Subgroups — #13129236

Order 1664.

1114 conjugacy classes of subgroups.

The subgroup lattice has 194 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	#11719431	C₂	1	4	yes	maximal, complemented
3	#11719405	C₂	1	4	yes	maximal, complemented
4	#11719334	C₂	1	8	yes	maximal, complemented
5	#11719284	C₂	1	4	yes	maximal, complemented
6	#11719282	C₂	1	4	yes	maximal, complemented
7	#11719240	C₂	1	4	yes	maximal, complemented
8	#11719182	C₂	1	2	yes	maximal, complemented
9	C₂×#6131	C₂²	1	8	yes	—
10	C₂×#6127	C₂²	1	4	yes	—
11	C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₁₃×#125	C₂²	1	2	yes	—
13	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
14	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
15	C₁₃×#117	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	#88200	C₂²	1	8	yes	—
18	#88184	C₂²	1	4	yes	—
19	#88184	C₂²	1	4	yes	—
20	#88150	C₂²	1	16	yes	—
21	#88149	C₂²	1	8	yes	—
22	#88149	C₂²	1	16	yes	—
23	#88148	C₂²	1	16	yes	—
24	C₄⋊C₄×D₁₃	C₂²	1	8	yes	—
25	#88138	C₂²	1	16	yes	—
26	#88129	C₂²	1	4	yes	—
27	#88128	C₂²	1	2	yes	—
28	#88128	C₂²	1	4	yes	—
29	C₄×D₅₂	C₂²	1	4	yes	—
30	C₄×D₅₂	C₂²	1	4	yes	—
31	C₄×D₅₂	C₂²	1	8	yes	—
32	#88124	C₂²	1	4	yes	—
33	#88123	C₂²	1	2	yes	—
34	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
38	#6131	—	4	16	no	—
39	#6127	—	4	8	no	—
40	C₂×D₅₂	C₂³	1	4	yes	—
41	C₂×D₅₂	C₂³	1	4	yes	—
42	C₂×D₅₂	C₂³	1	4	yes	—
43	C₂×D₅₂	C₂³	1	16	yes	—
44	C₂×C₄×D₁₃	C₂³	1	8	yes	—
45	C₂×C₄×D₁₃	C₂³	1	16	yes	—
46	C₂×Dic₂₆	C₂³	1	4	yes	—
47	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
48	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
50	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
51	C₄²×C₁₃	C₂³	1	2	yes	—
52	C₄²×C₁₃	C₂³	1	4	yes	—
53	C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	#6103	C₂³	1	8	yes	—
56	#6103	C₂³	1	8	yes	—
57	#6103	C₂³	1	16	yes	—
58	C₄⋊Dic₁₃	C₂³	1	4	yes	—
59	C₄⋊Dic₁₃	C₂³	1	4	yes	—
60	#6101	C₂³	1	16	yes	—
61	#3548	—	13	1	no	maximal, Sylow
62	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂²×D₁₃	C₂⁴	1	8	yes	—
64	Q₈×C₁₃	—	2	8	no	—
65	C₂×C₄×C₁₃	—	2	4	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
68	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
70	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
71	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₁₃	—	4	8	no	—
73	C₂×Dic₁₃	C₂⁴	1	8	yes	—
74	D₅₂	—	2	8	no	—
75	D₅₂	—	2	8	no	—
76	D₅₂	—	4	4	no	—
77	D₅₂	—	4	16	no	—
78	C₄×D₁₃	—	2	16	no	—
79	C₄×D₁₃	—	4	16	no	—
80	Dic₂₆	—	4	4	no	—
81	#562	—	13	4	no	—
82	#554	—	13	4	no	—
83	#550	—	13	4	no	—
84	#547	—	13	4	no	—
85	#540	—	13	4	no	—
86	#537	—	13	8	no	—
87	#533	D₁₃	1	1	yes	characteristic, complemented
88	#532	—	13	2	no	—
89	C₂²×C₁₃	—	2	2	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
92	C₂×D₁₃	—	2	8	no	—
93	C₂×D₁₃	—	2	8	no	—
94	C₂×D₁₃	—	2	8	no	—
95	C₄×C₁₃	—	2	2	no	—
96	C₄×C₁₃	—	2	8	no	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	H(5,𝔽₂)	1	4	yes	—
99	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
100	Dic₁₃	—	2	8	no	—
101	C₂×#41	—	13	8	no	—
