Subgroups — #13123865

Order 1664.

456 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	#11718868	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#1055	C₂	1	2	yes	maximal, complemented
4	#11718672	C₂	1	8	yes	maximal, complemented
5	#11718587	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₁₃	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₁₃	C₄	1	4	yes	—
10	C₄×D₄×C₁₃	C₂²	1	4	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₁₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
13	C₂×C₁₃×C₂²⋊C₄	C₄	1	1	yes	characteristic
14	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6099	C₂²	1	4	yes	complemented
16	C₂²×#1055	C₂²	1	4	yes	complemented
17	C₂²⋊#1055	C₂²	1	4	yes	—
18	C₂²⋊#1055	C₂²	1	4	yes	complemented
19	C₂²⋊#1055	C₂²	1	4	yes	complemented
20	C₄⋊#1055	C₂²	1	4	yes	complemented
21	#88043	C₂²	1	4	yes	complemented
22	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₁₃	—	2	4	no	—
25	C₂×D₄×C₁₃	C₂×C₄	1	4	yes	—
26	C₂²×C₄×C₁₃	—	4	2	no	—
27	C₂²×C₄×C₁₃	—	4	2	no	—
28	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₁₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
38	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
39	C₁₃×C₂²⋊C₄	C₂×C₄	1	4	yes	—
40	C₄²×C₁₃	C₂×C₄	1	2	yes	—
41	C₄²×C₁₃	C₂³	1	2	yes	—
42	#6099	—	4	4	no	—
43	C₂×#1055	—	4	4	no	—
44	C₂×#1055	C₂³	1	8	yes	—
45	#2973	—	13	1	no	maximal, Sylow
46	C₂³×C₁₃	—	2	2	no	—
47	C₂³×C₁₃	—	4	1	no	—
48	C₂³×C₁₃	—	4	1	no	—
49	C₂³×C₁₃	—	4	1	no	—
50	C₂³×C₁₃	—	4	1	no	—
51	C₂³×C₁₃	M₁₆	1	4	yes	complemented
52	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
53	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
54	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	D₄×C₁₃	—	2	8	no	—
56	C₂×C₄×C₁₃	—	2	2	no	—
57	C₂×C₄×C₁₃	—	2	4	no	—
58	C₂×C₄×C₁₃	—	2	4	no	—
59	C₂×C₄×C₁₃	—	2	2	no	—
60	C₂×C₄×C₁₃	—	4	2	no	—
61	C₂×C₄×C₁₃	—	4	2	no	—
62	C₂×C₄×C₁₃	—	4	1	no	—
63	C₂×C₄×C₁₃	—	4	1	no	—
64	C₂×C₄×C₁₃	—	4	1	no	—
65	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	#1055	—	2	8	no	—
72	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
73	#435	—	13	8	no	—
74	#432	—	13	2	no	—
75	#407	—	13	2	no	—
76	C₂×C₂²⋊C₈	—	13	2	no	—
77	C₂²×C₁₃	—	2	1	no	—
78	C₂²×C₁₃	—	2	2	no	—
79	C₂²×C₁₃	—	2	1	no	—
80	C₂²×C₁₃	—	2	2	no	—
81	C₂²×C₁₃	—	2	2	no	—
82	C₂²×C₁₃	—	2	2	no	—
83	C₂²×C₁₃	—	2	2	no	—
84	C₂²×C₁₃	—	4	4	no	—
85	C₂²×C₁₃	—	4	4	no	—
86	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
87	C₂²×C₁₃	C₂×M₁₆	1	4	yes	—
88	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
89	C₄×C₁₃	—	2	1	no	—
90	C₄×C₁₃	—	2	2	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	2	2	no	—
93	C₄×C₁₃	—	2	2	no	—
94	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
95	C₂²×D₄	Dic₁₃	1	1	yes	characteristic
96	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
97	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
98	C₂×M₁₆	—	13	4	no	—
99	C₂²×C₈	—	13	4	no	—
100	C₄×D₄	Dic₁₃	1	4	yes	—
101	C₄×D₄	C₂×D₁₃	1	4	yes	—
102	C₂×C₄⋊C₄	Dic₁₃	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	Dic₁₃	1	1	yes	characteristic
104	C₂×C₂²⋊C₄	Dic₁₃	1	1	yes	characteristic
105	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
106	C₄⋊C₈	—	13	4	no	—
