Subgroups — #13127239

Order 1664.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719270	C₂	1	2	yes	maximal, complemented
2	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718776	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718776	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718767	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#88058	C₂	1	1	yes	maximal, characteristic, complemented
7	#11718508	C₂	1	2	yes	maximal, complemented
8	#11718398	C₂	1	2	yes	maximal, complemented
9	#11718396	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₀₄	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718344	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
15	#88193	—	2	2	no	—
16	#88184	C₂²	1	2	yes	—
17	#88184	C₂²	1	2	yes	—
18	#88180	C₂²	1	2	yes	—
19	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂×D₁₀₄	C₂²	1	1	yes	characteristic
21	#88152	—	2	2	no	—
22	#88137	—	2	2	no	—
23	#88135	—	2	2	no	—
24	C₄×D₅₂	C₂²	1	2	yes	—
25	#88124	C₂²	1	2	yes	—
26	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	#88061	C₂²	1	2	yes	—
31	#88061	C₂²	1	2	yes	—
32	#88061	C₂²	1	2	yes	—
33	#88058	C₂²	1	2	yes	—
34	#88058	C₂²	1	1	yes	characteristic
35	#88058	C₂²	1	1	yes	characteristic
36	#88058	C₂²	1	1	yes	characteristic
37	#88057	C₂²	1	2	yes	—
38	#88057	C₂²	1	2	yes	—
39	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₂²⋊D₁₃	—	2	2	no	—
43	C₂×C₂²⋊D₁₃	—	2	2	no	—
44	C₂²×Dic₁₃	—	2	2	no	—
45	C₂×D₅₂	C₂³	1	2	yes	—
46	C₂×C₄×D₁₃	—	2	2	no	—
47	C₂×C₈×C₁₃	—	2	2	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
54	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
55	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂²⋊Dic₁₃	—	2	2	no	—
58	#6103	—	2	2	no	—
59	#6103	—	2	2	no	—
60	#6103	—	2	2	no	—
61	C₄⋊Dic₁₃	—	2	2	no	—
62	C₄⋊Dic₁₃	C₂³	1	2	yes	—
63	C₄⋊Dic₁₃	C₂³	1	2	yes	—
64	C₄⋊Dic₁₃	C₂³	1	2	yes	—
65	#6101	—	2	2	no	—
66	#6101	—	2	2	no	—
67	C₄×Dic₁₃	—	2	2	no	—
68	D₁₀₄	—	2	2	no	—
69	#3339	—	13	1	no	maximal, Sylow
70	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₁₃	—	2	2	no	—
73	D₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
83	C₂²⋊D₁₃	—	4	4	no	—
84	C₂²⋊D₁₃	—	8	2	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	—	2	2	no	—
88	C₂×Dic₁₃	—	8	2	no	—
89	D₅₂	—	2	2	no	—
90	D₅₂	—	2	2	no	—
91	C₄×D₁₃	—	8	2	no	—
92	C₈×C₁₃	—	2	1	no	—
93	C₈×C₁₃	—	2	1	no	—
94	C₈×C₁₃	—	2	1	no	—
95	C₈×C₁₃	#41	1	2	yes	—
96	#553	—	13	2	no	—
97	#499	—	13	1	no	—
98	#482	—	13	2	no	—
99	#481	—	13	2	no	—
100	#478	—	13	2	no	—
101	#466	—	13	1	no	—
102	#466	—	13	1	no	—
103	C₄×D₈	—	13	1	no	—
104	C₈×D₄	D₁₃	1	1	yes	characteristic, complemented
105	#427	—	13	1	no	—
106	C₂×#107	—	13	1	no	—
107	C₂²×C₁₃	—	2	1	no	—
108	C₂²×C₁₃	—	2	1	no	—
109	C₂²×C₁₃	—	2	1	no	—
110	C₂²×C₁₃	—	2	1	no	—
111	C₂²×C₁₃	—	2	1	no	—
112	C₂²×C₁₃	#135	1	2	yes	—
113	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₁₃	—	4	2	no	—
115	C₂×D₁₃	—	4	2	no	—
116	C₂×D₁₃	—	4	2	no	—
117	C₄×C₁₃	—	2	1	no	—
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
123	Dic₁₃	—	4	2	no	—
124	Dic₁₃	—	4	2	no	—
125	Dic₁₃	—	4	2	no	—
126	C₂×D₈	—	13	1	no	—
127	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
129	#126	—	26	2	no	—
130	#125	—	13	2	no	—
131	#123	—	26	2	no	—
132	#121	—	13	2	no	—
133	#121	—	13	2	no	—
134	#121	—	26	2	no	—
135	C₄×D₄	—	13	2	no	—
136	C₄×D₄	—	26	2	no	—
137	C₄×D₄	C₂×D₁₃	1	1	yes	characteristic
138	#117	—	13	2	no	—
139	C₂×C₄⋊C₄	—	13	2	no	—
140	#107	—	13	1	no	—
141	#107	—	13	1	no	—
142	#107	—	13	2	no	—
