Subgroups — #13128723

Order 1664.

992 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719433	C₂	1	4	yes	maximal, complemented
3	#11719402	C₂	1	4	yes	maximal, complemented
4	#11719347	C₂	1	8	yes	maximal, complemented
5	#11719289	C₂	1	4	yes	maximal, complemented
6	#11719275	C₂	1	4	yes	maximal, complemented
7	#11719231	C₂	1	2	yes	maximal
8	#11719221	C₂	1	2	yes	maximal, complemented
9	C₄×#6127	C₂	1	1	yes	maximal, characteristic
10	C₂×#88123	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#6127	C₂²	1	1	yes	characteristic
12	C₂²×Dic₂₆	C₂²	1	2	yes	—
13	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₁₃×#125	C₂²	1	2	yes	—
15	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
17	C₁₃×#117	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	#88200	C₂²	1	8	yes	—
20	Q₈×Dic₁₃	C₂²	1	8	yes	—
21	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
22	#88180	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	C₂²	1	4	yes	—
24	#88152	C₂²	1	16	yes	—
25	#88151	C₂²	1	8	yes	—
26	#88146	C₂²	1	8	yes	—
27	#88144	C₂²	1	8	yes	—
28	#88144	C₂²	1	16	yes	—
29	#88141	C₂²	1	8	yes	—
30	#88140	C₂²	1	8	yes	—
31	#88131	C₂²	1	8	yes	—
32	#88129	C₂²	1	4	yes	—
33	C₄×D₅₂	C₂²	1	2	yes	—
34	#88126	C₂²	1	2	yes	—
35	C₄²×D₁₃	C₂²	1	2	yes	—
36	#88123	C₂²	1	2	yes	—
37	#88123	C₂²	1	4	yes	—
38	#88123	C₂²	1	2	yes	—
39	#88123	C₂²	1	4	yes	—
40	C₄×Dic₂₆	C₂²	1	2	yes	—
41	C₄×Dic₂₆	C₂²	1	8	yes	—
42	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₁₃	C₂³	1	2	yes	—
46	C₂²×Dic₁₃	C₂³	1	4	yes	—
47	#6127	—	2	4	no	—
48	C₂×D₅₂	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₁₃	C₂³	1	2	yes	—
50	C₂×C₄×D₁₃	C₂³	1	4	yes	—
51	C₂×Dic₂₆	—	2	8	no	—
52	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
53	C₂×Dic₂₆	C₂³	1	4	yes	—
54	C₂×Dic₂₆	C₂³	1	4	yes	—
55	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
56	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
57	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
58	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
59	C₄²×C₁₃	C₂³	1	4	yes	—
60	C₄²×C₁₃	C₂³	1	2	yes	—
61	C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₄²×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²⋊Dic₁₃	C₂³	1	2	yes	—
64	#6103	C₂³	1	4	yes	—
65	#6103	C₂³	1	16	yes	—
66	C₄⋊Dic₁₃	C₂³	1	2	yes	—
67	C₄⋊Dic₁₃	C₂³	1	8	yes	—
68	C₄⋊Dic₁₃	C₂³	1	8	yes	—
69	C₄⋊Dic₁₃	C₂³	1	16	yes	—
70	#6101	C₂³	1	4	yes	—
71	#6101	C₂³	1	16	yes	—
72	C₄×Dic₁₃	C₂³	1	4	yes	—
73	C₄×Dic₁₃	C₂³	1	16	yes	—
74	C₄×Dic₁₃	C₂³	1	2	yes	—
75	#3509	—	13	1	no	maximal, Sylow
76	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂²×D₁₃	C₂⁴	1	2	yes	—
78	Q₈×C₁₃	—	2	8	no	—
79	C₂×C₄×C₁₃	—	2	4	no	—
80	C₂×C₄×C₁₃	#41	1	4	yes	—
81	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
82	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
86	C₂²⋊D₁₃	—	2	4	no	—
87	C₂×Dic₁₃	—	2	8	no	—
88	C₂×Dic₁₃	C₂⁴	1	4	yes	—
89	C₂×Dic₁₃	C₂⁴	1	2	yes	—
90	C₂×Dic₁₃	C₂⁴	1	8	yes	—
91	D₅₂	—	2	2	no	—
92	C₄×D₁₃	—	2	4	no	—
93	C₄×D₁₃	—	2	8	no	—
94	Dic₂₆	—	2	8	no	—
95	Dic₂₆	—	2	8	no	—
96	Dic₂₆	—	2	2	no	—
97	#557	—	13	4	no	—
98	#556	—	13	4	no	—
99	#552	—	13	8	no	—
100	#541	—	13	4	no	—
101	#536	—	13	2	no	—
102	#533	—	13	4	no	—
103	#533	D₁₃	1	1	yes	characteristic, complemented
104	C₂×C₄⋊Q₈	—	13	1	no	—
105	#519	—	13	2	no	—
106	C₄×#41	—	13	1	no	—
107	C₂²×C₁₃	—	2	2	no	—
108	C₂²×C₁₃	C₂×#41	1	2	yes	—
109	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
110	C₂×D₁₃	—	2	2	no	—
111	C₂×D₁₃	—	2	4	no	—
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	2	8	no	—
114	C₄×C₁₃	C₂×#41	1	2	yes	—
115	C₄×C₁₃	C₂×#41	1	2	yes	—
116	C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
117	Dic₁₃	—	2	8	no	—
118	Dic₁₃	—	2	2	no	—
