Subgroups — #15768175

Order 3328.

1422 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718868	C₂	1	2	yes	maximal
2	C₂×#11718144	C₂	1	4	yes	maximal, complemented
3	C₂⁴×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
5	C₂³×C₈×C₁₃	—	2	4	no	—
6	C₂³×C₈×C₁₃	—	2	8	no	—
7	C₂³×C₈×C₁₃	C₄	1	4	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂²×#6099	C₂²	1	4	yes	—
10	#11718868	—	2	4	no	—
11	#11718868	C₄	1	8	yes	—
12	C₂×C₂²⋊#1055	—	2	4	no	—
13	#11718144	—	2	8	no	—
14	C₂⁵×C₁₃	Q₈	1	1	yes	characteristic
15	C₂³×C₄×C₁₃	—	2	4	no	—
16	C₂³×C₄×C₁₃	—	2	4	no	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₃	D₄	1	2	yes	—
20	C₂³×C₄×C₁₃	D₄	1	4	yes	—
21	C₂³×C₄×C₁₃	Q₈	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	—	2	32	no	—
24	C₂²×C₈×C₁₃	—	2	8	no	—
25	C₂²×C₈×C₁₃	—	2	4	no	—
26	C₂²×C₈×C₁₃	—	2	4	no	—
27	C₂²×C₈×C₁₃	—	2	16	no	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₁₃	D₄	1	8	yes	—
30	C₂×#6099	—	2	4	no	—
31	C₂×#6099	—	2	8	no	—
32	C₂×#6099	—	2	8	no	—
33	C₂×#6099	C₂×C₄	1	8	yes	—
34	C₂²×#1055	—	2	4	no	—
35	C₂²⋊#1055	—	2	16	no	—
36	#10723	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	—	2	8	no	—
38	C₂⁴×C₁₃	—	2	4	no	—
39	C₂⁴×C₁₃	C₄⋊C₄	1	4	yes	—
40	C₂⁴×C₁₃	C₄⋊C₄	1	2	yes	—
41	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	2	16	no	—
46	C₂²×C₄×C₁₃	—	2	8	no	—
47	C₂²×C₄×C₁₃	—	2	8	no	—
48	C₂²×C₄×C₁₃	—	2	8	no	—
49	C₂²×C₄×C₁₃	C₄²	1	4	yes	—
50	C₂²×C₄×C₁₃	C₂²⋊C₄	1	8	yes	—
51	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
59	C₂²×C₄×C₁₃	#41	1	4	yes	—
60	C₂×C₈×C₁₃	—	2	32	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	16	no	—
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	8	yes	—
65	#6099	—	2	16	no	—
66	#6099	—	4	4	no	—
67	#6099	—	4	4	no	—
68	C₂×#1055	—	2	8	no	—
69	C₂×#1055	—	4	8	no	—
70	C₂⁴×C₈	D₁₃	1	1	yes	characteristic
71	C₂×#407	—	13	2	no	—
72	C₂×#341	—	13	4	no	—
73	C₂³×C₁₃	—	2	32	no	—
74	C₂³×C₁₃	—	2	16	no	—
75	C₂³×C₁₃	—	2	4	no	—
76	C₂³×C₁₃	—	2	8	no	—
77	C₂³×C₁₃	—	2	4	no	—
78	C₂³×C₁₃	—	2	4	no	—
79	C₂³×C₁₃	#95	1	4	yes	—
80	C₂³×C₁₃	#108	1	8	yes	—
81	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂³×C₁₃	C₂²⋊Q₈	1	4	yes	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	8	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	—	2	16	no	—
90	C₂×C₄×C₁₃	#95	1	4	yes	—
91	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₃	#117	1	2	yes	—
96	C₂×C₄×C₁₃	C₄×D₄	1	8	yes	—
97	C₂×C₄×C₁₃	#120	1	4	yes	—
98	C₂×C₄×C₁₃	#121	1	4	yes	—
99	C₂×C₄×C₁₃	#123	1	4	yes	—
100	C₈×C₁₃	—	2	8	no	—
101	#1055	—	4	8	no	—
102	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
103	C₂²×M₁₆	—	13	4	no	—
104	C₂³×C₈	—	2	4	no	—
105	C₂³×C₈	—	2	8	no	—
106	C₂³×C₈	Dic₁₃	1	4	yes	—
107	C₂³×C₈	C₂×D₁₃	1	2	yes	—
108	#407	—	13	8	no	—
109	#407	—	26	4	no	—
110	C₂×C₂²⋊C₈	—	26	4	no	—
111	#341	—	26	8	no	—
112	C₂²×C₁₃	—	2	32	no	—
113	C₂²×C₁₃	—	2	16	no	—
114	C₂²×C₁₃	—	2	8	no	—
115	C₂²×C₁₃	—	2	4	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	4	no	—
118	C₂²×C₁₃	#341	1	8	yes	complemented
119	C₂²×C₁₃	C₂×#95	1	1	yes	characteristic
120	C₂²×C₁₃	#380	1	2	yes	—
121	C₂²×C₁₃	#385	1	4	yes	—
122	C₂²×C₁₃	C₂×#108	1	4	yes	—
123	C₄×C₁₃	—	2	4	no	—
124	C₄×C₁₃	C₄×C₂²⋊C₄	1	2	yes	—
125	C₄×C₁₃	#379	1	1	yes	characteristic
126	C₄×C₁₃	#381	1	1	yes	characteristic
127	C₄×C₁₃	#386	1	4	yes	—
128	C₂⁵	Dic₂₆	1	1	yes	characteristic
129	C₂³×C₄	—	2	4	no	—
130	C₂³×C₄	—	2	4	no	—
131	C₂³×C₄	Dic₂₆	1	1	yes	characteristic
