no	—
142	Q₁₆	—	52	2	no	—
143	C₂×C₈	D₁₀₄	1	2	yes	—
144	C₂×C₈	D₁₀₄	1	2	yes	—
145	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
146	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
147	C₄⋊C₄	—	52	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	C₄⋊C₄	—	52	1	no	—
153	C₄⋊C₄	—	52	1	no	—
154	C₄⋊C₄	—	52	1	no	—
155	C₄²	C₂×D₅₂	1	1	yes	characteristic
156	C₁₆	—	2	2	no	—
157	C₁₆	—	2	2	no	—
158	C₁₃	#12533	1	1	yes	characteristic, complemented, Sylow
159	Q₈	—	104	2	no	—
160	Q₈	—	104	2	no	—
161	Q₈	—	104	2	no	—
162	Q₈	—	104	1	no	—
163	Q₈	—	104	1	no	—
164	C₂×C₄	—	104	1	no	—
165	C₂×C₄	—	104	1	no	—
166	C₂×C₄	—	104	1	no	—
167	C₂×C₄	—	104	1	no	—
168	C₂×C₄	#88128	1	1	yes	characteristic
169	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
170	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
171	C₈	—	2	2	no	—
172	C₈	—	2	2	no	—
173	C₈	#88163	1	2	yes	—
174	C₈	#88163	1	2	yes	—
175	C₂²	#11718356	1	1	yes	characteristic
176	C₄	—	208	1	no	—
177	C₄	—	208	1	no	—
178	C₄	—	208	1	no	—
179	C₄	—	208	1	no	—
180	C₄	#11718376	1	1	yes	characteristic
181	C₄	#11718376	1	1	yes	characteristic
182	C₄	#11718565	1	2	yes	—
183	C₄	#11718570	1	2	yes	—
184	C₂	#13114507	1	1	yes	characteristic
185	C₂	#13117097	1	1	yes	characteristic
186	C₂	#13117107	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1312	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#1312	D₁₃
hypocenter	C₁₃	#12533
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

Complements

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

N	H ≅ G/N	type
158. C₁₃	#12533	semidirect

Maximal subgroups

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

subgroup	index
#13117118	2
#13117111	2
#13117064	2
#13114446	2
#13114433	2
C₁₃×#1312	2
#12533	13