Subgroups — #15779352

Order 3328.

374 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718081	C₂	1	4	yes	maximal
2	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic
3	#13114996	C₂	1	1	yes	maximal, characteristic
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	#11718081	—	2	8	no	—
14	C₄×#6090	C₄	1	4	yes	complemented
15	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
18	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
19	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₁₃	—	2	2	no	—
21	C₂×C₄⋊Dic₁₃	—	4	2	no	—
22	C₁₃×#107	—	2	4	no	—
23	C₁₃×#107	—	4	2	no	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₃	Q₈	1	2	yes	—
26	#88058	—	2	4	no	—
27	#88058	—	4	2	no	—
28	C₂×#6090	C₂×C₄	1	4	yes	—
29	C₂×#6090	D₄	1	4	yes	—
30	C₂×#6090	Q₈	1	4	yes	—
31	#12905	—	13	1	no	maximal, Sylow
32	C₂²×C₄×C₁₃	—	4	2	no	—
33	C₂²×C₄×C₁₃	D₈	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×Dic₁₃	—	4	2	no	—
36	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₁₃	D₈	1	2	yes	—
38	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
39	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
41	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₁₃×C₄⋊C₄	—	4	2	no	—
45	C₁₃×C₄⋊C₄	—	8	4	no	—
46	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₁₃	Q₁₆	1	2	yes	—
48	C₄⋊Dic₁₃	—	4	2	no	—
49	C₄⋊Dic₁₃	—	4	2	no	—
50	C₄⋊Dic₁₃	—	8	4	no	—
51	#6090	C₄⋊C₄	1	8	yes	complemented
52	C₂×#366	—	13	4	no	—
53	C₂×C₄×C₁₆	—	13	1	no	—
54	#1845	—	13	1	no	—
55	#1845	D₁₃	1	1	yes	characteristic
56	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	—	8	2	no	—
58	C₂×C₄×C₁₃	—	8	4	no	—
59	C₂×C₄×C₁₃	#107	1	4	yes	—
60	C₂×C₄×C₁₃	D₁₆	1	4	yes	—
61	C₂×C₄×C₁₃	Q₃₂	1	4	yes	—
62	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×Dic₁₃	—	8	2	no	—
69	C₂×Dic₁₃	—	8	4	no	—
70	C₈×C₁₃	#107	1	4	yes	—
71	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
72	C₈×C₁₃	C₄⋊Q₈	1	2	yes	—
73	C₂²×C₁₆	—	13	2	no	—
74	C₂×#107	—	2	1	no	—
75	C₂×#107	—	13	2	no	—
76	C₂×#107	—	26	1	no	—
77	C₂×#107	C₂×D₁₃	1	2	yes	—
78	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
79	#384	—	2	1	no	—
80	#384	—	26	1	no	—
81	#366	—	26	8	no	—
82	C₄×C₁₆	—	13	4	no	—
83	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
84	C₂²×C₁₃	#493	1	1	yes	characteristic
85	C₂²×C₁₃	#500	1	1	yes	characteristic
86	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
87	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
88	C₄×C₁₃	—	16	2	no	—
89	C₄×C₁₃	#366	1	8	yes	complemented
90	C₄×C₁₃	#384	1	1	yes	characteristic
91	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
92	C₄×C₁₃	#494	1	1	yes	characteristic
93	C₄×C₁₃	#500	1	1	yes	characteristic
94	Dic₁₃	—	16	2	no	—
95	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	—	4	2	no	—
99	C₂×C₄⋊C₄	—	26	2	no	—
100	C₂×C₄⋊C₄	—	52	2	no	—
101	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
102	C₂×C₁₆	—	13	4	no	—
103	C₂×C₁₆	—	13	4	no	—
104	C₂×C₁₆	—	13	4	no	—
105	#107	—	2	4	no	—
106	#107	—	4	2	no	—
107	#107	—	26	4	no	—
108	#107	—	52	2	no	—
109	C₄×C₈	Dic₂₆	1	2	yes	—
110	C₄×C₈	C₄×D₁₃	1	2	yes	—
111	C₂×C₁₃	#1845	1	1	yes	characteristic
112	C₂×C₁₃	C₂×#366	1	2	yes	—
113	C₂×C₁₃	#2246	1	1	yes	characteristic
114	C₂×C₁₃	#2247	1	1	yes	characteristic
115	C₂×C₁₃	#2252	1	2	yes	—
116	C₂²×C₄	—	4	2	no	—
117	C₂²×C₄	—	52	2	no	—
118	C₂²×C₄	#6104	1	2	yes	—
119	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
120	C₂×C₈	#6101	1	2	yes	—
121	C₂×C₈	#6104	1	2	yes	—
122	C₂×C₈	#6107	1	2	yes	—
123	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
124	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
125	C₂×C₈	D₄×D₁₃	1	2	yes	—
126	C₂×C₈	Q₈×D₁₃	1	2	yes	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄⋊C₄	—	8	4	no	—
130	C₄⋊C₄	—	52	2	no	—
131	C₄⋊C₄	—	52	2	no	—
132	C₄⋊C₄	—	104	4	no	—
133	C₄²	#6101	1	2	yes	—
134	C₄²	#6107	1	2	yes	—
135	C₁₆	—	13	8	no	—
136	C₁₃	#12905	1	1	yes	characteristic, complemented, Sylow
137	C₂³	C₂×#6104	1	1	yes	characteristic
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	4	no	—
140	C₂×C₄	—	104	2	no	—
141	C₂×C₄	—	104	4	no	—
142	C₂×C₄	#88047	1	4	yes	—
143	C₂×C₄	#88066	1	4	yes	—
144	C₂×C₄	#88069	1	4	yes	—
145	C₂×C₄	C₂×#6101	1	1	yes	characteristic
146	C₂×C₄	C₂×#6104	1	1	yes	characteristic
147	C₂×C₄	#88194	1	1	yes	characteristic
148	C₂×C₄	C₂×#6107	1	1	yes	characteristic
149	C₂×C₄	C₂×#6107	1	1	yes	characteristic
150	C₂×C₄	#88198	1	1	yes	characteristic
151	C₈	#88047	1	4	yes	—
152	C₈	#88142	1	2	yes	—
153	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
154	C₂²	C₂×#88047	1	1	yes	characteristic
155	C₂²	#11718734	1	1	yes	characteristic
156	C₂²	#11718739	1	1	yes	characteristic
157	C₂²	C₂×#88066	1	2	yes	—
158	C₂²	C₂×#88069	1	2	yes	—
159	C₄	—	16	2	no	—
160	C₄	—	208	2	no	—
161	C₄	#11718081	1	8	yes	complemented
162	C₄	C₂×#88047	1	1	yes	characteristic
163	C₄	#11718637	1	1	yes	characteristic
164	C₄	#11718739	1	1	yes	characteristic
165	C₄	#11718753	1	1	yes	characteristic
166	C₂	#13114995	1	1	yes	characteristic
167	C₂	C₂×#11718081	1	2	yes	—
168	C₂	#13117085	1	1	yes	characteristic
169	C₂	#13117088	1	1	yes	characteristic
170	C₂	#13117109	1	2	yes	—

	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	#12905
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
14. C₄×#6090	C₄	semidirect
51. #6090	C₄⋊C₄	semidirect
89. C₄×C₁₃	#366	semidirect
136. C₁₃	#12905	semidirect
161. C₄	#11718081	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718081	2
C₂×C₄×#6090	2
#13114996	2
C₁₃×#1845	2
#12905	13