Subgroups — #15779298

Order 3328.

342 conjugacy classes of subgroups.

The subgroup lattice has 200 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	1	yes	maximal, characteristic
2	#13114996	C₂	1	1	yes	maximal, characteristic
3	#13112927	C₂	1	4	yes	maximal
4	C₁₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₃×#107	—	2	1	no	—
6	C₂×C₁₃×#107	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#384	—	2	1	no	—
9	C₂×#88058	—	2	1	no	—
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂²×#6090	C₂²	1	2	yes	—
12	#11718596	—	2	1	no	—
13	C₄×#6090	C₄	1	4	yes	complemented
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
17	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
18	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×C₄⋊Dic₁₃	—	4	1	no	—
22	C₂×C₄⋊Dic₁₃	—	4	1	no	—
23	C₁₃×#107	—	2	4	no	—
24	C₁₃×#107	—	4	2	no	—
25	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₁₃	Q₈	1	2	yes	—
27	#88058	—	2	4	no	—
28	#88058	—	4	2	no	—
29	C₂×#6090	—	2	2	no	—
30	C₂×#6090	—	2	2	no	—
31	C₂×#6090	C₂×C₄	1	4	yes	—
32	#12888	—	13	1	no	maximal, Sylow
33	C₂²×C₄×C₁₃	—	4	2	no	—
34	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₁₃	—	4	2	no	—
38	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
39	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
40	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈×C₁₃	#41	1	2	yes	—
44	C₂×C₈×C₁₃	#41	1	2	yes	—
45	C₁₃×C₄⋊C₄	—	4	2	no	—
46	C₁₃×C₄⋊C₄	—	4	2	no	—
47	C₁₃×C₄⋊C₄	—	8	2	no	—
48	C₁₃×C₄⋊C₄	—	8	2	no	—
49	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
50	C₄²×C₁₃	Q₁₆	1	2	yes	—
51	C₄⋊Dic₁₃	—	4	2	no	—
52	C₄⋊Dic₁₃	—	4	2	no	—
53	C₄⋊Dic₁₃	—	8	2	no	—
54	C₄⋊Dic₁₃	—	8	2	no	—
55	#6090	—	2	4	no	—
56	C₂×C₄×C₁₆	—	13	1	no	—
57	#1845	—	13	1	no	—
58	#1845	D₁₃	1	1	yes	characteristic
59	#1380	—	13	4	no	—
60	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	—	8	2	no	—
62	C₂×C₄×C₁₃	—	8	2	no	—
63	C₂×C₄×C₁₃	—	8	2	no	—
64	C₂×C₄×C₁₃	#107	1	2	yes	—
65	C₂×C₄×C₁₃	#107	1	2	yes	—
66	C₂×C₄×C₁₃	D₁₆	1	2	yes	—
67	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
68	C₂×C₄×C₁₃	SD₃₂	1	2	yes	—
69	C₂×C₄×C₁₃	Q₃₂	1	2	yes	—
70	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×Dic₁₃	—	8	2	no	—
77	C₂×Dic₁₃	—	8	2	no	—
78	C₂×Dic₁₃	—	8	2	no	—
79	C₈×C₁₃	#106	1	4	yes	—
80	C₈×C₁₃	#117	1	2	yes	—
81	C₈×C₁₃	#125	1	2	yes	—
82	C₂²×C₁₆	—	13	2	no	—
83	C₂×#107	—	2	1	no	—
84	C₂×#107	—	13	2	no	—
85	C₂×#107	—	26	1	no	—
86	C₂×#107	C₂×D₁₃	1	2	yes	—
87	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
88	#384	—	2	1	no	—
89	#384	—	26	1	no	—
90	C₄×C₁₆	—	13	4	no	—
91	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
92	C₂²×C₁₃	#486	1	1	yes	characteristic
93	C₂²×C₁₃	#486	1	1	yes	characteristic
94	C₂²×C₁₃	C₂×D₁₆	1	1	yes	characteristic
95	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
96	C₂²×C₁₃	C₂×SD₃₂	1	1	yes	characteristic
97	C₂²×C₁₃	C₂×Q₃₂	1	1	yes	characteristic
98	C₄×C₁₃	—	16	2	no	—
99	C₄×C₁₃	#366	1	4	yes	—
100	C₄×C₁₃	#367	1	4	yes	—
101	C₄×C₁₃	#382	1	1	yes	characteristic
102	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
103	C₄×C₁₃	#487	1	1	yes	characteristic
104	C₄×C₁₃	#487	1	1	yes	characteristic
105	Dic₁₃	—	16	2	no	—
106	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
107	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	4	1	no	—
110	C₂×C₄⋊C₄	—	4	1	no	—
