Subgroups — #15863524

Order 3328.

842 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13117234	C₂	1	2	yes	maximal
2	#13115000	C₂	1	2	yes	maximal, complemented
3	#13114967	C₂	1	2	yes	maximal, complemented
4	#13114869	C₂	1	4	yes	maximal
5	#13114733	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×M₁₆	C₄	1	8	yes	complemented
10	C₂×C₁₃×#97	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	#11718873	C₂²	1	4	yes	—
14	#11718872	—	2	2	no	—
15	C₄×C₂²⋊Dic₁₃	—	2	2	no	—
16	C₂×#88058	C₂²	1	4	yes	—
17	C₂×#88057	C₂²	1	4	yes	—
18	#11718598	—	2	2	no	—
19	#11718597	C₂²	1	4	yes	—
20	C₄×C₄⋊Dic₁₃	C₂²	1	4	yes	—
21	#11718593	—	2	2	no	—
22	#11718143	C₂²	1	8	yes	—
23	C₂³×C₄×C₁₃	D₄	1	2	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×C₁₃×M₁₆	—	2	4	no	—
26	C₂×C₁₃×M₁₆	—	2	4	no	—
27	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₃	D₄	1	2	yes	—
32	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₃	—	2	4	no	—
36	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
37	C₂×C₄×Dic₁₃	—	2	4	no	—
38	C₁₃×#97	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
41	#88071	—	2	4	no	—
42	#88071	—	2	4	no	—
43	#88071	—	2	4	no	—
44	#88071	—	2	4	no	—
45	#88058	—	2	8	no	—
46	#88057	—	2	8	no	—
47	#20825	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	4	4	no	—
52	C₂²×C₄×C₁₃	C₄⋊C₄	1	8	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₁₃	—	2	8	no	—
60	C₁₃×M₁₆	—	2	16	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₃	#41	1	4	yes	—
66	C₂×C₈×C₁₃	#41	1	4	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
70	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
71	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂²⋊Dic₁₃	—	4	8	no	—
73	C₄⋊Dic₁₃	—	2	8	no	—
74	C₄⋊Dic₁₃	—	2	8	no	—
75	C₄×Dic₁₃	—	4	8	no	—
76	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
77	#2283	—	13	2	no	—
78	#1846	—	13	2	no	—
79	#1835	—	13	2	no	—
80	#1809	—	13	4	no	—
81	#1776	—	13	4	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	—	2	4	no	—
84	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
85	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	—	2	4	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	8	no	—
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
99	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
100	C₂×Dic₁₃	—	4	8	no	—
101	C₂×Dic₁₃	—	4	8	no	—
102	C₂×Dic₁₃	—	4	8	no	—
103	C₈×C₁₃	—	2	4	no	—
104	C₈×C₁₃	#117	1	8	yes	—
105	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
106	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
107	C₂×#107	—	13	4	no	—
108	C₂×#106	—	13	4	no	—
109	C₄×M₁₆	Dic₁₃	1	8	yes	complemented
110	C₂×#97	C₂×D₁₃	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
113	#383	—	26	2	no	—
114	#382	—	13	4	no	—
115	#381	—	26	2	no	—
116	#380	—	13	4	no	—
117	C₄×C₄⋊C₄	—	13	4	no	—
118	C₄×C₂²⋊C₄	—	26	2	no	—
119	#376	—	26	2	no	—
120	#340	—	13	8	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	4	4	no	—
124	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₁₃	#532	1	2	yes	—
126	C₂²×C₁₃	#575	1	4	yes	—
127	C₄×C₁₃	—	2	4	no	—
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	C₂×#117	1	2	yes	—
131	C₄×C₁₃	#533	1	1	yes	characteristic
132	C₄×C₁₃	#533	1	1	yes	characteristic
133	Dic₁₃	—	8	8	no	—
134	C₂³×C₄	D₅₂	1	2	yes	—
135	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
136	C₂×M₁₆	—	2	4	no	—
137	C₂×M₁₆	—	2	4	no	—
138	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
139	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
140	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
141	C₂×C₄⋊C₄	—	13	8	no	—
142	C₂×C₂²⋊C₄	—	26	4	no	—
143	C₂×C₄²	—	26	4	no	—
144	C₂×C₄²	Dic₂₆	1	4	yes	—
145	C₂×C₄²	D₅₂	1	2	yes	—
146	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
147	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
148	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
149	#107	—	26	8	no	—
150	#106	—	26	8	no	—
151	#97	C₂×Dic₁₃	1	8	yes	—
152	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
153	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
154	#95	—	26	4	no	—
155	#95	—	26	4	no	—
156	#95	—	26	4	no	—
157	#95	—	26	4	no	—
158	C₂×C₁₃	—	4	2	no	—
159	C₂×C₁₃	#2285	1	1	yes	characteristic
160	C₂×C₁₃	#2911	1	2	yes	—
161	C₂×C₁₃	#3183	1	4	yes	—
162	C₂⁴	C₂×D₅₂	1	1	yes	characteristic
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	—	26	8	no	—
167	C₂²×C₄	C₄⋊Dic₁₃	1	8	yes	—
168	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
169	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₅₂	1	2	yes	—
171	C₂²×C₄	C₂×D₅₂	1	2	yes	—
172	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
173	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
174	M₁₆	—	2	16	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	#6127	1	4	yes	—
178	C₂×C₈	#6127	1	4	yes	—
179	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
180	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
181	C₄⋊C₄	—	26	8	no	—
182	C₄⋊C₄	—	26	8	no	—
183	C₂²⋊C₄	—	52	8	no	—
184	C₄²	—	52	8	no	—
185	C₄²	C₄⋊Dic₁₃	1	8	yes	—
186	C₄²	C₂×Dic₂₆	1	2	yes	—
187	C₄²	C₂×Dic₂₆	1	2	yes	—
188	C₄²	C₂²×Dic₁₃	1	2	yes	—
189	C₄²	C₂²×Dic₁₃	1	2	yes	—
190	C₁₃	#20825	1	1	yes	characteristic, complemented, Sylow
191	C₂³	—	2	2	no	—
192	C₂³	—	2	4	no	—
193	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
194	C₂³	C₂²×D₅₂	1	1	yes	characteristic
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	52	8	no	—
203	C₂×C₄	—	52	8	no	—
204	C₂×C₄	—	52	8	no	—
205	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
208	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
209	C₂×C₄	C₂×#6127	1	2	yes	—
210	C₂×C₄	C₂×#6127	1	2	yes	—
211	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	#88180	1	8	yes	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	4	no	—
217	C₂²	#11719182	1	2	yes	—
218	C₂²	#11719453	1	4	yes	—
219	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	2	4	no	—
222	C₄	—	2	4	no	—
223	C₄	—	104	8	no	—
224	C₄	#11719175	1	1	yes	characteristic
225	C₄	#11719175	1	1	yes	characteristic
226	C₄	C₂×#88180	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#13117243	1	1	yes	characteristic
229	C₂	#13123426	1	2	yes	—
230	C₂	#13125565	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂²×D₄
prime radical	C₂⁴×C₁₃	C₂×D₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#20825
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13115000	C₂	semidirect
3. #13114967	C₂	semidirect
5. #13114733	C₂	semidirect
9. C₄×C₁₃×M₁₆	C₄	semidirect
109. C₄×M₁₆	Dic₁₃	semidirect
190. C₁₃	#20825	semidirect

Maximal subgroups

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

subgroup	index
#13117234	2
#13115000	2
#13114967	2
#13114869	2
#13114733	2
C₂×C₄×C₁₃×M₁₆	2
#20825	13