Subgroups — #15780925

Order 3328.

338 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6090	C₂	1	2	yes	maximal
2	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
4	#13112914	C₂	1	2	yes	maximal
5	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
10	C₂²×#6090	C₂²	1	2	yes	—
11	C₂²×#6090	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	1	yes	characteristic
13	C₁₃×#336	C₄	1	2	yes	—
14	C₄⋊#6090	—	2	4	no	—
15	C₄⋊#6090	—	2	2	no	—
16	#11718053	—	2	2	no	—
17	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	C₈	1	2	yes	—
21	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	—	2	2	no	—
24	C₄×C₈×C₁₃	—	2	2	no	—
25	C₄×C₈×C₁₃	—	2	2	no	—
26	C₄×C₈×C₁₃	D₄	1	2	yes	—
27	C₄×C₈×C₁₃	Q₈	1	2	yes	—
28	C₁₃×#95	C₈	1	1	yes	characteristic
29	C₁₃×#95	C₈	1	1	yes	characteristic
30	C₂×#6090	—	2	1	no	—
31	C₂×#6090	—	2	1	no	—
32	C₂×#6090	—	2	1	no	—
33	C₂×#6090	—	2	1	no	—
34	C₂×#6090	—	2	2	no	—
35	C₂×#6090	—	2	2	no	—
36	C₂×#6090	—	2	2	no	—
37	C₂×#6090	D₄	1	2	yes	—
38	C₂×#6090	Q₈	1	2	yes	—
39	#13160	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	—	2	2	no	—
50	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
57	C₁₃×C₄⋊C₄	—	2	4	no	—
58	C₄²×C₁₃	—	2	2	no	—
59	C₄²×C₁₃	—	2	2	no	—
60	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
61	C₄²×C₁₃	M₁₆	1	2	yes	—
62	#6090	—	2	2	no	—
63	#6090	—	2	2	no	—
64	#6090	—	4	2	no	—
65	C₂×C₄⋊C₁₆	—	13	2	no	—
66	C₂×C₄⋊C₁₆	—	13	1	no	—
67	C₂×#346	—	13	1	no	—
68	C₈×C₄⋊C₄	D₁₃	1	1	yes	characteristic
69	#1374	—	13	2	no	—
70	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
79	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
80	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
81	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	—	4	2	no	—
89	C₈×C₁₃	#121	1	1	yes	characteristic
90	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₈×C₁₃	#125	1	1	yes	characteristic
92	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
93	C₂²×C₁₆	—	13	1	no	—
94	C₂²×C₁₆	—	13	1	no	—
95	C₂²×C₁₆	—	13	2	no	—
96	C₂×C₄⋊C₈	Dic₁₃	1	1	yes	characteristic
97	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
98	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
99	C₄×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
100	C₄⋊C₁₆	—	26	4	no	—
101	C₄⋊C₁₆	—	26	2	no	—
102	#346	—	26	2	no	—
103	#336	Dic₁₃	1	2	yes	—
104	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
105	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
106	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
108	C₂²×C₁₃	#504	1	2	yes	—
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	—	2	2	no	—
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	4	2	no	—
113	C₄×C₁₃	#389	1	1	yes	characteristic
114	C₄×C₁₃	#424	1	1	yes	characteristic
115	C₄×C₁₃	#436	1	1	yes	characteristic
116	C₄×C₁₃	#446	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
118	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	#1055	1	2	yes	—
121	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
122	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
123	C₂×C₁₆	—	13	2	no	—
124	C₂×C₁₆	—	13	2	no	—
125	C₂×C₁₆	—	26	1	no	—
126	C₂×C₁₆	—	26	1	no	—
127	C₂×C₁₆	—	26	1	no	—
128	C₂×C₁₆	—	26	1	no	—
129	C₂×C₁₆	—	26	2	no	—
130	C₂×C₁₆	—	26	2	no	—
131	C₂×C₁₆	—	26	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄×C₈	—	2	2	no	—
134	C₄×C₈	—	2	2	no	—
135	C₄×C₈	Dic₂₆	1	2	yes	—
136	C₄×C₈	D₅₂	1	2	yes	—
137	#95	#1055	1	1	yes	characteristic
138	#95	#1055	1	1	yes	characteristic
139	C₂×C₁₃	#1939	1	1	yes	characteristic
140	C₂×C₁₃	#2150	1	1	yes	characteristic
141	C₂×C₁₃	#2151	1	1	yes	characteristic
142	C₂×C₁₃	#2168	1	2	yes	—
143	C₂×C₁₃	#2183	1	2	yes	—
144	C₂²×C₄	C₂×#1055	1	2	yes	—
145	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
146	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
147	C₂²×C₄	C₂×#1055	1	2	yes	—
148	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
155	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
156	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
157	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
158	C₂×C₈	#6129	1	1	yes	characteristic
159	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
160	C₂×C₈	#6131	1	1	yes	characteristic
161	C₄⋊C₄	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	—	2	2	no	—
164	C₄²	#6099	1	2	yes	—
165	C₄²	C₄⋊Dic₁₃	1	2	yes	—
166	C₁₆	—	26	2	no	—
167	C₁₆	—	26	2	no	—
168	C₁₆	—	52	2	no	—
169	C₁₃	#13160	1	1	yes	characteristic, complemented, Sylow
170	C₂³	C₂²×#1055	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	C₄⋊#1055	1	2	yes	—
179	C₂×C₄	C₄⋊#1055	1	2	yes	—
180	C₂×C₄	#88052	1	4	yes	—
181	C₂×C₄	C₂×#6099	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
183	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
184	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
185	C₂×C₄	#88202	1	1	yes	characteristic
186	C₂×C₄	#88202	1	1	yes	characteristic
187	C₈	—	2	2	no	—
188	C₈	—	4	2	no	—
189	C₈	#88142	1	1	yes	characteristic
190	C₈	#88144	1	1	yes	characteristic
191	C₈	#88149	1	1	yes	characteristic
192	C₈	#88151	1	1	yes	characteristic
193	C₂²	C₂×C₄⋊#1055	1	1	yes	characteristic
194	C₂²	D₄×#1055	1	1	yes	characteristic
195	C₂²	Q₈×#1055	1	1	yes	characteristic
196	C₂²	C₂×#88052	1	2	yes	—
197	C₂²	#11718801	1	2	yes	—
198	C₄	—	2	2	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	4	no	—
201	C₄	—	4	2	no	—
202	C₄	#11718646	1	1	yes	characteristic
203	C₄	#11718660	1	1	yes	characteristic
204	C₄	#11718673	1	1	yes	characteristic
205	C₄	#11718685	1	1	yes	characteristic
206	C₂	#13115555	1	1	yes	characteristic
207	C₂	#13116592	1	1	yes	characteristic
208	C₂	#13116595	1	1	yes	characteristic
209	C₂	#13116662	1	2	yes	—
210	C₂	#13116745	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator, socle, prime radical	C₂³×C₁₃	C₂²×C₈
Fitting	C₈×C₁₃×C₄⋊C₄	C₂
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₈×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#13160
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
169. C₁₃	#13160	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6090	2
C₂×#11718053	2
#13112914	2
C₈×C₁₃×C₄⋊C₄	2
#13160	13