Subgroups — #15761962

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123234	C₂	1	1	yes	maximal, characteristic
2	#13114988	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718074	C₂	1	4	yes	maximal
4	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	1	no	—
6	C₂×#88123	—	2	1	no	—
7	C₂×#88123	C₂²	1	2	yes	—
8	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	—	2	1	no	—
10	C₄³×C₁₃	C₂²	1	1	yes	characteristic
11	#11718596	—	2	2	no	—
12	#11718596	C₄	1	4	yes	—
13	C₂×C₄⋊#1055	—	2	2	no	—
14	C₂×C₄⋊#1055	C₂²	1	2	yes	—
15	C₁₃×#336	—	2	1	no	—
16	#11718074	—	2	8	no	—
17	C₂²×Dic₂₆	—	2	2	no	—
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₁₃	D₄	1	2	yes	—
21	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₃	—	2	4	no	—
25	C₂×C₄⋊Dic₁₃	—	2	4	no	—
26	C₂×C₄⋊Dic₁₃	—	2	2	no	—
27	C₂×C₄⋊Dic₁₃	C₂×C₄	1	4	yes	—
28	C₂²×#1055	—	2	2	no	—
29	#88123	—	2	4	no	—
30	#88123	—	2	4	no	—
31	#88123	—	4	4	no	—
32	#88123	—	4	2	no	—
33	#88123	—	4	2	no	—
34	C₁₃×C₄⋊C₈	—	2	2	no	—
35	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
36	C₄×C₈×C₁₃	—	4	2	no	—
37	C₄⋊#1055	—	2	4	no	—
38	C₄⋊#1055	—	4	4	no	—
39	#9694	—	13	1	no	maximal, Sylow
40	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₃	—	2	4	no	—
46	C₂×Dic₂₆	—	4	4	no	—
47	C₂×Dic₂₆	—	4	2	no	—
48	C₂×Dic₂₆	—	4	2	no	—
49	C₂×Dic₂₆	—	4	4	no	—
50	C₂×C₈×C₁₃	—	2	4	no	—
51	C₂×C₈×C₁₃	—	4	2	no	—
52	C₂×C₈×C₁₃	—	4	2	no	—
53	C₄²×C₁₃	—	2	4	no	—
54	C₄²×C₁₃	—	2	2	no	—
55	C₄²×C₁₃	D₈	1	2	yes	—
56	C₄²×C₁₃	SD₁₆	1	4	yes	—
57	C₄²×C₁₃	SD₁₆	1	2	yes	—
58	C₄²×C₁₃	Q₁₆	1	4	yes	—
59	C₄²×C₁₃	C₂×D₄	1	2	yes	—
60	C₄²×C₁₃	#41	1	2	yes	—
61	C₄⋊Dic₁₃	—	2	8	no	—
62	C₄⋊Dic₁₃	—	4	8	no	—
63	C₄⋊Dic₁₃	—	4	8	no	—
64	C₄⋊Dic₁₃	—	4	2	no	—
65	C₄⋊Dic₁₃	—	4	2	no	—
66	C₂×#1055	—	4	2	no	—
67	C₂×#1055	—	4	4	no	—
68	#2868	—	13	1	no	—
69	#1842	—	13	1	no	—
70	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
71	C₂×#332	—	13	4	no	—
72	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	#101	1	4	yes	—
77	C₂×C₄×C₁₃	#102	1	4	yes	—
78	C₂×C₄×C₁₃	#103	1	4	yes	—
79	C₂×C₄×C₁₃	#103	1	4	yes	—
80	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
81	C₂×C₄×C₁₃	#121	1	2	yes	—
82	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×Dic₁₃	—	4	4	no	—
90	C₂×Dic₁₃	—	4	4	no	—
91	C₂×Dic₁₃	—	4	4	no	—
92	Dic₂₆	—	8	4	no	—
93	Dic₂₆	—	8	4	no	—
94	C₈×C₁₃	—	4	4	no	—
95	#1055	—	8	2	no	—
96	C₂×C₄⋊Q₈	—	13	2	no	—
97	C₂×C₄⋊Q₈	—	26	1	no	—
98	C₂×C₄⋊Q₈	—	26	1	no	—
99	C₂×C₄⋊C₈	—	13	2	no	—
100	C₂×C₄⋊C₈	—	26	2	no	—
101	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
102	C₂×C₄×C₈	—	2	1	no	—
103	#384	—	13	4	no	—
104	#384	—	26	2	no	—
105	C₄³	C₂×D₁₃	1	1	yes	characteristic
106	#336	—	2	1	no	—
107	#332	—	26	8	no	—
108	C₂²×C₁₃	#390	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×#101	1	2	yes	—
110	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×#103	1	2	yes	—
112	C₂²×C₁₃	#418	1	1	yes	characteristic
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	#332	1	8	yes	complemented
116	C₄×C₁₃	#459	1	2	yes	—
117	C₄×C₁₃	#460	1	2	yes	—
118	C₄×C₁₃	#487	1	2	yes	—
119	C₄×C₁₃	#492	1	1	yes	characteristic
120	C₄×C₁₃	#494	1	1	yes	characteristic
121	Dic₁₃	—	8	4	no	—
