Subgroups — #13123799

Order 1664.

412 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6129	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718788	C₂	1	2	yes	maximal, complemented
4	M₁₆×Dic₁₃	C₂	1	2	yes	maximal, complemented
5	#11718673	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718494	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718492	C₂	1	1	yes	maximal, characteristic, complemented
8	#11718389	C₂	1	2	yes	maximal, complemented
9	#11718379	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#6129	C₄	1	1	yes	characteristic
13	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	D₄×Dic₁₃	C₄	1	1	yes	characteristic
16	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
18	C₂×#6099	C₂²	1	2	yes	—
19	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₃	C₂²	1	1	yes	characteristic
21	#88146	C₄	1	1	yes	characteristic
22	#88141	C₄	1	1	yes	characteristic
23	#88135	C₄	1	2	yes	—
24	#88131	C₄	1	2	yes	—
25	C₄²×D₁₃	C₂²	1	1	yes	characteristic
26	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
27	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₂²⋊#1055	C₂²	1	2	yes	—
31	#88060	C₂²	1	2	yes	—
32	#88055	C₂²	1	2	yes	—
33	#88055	C₂²	1	2	yes	—
34	#88054	C₂²	1	2	yes	—
35	#88054	C₂²	1	2	yes	—
36	C₈×Dic₁₃	C₂²	1	2	yes	—
37	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
38	C₈×Dic₁₃	C₂²	1	1	yes	characteristic
39	C₄⋊#1055	C₂²	1	1	yes	characteristic
40	C₄×#1055	C₂²	1	1	yes	characteristic
41	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
44	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₂²×Dic₁₃	C₂×C₄	1	2	yes	—
46	#6129	—	2	4	no	—
47	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₁₃	C₂³	1	1	yes	characteristic
50	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
51	C₁₃×M₁₆	—	2	4	no	—
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂³	1	2	yes	—
54	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
56	C₁₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
57	C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
59	#6103	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
61	#6101	C₂×C₄	1	2	yes	—
62	#6101	C₂×C₄	1	2	yes	—
63	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
66	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₃	C₂³	1	2	yes	—
68	C₄×Dic₁₃	C₂³	1	1	yes	characteristic
69	#6099	—	2	4	no	—
70	C₂×#1055	C₂³	1	1	yes	characteristic
71	C₂×#1055	C₂³	1	2	yes	—
72	C₂×#1055	C₂³	1	1	yes	characteristic
73	C₈×D₁₃	—	2	2	no	—
74	C₈×D₁₃	—	2	2	no	—
75	#2965	—	13	1	no	maximal, Sylow
76	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂²×D₁₃	C₂²×C₄	1	1	yes	characteristic
78	D₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂²⋊D₁₃	—	2	4	no	—
86	C₂×Dic₁₃	—	2	4	no	—
87	C₂×Dic₁₃	—	2	4	no	—
88	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
90	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
91	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₁₃	—	2	1	no	—
94	C₄×D₁₃	—	2	1	no	—
95	C₄×D₁₃	—	2	2	no	—
96	C₄×D₁₃	M₁₆	1	4	yes	complemented
97	Dic₂₆	—	2	2	no	—
98	C₈×C₁₃	—	2	1	no	—
99	C₈×C₁₃	—	2	2	no	—
100	C₈×C₁₃	#41	1	2	yes	complemented
101	#1055	—	2	1	no	—
102	#1055	—	2	2	no	—
103	#1055	#41	1	2	yes	complemented
104	C₄×#41	—	13	1	no	—
105	#446	—	13	1	no	—
106	#446	—	13	1	no	—
107	#436	—	13	1	no	—
108	#436	D₁₃	1	1	yes	characteristic, complemented
109	#435	—	13	2	no	—
110	#435	—	13	2	no	—
111	#431	—	13	1	no	—
112	#431	—	13	2	no	—
113	C₄×M₁₆	—	13	2	no	—
114	C₂×C₄×C₈	—	13	1	no	—
115	C₂²×C₁₃	—	2	2	no	—
116	C₂²×C₁₃	—	2	2	no	—
117	C₂²×C₁₃	—	2	2	no	—
118	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₁₃	—	2	1	no	—
120	C₂×D₁₃	—	2	1	no	—
121	C₂×D₁₃	C₂×M₁₆	1	2	yes	—
122	C₄×C₁₃	—	2	2	no	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
125	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	Dic₁₃	—	2	1	no	—
128	Dic₁₃	—	2	2	no	—
129	Dic₁₃	—	2	2	no	—
130	Dic₁₃	C₂×M₁₆	1	2	yes	—
131	Dic₁₃	#131	1	2	yes	—
132	C₂×#41	—	13	1	no	—
133	C₂×M₁₆	—	13	2	no	—
