Subgroups — #15779358

Order 3328.

342 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718082	C₂	1	2	yes	maximal
2	C₂×#11718081	C₂	1	2	yes	maximal
3	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
4	#13115000	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#107	C₂²	1	2	yes	—
7	C₂×C₁₃×#106	—	2	1	no	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₁₃×#382	—	2	1	no	—
10	C₂×#88058	C₂²	1	2	yes	—
11	C₂×#88057	—	2	1	no	—
12	C₂²×#6090	C₂²	1	1	yes	characteristic
13	C₂²×#6090	C₂²	1	1	yes	characteristic
14	#11718597	—	2	1	no	—
15	#11718082	—	2	4	no	—
16	#11718081	—	2	4	no	—
17	C₄×#6090	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₁₃	—	2	2	no	—
23	C₁₃×#107	—	2	2	no	—
24	C₁₃×#107	—	2	2	no	—
25	C₁₃×#106	—	4	2	no	—
26	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₁₃	Q₈	1	2	yes	—
28	C₁₃×#95	—	4	2	no	—
29	#88071	—	4	2	no	—
30	#88058	—	2	2	no	—
31	#88058	—	2	2	no	—
32	#88057	—	4	2	no	—
33	C₂×#6090	C₂×C₄	1	2	yes	—
34	C₂×#6090	C₂×C₄	1	2	yes	—
35	C₂×#6090	D₄	1	2	yes	—
36	C₂×#6090	D₄	1	2	yes	—
37	C₂×#6090	Q₈	1	2	yes	—
38	C₂×#6090	Q₈	1	2	yes	—
39	#12907	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	—	4	2	no	—
41	C₂²×C₄×C₁₃	D₈	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	—	4	2	no	—
44	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₁₃	D₈	1	2	yes	—
46	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
47	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	—	4	2	no	—
54	C₁₃×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₄²×C₁₃	Q₁₆	1	2	yes	—
57	C₄⋊Dic₁₃	—	4	2	no	—
58	C₄⋊Dic₁₃	—	4	2	no	—
59	#6090	C₄⋊C₄	1	4	yes	—
60	#6090	C₄⋊C₄	1	4	yes	—
61	C₂×#367	—	13	2	no	—
62	C₂×#366	—	13	2	no	—
63	C₂×C₄×C₁₆	—	13	1	no	—
64	#1846	—	13	1	no	—
65	#1846	D₁₃	1	1	yes	characteristic
66	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₁₃	—	2	4	no	—
68	C₂×C₄×C₁₃	—	8	2	no	—
69	C₂×C₄×C₁₃	—	8	4	no	—
70	C₂×C₄×C₁₃	#107	1	4	yes	—
71	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×Dic₁₃	—	8	2	no	—
78	C₂×Dic₁₃	—	8	4	no	—
79	C₈×C₁₃	#107	1	4	yes	—
80	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
83	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
84	C₂²×C₁₆	—	13	1	no	—
85	C₂²×C₁₆	—	13	1	no	—
86	C₂×#107	—	13	2	no	—
87	C₂×#107	C₂×D₁₃	1	2	yes	—
88	C₂×#106	—	2	1	no	—
89	C₂×#106	—	26	1	no	—
90	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
91	#382	—	2	1	no	—
92	#382	—	26	1	no	—
93	#367	—	26	4	no	—
94	#366	—	26	4	no	—
95	C₄×C₁₆	—	13	4	no	—
96	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
97	C₂²×C₁₃	#493	1	1	yes	characteristic
98	C₂²×C₁₃	#500	1	1	yes	characteristic
99	C₂²×C₁₃	#508	1	4	yes	—
100	C₄×C₁₃	—	2	4	no	—
101	C₄×C₁₃	—	16	2	no	—
102	C₄×C₁₃	#384	1	1	yes	characteristic
103	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
104	C₄×C₁₃	#494	1	1	yes	characteristic
105	C₄×C₁₃	#500	1	1	yes	characteristic
106	Dic₁₃	—	16	2	no	—
107	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