111	C₂×C₄⋊C₄	—	26	2	no	—
112	C₂×C₄⋊C₄	—	52	1	no	—
113	C₂×C₄⋊C₄	—	52	1	no	—
114	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
115	C₂×C₁₆	—	13	4	no	—
116	C₂×C₁₆	—	26	2	no	—
117	C₂×C₁₆	—	26	2	no	—
118	#107	—	2	4	no	—
119	#107	—	4	2	no	—
120	#107	—	26	4	no	—
121	#107	—	52	2	no	—
122	C₄×C₈	Dic₂₆	1	2	yes	—
123	C₄×C₈	C₄×D₁₃	1	2	yes	—
124	C₂×C₁₃	#1834	1	1	yes	characteristic
125	C₂×C₁₃	C₂×#366	1	1	yes	characteristic
126	C₂×C₁₃	C₂×#367	1	1	yes	characteristic
127	C₂×C₁₃	#2240	1	1	yes	characteristic
128	C₂×C₁₃	#2240	1	1	yes	characteristic
129	C₂×C₁₃	#2241	1	1	yes	characteristic
130	C₂×C₁₃	#2241	1	1	yes	characteristic
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	52	2	no	—
133	C₂²×C₄	#6104	1	1	yes	characteristic
134	C₂²×C₄	#6104	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
136	C₂×C₈	#6101	1	2	yes	—
137	C₂×C₈	#6105	1	2	yes	—
138	C₂×C₈	#6106	1	2	yes	—
139	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
140	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
141	C₂×C₈	#6129	1	2	yes	—
142	C₂×C₈	#6131	1	2	yes	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	8	2	no	—
146	C₄⋊C₄	—	8	2	no	—
147	C₄⋊C₄	—	52	2	no	—
148	C₄⋊C₄	—	52	2	no	—
149	C₄⋊C₄	—	104	2	no	—
150	C₄⋊C₄	—	104	2	no	—
151	C₄²	#6101	1	2	yes	—
152	C₄²	#6107	1	2	yes	—
153	C₁₆	—	26	4	no	—
154	C₁₃	#12888	1	1	yes	characteristic, complemented, Sylow
155	C₂³	C₂×#6104	1	1	yes	characteristic
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	2	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	—	104	2	no	—
160	C₂×C₄	—	104	2	no	—
161	C₂×C₄	—	104	2	no	—
162	C₂×C₄	#88047	1	2	yes	—
163	C₂×C₄	#88047	1	2	yes	—
164	C₂×C₄	#88066	1	2	yes	—
165	C₂×C₄	#88067	1	2	yes	—
166	C₂×C₄	#88068	1	2	yes	—
167	C₂×C₄	#88069	1	2	yes	—
168	C₂×C₄	C₂×#6101	1	1	yes	characteristic
169	C₂×C₄	C₂×#6105	1	1	yes	characteristic
170	C₂×C₄	#88190	1	1	yes	characteristic
171	C₂×C₄	C₂×#6106	1	1	yes	characteristic
172	C₂×C₄	C₂×#6107	1	1	yes	characteristic
173	C₂×C₄	#88201	1	1	yes	characteristic
174	C₈	#88048	1	4	yes	—
175	C₈	#88144	1	2	yes	—
176	C₈	#88146	1	2	yes	—
177	C₂²	C₂×#88047	1	1	yes	characteristic
178	C₂²	#11718731	1	1	yes	characteristic
179	C₂²	#11718749	1	1	yes	characteristic
180	C₂²	C₂×#88066	1	1	yes	characteristic
181	C₂²	C₂×#88067	1	1	yes	characteristic
182	C₂²	C₂×#88068	1	1	yes	characteristic
183	C₂²	C₂×#88069	1	1	yes	characteristic
184	C₄	—	16	2	no	—
185	C₄	—	208	2	no	—
186	C₄	#11718081	1	4	yes	—
187	C₄	#11718082	1	4	yes	—
188	C₄	C₂×#88048	1	1	yes	characteristic
189	C₄	#11718638	1	1	yes	characteristic
190	C₄	#11718742	1	1	yes	characteristic
191	C₄	#11718754	1	1	yes	characteristic
192	C₂	#13114963	1	1	yes	characteristic
193	C₂	C₂×#11718081	1	1	yes	characteristic
194	C₂	C₂×#11718082	1	1	yes	characteristic
195	C₂	#13117059	1	1	yes	characteristic
196	C₂	#13117061	1	1	yes	characteristic
197	C₂	#13117063	1	1	yes	characteristic
198	C₂	#13117065	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1845	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1845	D₁₃
hypocenter	C₁₃	#12888
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

Complements

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

N	H ≅ G/N	type
13. C₄×#6090	C₄	semidirect
154. C₁₃	#12888	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
#13114996	2
#13112927	2
C₁₃×#1845	2
#12888	13