Subgroups — #15754116

Order 3328.

446 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	C₄×C₄⋊#1055	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718049	C₂	1	1	yes	maximal, characteristic
4	#13112671	C₂	1	2	yes	maximal
5	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#97	C₄	1	2	yes	—
7	C₂×C₁₃×#97	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#1055	C₂²	1	2	yes	—
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₂×C₄×#1055	—	2	2	no	—
14	#11718116	—	2	2	no	—
15	#11718050	C₄	1	4	yes	complemented
16	#11718049	C₄	1	4	yes	complemented
17	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃	D₄	1	2	yes	—
22	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂²×#1055	—	2	2	no	—
25	C₂²×#1055	—	2	2	no	—
26	C₁₃×#97	—	2	4	no	—
27	C₁₃×#97	C₈	1	8	yes	complemented
28	C₄×C₈×C₁₃	—	2	1	no	—
29	C₄×C₈×C₁₃	—	2	1	no	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₄⋊#1055	—	2	2	no	—
33	C₄⋊#1055	—	2	2	no	—
34	C₄⋊#1055	C₂×C₄	1	4	yes	—
35	C₄⋊#1055	C₂×C₄	1	4	yes	—
36	C₄×#1055	—	4	4	no	—
37	#8275	—	13	1	no	maximal, Sylow
38	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₁₃	—	2	2	no	—
43	C₂×C₈×C₁₃	—	2	2	no	—
44	C₂×C₈×C₁₃	—	2	2	no	—
45	C₂×C₈×C₁₃	C₄²	1	2	yes	—
46	C₂×C₈×C₁₃	C₄²	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
48	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
49	C₄²×C₁₃	—	2	2	no	—
50	C₄²×C₁₃	—	2	2	no	—
51	C₄²×C₁₃	—	4	2	no	—
52	C₄²×C₁₃	C₂×C₈	1	4	yes	—
53	C₄²×C₁₃	M₁₆	1	4	yes	—
54	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	#41	1	1	yes	characteristic
57	C₄²×C₁₃	#41	1	1	yes	characteristic
58	C₂×#1055	—	2	4	no	—
59	C₂×#1055	—	2	4	no	—
60	C₂×#1055	—	4	2	no	—
61	C₂×#1055	—	4	2	no	—
62	C₂×#1055	—	4	2	no	—
63	C₂×#1055	—	4	2	no	—
64	C₄×C₄⋊C₈	—	13	2	no	—
65	C₄×#97	D₁₃	1	1	yes	characteristic
66	C₂×#335	—	13	1	no	—
67	C₂×#334	—	13	1	no	—
68	#1277	—	13	2	no	—
69	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	C₄⋊C₈	1	8	yes	—
74	C₂×C₄×C₁₃	C₂×C₄²	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×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
82	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₄×C₁₃	#137	1	2	yes	—
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	—	2	4	no	—
86	C₈×C₁₃	C₄×C₈	1	4	yes	complemented
87	#1055	—	4	4	no	—
88	#1055	—	4	4	no	—
89	C₂×C₄⋊C₈	—	13	2	no	—
90	C₂×C₄⋊C₈	—	13	2	no	—
91	C₂×#97	Dic₁₃	1	2	yes	—
92	C₂×#97	Dic₁₃	1	2	yes	—
93	C₂×C₄×C₈	—	26	2	no	—
94	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
96	C₄³	C₂×D₁₃	1	1	yes	characteristic
97	#336	—	26	2	no	—
98	#335	—	13	4	no	—
99	#334	—	13	4	no	—
100	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
101	C₂²×C₁₃	C₂×C₄⋊C₈	1	2	yes	—
102	C₂²×C₁₃	#428	1	2	yes	—
103	C₂²×C₁₃	#430	1	2	yes	—
104	C₄×C₁₃	—	2	8	no	—
105	C₄×C₁₃	—	2	1	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
108	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
109	C₄×C₁₃	#431	1	2	yes	—
110	C₄×C₁₃	#440	1	2	yes	—
111	C₄×C₁₃	#441	1	1	yes	characteristic
112	C₄×C₁₃	#442	1	1	yes	characteristic
113	C₂²×C₈	—	26	2	no	—
114	C₂²×C₈	—	26	2	no	—
115	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
116	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
117	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂×C₄²	Dic₂₆	1	2	yes	—