134	C₂×M₁₆	C₂×D₁₃	1	2	yes	—
135	C₂²×C₈	—	13	1	no	—
136	C₂²×C₈	—	13	1	no	—
137	C₄×Q₈	—	13	1	no	—
138	C₄×Q₈	—	13	1	no	—
139	C₄×D₄	—	13	1	no	—
140	C₄×D₄	—	13	2	no	—
141	C₄×D₄	—	13	2	no	—
142	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
143	#117	—	13	1	no	—
144	#117	—	13	2	no	—
145	C₂×C₄²	—	13	2	no	—
146	C₂×C₄²	—	13	1	no	—
147	C₄⋊C₈	—	13	2	no	—
148	C₄⋊C₈	—	13	2	no	—
149	C₄⋊C₈	—	13	1	no	—
150	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂²⋊C₈	—	13	2	no	—
152	C₂²⋊C₈	—	13	2	no	—
153	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
154	#97	—	13	2	no	—
155	#97	—	13	2	no	—
156	C₄×C₈	—	13	1	no	—
157	C₄×C₈	—	13	1	no	—
158	C₄×C₈	—	13	2	no	—
159	C₄×C₈	—	13	1	no	—
160	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
161	C₂×C₁₃	—	4	2	no	—
162	C₂×C₁₃	C₄×#41	1	1	yes	characteristic
163	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
164	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
165	D₁₃	—	2	2	no	—
166	#41	—	26	4	no	—
167	C₂×Q₈	—	13	1	no	—
168	C₂×D₄	—	13	2	no	—
169	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	13	1	no	—
171	C₂²×C₄	—	13	2	no	—
172	C₂²×C₄	—	13	2	no	—
173	C₂²×C₄	—	13	1	no	—
174	C₂²×C₄	—	13	1	no	—
175	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
176	M₁₆	—	2	4	no	—
177	M₁₆	—	26	4	no	—
178	C₂×C₈	—	13	1	no	—
179	C₂×C₈	—	13	1	no	—
180	C₂×C₈	—	13	2	no	—
181	C₂×C₈	—	26	2	no	—
182	C₂×C₈	—	26	2	no	—
183	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
184	C₂×C₈	C₂²×D₁₃	1	2	yes	—
185	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
186	C₄⋊C₄	—	13	2	no	—
187	C₄⋊C₄	—	13	2	no	—
188	C₄⋊C₄	—	13	1	no	—
189	C₄⋊C₄	C₄×D₁₃	1	1	yes	characteristic
190	C₂²⋊C₄	—	13	2	no	—
191	C₂²⋊C₄	—	13	2	no	—
192	C₂²⋊C₄	C₄×D₁₃	1	2	yes	—
193	C₄²	—	13	2	no	—
194	C₄²	—	13	1	no	—
195	C₄²	—	13	1	no	—
196	C₄²	—	13	2	no	—
197	C₄²	—	13	1	no	—
198	C₄²	—	13	2	no	—
199	C₄²	C₂²×D₁₃	1	1	yes	characteristic
200	C₁₃	#2965	1	1	yes	characteristic, complemented, Sylow
201	C₂³	—	13	1	no	—
202	C₂³	C₂×C₄×D₁₃	1	2	yes	—
203	Q₈	—	26	2	no	—
204	D₄	—	2	2	no	—
205	D₄	—	26	4	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	13	1	no	—
209	C₂×C₄	—	13	1	no	—
210	C₂×C₄	—	13	4	no	—
211	C₂×C₄	—	13	2	no	—
212	C₂×C₄	—	13	1	no	—
213	C₂×C₄	—	13	2	no	—
214	C₂×C₄	—	26	2	no	—
215	C₂×C₄	—	26	4	no	—
216	C₂×C₄	—	26	4	no	—
217	C₂×C₄	—	26	1	no	—
218	C₂×C₄	—	26	1	no	—
219	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
220	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
222	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	2	1	no	—
225	C₈	—	13	2	no	—
226	C₈	—	26	1	no	—
227	C₈	—	26	2	no	—
228	C₈	#6129	1	2	yes	complemented
229	C₂²	—	2	2	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	13	2	no	—
233	C₂²	—	26	1	no	—
234	C₂²	—	26	1	no	—
235	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	13	2	no	—
239	C₄	—	13	2	no	—
240	C₄	—	26	2	no	—
241	C₄	—	26	2	no	—
242	C₄	—	26	1	no	—
243	C₄	M₁₆×D₁₃	1	2	yes	—
244	C₄	C₂×#6129	1	1	yes	characteristic
245	C₄	#41×D₁₃	1	1	yes	characteristic
246	C₂	—	4	2	no	—
247	C₂	—	26	2	no	—
248	C₂	C₄×#6129	1	1	yes	characteristic
249	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
250	C₂	D₁₃×#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#436	C₂
prime radical	C₂×#6129	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#436	D₁₃
hypocenter	C₁₃	#2965
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
2. C₁₃×#436	C₂	semidirect
3. #11718788	C₂	semidirect
4. M₁₆×Dic₁₃	C₂	semidirect
5. #11718673	C₂	semidirect
6. #11718494	C₂	semidirect
7. #11718492	C₂	semidirect
8. #11718389	C₂	semidirect
9. #11718379	C₂	semidirect
10. C₄×C₈×D₁₃	C₂	semidirect
11. #11718340	C₂	semidirect
96. C₄×D₁₃	M₁₆	semidirect
100. C₈×C₁₃	#41	semidirect
103. #1055	#41	semidirect
108. #436	D₁₃	semidirect
200. C₁₃	#2965	semidirect
228. C₈	#6129	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6129	2
C₁₃×#436	2
#11718788	2
M₁₆×Dic₁₃	2
#11718673	2
#11718494	2
#11718492	2
#11718389	2
#11718379	2
C₄×C₈×D₁₃	2
#11718340	2
#2965	13