Subgroups — #13123580

Order 1664.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×#88202	C₂	1	2	yes	maximal, complemented
3	#11718895	C₂	1	2	yes	maximal, complemented
4	#11718673	C₂	1	2	yes	maximal, complemented
5	#11718672	C₂	1	4	yes	maximal, complemented
6	D₄×#1055	C₂	1	2	yes	maximal, complemented
7	C₂×C₄⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6099	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#41	C₂²	1	2	yes	—
10	C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₁₃×#127	C₄	1	1	yes	characteristic
12	C₁₃×#124	C₄	1	2	yes	—
13	C₁₃×#121	C₄	1	2	yes	—
14	C₁₃×#121	C₄	1	2	yes	—
15	C₄×D₄×C₁₃	C₂²	1	2	yes	—
16	C₄×D₄×C₁₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	#88202	—	2	8	no	—
19	C₂×#6099	C₂²	1	4	yes	—
20	C₂×#6099	C₂²	1	1	yes	characteristic
21	C₂×#6099	C₂²	1	1	yes	characteristic
22	C₂²×#1055	C₂²	1	4	yes	—
23	C₂²×#1055	C₂²	1	2	yes	—
24	C₂²⋊#1055	C₂²	1	4	yes	—
25	C₂²⋊#1055	C₂²	1	4	yes	—
26	C₄⋊#1055	C₂²	1	1	yes	characteristic
27	C₄⋊#1055	C₂²	1	2	yes	—
28	C₄⋊#1055	C₂²	1	1	yes	characteristic
29	#88043	C₂²	1	2	yes	—
30	C₄×#1055	C₂²	1	2	yes	—
31	C₁₃×#41	—	2	4	no	—
32	C₁₃×#41	—	2	4	no	—
33	C₂×Q₈×C₁₃	C₂×C₄	1	2	yes	—
34	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
35	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
36	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
37	C₂×D₄×C₁₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
43	C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
45	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
46	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₄²×C₁₃	C₂³	1	1	yes	characteristic
49	C₄²×C₁₃	C₂³	1	1	yes	characteristic
50	#6099	—	2	2	no	—
51	#6099	—	2	8	no	—
52	#6099	D₄	1	4	yes	complemented
53	C₂×#1055	—	2	4	no	—
54	C₂×#1055	—	2	8	no	—
55	C₂×#1055	C₂³	1	2	yes	—
56	C₂×#1055	C₂³	1	2	yes	—
57	C₂×#1055	C₂³	1	2	yes	—
58	C₂×#1055	C₂³	1	2	yes	—
59	#2935	—	13	1	no	maximal, Sylow
60	C₂³×C₁₃	C₂²×C₄	1	4	yes	—
61	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	Q₈×C₁₃	—	2	4	no	—
63	D₄×C₁₃	—	2	8	no	—
64	D₄×C₁₃	—	2	4	no	—
65	D₄×C₁₃	—	2	4	no	—
66	D₄×C₁₃	—	4	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	4	1	no	—
71	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₃	#41	1	2	yes	—
79	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
80	#1055	—	2	2	no	—
81	#1055	—	2	2	no	—
82	#1055	—	2	2	no	—
83	#1055	C₂×D₄	1	4	yes	—
84	C₂×#131	—	13	2	no	—
85	#532	D₁₃	1	1	yes	characteristic
86	#436	—	13	2	no	—
87	#435	—	13	4	no	—
88	C₈×D₄	—	13	2	no	—
89	C₂×C₄⋊C₈	—	13	1	no	—
90	#408	—	13	2	no	—
91	C₄×M₁₆	—	13	1	no	—
92	C₂²×C₁₃	—	2	1	no	—
93	C₂²×C₁₃	—	2	1	no	—
94	C₂²×C₁₃	—	2	4	no	—
95	C₂²×C₁₃	—	2	4	no	—
96	C₂²×C₁₃	—	2	4	no	—
97	C₂²×C₁₃	C₄×D₄	1	2	yes	—
98	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₄×C₁₃	—	2	4	no	—
100	C₄×C₁₃	—	2	1	no	—
101	C₄×C₁₃	—	2	1	no	—
102	C₄×C₁₃	—	2	1	no	—
103	C₄×C₁₃	C₄×D₄	1	2	yes	—
104	C₄×C₁₃	#131	1	2	yes	complemented
105	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
107	C₂×#41	C₂×D₁₃	1	2	yes	—
108	#131	—	26	8	no	—
109	C₂×M₁₆	—	13	1	no	—
110	C₂×M₁₆	—	13	4	no	—
111	C₂×M₁₆	—	13	1	no	—
112	C₂²×C₈	—	13	2	no	—
113	C₂²×C₈	—	13	4	no	—
114	C₄⋊Q₈	Dic₁₃	1	1	yes	characteristic
115	#127	Dic₁₃	1	1	yes	characteristic
116	#124	Dic₁₃	1	2	yes	—
117	#121	Dic₁₃	1	2	yes	—
