Subgroups — C₂×#13123199 #16068899

Order 3328.

2558 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88124	C₂	1	2	yes	maximal
2	#13123199	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718589	C₂	1	8	yes	maximal
4	C₂×#11718596	C₂	1	2	yes	maximal
5	C₂×#11718597	C₂	1	2	yes	maximal
6	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
12	C₂²×C₄⋊Dic₁₃	C₂²	1	8	yes	—
13	C₂²×#6101	C₂²	1	8	yes	—
14	C₂×#88124	—	2	16	no	—
15	C₂×#88124	C₂²	1	16	yes	—
16	C₄³×C₁₃	C₂²	1	8	yes	—
17	C₂×#88071	C₂²	1	8	yes	—
18	#11718597	C₂²	1	16	yes	—
19	#11718596	C₂²	1	16	yes	—
20	#11718589	C₂²	1	64	yes	—
21	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
22	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
25	C₂³×Dic₁₃	C₂³	1	8	yes	—
26	C₂×C₄²×C₁₃	—	2	16	no	—
27	C₂×C₄²×C₁₃	D₄	1	8	yes	—
28	C₂×C₄²×C₁₃	Q₈	1	8	yes	—
29	C₂×C₄²×C₁₃	Q₈	1	8	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	16	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₁₃	—	2	16	no	—
35	C₂×C₄⋊Dic₁₃	—	2	32	no	—
36	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
37	C₂×C₄⋊Dic₁₃	C₂³	1	32	yes	—
38	C₂×#6101	—	2	32	no	—
39	C₂×#6101	C₂³	1	32	yes	—
40	#88124	—	2	64	no	—
41	#88071	C₂³	1	32	yes	—
42	C₂×#2859	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	—	2	8	no	—
45	C₂²×C₄×C₁₃	—	2	8	no	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
47	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
49	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
50	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
52	C₂²×C₄×C₁₃	#41	1	16	yes	—
53	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
57	C₂²×Dic₁₃	—	2	16	no	—
58	C₂²×Dic₁₃	—	2	32	no	—
59	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
60	C₄²×C₁₃	—	2	32	no	—
61	C₄²×C₁₃	C₂×D₄	1	16	yes	—
62	C₄²×C₁₃	C₂×Q₈	1	16	yes	—
63	C₄²×C₁₃	C₂×Q₈	1	16	yes	—
64	C₄⋊Dic₁₃	—	2	32	no	—
65	C₄⋊Dic₁₃	—	2	64	no	—
66	#6101	—	2	64	no	—
67	C₂²×#125	—	13	2	no	—
68	#2859	—	13	16	no	—
69	C₂×#389	—	13	8	no	—
70	C₂×#384	—	13	2	no	—
71	C₂×#382	—	13	2	no	—
72	C₂×C₄³	D₁₃	1	1	yes	characteristic
73	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
74	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
75	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
76	C₂³×C₁₃	C₂×#41	1	8	yes	—
77	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	16	no	—
79	C₂×C₄×C₁₃	—	2	16	no	—
80	C₂×C₄×C₁₃	—	2	8	no	—
81	C₂×C₄×C₁₃	—	2	8	no	—
82	C₂×C₄×C₁₃	#125	1	32	yes	complemented
83	C₂×C₄×C₁₃	C₄⋊Q₈	1	16	yes	—
84	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
85	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
86	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
87	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
88	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
89	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
90	C₂×C₄×C₁₃	C₂×#41	1	16	yes	—
91	C₂×Dic₁₃	—	2	16	no	—
92	C₂×Dic₁₃	—	2	32	no	—
93	C₂×Dic₁₃	—	4	32	no	—
94	C₂×#125	—	13	16	no	—
95	C₂×#125	—	26	16	no	—
96	C₂²×C₄⋊C₄	—	13	8	no	—
97	C₂²×C₄⋊C₄	—	13	4	no	—
98	C₂²×C₄⋊C₄	—	13	8	no	—
99	C₂²×C₄²	C₂×D₁₃	1	4	yes	—
100	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
101	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
102	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
103	#389	—	13	64	no	—
104	#384	—	13	16	no	—
105	#382	—	13	16	no	—
106	C₂×#95	—	13	8	no	—
107	C₄³	C₂×D₁₃	1	8	yes	—
108	C₂²×C₁₃	C₂×#125	1	8	yes	—
109	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
110	C₂²×C₁₃	#532	1	8	yes	—
111	C₂²×C₁₃	#533	1	8	yes	—
112	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
113	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
114	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
115	C₂²×C₁₃	C₂²×#41	1	4	yes	—
116	C₄×C₁₃	—	2	8	no	—
117	C₄×C₁₃	—	2	8	no	—
118	C₄×C₁₃	C₂×#125	1	16	yes	complemented
119	C₄×C₁₃	C₂×C₄⋊Q₈	1	8	yes	—
120	Dic₁₃	—	4	16	no	—