143	#107	—	13	1	no	—
144	#106	—	13	2	no	—
145	#106	—	13	2	no	—
146	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
147	#102	—	13	2	no	—
148	#102	—	13	2	no	—
149	#102	—	13	2	no	—
150	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
151	C₂²⋊C₈	C₂×D₁₃	1	1	yes	characteristic
152	C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
153	C₂×C₁₃	—	2	2	no	—
154	C₂×C₁₃	—	4	1	no	—
155	C₂×C₁₃	#547	1	1	yes	characteristic
156	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
157	C₂×C₁₃	#576	1	1	yes	characteristic
158	D₁₃	—	8	2	no	—
159	C₂×D₄	—	13	2	no	—
160	C₂×D₄	—	26	2	no	—
161	C₂×D₄	—	26	2	no	—
162	C₂×D₄	D₅₂	1	1	yes	characteristic
163	C₂²×C₄	—	26	2	no	—
164	C₂²×C₄	—	26	2	no	—
165	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
166	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
167	D₈	—	26	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
171	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
174	C₄⋊C₄	—	13	2	no	—
175	C₄⋊C₄	—	13	2	no	—
176	C₄⋊C₄	—	13	2	no	—
177	C₄⋊C₄	—	26	2	no	—
178	C₄⋊C₄	—	26	2	no	—
179	C₄⋊C₄	—	26	2	no	—
180	C₄⋊C₄	D₅₂	1	1	yes	characteristic
181	C₂²⋊C₄	—	26	2	no	—
182	C₂²⋊C₄	—	26	2	no	—
183	C₂²⋊C₄	—	26	2	no	—
184	C₂²⋊C₄	—	26	2	no	—
185	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
186	C₂²⋊C₄	D₅₂	1	1	yes	characteristic
187	C₄²	—	26	2	no	—
188	C₄²	C₂²×D₁₃	1	1	yes	characteristic
189	C₁₃	#3339	1	1	yes	characteristic, complemented, Sylow
190	C₂³	—	26	2	no	—
191	C₂³	C₂×D₅₂	1	1	yes	characteristic
192	C₂³	C₂×D₅₂	1	1	yes	characteristic
193	D₄	—	2	2	no	—
194	D₄	—	26	2	no	—
195	D₄	—	26	2	no	—
196	D₄	—	52	4	no	—
197	D₄	—	104	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	26	2	no	—
203	C₂×C₄	—	26	2	no	—
204	C₂×C₄	—	26	2	no	—
205	C₂×C₄	—	104	2	no	—
206	C₂×C₄	—	104	2	no	—
207	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
208	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
209	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
210	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
211	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
212	C₈	—	2	1	no	—
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	#6129	1	2	yes	—
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	52	2	no	—
222	C₂²	—	52	2	no	—
223	C₂²	—	52	2	no	—
224	C₂²	#88158	1	2	yes	—
225	C₂²	C₂²×D₅₂	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	52	2	no	—
231	C₄	—	52	2	no	—
232	C₄	—	52	2	no	—
233	C₄	C₂×#6129	1	1	yes	characteristic
234	C₄	#88257	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	—	4	1	no	—
237	C₂	—	104	2	no	—
238	C₂	#11719260	1	1	yes	characteristic
239	C₂	C₂×#88158	1	1	yes	characteristic
240	C₂	#11719457	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₂
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₈×D₄×C₁₃	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	C₈×D₄	D₁₃
hypocenter	C₁₃	#3339
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #11719270	C₂	semidirect
2. C₈×D₄×C₁₃	C₂	semidirect
3. #11718776	C₂	semidirect
4. #11718776	C₂	semidirect
5. #11718767	C₂	semidirect
6. C₂×#88058	C₂	semidirect
7. #11718508	C₂	semidirect
8. #11718398	C₂	semidirect
9. #11718396	C₂	semidirect
10. C₄×D₁₀₄	C₂	semidirect
11. #11718344	C₂	semidirect
104. C₈×D₄	D₁₃	semidirect
189. C₁₃	#3339	semidirect

Maximal subgroups

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

subgroup	index
#11719270	2
C₈×D₄×C₁₃	2
#11718776	2
#11718767	2
C₂×#88058	2
#11718508	2
#11718398	2
#11718396	2
C₄×D₁₀₄	2
#11718344	2
#3339	13