Subgroups — C₂×#13123917 #16727858

Order 3328.

2234 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	#13123917	C₂	1	16	yes	maximal, direct product factor
2	C₂×Q₈×#1055	C₂	1	8	yes	maximal, complemented
3	C₂×#11718586	C₂	1	6	yes	maximal
4	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₁₃	C₄	1	4	yes	—
7	C₂×C₄×Q₈×C₁₃	C₄	1	8	yes	—
8	C₂×C₄×Q₈×C₁₃	C₂²	1	8	yes	—
9	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₄	1	3	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
12	C₂×C₂²⋊#1055	C₂²	1	4	yes	—
13	Q₈×#1055	C₂²	1	64	yes	complemented
14	#11718586	C₂²	1	48	yes	—
15	C₂×C₄⋊#1055	C₂²	1	12	yes	—
16	C₂×C₄×#1055	C₂²	1	12	yes	—
17	C₂²×Q₈×C₁₃	C₈	1	16	yes	complemented
18	C₂²×Q₈×C₁₃	C₂×C₄	1	4	yes	—
19	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
20	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
21	C₂³×C₄×C₁₃	C₂×C₄	1	3	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
23	C₄×Q₈×C₁₃	C₂×C₄	1	32	yes	—
24	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
25	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
26	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	12	yes	—
27	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	12	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
32	C₂²×#1055	C₂³	1	8	yes	—
33	C₂²⋊#1055	C₂³	1	16	yes	—
34	C₄⋊#1055	C₂³	1	48	yes	—
35	C₄×#1055	C₂³	1	48	yes	—
36	C₂×#2982	—	13	1	no	maximal, Sylow
37	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×Q₈×C₁₃	C₂×C₈	1	32	yes	—
39	C₂×Q₈×C₁₃	C₂×C₈	1	64	yes	complemented
40	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	4	6	no	—
43	C₂²×C₄×C₁₃	C₂×C₈	1	24	yes	—
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
47	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
50	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	C₂²×C₄	1	48	yes	—
53	C₄²×C₁₃	C₂²×C₄	1	24	yes	—
54	C₄²×C₁₃	C₂²×C₄	1	12	yes	—
55	C₄²×C₁₃	C₂⁴	1	12	yes	—
56	C₂×#1055	—	2	16	no	—
57	C₂×#1055	C₂⁴	1	16	yes	—
58	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
59	#2982	—	13	16	no	—
60	C₂×C₈×Q₈	—	13	8	no	—
61	C₂×#431	—	13	6	no	—
62	C₂³×C₁₃	—	2	2	no	—
63	C₂³×C₁₃	—	2	2	no	—
64	C₂³×C₁₃	C₂²×C₈	1	4	yes	—
65	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
66	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
67	Q₈×C₁₃	C₂²×C₈	1	64	yes	—
68	C₂×C₄×C₁₃	—	2	2	no	—
69	C₂×C₄×C₁₃	—	2	12	no	—
70	C₂×C₄×C₁₃	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	4	12	no	—
73	C₂×C₄×C₁₃	C₂²×C₈	1	24	yes	—
74	C₂×C₄×C₁₃	C₂²×C₈	1	48	yes	—
75	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
77	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
79	C₂×C₄×C₁₃	2₋¹⁺⁴	1	4	yes	—
80	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
81	#1055	—	2	16	no	—
82	C₂³×Q₈	Dic₁₃	1	1	yes	characteristic
83	C₂×C₄×Q₈	Dic₁₃	1	4	yes	—
84	C₂×C₄×Q₈	Dic₁₃	1	8	yes	—
85	C₂×C₄×Q₈	C₂×D₁₃	1	4	yes	—
86	C₂×C₄×Q₈	C₂×D₁₃	1	8	yes	—
87	C₂²×C₄⋊C₄	Dic₁₃	1	3	yes	—
88	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
89	C₈×Q₈	—	13	64	no	—
90	#431	—	13	48	no	—
91	C₂×C₄⋊C₈	—	13	12	no	—
92	C₂×C₂²⋊C₈	—	13	4	no	—
93	C₂×C₄×C₈	—	13	12	no	—
94	C₂²×C₁₃	—	2	2	no	—
95	C₂²×C₁₃	—	2	8	no	—
96	C₂²×C₁₃	—	2	2	no	—
97	C₂²×C₁₃	#519	1	2	yes	—
98	C₂²×C₁₃	C₂³×C₈	1	4	yes	—
99	C₂²×C₁₃	C₂³×C₈	1	2	yes	—
100	C₂²×C₁₃	#568	1	2	yes	—
101	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
102	C₄×C₁₃	—	2	2	no	—
103	C₄×C₁₃	—	2	12	no	—
104	C₄×C₁₃	#519	1	2	yes	—
105	C₄×C₁₃	C₂³×C₈	1	24	yes	—
106	C₄×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
107	C₂²×Q₈	#1055	1	16	yes	complemented
108	C₂²×Q₈	C₂×Dic₁₃	1	4	yes	—
109	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
