Subgroups — #13123965

Order 1664.

394 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6127	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718585	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718498	C₂	1	2	yes	maximal, complemented
5	#11718494	C₂	1	2	yes	maximal, complemented
6	#11718390	C₂	1	4	yes	maximal, complemented
7	#11718349	C₂	1	2	yes	maximal, complemented
8	C₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
9	C₂×#6127	C₄	1	1	yes	characteristic
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₄×C₂²⋊D₁₃	C₄	1	2	yes	—
12	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
13	#88180	C₄	1	1	yes	characteristic
14	C₂×#6099	C₂²	1	2	yes	—
15	C₂×#6094	C₂²	1	4	yes	—
16	C₄×D₅₂	C₄	1	1	yes	characteristic
17	C₄×D₅₂	C₂²	1	1	yes	characteristic
18	#88126	C₄	1	2	yes	—
19	C₄²×D₁₃	C₂²	1	2	yes	—
20	C₄×Dic₂₆	C₄	1	1	yes	characteristic
21	C₄×Dic₂₆	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
25	C₄×C₈×C₁₃	C₂²	1	2	yes	—
26	#88060	C₂²	1	4	yes	—
27	#88054	C₂²	1	4	yes	—
28	C₈×Dic₁₃	C₂²	1	4	yes	—
29	C₄⋊#1055	C₂²	1	2	yes	—
30	C₄⋊#1055	C₂²	1	2	yes	—
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
35	#6127	—	2	4	no	—
36	C₂×D₅₂	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₃	C₂³	1	2	yes	—
40	C₂×Dic₂₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂×C₈×C₁₃	C₂³	1	2	yes	—
43	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
48	#6103	C₂×C₄	1	2	yes	—
49	#6103	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
52	#6101	C₂×C₄	1	2	yes	—
53	#6101	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₃	C₂³	1	2	yes	—
57	#6099	—	4	2	no	—
58	C₂×#1055	C₂³	1	4	yes	—
59	#6094	—	4	4	no	—
60	#2991	—	13	1	no	maximal, Sylow
61	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	1	no	—
66	C₂×C₄×C₁₃	—	2	1	no	—
67	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²⋊D₁₃	—	2	4	no	—
74	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
77	D₅₂	—	2	2	no	—
78	C₄×D₁₃	—	2	4	no	—
79	C₄×D₁₃	—	2	2	no	—
80	C₄×D₁₃	—	2	2	no	—
81	C₄×D₁₃	—	2	4	no	—
82	Dic₂₆	M₁₆	1	4	yes	complemented
83	C₈×C₁₃	—	2	2	no	—
84	C₈×C₁₃	—	2	2	no	—
85	#1055	—	2	4	no	—
86	C₄×#41	—	13	1	no	—
87	C₈×Q₈	—	13	2	no	—
88	#436	—	13	4	no	—
89	#436	—	13	2	no	—
90	#431	—	13	2	no	—
91	#431	D₁₃	1	1	yes	characteristic, complemented
92	#423	—	13	1	no	—
93	#423	—	13	2	no	—
94	C₂²×C₁₃	—	2	1	no	—
95	C₂²×C₁₃	—	2	1	no	—
96	C₂²×C₁₃	—	2	1	no	—
97	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
98	C₂×D₁₃	—	2	2	no	—
99	C₂×D₁₃	—	2	2	no	—
100	C₂×D₁₃	—	2	2	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	—	2	1	no	—
104	C₄×C₁₃	C₂×M₁₆	1	2	yes	—
105	C₄×C₁₃	#131	1	2	yes	—
106	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
107	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
108	Dic₁₃	—	2	2	no	—
109	Dic₁₃	—	2	2	no	—
110	Dic₁₃	C₂×M₁₆	1	4	yes	—
111	C₂×#41	—	13	1	no	—
112	C₂×M₁₆	—	13	2	no	—
113	C₂×M₁₆	—	13	4	no	—
114	C₄×Q₈	—	13	1	no	—
115	C₄×Q₈	—	13	1	no	—
116	C₄×D₄	—	13	2	no	—