108	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	26	2	no	—
111	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
112	C₂×C₁₆	—	13	2	no	—
113	C₂×C₁₆	—	13	2	no	—
114	C₂×C₁₆	—	13	2	no	—
115	C₂×C₁₆	—	13	2	no	—
116	C₂×C₁₆	—	13	2	no	—
117	C₂×C₁₆	—	13	2	no	—
118	#107	—	2	2	no	—
119	#107	—	2	2	no	—
120	#107	—	26	2	no	—
121	#107	—	26	2	no	—
122	#106	—	4	2	no	—
123	#106	—	52	2	no	—
124	C₄×C₈	Dic₂₆	1	2	yes	—
125	C₄×C₈	C₄×D₁₃	1	2	yes	—
126	#95	—	4	2	no	—
127	#95	—	52	2	no	—
128	C₂×C₁₃	#1845	1	1	yes	characteristic
129	C₂×C₁₃	#2158	1	2	yes	—
130	C₂×C₁₃	#2249	1	2	yes	—
131	C₂×C₁₃	#2253	1	2	yes	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	—	52	2	no	—
134	C₂²×C₄	#6104	1	2	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
136	C₂×C₈	#6101	1	2	yes	—
137	C₂×C₈	#6104	1	2	yes	—
138	C₂×C₈	#6107	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	D₄×D₁₃	1	1	yes	characteristic
142	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
143	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
144	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	52	2	no	—
148	C₄⋊C₄	—	52	2	no	—
149	C₄²	#6101	1	2	yes	—
150	C₄²	#6107	1	2	yes	—
151	C₁₆	—	13	4	no	—
152	C₁₆	—	13	4	no	—
153	C₁₃	#12907	1	1	yes	characteristic, complemented, Sylow
154	C₂³	C₂×#6104	1	1	yes	characteristic
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	104	2	no	—
159	C₂×C₄	—	104	4	no	—
160	C₂×C₄	#88047	1	4	yes	—
161	C₂×C₄	C₂×#6101	1	1	yes	characteristic
162	C₂×C₄	C₂×#6104	1	1	yes	characteristic
163	C₂×C₄	#88194	1	1	yes	characteristic
164	C₂×C₄	C₂×#6107	1	1	yes	characteristic
165	C₂×C₄	C₂×#6107	1	1	yes	characteristic
166	C₂×C₄	#88198	1	1	yes	characteristic
167	C₈	#88047	1	4	yes	—
168	C₈	#88142	1	1	yes	characteristic
169	C₈	#88142	1	1	yes	characteristic
170	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
171	C₈	C₄⋊C₄×D₁₃	1	1	yes	characteristic
172	C₂²	C₂×#88047	1	1	yes	characteristic
173	C₂²	#11718734	1	1	yes	characteristic
174	C₂²	#11718739	1	1	yes	characteristic
175	C₂²	#11718855	1	4	yes	—
176	C₄	—	2	4	no	—
177	C₄	—	16	2	no	—
178	C₄	—	208	2	no	—
179	C₄	C₂×#88047	1	1	yes	characteristic
180	C₄	#11718637	1	1	yes	characteristic
181	C₄	#11718739	1	1	yes	characteristic
182	C₄	#11718753	1	1	yes	characteristic
183	C₂	#13114995	1	1	yes	characteristic
184	C₂	#13116618	1	2	yes	—
185	C₂	#13117095	1	2	yes	—
186	C₂	#13117112	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#6104
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₁₃×#1846	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1846	D₁₃
hypocenter	C₁₃	#12907
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

Complements

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

N	H ≅ G/N	type
17. C₄×#6090	C₄	semidirect
153. C₁₃	#12907	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718082	2
C₂×#11718081	2
C₂×C₄×#6090	2
#13115000	2
C₁₃×#1846	2
#12907	13