119	Dic₁₃	—	2	4	no	—
120	C₂×#41	—	13	1	no	—
121	C₂²×Q₈	—	13	2	no	—
122	C₄⋊Q₈	—	13	2	no	—
123	C₄⋊Q₈	—	13	2	no	—
124	C₄⋊Q₈	—	13	4	no	—
125	C₄⋊Q₈	—	13	4	no	—
126	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
127	#126	—	13	16	no	—
128	#125	—	13	8	no	—
129	#125	—	13	16	no	—
130	#125	C₂×D₁₃	1	2	yes	—
131	#124	—	13	4	no	—
132	#123	—	13	8	no	—
133	C₂²⋊Q₈	—	13	8	no	—
134	C₂²⋊Q₈	—	13	8	no	—
135	C₂²⋊Q₈	C₂×D₁₃	1	4	yes	—
136	C₄×Q₈	—	13	8	no	—
137	C₄×Q₈	—	13	8	no	—
138	C₄×Q₈	—	13	2	no	—
139	C₄×Q₈	—	13	8	no	—
140	C₄×Q₈	C₂×D₁₃	1	4	yes	—
141	C₄×D₄	—	13	4	no	—
142	C₄×D₄	—	13	2	no	—
143	#117	—	13	1	no	—
144	#117	—	13	8	no	—
145	#117	—	13	8	no	—
146	#117	—	13	2	no	—
147	#117	C₂×D₁₃	1	2	yes	—
148	C₂×C₄⋊C₄	—	13	4	no	—
149	C₂×C₄²	—	13	2	no	—
150	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
151	C₂×C₁₃	—	2	2	no	—
152	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
153	C₂×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
154	D₁₃	—	4	2	no	—
155	#41	—	26	4	no	—
156	C₂×Q₈	—	13	4	no	—
157	C₂×Q₈	—	13	4	no	—
158	C₂×Q₈	—	13	1	no	—
159	C₂×Q₈	—	26	8	no	—
160	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
161	C₂×D₄	—	13	2	no	—
162	C₂×D₄	—	13	1	no	—
163	C₂²×C₄	—	13	4	no	—
164	C₂²×C₄	—	13	4	no	—
165	C₂²×C₄	—	13	2	no	—
166	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
167	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
168	C₄⋊C₄	—	13	8	no	—
169	C₄⋊C₄	—	13	8	no	—
170	C₄⋊C₄	—	13	2	no	—
171	C₄⋊C₄	—	13	4	no	—
172	C₄⋊C₄	—	13	16	no	—
173	C₄⋊C₄	—	13	16	no	—
174	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
175	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
176	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
177	C₂²⋊C₄	—	13	2	no	—
178	C₂²⋊C₄	—	13	4	no	—
179	C₂²⋊C₄	—	13	16	no	—
180	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
181	C₄²	—	13	2	no	—
182	C₄²	—	13	4	no	—
183	C₄²	—	13	16	no	—
184	C₄²	C₂²×D₁₃	1	1	yes	characteristic
185	C₄²	C₂²×D₁₃	1	1	yes	characteristic
186	C₄²	C₂²×D₁₃	1	2	yes	—
187	C₄²	C₂²×D₁₃	1	4	yes	—
188	C₁₃	#3509	1	1	yes	characteristic, complemented, Sylow
189	C₂³	—	13	2	no	—
190	C₂³	C₂³×D₁₃	1	1	yes	characteristic
191	Q₈	—	2	8	no	—
192	Q₈	—	26	8	no	—
193	Q₈	—	26	8	no	—
194	Q₈	—	26	2	no	—
195	D₄	—	26	4	no	—
196	D₄	—	26	2	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	13	4	no	—
199	C₂×C₄	—	13	8	no	—
200	C₂×C₄	—	13	2	no	—
201	C₂×C₄	—	26	8	no	—
202	C₂×C₄	—	26	4	no	—
203	C₂×C₄	—	26	8	no	—
204	C₂×C₄	#6131	1	4	yes	—
205	C₂×C₄	C₂³×D₁₃	1	8	yes	—
206	C₂×C₄	C₂³×D₁₃	1	2	yes	—
207	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
208	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₁₃	1	2	yes	—
210	C₂²	—	2	2	no	—
211	C₂²	—	26	2	no	—
212	C₂²	—	26	4	no	—
213	C₂²	C₂×#6131	1	2	yes	—
214	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	8	no	—
217	C₄	—	26	2	no	—
218	C₄	—	26	8	no	—
219	C₄	—	26	4	no	—
220	C₄	C₂×#6131	1	2	yes	—
221	C₄	C₂×#6131	1	2	yes	—
222	C₄	#88257	1	4	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	52	2	no	—
225	C₂	C₂²×#6131	1	1	yes	characteristic
226	C₂	C₂×#88257	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
prime radical	C₂×#6127	C₂²
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3509
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
2. #11719433	C₂	semidirect
3. #11719402	C₂	semidirect
4. #11719347	C₂	semidirect
5. #11719289	C₂	semidirect
6. #11719275	C₂	semidirect
8. #11719221	C₂	semidirect
10. C₂×#88123	C₂	semidirect
103. #533	D₁₃	semidirect
188. C₁₃	#3509	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
#11719433	2
#11719402	2
#11719347	2
#11719289	2
#11719275	2
#11719231	2
#11719221	2
C₄×#6127	2
C₂×#88123	2
#3509	13