102	C₂×#41	—	13	4	no	—
103	C₄⋊Q₈	—	13	2	no	—
104	C₄⋊Q₈	C₂×D₁₃	1	2	yes	—
105	#127	—	13	2	no	—
106	#127	—	13	4	no	—
107	#125	—	13	4	no	—
108	#125	C₂×D₁₃	1	2	yes	—
109	#124	—	13	4	no	—
110	#123	—	13	16	no	—
111	C₂²⋊Q₈	—	13	8	no	—
112	C₂²⋊Q₈	—	13	16	no	—
113	C₂²⋊Q₈	C₂×D₁₃	1	4	yes	—
114	#121	—	13	4	no	—
115	#121	—	13	4	no	—
116	#121	—	13	8	no	—
117	#121	—	13	16	no	—
118	#121	—	13	16	no	—
119	C₄×Q₈	C₂×D₁₃	1	4	yes	—
120	C₄×D₄	—	13	4	no	—
121	C₄×D₄	—	13	4	no	—
122	C₄×D₄	—	13	8	no	—
123	#117	C₂×D₁₃	1	2	yes	—
124	C₂×C₄⋊C₄	—	13	8	no	—
125	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
126	C₂×C₁₃	—	4	1	no	—
127	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
128	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
129	D₁₃	—	4	8	no	—
130	#41	—	52	8	no	—
131	#41	—	52	16	no	—
132	C₂×Q₈	—	13	4	no	—
133	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
134	C₂×D₄	—	13	8	no	—
135	C₂×D₄	—	13	4	no	—
136	C₂×D₄	—	13	4	no	—
137	C₂×D₄	—	13	16	no	—
138	C₂×D₄	—	13	4	no	—
139	C₂²×C₄	—	13	8	no	—
140	C₂²×C₄	—	13	16	no	—
141	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
143	C₄⋊C₄	—	13	4	no	—
144	C₄⋊C₄	—	13	16	no	—
145	C₄⋊C₄	—	13	4	no	—
146	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
147	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
148	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
149	C₂²⋊C₄	—	13	16	no	—
150	C₂²⋊C₄	—	13	8	no	—
151	C₂²⋊C₄	—	13	8	no	—
152	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
153	C₄²	C₂²×D₁₃	1	1	yes	characteristic
154	C₄²	C₂²×D₁₃	1	2	yes	—
155	C₄²	C₂²×D₁₃	1	1	yes	characteristic
156	C₄²	C₂²×D₁₃	1	4	yes	—
157	C₁₃	#3548	1	1	yes	characteristic, complemented, Sylow
158	C₂³	—	13	8	no	—
159	C₂³	C₂³×D₁₃	1	1	yes	characteristic
160	Q₈	—	2	8	no	—
161	Q₈	—	52	4	no	—
162	D₄	—	26	8	no	—
163	D₄	—	26	8	no	—
164	D₄	—	52	4	no	—
165	D₄	—	52	16	no	—
166	D₄	—	52	8	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	13	8	no	—
170	C₂×C₄	—	26	16	no	—
171	C₂×C₄	—	52	16	no	—
172	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
173	C₂×C₄	C₂³×D₁₃	1	2	yes	—
174	C₂×C₄	C₂³×D₁₃	1	8	yes	—
175	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
176	C₂×C₄	C₂³×D₁₃	1	2	yes	—
177	C₂²	—	2	1	no	—
178	C₂²	—	2	2	no	—
179	C₂²	—	26	8	no	—
180	C₂²	—	26	8	no	—
181	C₂²	—	26	8	no	—
182	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
183	C₄	—	2	1	no	—
184	C₄	—	2	2	no	—
185	C₄	—	2	8	no	—
186	C₄	—	26	8	no	—
187	C₄	#88254	1	2	yes	—
188	C₄	#88256	1	4	yes	—
189	C₂	—	4	1	no	—
190	C₂	—	52	8	no	—
191	C₂	C₂×#88254	1	1	yes	characteristic
192	C₂	C₂×#88256	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3548
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. #11719431	C₂	semidirect
3. #11719405	C₂	semidirect
4. #11719334	C₂	semidirect
5. #11719284	C₂	semidirect
6. #11719282	C₂	semidirect
7. #11719240	C₂	semidirect
8. #11719182	C₂	semidirect
87. #533	D₁₃	semidirect
157. C₁₃	#3548	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
#11719431	2
#11719405	2
#11719334	2
#11719284	2
#11719282	2
#11719240	2
#11719182	2
#3548	13