117	C₄×D₄	—	13	2	no	—
118	C₄×D₄	—	13	1	no	—
119	C₄×D₄	—	13	1	no	—
120	#117	—	13	1	no	—
121	#117	—	13	2	no	—
122	C₂×C₄²	—	13	2	no	—
123	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
124	C₄⋊C₈	—	13	4	no	—
125	C₄⋊C₈	—	13	2	no	—
126	C₄⋊C₈	—	13	2	no	—
127	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
129	C₂²⋊C₈	—	13	4	no	—
130	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
131	C₄×C₈	—	13	4	no	—
132	C₄×C₈	C₂×D₁₃	1	2	yes	—
133	C₂×C₁₃	—	4	1	no	—
134	C₂×C₁₃	#520	1	1	yes	characteristic
135	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
136	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
137	D₁₃	—	4	2	no	—
138	#41	—	26	4	no	—
139	C₂×Q₈	—	13	1	no	—
140	C₂×D₄	—	13	2	no	—
141	C₂×D₄	—	13	1	no	—
142	C₂²×C₄	—	13	2	no	—
143	C₂²×C₄	—	13	2	no	—
144	C₂²×C₄	—	13	2	no	—
145	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
146	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
148	M₁₆	—	52	4	no	—
149	M₁₆	—	52	2	no	—
150	C₂×C₈	—	13	4	no	—
151	C₂×C₈	C₂²×D₁₃	1	2	yes	—
152	C₂×C₈	C₂²×D₁₃	1	2	yes	—
153	C₄⋊C₄	—	13	2	no	—
154	C₄⋊C₄	—	13	2	no	—
155	C₄⋊C₄	—	13	1	no	—
156	C₄⋊C₄	—	13	1	no	—
157	C₂²⋊C₄	—	13	2	no	—
158	C₂²⋊C₄	—	13	2	no	—
159	C₂²⋊C₄	—	13	2	no	—
160	C₄²	—	13	2	no	—
161	C₄²	—	13	2	no	—
162	C₄²	—	13	2	no	—
163	C₄²	C₄×D₁₃	1	1	yes	characteristic
164	C₄²	C₄×D₁₃	1	1	yes	characteristic
165	C₄²	C₂²×D₁₃	1	1	yes	characteristic
166	C₄²	C₂²×D₁₃	1	1	yes	characteristic
167	C₁₃	#2991	1	1	yes	characteristic, complemented, Sylow
168	C₂³	—	13	2	no	—
169	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
170	Q₈	—	13	4	no	—
171	D₄	—	26	4	no	—
172	D₄	—	26	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	13	2	no	—
178	C₂×C₄	—	13	2	no	—
179	C₂×C₄	—	13	2	no	—
180	C₂×C₄	—	26	4	no	—
181	C₂×C₄	—	26	4	no	—
182	C₂×C₄	—	26	2	no	—
183	C₂×C₄	—	26	2	no	—
184	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	26	4	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	26	2	no	—
197	C₂²	—	26	2	no	—
198	C₂²	—	26	2	no	—
199	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	13	4	no	—
204	C₄	—	26	2	no	—
205	C₄	—	26	2	no	—
206	C₄	#88155	1	2	yes	—
207	C₄	M₁₆×D₁₃	1	2	yes	—
208	C₄	#88256	1	1	yes	characteristic
209	C₄	#88257	1	1	yes	characteristic
210	C₂	—	4	1	no	—
211	C₂	—	52	2	no	—
212	C₂	#11719228	1	1	yes	characteristic
213	C₂	C₂×#88155	1	1	yes	characteristic
214	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#431	C₂
prime radical	C₂×#6127	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#431	D₁₃
hypocenter	C₁₃	#2991
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. C₁₃×#431	C₂	semidirect
3. #11718585	C₂	semidirect
4. #11718498	C₂	semidirect
5. #11718494	C₂	semidirect
6. #11718390	C₂	semidirect
7. #11718349	C₂	semidirect
8. C₈×Dic₂₆	C₂	semidirect
82. Dic₂₆	M₁₆	semidirect
91. #431	D₁₃	semidirect
167. C₁₃	#2991	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6127	2
C₁₃×#431	2
#11718585	2
#11718498	2
#11718494	2
#11718390	2
#11718349	2
C₈×Dic₂₆	2
#2991	13