122	C₂²×Q₈	—	26	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	26	2	no	—
125	C₄⋊Q₈	—	26	4	no	—
126	C₄⋊Q₈	—	26	4	no	—
127	C₄⋊Q₈	—	52	4	no	—
128	C₄⋊Q₈	—	52	2	no	—
129	C₄⋊Q₈	—	52	2	no	—
130	C₂×C₄⋊C₄	—	13	4	no	—
131	C₂×C₄⋊C₄	—	26	4	no	—
132	C₂×C₄⋊C₄	—	26	4	no	—
133	C₂×C₄⋊C₄	—	26	2	no	—
134	C₂×C₄²	C₄×D₁₃	1	2	yes	—
135	C₂×C₄²	D₅₂	1	2	yes	—
136	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
137	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
138	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	26	4	no	—
141	C₄⋊C₈	—	52	4	no	—
142	C₄⋊C₈	D₅₂	1	4	yes	—
143	C₄×C₈	—	4	2	no	—
144	C₂×C₁₃	C₂×#332	1	2	yes	—
145	C₂×C₁₃	#1549	1	2	yes	—
146	C₂×C₁₃	#1956	1	1	yes	characteristic
147	C₂×C₁₃	#1966	1	1	yes	characteristic
148	C₂×C₁₃	#1982	1	1	yes	characteristic
149	C₂×Q₈	—	52	4	no	—
150	C₂×Q₈	—	52	2	no	—
151	C₂×Q₈	—	52	2	no	—
152	C₂×Q₈	—	52	4	no	—
153	C₂²×C₄	—	26	4	no	—
154	C₂²×C₄	#6103	1	2	yes	—
155	C₂²×C₄	#6103	1	2	yes	—
156	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	4	2	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	52	2	no	—
163	C₂×C₈	—	52	4	no	—
164	C₄⋊C₄	—	26	8	no	—
165	C₄⋊C₄	—	52	8	no	—
166	C₄⋊C₄	—	52	2	no	—
167	C₄⋊C₄	—	52	2	no	—
168	C₄⋊C₄	—	52	8	no	—
169	C₄²	—	2	4	no	—
170	C₄²	—	2	2	no	—
171	C₄²	#6095	1	4	yes	—
172	C₄²	Dic₅₂	1	4	yes	—
173	C₄²	#6104	1	2	yes	—
174	C₄²	#6106	1	2	yes	—
175	C₄²	C₂×D₅₂	1	2	yes	—
176	C₄²	#6127	1	2	yes	—
177	C₁₃	#9694	1	1	yes	characteristic, complemented, Sylow
178	C₂³	C₂×#6103	1	1	yes	characteristic
179	Q₈	—	104	4	no	—
180	Q₈	—	104	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	52	4	no	—
185	C₂×C₄	—	52	4	no	—
186	C₂×C₄	—	52	4	no	—
187	C₂×C₄	#88049	1	4	yes	—
188	C₂×C₄	#88056	1	4	yes	—
189	C₂×C₄	#88056	1	4	yes	—
190	C₂×C₄	#88064	1	4	yes	—
191	C₂×C₄	C₄×D₅₂	1	2	yes	—
192	C₂×C₄	#88128	1	1	yes	characteristic
193	C₂×C₄	#88129	1	1	yes	characteristic
194	C₂×C₄	C₂×#6095	1	2	yes	—
195	C₂×C₄	C₂×Dic₅₂	1	2	yes	—
196	C₂×C₄	#88184	1	2	yes	—
197	C₂×C₄	C₂×#6104	1	1	yes	characteristic
198	C₂×C₄	C₂×#6106	1	1	yes	characteristic
199	C₂×C₄	#88203	1	2	yes	—
200	C₈	—	4	4	no	—
201	C₈	—	104	2	no	—
202	C₂²	#11718600	1	1	yes	characteristic
203	C₂²	C₂×#88049	1	1	yes	characteristic
204	C₂²	#11718662	1	1	yes	characteristic
205	C₂²	C₂×#88056	1	2	yes	—
206	C₂²	C₂×#88064	1	2	yes	—
207	C₄	—	2	2	no	—
208	C₄	—	2	4	no	—
209	C₄	—	104	4	no	—
210	C₄	#11718074	1	8	yes	complemented
211	C₄	#11718342	1	2	yes	—
212	C₄	#11718354	1	1	yes	characteristic
213	C₄	#11718360	1	1	yes	characteristic
214	C₄	#11718676	1	2	yes	—
215	C₄	#11718688	1	2	yes	—
216	C₂	C₂×#11718074	1	2	yes	—
217	C₂	#13113520	1	2	yes	—
218	C₂	#13115638	1	1	yes	characteristic
219	C₂	#13115680	1	1	yes	characteristic
220	C₂	#13115763	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#6103
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×C₄⋊C₈	D₁₃
hypocenter	C₁₃	#9694
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
115. C₄×C₁₃	#332	semidirect
177. C₁₃	#9694	semidirect
210. C₄	#11718074	semidirect

Maximal subgroups

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

subgroup	index
#13123234	2
#13114988	2
C₂×#11718074	2
C₄×C₁₃×C₄⋊C₈	2
#9694	13