132	C₂³×C₄	C₄×D₁₃	1	4	yes	—
133	C₂³×C₄	D₅₂	1	2	yes	—
134	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
135	C₂³×C₄	C₂²⋊D₁₃	1	4	yes	—
136	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
137	C₂×M₁₆	—	13	8	no	—
138	C₂×M₁₆	—	26	4	no	—
139	C₂×M₁₆	—	26	8	no	—
140	C₂×M₁₆	—	26	8	no	—
141	C₂²×C₈	—	2	32	no	—
142	C₂²×C₈	—	2	8	no	—
143	C₂²×C₈	—	2	4	no	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	—	2	16	no	—
146	C₂²×C₈	—	26	4	no	—
147	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
148	C₂²×C₈	C₂²⋊D₁₃	1	8	yes	—
149	C₂²⋊C₈	—	26	16	no	—
150	C₂×C₁₃	—	2	8	no	—
151	C₂×C₁₃	—	2	4	no	—
152	C₂×C₁₃	C₂²⋊#95	1	1	yes	characteristic
153	C₂×C₁₃	C₂×#341	1	2	yes	—
154	C₂×C₁₃	#1810	1	4	yes	—
155	C₂⁴	—	2	4	no	—
156	C₂⁴	—	2	8	no	—
157	C₂⁴	#6101	1	4	yes	—
158	C₂⁴	C₄⋊Dic₁₃	1	2	yes	—
159	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	16	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	C₄×Dic₁₃	1	4	yes	—
168	C₂²×C₄	#6101	1	4	yes	—
169	C₂²×C₄	C₄⋊Dic₁₃	1	2	yes	—
170	C₂²×C₄	#6103	1	8	yes	—
171	C₂²×C₄	C₂²⋊Dic₁₃	1	4	yes	—
172	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
173	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
174	C₂²×C₄	#6127	1	4	yes	—
175	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
177	C₂²×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
178	M₁₆	—	26	16	no	—
179	M₁₆	—	52	4	no	—
180	M₁₆	—	52	4	no	—
181	C₂×C₈	—	2	32	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	16	no	—
185	C₂×C₈	—	26	8	no	—
186	C₂×C₈	—	52	8	no	—
187	C₂×C₈	C₂²⋊Dic₁₃	1	8	yes	—
188	C₁₃	#10723	1	1	yes	characteristic, complemented, Sylow
189	C₂³	—	2	32	no	—
190	C₂³	—	2	16	no	—
191	C₂³	—	2	4	no	—
192	C₂³	—	2	4	no	—
193	C₂³	—	2	8	no	—
194	C₂³	—	2	4	no	—
195	C₂³	#88059	1	8	yes	—
196	C₂³	#88071	1	4	yes	—
197	C₂³	C₂×#6101	1	2	yes	—
198	C₂³	#88178	1	4	yes	—
199	C₂³	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	16	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	#88071	1	4	yes	—
207	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
208	C₂×C₄	#88180	1	2	yes	—
209	C₂×C₄	C₂×#6103	1	2	yes	—
210	C₂×C₄	C₄×C₂²⋊D₁₃	1	8	yes	—
211	C₂×C₄	#88183	1	4	yes	—
212	C₂×C₄	#88184	1	4	yes	—
213	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	2	yes	—
214	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
215	C₂×C₄	#88207	1	4	yes	—
216	C₈	—	2	8	no	—
217	C₈	—	52	8	no	—
218	C₂²	—	2	32	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	—	2	16	no	—
224	C₂²	#11718144	1	8	yes	complemented
225	C₂²	C₂×#88059	1	4	yes	—
226	C₂²	C₂×#88071	1	1	yes	characteristic
227	C₂²	#11718871	1	4	yes	—
228	C₂²	#11718873	1	2	yes	—
229	C₄	—	2	4	no	—
230	C₄	C₄×C₂²⋊Dic₁₃	1	2	yes	—
231	C₄	#11718872	1	1	yes	characteristic
232	C₄	#11718874	1	4	yes	—
233	C₄	#11718908	1	1	yes	characteristic
234	C₂	—	2	8	no	—
235	C₂	—	2	4	no	—
236	C₂	#13113059	1	1	yes	characteristic
237	C₂	C₂×#11718144	1	2	yes	—
238	C₂	#13114874	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₅₂
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂⁴×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁵×C₁₃	Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂⁴×C₈	D₁₃
hypocenter	C₁₃	#10723
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#11718144	C₂	semidirect
118. C₂²×C₁₃	#341	semidirect
188. C₁₃	#10723	semidirect
224. C₂²	#11718144	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718868	2
C₂×#11718144	2
C₂⁴×C₈×C₁₃	2
#10723	13