Subgroups — C₂×#11719290 #13128424

Order 1664.

1240 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88201	C₂	1	2	yes	maximal, complemented
3	#11719290	C₂	1	16	yes	maximal, direct product factor
4	C₂×#88148	C₂	1	4	yes	maximal, complemented
5	C₂×#88147	C₂	1	2	yes	maximal, complemented
6	C₂×#88129	C₂	1	2	yes	maximal, complemented
7	C₂×#88128	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×D₅₂	C₂	1	2	yes	maximal, complemented
9	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×D₅₂	C₂²	1	2	yes	—
12	C₂²×D₅₂	C₂²	1	2	yes	—
13	C₂²×D₅₂	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₃	C₂²	1	4	yes	—
15	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₁₃	C₂²	1	8	yes	complemented
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
21	#88201	C₂²	1	16	yes	complemented
22	C₂×#6103	C₂²	1	4	yes	—
23	C₂×#6103	C₂²	1	4	yes	—
24	C₂×#6103	C₂²	1	4	yes	—
25	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	C₂×#6101	C₂²	1	2	yes	—
27	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
28	#88148	C₂²	1	32	yes	complemented
29	#88147	C₂²	1	16	yes	complemented
30	#88129	C₂²	1	16	yes	complemented
31	#88128	C₂²	1	8	yes	—
32	C₄×D₅₂	C₂²	1	16	yes	complemented
33	C₄×Dic₂₆	C₂²	1	8	yes	complemented
34	C₂³×D₁₃	C₂³	1	4	yes	—
35	C₂×Q₈×C₁₃	—	2	4	no	—
36	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂³	1	2	yes	—
43	C₂²×Dic₁₃	C₂³	1	2	yes	—
44	C₂×D₅₂	—	2	8	no	—
45	C₂×D₅₂	—	2	8	no	—
46	C₂×D₅₂	—	4	4	no	—
47	C₂×D₅₂	C₂³	1	8	yes	—
48	C₂×D₅₂	C₂³	1	8	yes	—
49	C₂×D₅₂	C₂³	1	8	yes	—
50	C₂×C₄×D₁₃	—	4	8	no	—
51	C₂×C₄×D₁₃	C₂³	1	16	yes	—
52	C₂×Dic₂₆	—	2	4	no	—
53	C₂×Dic₂₆	C₂³	1	4	yes	—
54	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
55	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
56	C₄²×C₁₃	C₂³	1	4	yes	—
57	C₄²×C₁₃	C₂³	1	8	yes	—
58	#6103	C₂³	1	16	yes	—
59	#6103	C₂³	1	16	yes	—
60	#6103	C₂³	1	16	yes	—
61	C₄⋊Dic₁₃	C₂³	1	4	yes	—
62	#6101	C₂³	1	8	yes	—
63	C₄×Dic₁₃	C₂³	1	8	yes	—
64	C₂×#559	—	13	1	no	maximal, Sylow
65	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂²×D₁₃	—	2	8	no	—
67	C₂²×D₁₃	—	2	8	no	—
68	C₂²×D₁₃	—	2	8	no	—
69	C₂²×D₁₃	C₂⁴	1	8	yes	—
70	Q₈×C₁₃	—	2	8	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	#41	1	4	yes	—
75	C₂×C₄×C₁₃	#41	1	4	yes	—
76	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
78	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
79	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
80	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
81	C₂×Dic₁₃	—	2	4	no	—
82	C₂×Dic₁₃	—	2	4	no	—
83	C₂×Dic₁₃	C₂⁴	1	4	yes	—
84	C₂×Dic₁₃	C₂⁴	1	4	yes	—
85	D₅₂	—	2	16	no	—
86	D₅₂	—	2	16	no	—
87	D₅₂	—	4	8	no	—
88	C₄×D₁₃	—	4	16	no	—
89	Dic₂₆	—	2	8	no	—
90	#559	—	13	16	no	—
91	C₂×#127	—	13	1	no	—
92	C₂×#124	—	13	2	no	—
93	C₂×#124	—	13	2	no	—
94	C₂×#123	—	13	4	no	—
95	C₂×C₄×Q₈	—	13	1	no	—
96	C₂×C₄×Q₈	D₁₃	1	1	yes	characteristic, complemented
97	C₂×C₄×D₄	—	13	2	no	—
98	C₂×C₄×D₄	—	13	2	no	—
99	C₂²×C₁₃	C₂×#41	1	2	yes	—
100	C₂²×C₁₃	C₂×#41	1	2	yes	—
101	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
103	C₂×D₁₃	—	2	8	no	—
104	C₂×D₁₃	—	2	8	no	—
105	C₂×D₁₃	—	2	8	no	—
106	C₂×D₁₃	—	4	16	no	—
107	C₄×C₁₃	—	2	4	no	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	C₂×#41	1	4	yes	—
111	C₄×C₁₃	C₂×#41	1	4	yes	—
112	Dic₁₃	—	2	4	no	—
113	Dic₁₃	—	2	4	no	—
114	C₂²×Q₈	—	13	1	no	—
115	C₂²×Q₈	C₂×D₁₃	1	1	yes	characteristic
116	C₂²×D₄	—	13	2	no	—
117	C₂²×D₄	—	13	2	no	—
118	C₂²×D₄	—	13	2	no	—
119	C₂³×C₄	—	13	4	no	—
120	#127	—	13	8	no	—
121	#124	—	13	16	no	—
122	#124	—	13	16	no	—
123	#123	—	13	32	no	—
124	C₄×Q₈	—	13	8	no	—
125	C₄×Q₈	C₂×D₁₃	1	8	yes	complemented
126	C₄×D₄	—	13	16	no	—
