Subgroups — #15754160

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13114705	C₂	1	2	yes	maximal
2	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#1055	C₂	1	1	yes	maximal, characteristic
4	#13112677	C₂	1	2	yes	maximal
5	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#106	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊#1055	C₂²	1	2	yes	—
10	C₂×#88043	—	2	2	no	—
11	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	C₁₃×#340	C₄	1	2	yes	—
14	#11718116	—	2	2	no	—
15	#11718049	C₄	1	4	yes	complemented
16	C₈×#1055	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₁₃	—	2	2	no	—
21	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×#1055	—	2	1	no	—
23	C₂²×#1055	—	2	2	no	—
24	C₂²×#1055	D₄	1	1	yes	characteristic
25	C₂²×#1055	D₄	1	1	yes	characteristic
26	C₁₃×#106	—	2	4	no	—
27	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
29	C₁₃×#95	—	2	2	no	—
30	C₄⋊#1055	—	2	2	no	—
31	C₄⋊#1055	C₂×C₄	1	4	yes	—
32	#88043	—	4	4	no	—
33	C₄×#1055	—	2	1	no	—
34	C₄×#1055	C₂×C₄	1	2	yes	—
35	C₄×#1055	C₂×C₄	1	2	yes	—
36	C₄×#1055	Q₈	1	2	yes	—
37	#8283	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	—	2	4	no	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	C₄²	1	2	yes	—
44	C₂×C₈×C₁₃	C₄²	1	2	yes	—
45	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
46	C₁₃×C₄⋊C₄	—	2	4	no	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₄²×C₁₃	—	4	2	no	—
49	C₄²×C₁₃	—	4	2	no	—
50	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₁₃	#41	1	1	yes	characteristic
54	C₂×#1055	—	2	4	no	—
55	C₂×#1055	—	2	2	no	—
56	C₂×#1055	—	2	2	no	—
57	C₂×#1055	—	2	4	no	—
58	C₂×#1055	—	4	2	no	—
59	C₂×#1055	—	4	2	no	—
60	C₂×#1055	C₄⋊C₄	1	2	yes	—
61	C₂×#1055	C₄⋊C₄	1	2	yes	—
62	C₂×#1055	SD₁₆	1	4	yes	—
63	C₄×#106	D₁₃	1	1	yes	characteristic
64	#1770	—	13	2	no	—
65	C₂×#334	—	13	1	no	—
66	C₂×C₈²	—	13	1	no	—
67	#1279	—	13	2	no	—
68	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	2	no	—
70	C₂×C₄×C₁₃	—	4	4	no	—
71	C₂×C₄×C₁₃	—	4	4	no	—
72	C₂×C₄×C₁₃	—	4	4	no	—
73	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
80	C₂×C₄×C₁₃	#131	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	#135	1	2	yes	—
83	C₈×C₁₃	—	2	2	no	—
84	C₈×C₁₃	#97	1	4	yes	complemented
85	#1055	—	2	4	no	—
86	#1055	—	2	2	no	—
87	#1055	—	4	4	no	—
88	#1055	#106	1	4	yes	complemented
89	C₂×#106	Dic₁₃	1	2	yes	—
90	C₂×C₄⋊C₈	—	13	2	no	—
91	C₂×#97	—	26	2	no	—
92	C₂×C₄×C₈	—	13	1	no	—
93	C₂×C₄×C₈	—	13	1	no	—
94	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
95	C₄×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
96	#340	Dic₁₃	1	2	yes	—
97	#336	—	26	2	no	—
98	#334	—	13	4	no	—
99	C₈²	—	13	4	no	—
100	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₃	#436	1	2	yes	—
102	C₂²×C₁₃	C₄×SD₁₆	1	2	yes	—
103	C₂²×C₁₃	#443	1	2	yes	—
104	C₄×C₁₃	—	2	1	no	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	8	4	no	—
107	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
108	C₄×C₁₃	#405	1	1	yes	characteristic
109	C₄×C₁₃	C₂×#106	1	1	yes	characteristic
110	C₄×C₁₃	#427	1	1	yes	characteristic
111	C₄×C₁₃	C₄×SD₁₆	1	2	yes	—
112	C₄×C₁₃	#446	1	2	yes	—