119	C₂×C₄²	D₅₂	1	2	yes	—
120	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
121	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
122	C₄⋊C₈	—	13	4	no	—
123	C₄⋊C₈	—	13	4	no	—
124	C₄⋊C₈	—	26	2	no	—
125	C₄⋊C₈	—	26	2	no	—
126	#97	—	2	4	no	—
127	#97	#1055	1	8	yes	complemented
128	C₄×C₈	—	2	1	no	—
129	C₄×C₈	—	2	1	no	—
130	C₄×C₈	—	52	4	no	—
131	C₄×C₈	C₄×D₁₃	1	2	yes	—
132	C₄×C₈	C₄×D₁₃	1	2	yes	—
133	C₂×C₁₃	#1565	1	4	yes	—
134	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
135	C₂×C₁₃	#1776	1	1	yes	characteristic
136	C₂×C₁₃	#1778	1	1	yes	characteristic
137	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
138	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
139	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
140	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
141	C₂×C₈	—	2	2	no	—
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	26	4	no	—
145	C₂×C₈	—	26	4	no	—
146	C₂×C₈	—	52	2	no	—
147	C₂×C₈	—	52	2	no	—
148	C₂×C₈	—	52	2	no	—
149	C₂×C₈	—	52	2	no	—
150	C₂×C₈	C₂×#1055	1	4	yes	—
151	C₂×C₈	C₂×#1055	1	4	yes	—
152	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
153	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
154	C₄²	—	2	2	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	4	2	no	—
157	C₄²	C₂×#1055	1	4	yes	—
158	C₄²	#6099	1	4	yes	—
159	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
161	C₄²	#6127	1	1	yes	characteristic
162	C₄²	#6127	1	1	yes	characteristic
163	C₁₃	#8275	1	1	yes	characteristic, complemented, Sylow
164	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	C₄⋊#1055	1	8	yes	—
169	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
170	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
171	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
172	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
173	C₂×C₄	#88162	1	2	yes	—
174	C₂×C₄	#88163	1	2	yes	—
175	C₂×C₄	C₂²×#1055	1	2	yes	—
176	C₂×C₄	C₂×#6099	1	2	yes	—
177	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
178	C₂×C₄	#88180	1	1	yes	characteristic
179	C₈	—	2	2	no	—
180	C₈	—	2	4	no	—
181	C₈	—	52	4	no	—
182	C₈	—	52	4	no	—
183	C₈	C₄×#1055	1	4	yes	complemented
184	C₂²	C₂×C₄⋊#1055	1	2	yes	—
185	C₂²	C₄×C₄⋊Dic₁₃	1	1	yes	characteristic
186	C₂²	#11718782	1	2	yes	—
187	C₂²	#11718786	1	2	yes	—
188	C₄	—	2	1	no	—
189	C₄	—	2	1	no	—
190	C₄	—	2	8	no	—
191	C₄	#11718371	1	2	yes	—
192	C₄	#11718372	1	1	yes	characteristic
193	C₄	#11718377	1	1	yes	characteristic
194	C₄	C₂×C₄×#1055	1	1	yes	characteristic
195	C₄	C₄×#6099	1	1	yes	characteristic
196	C₄	#11718586	1	2	yes	—
197	C₂	#13113588	1	4	yes	—
198	C₂	C₄×C₄⋊#1055	1	1	yes	characteristic
199	C₂	#13114733	1	1	yes	characteristic
200	C₂	#13114742	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₄×C₁₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#97	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#97	D₁₃
hypocenter	C₁₃	#8275
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
15. #11718050	C₄	semidirect
16. #11718049	C₄	semidirect
27. C₁₃×#97	C₈	semidirect
86. C₈×C₁₃	C₄×C₈	semidirect
127. #97	#1055	semidirect
163. C₁₃	#8275	semidirect
183. C₈	C₄×#1055	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊#1055	2
C₂×#11718050	2
C₂×#11718049	2
#13112671	2
C₄×C₁₃×#97	2
#8275	13