118	#121	Dic₁₃	1	2	yes	—
119	C₄×D₄	C₂×D₁₃	1	2	yes	—
120	C₄×D₄	C₂×D₁₃	1	2	yes	—
121	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
122	C₄⋊C₈	—	13	1	no	—
123	C₄⋊C₈	—	13	2	no	—
124	C₄⋊C₈	—	13	1	no	—
125	C₂²⋊C₈	—	13	4	no	—
126	C₂²⋊C₈	—	13	4	no	—
127	#97	—	13	2	no	—
128	C₄×C₈	—	13	2	no	—
129	C₂×C₁₃	—	2	2	no	—
130	C₂×C₁₃	—	4	4	no	—
131	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
133	C₂×C₁₃	#568	1	1	yes	characteristic
134	#41	—	2	4	no	—
135	#41	—	2	4	no	—
136	C₂×Q₈	C₂×Dic₁₃	1	2	yes	—
137	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
138	C₂×D₄	C₂×Dic₁₃	1	4	yes	—
139	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
140	C₂×D₄	C₂×Dic₁₃	1	2	yes	—
141	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
144	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
145	M₁₆	—	13	4	no	—
146	M₁₆	—	26	8	no	—
147	M₁₆	—	26	2	no	—
148	C₂×C₈	—	13	2	no	—
149	C₂×C₈	—	13	2	no	—
150	C₂×C₈	—	13	2	no	—
151	C₂×C₈	—	13	2	no	—
152	C₂×C₈	—	26	8	no	—
153	C₂×C₈	—	26	4	no	—
154	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
155	C₄⋊C₄	C₂×Dic₁₃	1	2	yes	—
156	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
157	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
158	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
159	C₄²	C₂×Dic₁₃	1	1	yes	characteristic
160	C₄²	C₂²×D₁₃	1	1	yes	characteristic
161	C₄²	C₂²×D₁₃	1	1	yes	characteristic
162	C₁₃	#2935	1	1	yes	characteristic, complemented, Sylow
163	C₂³	C₂²×Dic₁₃	1	4	yes	—
164	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
165	Q₈	—	2	4	no	—
166	D₄	—	2	8	no	—
167	D₄	—	2	4	no	—
168	D₄	—	2	4	no	—
169	D₄	—	4	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	1	no	—
174	C₂×C₄	D₄×D₁₃	1	2	yes	—
175	C₂×C₄	#6129	1	2	yes	—
176	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
177	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
178	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
179	C₂×C₄	C₂²×Dic₁₃	1	4	yes	—
180	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
181	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
182	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
183	C₈	—	13	4	no	—
184	C₈	—	26	2	no	—
185	C₈	—	26	2	no	—
186	C₈	—	26	2	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	D₄×Dic₁₃	1	2	yes	—
193	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	D₄×Dic₁₃	1	2	yes	—
199	C₄	#88202	1	2	yes	complemented
200	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
201	C₄	C₂×#6129	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	4	4	no	—
204	C₂	C₂×D₄×Dic₁₃	1	1	yes	characteristic
205	C₂	C₂×#88202	1	1	yes	characteristic
206	C₂	#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting, prime radical	C₁₃×#532	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#532	D₁₃
hypocenter	C₁₃	#2935
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
2. C₂×#88202	C₂	semidirect
3. #11718895	C₂	semidirect
4. #11718673	C₂	semidirect
5. #11718672	C₂	semidirect
6. D₄×#1055	C₂	semidirect
7. C₂×C₄⋊#1055	C₂	semidirect
8. C₄×#6099	C₂	semidirect
52. #6099	D₄	semidirect
104. C₄×C₁₃	#131	semidirect
162. C₁₃	#2935	semidirect
199. C₄	#88202	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#532	2
C₂×#88202	2
#11718895	2
#11718673	2
#11718672	2
D₄×#1055	2
C₂×C₄⋊#1055	2
C₄×#6099	2
#2935	13