107	C₂²⋊C₈	—	13	4	no	—
108	C₂²⋊C₈	—	13	4	no	—
109	C₂²⋊C₈	—	13	4	no	—
110	#97	—	13	4	no	—
111	C₂×C₁₃	—	2	2	no	—
112	C₂×C₁₃	—	2	4	no	—
113	C₂×C₁₃	—	4	2	no	—
114	C₂×C₁₃	#518	1	1	yes	characteristic
115	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
116	C₂×C₁₃	#568	1	1	yes	characteristic
117	C₂⁴	C₂×Dic₁₃	1	1	yes	characteristic
118	C₂⁴	C₂×Dic₁₃	1	1	yes	characteristic
119	C₂×D₄	—	2	4	no	—
120	C₂×D₄	C₂×Dic₁₃	1	4	yes	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	4	2	no	—
123	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
124	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
125	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
126	C₂²×C₄	C₂×Dic₁₃	1	1	yes	characteristic
127	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
128	C₂²×C₄	C₂×Dic₁₃	1	2	yes	—
129	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
131	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
132	M₁₆	—	52	4	no	—
133	C₂×C₈	—	13	8	no	—
134	C₂×C₈	—	52	4	no	—
135	C₄⋊C₄	C₂×Dic₁₃	1	4	yes	—
136	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
137	C₂²⋊C₄	C₂×Dic₁₃	1	4	yes	—
138	C₄²	C₂×Dic₁₃	1	2	yes	—
139	C₄²	C₂²×D₁₃	1	2	yes	—
140	C₁₃	#2973	1	1	yes	characteristic, complemented, Sylow
141	C₂³	—	2	2	no	—
142	C₂³	—	4	1	no	—
143	C₂³	—	4	1	no	—
144	C₂³	—	4	1	no	—
145	C₂³	—	4	1	no	—
146	C₂³	#6099	1	4	yes	complemented
147	C₂³	C₂²×Dic₁₃	1	2	yes	—
148	C₂³	C₂²×Dic₁₃	1	2	yes	—
149	C₂³	C₂²×Dic₁₃	1	1	yes	characteristic
150	D₄	—	2	8	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	—	4	2	no	—
157	C₂×C₄	—	4	1	no	—
158	C₂×C₄	—	4	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
161	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
162	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
163	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
164	C₂×C₄	C₂²×Dic₁₃	1	2	yes	—
165	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
166	C₈	—	26	8	no	—
167	C₂²	—	2	1	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	1	no	—
170	C₂²	—	2	2	no	—
171	C₂²	—	2	2	no	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	4	4	no	—
175	C₂²	—	4	4	no	—
176	C₂²	C₂×#6099	1	2	yes	—
177	C₂²	C₂×#6099	1	4	yes	—
178	C₂²	C₂³×Dic₁₃	1	1	yes	characteristic
179	C₄	—	2	2	no	—
180	C₄	—	2	1	no	—
181	C₄	—	2	2	no	—
182	C₄	—	2	2	no	—
183	C₄	—	2	2	no	—
184	C₄	#88251	1	2	yes	—
185	C₂	—	2	2	no	—
186	C₂	—	2	4	no	—
187	C₂	—	4	2	no	—
188	C₂	C₂²×#6099	1	1	yes	characteristic
189	C₂	#11719503	1	1	yes	characteristic
190	C₂	#11719524	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₂×C₄×D₄×C₁₃	C₂
prime radical	C₂²×D₄×C₁₃	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	#2973
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. #11718868	C₂	semidirect
3. C₂×C₂²⋊#1055	C₂	semidirect
4. #11718672	C₂	semidirect
5. #11718587	C₂	semidirect
15. C₂×#6099	C₂²	semidirect
16. C₂²×#1055	C₂²	semidirect
18. C₂²⋊#1055	C₂²	semidirect
19. C₂²⋊#1055	C₂²	semidirect
20. C₄⋊#1055	C₂²	semidirect
21. #88043	C₂²	semidirect
51. C₂³×C₁₃	M₁₆	semidirect
140. C₁₃	#2973	semidirect
146. C₂³	#6099	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₁₃	2
#11718868	2
C₂×C₂²⋊#1055	2
#11718672	2
#11718587	2
#2973	13