Subgroups — #15779360

Order 3328.

342 conjugacy classes of subgroups.

The subgroup lattice has 158 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718097	C₂	1	2	yes	maximal
2	C₂×#11718096	C₂	1	2	yes	maximal
3	#13115000	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88058	C₂²	1	4	yes	—
9	C₂×#88057	—	2	2	no	—
10	#11718597	—	2	2	no	—
11	C₄×C₁₃×C₁₆	C₄	1	4	yes	complemented
12	#11718097	—	2	4	no	—
13	#11718096	—	2	4	no	—
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₃	—	2	4	no	—
18	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
21	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
22	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
23	C₂×C₁₃×C₁₆	Q₈	1	2	yes	—
24	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₃	Q₈	1	2	yes	—
26	#88071	—	4	4	no	—
27	#88058	—	2	4	no	—
28	#88058	—	2	4	no	—
29	#88057	—	4	4	no	—
30	#12907	—	13	1	no	maximal, Sylow
31	C₂²×C₄×C₁₃	D₈	1	2	yes	—
32	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₁₃	—	4	4	no	—
34	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₁₃	D₈	1	2	yes	—
36	C₂×C₈×C₁₃	Q₁₆	1	2	yes	—
37	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₁₃	Q₁₆	1	2	yes	—
45	C₄⋊Dic₁₃	—	4	4	no	—
46	C₄⋊Dic₁₃	—	4	4	no	—
47	C₁₃×C₁₆	C₄⋊C₄	1	4	yes	—
48	C₁₃×C₁₆	C₄⋊C₄	1	4	yes	—
49	C₂×#367	—	13	2	no	—
50	C₂×#366	—	13	2	no	—
51	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
52	#1846	—	13	2	no	—
53	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃	—	2	4	no	—
55	C₂×C₄×C₁₃	#107	1	4	yes	—
56	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×Dic₁₃	—	8	4	no	—
63	C₂×Dic₁₃	—	8	8	no	—
64	C₈×C₁₃	#107	1	4	yes	—
65	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
69	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
71	C₂×#107	—	13	4	no	—
72	C₂×#106	—	26	2	no	—
73	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
74	#382	—	26	2	no	—
75	#367	—	26	4	no	—
76	#366	—	26	4	no	—
77	C₄×C₁₆	Dic₁₃	1	4	yes	complemented
78	C₂²×C₁₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₁₃	#493	1	1	yes	characteristic
80	C₂²×C₁₃	#500	1	1	yes	characteristic
81	C₂²×C₁₃	#508	1	4	yes	—
82	C₄×C₁₃	—	2	4	no	—
83	C₄×C₁₃	#384	1	1	yes	characteristic
84	C₄×C₁₃	C₂×#107	1	1	yes	characteristic
85	C₄×C₁₃	#494	1	1	yes	characteristic
86	C₄×C₁₃	#500	1	1	yes	characteristic
87	Dic₁₃	—	16	4	no	—
88	C₂²×C₈	D₅₂	1	1	yes	characteristic
89	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	26	4	no	—
91	C₂×C₄²	D₅₂	1	1	yes	characteristic
92	C₂×C₁₆	Dic₂₆	1	2	yes	—
93	C₂×C₁₆	Dic₂₆	1	2	yes	—
94	C₂×C₁₆	D₅₂	1	2	yes	—
95	C₂×C₁₆	D₅₂	1	2	yes	—
96	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
97	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
98	#107	—	26	4	no	—
99	#107	—	26	4	no	—
100	#106	—	52	4	no	—
101	C₄×C₈	Dic₂₆	1	2	yes	—
102	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
103	#95	—	52	4	no	—
104	C₂×C₁₃	#1845	1	1	yes	characteristic
105	C₂×C₁₃	#2158	1	2	yes	—
106	C₂×C₁₃	#2249	1	2	yes	—
107	C₂×C₁₃	#2253	1	2	yes	—
108	C₂²×C₄	—	52	4	no	—
109	C₂²×C₄	D₁₀₄	1	2	yes	—
110	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
111	C₂×C₈	D₁₀₄	1	2	yes	—
112	C₂×C₈	Dic₅₂	1	2	yes	—
113	C₂×C₈	C₄⋊Dic₁₃	1	2	yes	—
114	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
115	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
117	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
118	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
119	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
120	C₄⋊C₄	—	52	4	no	—
121	C₄⋊C₄	—	52	4	no	—
122	C₄²	Dic₅₂	1	2	yes	—
123	C₄²	C₄⋊Dic₁₃	1	2	yes	—
124	C₁₆	C₄⋊Dic₁₃	1	4	yes	—
125	C₁₆	C₄⋊Dic₁₃	1	4	yes	—
126	C₁₃	#12907	1	1	yes	characteristic, complemented, Sylow
127	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	104	8	no	—
130	C₂×C₄	—	104	4	no	—
131	C₂×C₄	#88058	1	4	yes	—
132	C₂×C₄	#88123	1	1	yes	characteristic
133	C₂×C₄	#88128	1	1	yes	characteristic
134	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₅₂	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
138	C₈	#88058	1	4	yes	—
139	C₈	#88123	1	1	yes	characteristic
140	C₈	#88123	1	1	yes	characteristic
141	C₈	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
142	C₈	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
143	C₂²	#11718343	1	1	yes	characteristic
144	C₂²	#11718356	1	1	yes	characteristic
145	C₂²	#11718565	1	4	yes	—
146	C₂²	C₂×#88058	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	208	4	no	—
149	C₄	#11718343	1	1	yes	characteristic
150	C₄	#11718360	1	1	yes	characteristic
151	C₄	#11718596	1	1	yes	characteristic
152	C₄	C₂×#88058	1	1	yes	characteristic
153	C₂	#13114996	1	1	yes	characteristic
154	C₂	#13116620	1	2	yes	—
155	C₂	#13117097	1	2	yes	—
156	C₂	#13117115	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂×D₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#12907
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₁₃×C₁₆	C₄	semidirect
77. C₄×C₁₆	Dic₁₃	semidirect
126. 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×#11718097	2
C₂×#11718096	2
#13115000	2
C₂×C₄×C₁₃×C₁₆	2
#12907	13