Subgroups — #15761953

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 218 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×#11718078	C₂	1	4	yes	maximal
4	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×#88123	—	2	2	no	—
6	C₂×#88123	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₁₃	—	2	1	no	—
9	C₄³×C₁₃	C₂²	1	1	yes	characteristic
10	#11718596	—	2	1	no	—
11	#11718596	—	2	1	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	#11718078	—	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	4	no	—
33	C₁₃×C₄⋊C₈	—	2	2	no	—
34	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
35	C₄×C₈×C₁₃	—	4	2	no	—
36	C₄⋊#1055	—	2	4	no	—
37	C₄⋊#1055	—	4	4	no	—
38	#9692	—	13	1	no	maximal, Sylow
39	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×Dic₁₃	—	2	4	no	—
45	C₂×Dic₂₆	—	4	2	no	—
46	C₂×Dic₂₆	—	4	2	no	—
47	C₂×Dic₂₆	—	4	4	no	—
48	C₂×Dic₂₆	—	4	4	no	—
49	C₂×C₈×C₁₃	—	2	4	no	—
50	C₂×C₈×C₁₃	—	4	2	no	—
51	C₂×C₈×C₁₃	—	4	2	no	—
52	C₄²×C₁₃	—	2	4	no	—
53	C₄²×C₁₃	—	2	2	no	—
54	C₄²×C₁₃	D₈	1	4	yes	—
55	C₄²×C₁₃	SD₁₆	1	4	yes	—
56	C₄²×C₁₃	SD₁₆	1	2	yes	—
57	C₄²×C₁₃	Q₁₆	1	2	yes	—
58	C₄²×C₁₃	C₂×D₄	1	2	yes	—
59	C₄²×C₁₃	#41	1	2	yes	—
60	C₄⋊Dic₁₃	—	2	8	no	—
61	C₄⋊Dic₁₃	—	4	8	no	—
62	C₄⋊Dic₁₃	—	4	8	no	—
63	C₄⋊Dic₁₃	—	4	4	no	—
64	C₂×#1055	—	4	2	no	—
65	C₂×#1055	—	4	4	no	—
66	#2868	—	13	1	no	—
67	#1842	—	13	1	no	—
68	C₄×C₄⋊C₈	D₁₃	1	1	yes	characteristic
69	C₂×#332	—	13	4	no	—
70	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	—	2	2	no	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	#101	1	4	yes	—
75	C₂×C₄×C₁₃	#102	1	4	yes	—
76	C₂×C₄×C₁₃	#102	1	4	yes	—
77	C₂×C₄×C₁₃	#103	1	4	yes	—
78	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
79	C₂×C₄×C₁₃	#121	1	2	yes	—
80	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	#137	1	2	yes	—
87	C₂×Dic₁₃	—	4	4	no	—
88	C₂×Dic₁₃	—	4	8	no	—
89	Dic₂₆	—	8	4	no	—
90	Dic₂₆	—	8	2	no	—
91	Dic₂₆	—	8	2	no	—
92	C₈×C₁₃	—	4	4	no	—
93	#1055	—	8	2	no	—
94	C₂×C₄⋊Q₈	—	13	2	no	—
95	C₂×C₄⋊Q₈	—	26	2	no	—
96	C₂×C₄⋊C₈	—	13	2	no	—
97	C₂×C₄⋊C₈	—	26	2	no	—
98	C₂×C₄⋊C₈	C₂×D₁₃	1	2	yes	—
99	C₂×C₄×C₈	—	2	1	no	—
100	#384	—	13	4	no	—
101	#384	—	26	1	no	—
102	#384	—	26	1	no	—
103	C₄³	C₂×D₁₃	1	1	yes	characteristic
104	#336	—	2	1	no	—
105	#332	—	26	8	no	—
106	C₂²×C₁₃	#390	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×#101	1	2	yes	—
108	C₂²×C₁₃	C₂×#102	1	2	yes	—
109	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
110	C₂²×C₁₃	#419	1	1	yes	characteristic
111	C₄×C₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	#332	1	8	yes	complemented
114	C₄×C₁₃	#461	1	2	yes	—
115	C₄×C₁₃	#462	1	2	yes	—
116	C₄×C₁₃	#486	1	2	yes	—
117	C₄×C₁₃	#492	1	1	yes	characteristic
118	C₄×C₁₃	#493	1	1	yes	characteristic
119	Dic₁₃	—	8	4	no	—
120	C₂²×Q₈	—	26	2	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	26	2	no	—
123	C₄⋊Q₈	—	26	4	no	—
124	C₄⋊Q₈	—	26	4	no	—