113	C₂²×C₈	—	13	1	no	—
114	C₂²×C₈	—	13	1	no	—
115	C₂²×C₈	—	26	1	no	—
116	C₂²×C₈	—	26	2	no	—
117	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
120	C₂×C₄²	—	2	2	no	—
121	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	#106	—	2	4	no	—
123	C₄⋊C₈	—	13	4	no	—
124	C₄⋊C₈	—	26	2	no	—
125	#97	—	52	4	no	—
126	C₄×C₈	—	13	2	no	—
127	C₄×C₈	—	13	2	no	—
128	C₄×C₈	—	13	2	no	—
129	C₄×C₈	—	26	1	no	—
130	C₄×C₈	C₄×D₁₃	1	2	yes	—
131	C₄×C₈	C₄×D₁₃	1	2	yes	—
132	#95	—	2	2	no	—
133	C₂×C₁₃	#1590	1	4	yes	—
134	C₂×C₁₃	#1770	1	1	yes	characteristic
135	C₂×C₁₃	#1772	1	1	yes	characteristic
136	C₂×C₁₃	C₄×#106	1	1	yes	characteristic
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
139	C₂²×C₄	#6129	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	13	2	no	—
143	C₂×C₈	—	13	4	no	—
144	C₂×C₈	—	13	2	no	—
145	C₂×C₈	—	26	4	no	—
146	C₂×C₈	—	26	2	no	—
147	C₂×C₈	—	26	2	no	—
148	C₂×C₈	—	26	4	no	—
149	C₂×C₈	—	52	2	no	—
150	C₂×C₈	—	52	2	no	—
151	C₂×C₈	#6099	1	4	yes	—
152	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
153	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	2	4	no	—
156	C₄²	—	4	2	no	—
157	C₄²	—	4	2	no	—
158	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
159	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
160	C₄²	Q₈×D₁₃	1	1	yes	characteristic
161	C₄²	#6131	1	1	yes	characteristic
162	C₁₃	#8283	1	1	yes	characteristic, complemented, Sylow
163	C₂³	D₄×Dic₁₃	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	—	4	4	no	—
168	C₂×C₄	#88141	1	1	yes	characteristic
169	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
170	C₂×C₄	#88146	1	1	yes	characteristic
171	C₂×C₄	#88147	1	1	yes	characteristic
172	C₂×C₄	SD₁₆×D₁₃	1	2	yes	—
173	C₂×C₄	#88170	1	2	yes	—
174	C₂×C₄	C₂×#6099	1	2	yes	—
175	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
176	C₂×C₄	Q₈×Dic₁₃	1	1	yes	characteristic
177	C₂×C₄	#88202	1	2	yes	—
178	C₈	—	2	2	no	—
179	C₈	—	13	4	no	—
180	C₈	—	26	4	no	—
181	C₈	—	26	2	no	—
182	C₈	—	52	4	no	—
183	C₈	#88043	1	4	yes	complemented
184	C₂²	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
185	C₂²	#11718673	1	2	yes	—
186	C₂²	SD₁₆×Dic₁₃	1	2	yes	—
187	C₂²	#11718857	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	8	4	no	—
191	C₄	#11718513	1	2	yes	—
192	C₄	D₁₃×#106	1	1	yes	characteristic
193	C₄	#11718521	1	1	yes	characteristic
194	C₄	C₂×#88043	1	1	yes	characteristic
195	C₄	#11718658	1	1	yes	characteristic
196	C₄	#11718685	1	2	yes	—
197	C₂	#13113727	1	4	yes	—
198	C₂	#13114705	1	1	yes	characteristic
199	C₂	#13114717	1	1	yes	characteristic
200	C₂	#106×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#106	C₂
socle, prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#106	D₁₃
hypocenter	C₁₃	#8283
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
15. #11718049	C₄	semidirect
16. C₈×#1055	C₄	semidirect
84. C₈×C₁₃	#97	semidirect
88. #1055	#106	semidirect
162. C₁₃	#8283	semidirect
183. C₈	#88043	semidirect

Maximal subgroups

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

subgroup	index
#13114705	2
C₂×#11718049	2
C₂×C₈×#1055	2
#13112677	2
C₄×C₁₃×#106	2
#8283	13