127	C₄×D₄	—	13	16	no	—
128	C₂×C₄⋊C₄	—	13	1	no	—
129	C₂×C₄⋊C₄	—	13	2	no	—
130	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
131	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
132	C₂×C₂²⋊C₄	—	13	4	no	—
133	C₂×C₂²⋊C₄	—	13	4	no	—
134	C₂×C₂²⋊C₄	—	13	4	no	—
135	C₂×C₄²	—	13	2	no	—
136	C₂×C₄²	C₂×D₁₃	1	2	yes	—
137	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
138	C₂×C₁₃	#559	1	4	yes	complemented
139	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
140	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
141	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
142	D₁₃	—	4	8	no	—
143	C₂⁴	—	13	4	no	—
144	C₂×Q₈	—	2	4	no	—
145	C₂×Q₈	—	13	4	no	—
146	C₂×Q₈	—	26	4	no	—
147	C₂×Q₈	C₂²×D₁₃	1	4	yes	—
148	C₂×D₄	—	13	8	no	—
149	C₂×D₄	—	13	8	no	—
150	C₂×D₄	—	13	8	no	—
151	C₂×D₄	—	26	8	no	—
152	C₂×D₄	—	26	8	no	—
153	C₂×D₄	—	52	4	no	—
154	C₂²×C₄	—	13	2	no	—
155	C₂²×C₄	—	13	2	no	—
156	C₂²×C₄	—	13	16	no	—
157	C₂²×C₄	—	52	8	no	—
158	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
162	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
163	C₄⋊C₄	—	13	4	no	—
164	C₄⋊C₄	—	13	8	no	—
165	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
166	C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
167	C₂²⋊C₄	—	13	16	no	—
168	C₂²⋊C₄	—	13	16	no	—
169	C₂²⋊C₄	—	13	16	no	—
170	C₄²	—	13	8	no	—
171	C₄²	C₂²×D₁₃	1	8	yes	—
172	C₄²	C₂²×D₁₃	1	4	yes	—
173	C₁₃	C₂×#559	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	13	8	no	—
175	C₂³	—	26	8	no	—
176	C₂³	—	26	8	no	—
177	C₂³	—	26	8	no	—
178	C₂³	C₂³×D₁₃	1	1	yes	characteristic
179	Q₈	—	2	8	no	—
180	Q₈	—	26	8	no	—
181	D₄	—	26	16	no	—
182	D₄	—	26	16	no	—
183	D₄	—	52	8	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	13	4	no	—
188	C₂×C₄	—	13	4	no	—
189	C₂×C₄	—	26	4	no	—
190	C₂×C₄	—	26	4	no	—
191	C₂×C₄	—	52	16	no	—
192	C₂×C₄	#6127	1	4	yes	—
193	C₂×C₄	#6131	1	4	yes	—
194	C₂×C₄	C₂³×D₁₃	1	4	yes	—
195	C₂×C₄	C₂³×D₁₃	1	4	yes	—
196	C₂×C₄	C₂³×D₁₃	1	2	yes	—
197	C₂×C₄	C₂³×D₁₃	1	2	yes	—
198	C₂×C₄	C₂³×D₁₃	1	2	yes	—
199	C₂²	—	26	8	no	—
200	C₂²	—	26	8	no	—
201	C₂²	—	26	8	no	—
202	C₂²	—	52	16	no	—
203	C₂²	C₂×#6127	1	2	yes	—
204	C₂²	C₂×#6131	1	2	yes	—
205	C₂²	#88256	1	2	yes	—
206	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
207	C₄	—	2	4	no	—
208	C₄	—	2	4	no	—
209	C₄	—	2	2	no	—
210	C₄	—	26	4	no	—
211	C₄	—	26	4	no	—
212	C₄	C₂×#6127	1	4	yes	—
213	C₄	C₂×#6131	1	4	yes	—
214	C₂	—	52	8	no	—
215	C₂	#11719290	1	4	yes	direct product factor
216	C₂	C₂²×#6127	1	1	yes	characteristic
217	C₂	C₂²×#6131	1	1	yes	characteristic
218	C₂	C₂×#88256	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₄×Q₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂×#88128	C₂
hypercenter	C₂×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#559
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×Q₈×C₁₃	C₂	semidirect
2. C₂×#88201	C₂	semidirect
3. #11719290	C₂	direct (N × H)
4. C₂×#88148	C₂	semidirect
5. C₂×#88147	C₂	semidirect
6. C₂×#88129	C₂	semidirect
8. C₂×C₄×D₅₂	C₂	semidirect
9. C₂×C₄×Dic₂₆	C₂	semidirect
16. C₄×Q₈×C₁₃	C₂²	semidirect
21. #88201	C₂²	semidirect
28. #88148	C₂²	semidirect
29. #88147	C₂²	semidirect
30. #88129	C₂²	semidirect
32. C₄×D₅₂	C₂²	semidirect
33. C₄×Dic₂₆	C₂²	semidirect
96. C₂×C₄×Q₈	D₁₃	semidirect
125. C₄×Q₈	C₂×D₁₃	semidirect
138. C₂×C₁₃	#559	semidirect
173. C₁₃	C₂×#559	semidirect
215. C₂	#11719290	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Q₈×C₁₃	2
C₂×#88201	2
#11719290	2
C₂×#88148	2
C₂×#88147	2
C₂×#88129	2
C₂×#88128	2
C₂×C₄×D₅₂	2
C₂×C₄×Dic₂₆	2
C₂×#559	13

Subgroups — C2×#11719290 #13128424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719290 #13128424