Subgroups — C₂×#13123233 #16069016

Order 3328.

3774 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	C₂²×#88142	C₂	1	6	yes	maximal
2	#13123233	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718596	C₂	1	1	yes	maximal, characteristic
4	C₂×#11718637	C₂	1	6	yes	maximal
5	C₂×C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	C₂²	1	4	yes	—
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	6	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₁₃	C₂²	1	6	yes	—
11	C₂²×#6101	C₂²	1	12	yes	—
12	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	—
13	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	—
14	C₂×#88142	—	2	48	no	—
15	C₂×#88142	C₂²	1	48	yes	—
16	C₁₃×#384	C₂²	1	8	yes	—
17	#11718637	C₂²	1	48	yes	—
18	#11718596	C₂²	1	8	yes	—
19	C₄²×Dic₁₃	C₂²	1	8	yes	—
20	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
22	C₂³×Dic₁₃	C₂³	1	4	yes	—
23	C₂³×Dic₁₃	C₂³	1	1	yes	characteristic
24	C₂³×Dic₁₃	C₂³	1	3	yes	—
25	C₂²×Dic₂₆	—	2	48	no	—
26	C₂²×Dic₂₆	C₂³	1	16	yes	—
27	C₂×C₁₃×C₄⋊C₄	—	2	24	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
29	C₂×C₄²×C₁₃	Q₈	1	8	yes	complemented
30	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₁₃	—	2	24	no	—
32	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
33	C₂×#6101	—	2	48	no	—
34	C₂×#6101	C₂³	1	48	yes	—
35	C₂×C₄×Dic₁₃	D₄	1	24	yes	—
36	C₂×C₄×Dic₁₃	Q₈	1	24	yes	complemented
37	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
38	C₂×C₄×Dic₁₃	C₂³	1	12	yes	—
39	#88142	—	2	192	no	—
40	C₂×#2868	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	24	no	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
44	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	12	yes	—
46	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
47	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
48	C₂²×Dic₁₃	—	2	24	no	—
49	C₂²×Dic₁₃	C₂×D₄	1	12	yes	—
50	C₂²×Dic₁₃	C₂×D₄	1	12	yes	—
51	C₂²×Dic₁₃	C₂×Q₈	1	24	yes	—
52	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
53	C₂²×Dic₁₃	C₂⁴	1	2	yes	—
54	C₂²×Dic₁₃	C₂⁴	1	6	yes	—
55	C₂×Dic₂₆	—	2	96	no	—
56	C₂×Dic₂₆	—	4	64	no	—
57	C₁₃×C₄⋊C₄	—	2	48	no	—
58	C₄²×C₁₃	C₂×Q₈	1	16	yes	complemented
59	C₄⋊Dic₁₃	—	2	48	no	—
60	#6101	—	2	96	no	—
61	C₄×Dic₁₃	C₂×D₄	1	48	yes	—
62	C₄×Dic₁₃	C₂×Q₈	1	48	yes	complemented
63	C₂²×C₄⋊Q₈	—	13	6	no	—
64	#2868	—	13	16	no	—
65	C₂×#384	—	13	1	no	—
66	C₂×#384	—	13	6	no	—
67	C₂×#384	D₁₃	1	1	yes	characteristic
68	C₂×C₄³	—	13	1	no	—
69	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
70	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
71	C₂³×C₁₃	C₂²×Q₈	1	6	yes	—
72	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	24	no	—
74	C₂×C₄×C₁₃	—	4	16	no	—
75	C₂×C₄×C₁₃	C₄⋊Q₈	1	48	yes	complemented
76	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
77	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
78	C₂×C₄×C₁₃	C₂²×Q₈	1	12	yes	—
79	C₂×Dic₁₃	—	2	24	no	—
80	C₂×Dic₁₃	—	4	16	no	—
81	C₂×Dic₁₃	#127	1	16	yes	—
82	C₂×Dic₁₃	C₄⋊Q₈	1	48	yes	complemented
83	C₂×Dic₁₃	C₂²×D₄	1	12	yes	—
84	C₂×Dic₁₃	C₂²×D₄	1	12	yes	—
85	C₂×Dic₁₃	C₂²×Q₈	1	24	yes	—
86	Dic₂₆	—	4	64	no	—
87	C₂³×Q₈	—	13	4	no	—
88	C₂×C₄⋊Q₈	—	13	48	no	—
89	C₂×C₄⋊Q₈	—	26	48	no	—
90	C₂²×C₄⋊C₄	—	13	12	no	—
91	C₂²×C₄⋊C₄	—	13	6	no	—
92	C₂²×C₄⋊C₄	C₂×D₁₃	1	6	yes	—
93	C₂²×C₄²	—	13	3	no	—
94	C₂²×C₄²	—	13	3	no	—
95	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
96	#384	—	13	48	no	—
97	#384	—	13	8	no	—
98	#384	C₂×D₁₃	1	8	yes	—
99	C₄³	—	13	8	no	—
100	C₂²×C₁₃	C₂×#127	1	4	yes	—
101	C₂²×C₁₃	C₂×C₄⋊Q₈	1	12	yes	—
102	C₂²×C₁₃	C₂×C₄⋊Q₈	1	12	yes	—
103	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
104	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