121	C₂³×C₄	—	13	8	no	—
122	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
123	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
124	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
125	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
126	#125	—	26	64	no	—
127	C₂×C₄⋊C₄	—	13	32	no	—
128	C₂×C₄⋊C₄	—	13	16	no	—
129	C₂×C₄⋊C₄	—	13	32	no	—
130	C₂×C₄⋊C₄	—	26	32	no	—
131	C₂×C₄⋊C₄	—	26	16	no	—
132	C₂×C₄⋊C₄	—	26	32	no	—
133	C₂×C₄²	—	2	16	no	—
134	C₂×C₄²	Dic₂₆	1	8	yes	—
135	C₂×C₄²	Dic₂₆	1	8	yes	—
136	C₂×C₄²	D₅₂	1	8	yes	—
137	C₂×C₄²	C₂²×D₁₃	1	16	yes	—
138	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
139	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
140	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
141	#95	—	13	32	no	—
142	C₂×C₁₃	#2859	1	8	yes	complemented
143	C₂×C₁₃	C₂²×#125	1	2	yes	—
144	C₂×C₁₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₁₃	C₂×#532	1	2	yes	—
146	C₂×C₁₃	C₂×#533	1	2	yes	—
147	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	—	2	8	no	—
150	C₂²×C₄	—	13	16	no	—
151	C₂²×C₄	—	26	16	no	—
152	C₂²×C₄	—	26	32	no	—
153	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
154	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
155	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
156	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
157	C₂²×C₄	C₂×D₅₂	1	8	yes	—
158	C₂²×C₄	C₂×D₅₂	1	4	yes	—
159	C₂²×C₄	#6127	1	16	yes	—
160	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
161	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
162	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
163	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
164	C₄⋊C₄	—	26	64	no	—
165	C₄⋊C₄	—	26	32	no	—
166	C₄⋊C₄	—	26	64	no	—
167	C₄²	—	2	32	no	—
168	C₄²	C₂×Dic₂₆	1	16	yes	—
169	C₄²	C₂×Dic₂₆	1	16	yes	—
170	C₄²	C₂×D₅₂	1	16	yes	—
171	C₁₃	C₂×#2859	1	1	yes	characteristic, complemented, Sylow
172	C₂³	C₂²×Dic₂₆	1	2	yes	—
173	C₂³	C₂²×Dic₂₆	1	2	yes	—
174	C₂³	C₂²×D₅₂	1	2	yes	—
175	C₂³	C₂×#6127	1	8	yes	—
176	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
177	C₂×C₄	—	2	16	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	2	8	no	—
181	C₂×C₄	—	26	16	no	—
182	C₂×C₄	—	26	32	no	—
183	C₂×C₄	—	52	32	no	—
184	C₂×C₄	#88123	1	16	yes	—
185	C₂×C₄	#88124	1	32	yes	complemented
186	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
187	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
188	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
189	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
190	C₂×C₄	C₂²×D₅₂	1	8	yes	—
191	C₂×C₄	C₂²×D₅₂	1	4	yes	—
192	C₂×C₄	C₂×#6127	1	16	yes	—
193	C₂²	C₂×#88123	1	4	yes	—
194	C₂²	C₂×#88124	1	8	yes	—
195	C₂²	#11719175	1	8	yes	—
196	C₂²	#11719182	1	8	yes	—
197	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
198	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
199	C₂²	C₂³×D₅₂	1	1	yes	characteristic
200	C₂²	C₂²×#6127	1	4	yes	—
201	C₄	—	2	8	no	—
202	C₄	—	2	8	no	—
203	C₄	—	52	16	no	—
204	C₄	C₂×#88123	1	8	yes	—
205	C₄	C₂×#88124	1	16	yes	complemented
206	C₂	#13123199	1	8	yes	direct product factor
207	C₂	C₂²×#88124	1	2	yes	—
208	C₂	C₂²×#88123	1	1	yes	characteristic
209	C₂	C₂×#11719182	1	2	yes	—
210	C₂	C₂×#11719175	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₄³×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	C₂×#2859
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #13123199	C₂	direct (N × H)
82. C₂×C₄×C₁₃	#125	semidirect
118. C₄×C₁₃	C₂×#125	semidirect
142. C₂×C₁₃	#2859	semidirect
171. C₁₃	C₂×#2859	semidirect
185. C₂×C₄	#88124	semidirect
205. C₄	C₂×#88124	semidirect
206. C₂	#13123199	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×#88124	2
#13123199	2
C₂×#11718589	2
C₂×#11718596	2
C₂×#11718597	2
C₂×C₄³×C₁₃	2
C₂×#2859	13

Subgroups — C2×#13123199 #16068899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123199 #16068899