125	C₄⋊Q₈	—	52	4	no	—
126	C₄⋊Q₈	—	52	4	no	—
127	C₂×C₄⋊C₄	—	13	4	no	—
128	C₂×C₄⋊C₄	—	26	4	no	—
129	C₂×C₄⋊C₄	—	26	4	no	—
130	C₂×C₄⋊C₄	—	26	2	no	—
131	C₂×C₄²	C₄×D₁₃	1	2	yes	—
132	C₂×C₄²	D₅₂	1	2	yes	—
133	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
134	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
135	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	26	4	no	—
138	C₄⋊C₈	—	52	4	no	—
139	C₄⋊C₈	D₅₂	1	4	yes	—
140	C₄×C₈	—	4	2	no	—
141	C₂×C₁₃	C₂×#332	1	2	yes	—
142	C₂×C₁₃	#1546	1	2	yes	—
143	C₂×C₁₃	#1957	1	1	yes	characteristic
144	C₂×C₁₃	#1966	1	1	yes	characteristic
145	C₂×C₁₃	#1981	1	1	yes	characteristic
146	C₂×Q₈	—	52	2	no	—
147	C₂×Q₈	—	52	2	no	—
148	C₂×Q₈	—	52	4	no	—
149	C₂×Q₈	—	52	4	no	—
150	C₂²×C₄	—	26	4	no	—
151	C₂²×C₄	#6103	1	2	yes	—
152	C₂²×C₄	#6103	1	2	yes	—
153	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
156	C₂×C₈	—	2	4	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	4	2	no	—
159	C₂×C₈	—	52	2	no	—
160	C₂×C₈	—	52	4	no	—
161	C₄⋊C₄	—	26	8	no	—
162	C₄⋊C₄	—	52	4	no	—
163	C₄⋊C₄	—	52	8	no	—
164	C₄⋊C₄	—	52	8	no	—
165	C₄²	—	2	4	no	—
166	C₄²	—	2	2	no	—
167	C₄²	#6095	1	4	yes	—
168	C₄²	D₁₀₄	1	4	yes	—
169	C₄²	#6105	1	2	yes	—
170	C₄²	#6107	1	2	yes	—
171	C₄²	C₂×D₅₂	1	2	yes	—
172	C₄²	#6127	1	2	yes	—
173	C₁₃	#9692	1	1	yes	characteristic, complemented, Sylow
174	C₂³	C₂×#6103	1	1	yes	characteristic
175	Q₈	—	104	2	no	—
176	Q₈	—	104	4	no	—
177	Q₈	—	104	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	52	8	no	—
182	C₂×C₄	—	52	4	no	—
183	C₂×C₄	#88050	1	4	yes	—
184	C₂×C₄	#88061	1	4	yes	—
185	C₂×C₄	#88061	1	4	yes	—
186	C₂×C₄	#88064	1	4	yes	—
187	C₂×C₄	C₄×D₅₂	1	2	yes	—
188	C₂×C₄	#88128	1	1	yes	characteristic
189	C₂×C₄	#88129	1	1	yes	characteristic
190	C₂×C₄	C₂×#6095	1	2	yes	—
191	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
192	C₂×C₄	#88184	1	2	yes	—
193	C₂×C₄	C₂×#6105	1	1	yes	characteristic
194	C₂×C₄	C₂×#6107	1	1	yes	characteristic
195	C₂×C₄	#88205	1	2	yes	—
196	C₈	—	4	4	no	—
197	C₈	—	104	2	no	—
198	C₂²	#11718600	1	1	yes	characteristic
199	C₂²	C₂×#88050	1	1	yes	characteristic
200	C₂²	#11718665	1	1	yes	characteristic
201	C₂²	C₂×#88061	1	2	yes	—
202	C₂²	C₂×#88064	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	104	4	no	—
206	C₄	#11718078	1	8	yes	complemented
207	C₄	#11718354	1	1	yes	characteristic
208	C₄	#11718355	1	2	yes	—
209	C₄	#11718356	1	1	yes	characteristic
210	C₄	#11718680	1	2	yes	—
211	C₄	#11718692	1	2	yes	—
212	C₂	C₂×#11718078	1	2	yes	—
213	C₂	#13113507	1	2	yes	—
214	C₂	#13115642	1	1	yes	characteristic
215	C₂	#13115680	1	1	yes	characteristic
216	C₂	#13115759	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	#9692
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
113. C₄×C₁₃	#332	semidirect
173. C₁₃	#9692	semidirect
206. C₄	#11718078	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×#11718078	2
C₄×C₁₃×C₄⋊C₈	2
#9692	13