110	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
111	C₂³×C₄	C₂×Dic₁₃	1	3	yes	—
112	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
113	C₂²×C₈	—	13	8	no	—
114	C₄×Q₈	C₂×Dic₁₃	1	16	yes	—
115	C₄×Q₈	C₂×Dic₁₃	1	32	yes	—
116	C₄×Q₈	C₂²×D₁₃	1	16	yes	—
117	C₂×C₄⋊C₄	C₂×Dic₁₃	1	12	yes	—
118	C₂×C₄⋊C₄	C₂×Dic₁₃	1	24	yes	—
119	C₂×C₄²	C₂×Dic₁₃	1	6	yes	—
120	C₂×C₄²	C₂×Dic₁₃	1	12	yes	—
121	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
122	C₂×C₄²	C₂²×D₁₃	1	6	yes	—
123	C₄⋊C₈	—	13	48	no	—
124	C₂²⋊C₈	—	13	16	no	—
125	C₄×C₈	—	13	48	no	—
126	C₂×C₁₃	—	2	4	no	—
127	C₂×C₁₃	#2982	1	4	yes	complemented
128	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
129	C₂×C₁₃	C₂⁴×C₈	1	1	yes	characteristic
130	C₂×C₁₃	C₂×#568	1	1	yes	characteristic
131	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
132	C₂×Q₈	C₂×#1055	1	32	yes	—
133	C₂×Q₈	C₂×#1055	1	64	yes	complemented
134	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	4	6	no	—
137	C₂²×C₄	C₂×#1055	1	24	yes	—
138	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
139	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
140	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
141	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
142	C₂²×C₄	C₂²×Dic₁₃	1	6	yes	—
143	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
144	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
145	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
146	C₂×C₈	—	13	16	no	—
147	C₂×C₈	—	26	16	no	—
148	C₄⋊C₄	C₂²×Dic₁₃	1	48	yes	—
149	C₄²	C₂²×Dic₁₃	1	24	yes	—
150	C₄²	C₂²×Dic₁₃	1	12	yes	—
151	C₄²	C₂³×D₁₃	1	12	yes	—
152	C₁₃	C₂×#2982	1	1	yes	characteristic, complemented, Sylow
153	C₂³	—	2	2	no	—
154	C₂³	—	2	2	no	—
155	C₂³	C₂²×#1055	1	4	yes	—
156	C₂³	C₂³×Dic₁₃	1	2	yes	—
157	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
158	Q₈	C₂²×#1055	1	64	yes	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	12	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	4	12	no	—
164	C₂×C₄	C₂²×#1055	1	24	yes	—
165	C₂×C₄	C₂²×#1055	1	48	yes	—
166	C₂×C₄	#88254	1	4	yes	—
167	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
168	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
169	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
170	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
171	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
172	C₈	—	26	16	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	8	no	—
175	C₂²	—	2	2	no	—
176	C₂²	C₂³×#1055	1	4	yes	—
177	C₂²	C₂³×#1055	1	2	yes	—
178	C₂²	#11719517	1	2	yes	—
179	C₂²	#11719524	1	2	yes	—
180	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
181	C₄	—	2	12	no	—
182	C₄	—	2	2	no	—
183	C₄	C₂³×#1055	1	24	yes	—
184	C₄	#11719517	1	2	yes	—
185	C₄	C₂×#88254	1	2	yes	—
186	C₂	—	2	4	no	—
187	C₂	#13123917	1	4	yes	direct product factor
188	C₂	C₂×#11719517	1	1	yes	characteristic
189	C₂	C₂⁴×#1055	1	1	yes	characteristic
190	C₂	C₂×#11719524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#2982
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #13123917	C₂	direct (N × H)
2. C₂×Q₈×#1055	C₂	semidirect
13. Q₈×#1055	C₂²	semidirect
17. C₂²×Q₈×C₁₃	C₈	semidirect
39. C₂×Q₈×C₁₃	C₂×C₈	semidirect
107. C₂²×Q₈	#1055	semidirect
127. C₂×C₁₃	#2982	semidirect
133. C₂×Q₈	C₂×#1055	semidirect
152. C₁₃	C₂×#2982	semidirect
187. C₂	#13123917	direct (N × H)

Maximal subgroups

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

subgroup	index
#13123917	2
C₂×Q₈×#1055	2
C₂×#11718586	2
C₂²×C₄×Q₈×C₁₃	2
C₂×#2982	13

Subgroups — C2×#13123917 #16727858

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123917 #16727858