Subgroups — #13123902

Order 1664.

410 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719182	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718586	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718499	C₂	1	2	yes	maximal, complemented
5	#11718495	C₂	1	2	yes	maximal, complemented
6	#11718388	C₂	1	4	yes	maximal, complemented
7	#11718349	C₂	1	2	yes	maximal, complemented
8	C₈×D₅₂	C₂	1	2	yes	maximal, complemented
9	C₂×#6127	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	#88184	C₄	1	2	yes	—
12	#88184	C₄	1	2	yes	—
13	C₂×#6094	C₂²	1	4	yes	—
14	C₂×C₈×D₁₃	C₂²	1	4	yes	—
15	#88129	C₄	1	2	yes	—
16	#88128	C₄	1	1	yes	characteristic
17	C₄×D₅₂	C₂²	1	2	yes	—
18	C₄×D₅₂	C₂²	1	2	yes	—
19	#88123	C₄	1	1	yes	characteristic
20	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₄×C₈×C₁₃	C₂²	1	2	yes	—
24	C₂²⋊#1055	C₂²	1	2	yes	—
25	#88060	C₂²	1	4	yes	—
26	#88060	C₂²	1	4	yes	—
27	C₄⋊#1055	C₂²	1	1	yes	characteristic
28	C₄⋊#1055	C₂²	1	1	yes	characteristic
29	C₄×#1055	C₂²	1	2	yes	—
30	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
34	#6127	—	4	2	no	—
35	#6127	—	4	2	no	—
36	C₂×D₅₂	C₂×C₄	1	2	yes	—
37	C₂×D₅₂	C₂×C₄	1	2	yes	—
38	C₂×D₅₂	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₃	C₂³	1	4	yes	—
40	C₂×Dic₂₆	C₂×C₄	1	2	yes	—
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	#6103	C₂×C₄	1	4	yes	—
48	#6103	C₂×C₄	1	4	yes	—
49	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
50	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
51	C₂×#1055	C₂³	1	2	yes	—
52	C₂×#1055	C₂³	1	2	yes	—
53	#6094	—	4	4	no	—
54	C₈×D₁₃	—	4	4	no	—
55	#2979	—	13	1	no	maximal, Sylow
56	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×D₁₃	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₃	—	2	2	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	2	1	no	—
61	C₂×C₄×C₁₃	—	2	1	no	—
62	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²⋊D₁₃	—	4	4	no	—
69	C₂×Dic₁₃	C₂²×C₄	1	4	yes	—
70	D₅₂	—	2	4	no	—
71	D₅₂	—	2	4	no	—
72	D₅₂	—	4	2	no	—
73	C₄×D₁₃	—	2	4	no	—
74	C₄×D₁₃	—	2	4	no	—
75	Dic₂₆	—	4	2	no	—
76	C₈×C₁₃	—	2	2	no	—
77	C₈×C₁₃	—	2	2	no	—
78	#1055	—	2	2	no	—
79	#1055	—	2	2	no	—
80	#532	—	13	1	no	—
81	#436	—	13	2	no	—
82	#436	—	13	2	no	—
83	C₈×D₄	—	13	2	no	—
84	C₈×D₄	—	13	2	no	—
85	#431	—	13	1	no	—
86	#431	D₁₃	1	1	yes	characteristic, complemented
87	#408	—	13	4	no	—
88	C₂²×C₁₃	—	2	1	no	—
89	C₂²×C₁₃	—	2	1	no	—
90	C₂²×C₁₃	—	2	1	no	—
91	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
92	C₂×D₁₃	—	2	4	no	—
93	C₂×D₁₃	—	2	4	no	—
94	C₂×D₁₃	—	2	4	no	—
95	C₄×C₁₃	—	2	1	no	—
96	C₄×C₁₃	—	2	1	no	—
97	C₄×C₁₃	—	2	1	no	—
98	C₄×C₁₃	#131	1	2	yes	—
99	C₄×C₁₃	#131	1	2	yes	—
100	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
101	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
102	Dic₁₃	—	2	4	no	—
103	C₂×#41	—	13	2	no	—
104	C₂×M₁₆	—	13	4	no	—
105	C₂²×C₈	—	13	4	no	—
106	C₄⋊Q₈	—	13	1	no	—
107	#127	—	13	1	no	—
108	#124	—	13	2	no	—
109	#121	—	13	2	no	—
110	#121	—	13	2	no	—
111	C₄×D₄	—	13	2	no	—
112	C₄×D₄	—	13	2	no	—