105	C₂²×C₁₃	C₂³×Q₈	1	3	yes	—
106	C₄×C₁₃	—	4	8	no	—
107	C₄×C₁₃	C₂×C₄⋊Q₈	1	24	yes	complemented
108	Dic₁₃	—	4	8	no	—
109	Dic₁₃	C₂×#127	1	8	yes	—
110	Dic₁₃	C₂×C₄⋊Q₈	1	24	yes	complemented
111	C₂²×Q₈	—	13	16	no	—
112	C₂²×Q₈	—	26	48	no	—
113	C₂³×C₄	—	13	3	no	—
114	C₂³×C₄	—	13	1	no	—
115	C₂³×C₄	—	13	4	no	—
116	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
117	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
118	C₄⋊Q₈	—	26	192	no	—
119	C₂×C₄⋊C₄	—	2	24	no	—
120	C₂×C₄⋊C₄	—	13	48	no	—
121	C₂×C₄⋊C₄	—	13	24	no	—
122	C₂×C₄⋊C₄	—	26	48	no	—
123	C₂×C₄⋊C₄	—	26	24	no	—
124	C₂×C₄⋊C₄	C₂²×D₁₃	1	24	yes	—
125	C₂×C₄²	—	13	24	no	—
126	C₂×C₄²	—	13	24	no	—
127	C₂×C₄²	—	13	12	no	—
128	C₂×C₄²	—	13	12	no	—
129	C₂×C₄²	Dic₂₆	1	8	yes	complemented
130	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
131	C₂×C₁₃	#2868	1	8	yes	complemented
132	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
133	C₂×C₁₃	C₂²×C₄⋊Q₈	1	3	yes	—
134	C₂×C₁₃	C₂²×C₄⋊Q₈	1	3	yes	—
135	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
136	C₂×Q₈	—	26	96	no	—
137	C₂×Q₈	—	52	64	no	—
138	C₂²×C₄	—	2	24	no	—
139	C₂²×C₄	—	13	12	no	—
140	C₂²×C₄	—	13	12	no	—
141	C₂²×C₄	—	13	24	no	—
142	C₂²×C₄	—	13	6	no	—
143	C₂²×C₄	—	13	2	no	—
144	C₂²×C₄	—	13	8	no	—
145	C₂²×C₄	—	26	24	no	—
146	C₂²×C₄	C₂×Dic₂₆	1	12	yes	—
147	C₂²×C₄	D₄×D₁₃	1	12	yes	—
148	C₂²×C₄	Q₈×D₁₃	1	12	yes	—
149	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
150	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
151	C₄⋊C₄	—	2	48	no	—
152	C₄⋊C₄	—	26	96	no	—
153	C₄⋊C₄	—	26	48	no	—
154	C₄²	—	13	48	no	—
155	C₄²	—	13	48	no	—
156	C₄²	C₂×Dic₂₆	1	16	yes	complemented
157	C₁₃	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
158	C₂³	C₂²×Dic₂₆	1	2	yes	—
159	C₂³	C₂×D₄×D₁₃	1	6	yes	—
160	C₂³	C₂×Q₈×D₁₃	1	6	yes	—
161	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
162	Q₈	—	52	64	no	—
163	C₂×C₄	—	2	24	no	—
164	C₂×C₄	—	4	16	no	—
165	C₂×C₄	—	13	48	no	—
166	C₂×C₄	—	13	12	no	—
167	C₂×C₄	—	13	12	no	—
168	C₂×C₄	—	13	24	no	—
169	C₂×C₄	—	13	16	no	—
170	C₂×C₄	—	26	24	no	—
171	C₂×C₄	—	52	16	no	—
172	C₂×C₄	#88142	1	48	yes	complemented
173	C₂×C₄	C₂²×Dic₂₆	1	12	yes	—
174	C₂×C₄	C₂×D₄×D₁₃	1	12	yes	—
175	C₂×C₄	C₂×Q₈×D₁₃	1	12	yes	—
176	C₂²	C₂×#88142	1	12	yes	—
177	C₂²	D₁₃×#127	1	4	yes	—
178	C₂²	D₁₃×C₄⋊Q₈	1	12	yes	—
179	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
180	C₂²	C₂²×D₄×D₁₃	1	3	yes	—
181	C₂²	C₂²×Q₈×D₁₃	1	3	yes	—
182	C₄	—	4	8	no	—
183	C₄	—	13	8	no	—
184	C₄	—	13	24	no	—
185	C₄	—	52	8	no	—
186	C₄	C₂×#88142	1	24	yes	complemented
187	C₂	#13123233	1	8	yes	direct product factor
188	C₂	C₂×D₁₃×#127	1	1	yes	characteristic
189	C₂	C₂×D₁₃×C₄⋊Q₈	1	3	yes	—
190	C₂	C₂²×#88142	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#384	C₂
socle, prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×#384	D₁₃
hypocenter	C₁₃	C₂×#2868
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. #13123233	C₂	direct (N × H)
29. C₂×C₄²×C₁₃	Q₈	semidirect
36. C₂×C₄×Dic₁₃	Q₈	semidirect
58. C₄²×C₁₃	C₂×Q₈	semidirect
62. C₄×Dic₁₃	C₂×Q₈	semidirect
75. C₂×C₄×C₁₃	C₄⋊Q₈	semidirect
82. C₂×Dic₁₃	C₄⋊Q₈	semidirect
107. C₄×C₁₃	C₂×C₄⋊Q₈	semidirect
110. Dic₁₃	C₂×C₄⋊Q₈	semidirect
129. C₂×C₄²	Dic₂₆	semidirect
131. C₂×C₁₃	#2868	semidirect
156. C₄²	C₂×Dic₂₆	semidirect
157. C₁₃	C₂×#2868	semidirect
172. C₂×C₄	#88142	semidirect
186. C₄	C₂×#88142	semidirect
187. C₂	#13123233	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88142	2
#13123233	2
C₂×#11718596	2
C₂×#11718637	2
C₂×C₄²×Dic₁₃	2
C₂×C₁₃×#384	2
C₂×#2868	13

Subgroups — C2×#13123233 #16069016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123233 #16069016