113	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
114	C₄⋊C₈	—	13	1	no	—
115	C₄⋊C₈	—	13	1	no	—
116	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₄⋊C₈	C₂×D₁₃	1	1	yes	characteristic
118	C₂²⋊C₈	—	13	2	no	—
119	C₂²⋊C₈	—	13	4	no	—
120	C₂²⋊C₈	—	13	4	no	—
121	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
122	C₄×C₈	—	13	2	no	—
123	C₄×C₈	C₂×D₁₃	1	2	yes	—
124	C₂×C₁₃	—	4	1	no	—
125	C₂×C₁₃	#518	1	1	yes	characteristic
126	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
127	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
128	D₁₃	—	4	4	no	—
129	#41	—	52	2	no	—
130	#41	—	52	2	no	—
131	C₂×Q₈	—	13	2	no	—
132	C₂×D₄	—	13	4	no	—
133	C₂×D₄	—	13	2	no	—
134	C₂×D₄	—	13	2	no	—
135	C₂×D₄	—	13	2	no	—
136	C₂²×C₄	—	13	4	no	—
137	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
138	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
140	M₁₆	—	52	4	no	—
141	C₂×C₈	—	13	2	no	—
142	C₂×C₈	—	13	2	no	—
143	C₂×C₈	—	52	4	no	—
144	C₂×C₈	C₂²×D₁₃	1	2	yes	—
145	C₂×C₈	C₂²×D₁₃	1	2	yes	—
146	C₄⋊C₄	—	13	2	no	—
147	C₄⋊C₄	—	13	2	no	—
148	C₂²⋊C₄	—	13	4	no	—
149	C₂²⋊C₄	—	13	4	no	—
150	C₄²	C₄×D₁₃	1	1	yes	characteristic
151	C₄²	C₄×D₁₃	1	1	yes	characteristic
152	C₄²	C₂²×D₁₃	1	1	yes	characteristic
153	C₄²	C₂²×D₁₃	1	1	yes	characteristic
154	C₁₃	#2979	1	1	yes	characteristic, complemented, Sylow
155	C₂³	—	13	4	no	—
156	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
157	Q₈	—	52	2	no	—
158	D₄	—	26	4	no	—
159	D₄	—	26	4	no	—
160	D₄	—	52	2	no	—
161	D₄	—	52	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	1	no	—
166	C₂×C₄	—	13	4	no	—
167	C₂×C₄	—	26	4	no	—
168	C₂×C₄	—	26	4	no	—
169	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
170	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
172	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
173	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
174	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
175	C₈	—	2	2	no	—
176	C₈	—	2	2	no	—
177	C₈	—	26	2	no	—
178	C₈	—	26	2	no	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	26	4	no	—
183	C₂²	—	26	4	no	—
184	C₂²	—	26	4	no	—
185	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	2	1	no	—
189	C₄	—	26	4	no	—
190	C₄	#88155	1	2	yes	—
191	C₄	#88161	1	2	yes	—
192	C₄	#88256	1	1	yes	characteristic
193	C₄	#88256	1	1	yes	characteristic
194	C₂	—	4	1	no	—
195	C₂	—	52	4	no	—
196	C₂	#11719226	1	1	yes	characteristic
197	C₂	C₂×#88155	1	1	yes	characteristic
198	C₂	C₂×#88161	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	#11719182	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#431	D₁₃
hypocenter	C₁₃	#2979
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. #11718586	C₂	semidirect
4. #11718499	C₂	semidirect
5. #11718495	C₂	semidirect
6. #11718388	C₂	semidirect
7. #11718349	C₂	semidirect
8. C₈×D₅₂	C₂	semidirect
86. #431	D₁₃	semidirect
154. C₁₃	#2979	semidirect

Maximal subgroups

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

subgroup	index
#11719182	2
C₁₃×#431	2
#11718586	2
#11718499	2
#11718495	2
#11718388	2
#11718349	2
C₈×D₅₂	2
#2979	13