Subgroups — C₂×#11719247 #13128382

Order 1664.

1112 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	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#88190	C₂	1	2	yes	maximal
3	C₂×#88180	C₂	1	2	yes	maximal, complemented
4	C₂×#88178	C₂	1	2	yes	maximal, complemented
5	#11719247	C₂	1	16	yes	maximal, direct product factor
6	C₂×#88141	C₂	1	2	yes	maximal, complemented
7	C₂×#88133	C₂	1	4	yes	maximal, complemented
8	C₂×#88123	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×Dic₂₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
12	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
13	C₂²×Dic₂₆	C₂²	1	2	yes	—
14	C₄×D₄×C₁₃	C₂²	1	8	yes	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
19	C₂×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
20	#88190	C₂²	1	16	yes	—
21	#88180	C₂²	1	16	yes	complemented
22	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₃	C₂²	1	2	yes	—
25	#88178	C₂²	1	16	yes	complemented
26	C₂×#6101	C₂²	1	2	yes	—
27	C₂×#6101	C₂²	1	4	yes	—
28	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
29	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
30	#88141	C₂²	1	16	yes	complemented
31	#88133	C₂²	1	32	yes	complemented
32	#88123	C₂²	1	8	yes	complemented
33	C₄×Dic₂₆	C₂²	1	8	yes	complemented
34	C₂⁴×C₁₃	C₂³	1	2	yes	—
35	C₂×D₄×C₁₃	—	2	4	no	—
36	C₂×D₄×C₁₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₁₃	—	2	8	no	—
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂³	1	2	yes	—
44	C₂²×Dic₁₃	C₂³	1	2	yes	—
45	C₂²×Dic₁₃	C₂³	1	4	yes	—
46	C₂×Dic₂₆	—	2	8	no	—
47	C₂×Dic₂₆	—	2	4	no	—
48	C₂×Dic₂₆	C₂³	1	4	yes	—
49	C₂×Dic₂₆	C₂³	1	8	yes	—
50	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
51	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
52	C₄²×C₁₃	C₂³	1	4	yes	—
53	C₂²⋊Dic₁₃	C₂³	1	16	yes	—
54	C₂²⋊Dic₁₃	C₂³	1	16	yes	—
55	C₄⋊Dic₁₃	C₂³	1	4	yes	—
56	C₄⋊Dic₁₃	C₂³	1	8	yes	—
57	C₄⋊Dic₁₃	C₂³	1	8	yes	—
58	#6101	C₂³	1	8	yes	—
59	#6101	C₂³	1	16	yes	—
60	C₄×Dic₁₃	C₂³	1	8	yes	—
61	C₄×Dic₁₃	C₂³	1	16	yes	—
62	C₂×#556	—	13	1	no	maximal, Sylow
63	C₂³×C₁₃	—	2	4	no	—
64	C₂³×C₁₃	—	2	4	no	—
65	C₂³×C₁₃	—	2	4	no	—
66	C₂³×C₁₃	C₂⁴	1	4	yes	—
67	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
68	D₄×C₁₃	—	2	8	no	—
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	—	2	2	no	—
71	C₂×C₄×C₁₃	—	2	16	no	—
72	C₂×C₄×C₁₃	#41	1	4	yes	—
73	C₂×C₄×C₁₃	#41	1	4	yes	—
74	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
75	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
76	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
77	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
78	C₂×Dic₁₃	—	2	8	no	—
79	C₂×Dic₁₃	—	2	4	no	—
80	C₂×Dic₁₃	—	2	4	no	—
81	C₂×Dic₁₃	C₂⁴	1	4	yes	—
82	C₂×Dic₁₃	C₂⁴	1	4	yes	—
83	C₂×Dic₁₃	C₂⁴	1	8	yes	—
84	Dic₂₆	—	2	16	no	—
85	Dic₂₆	—	2	8	no	—
86	#556	—	13	16	no	—
87	C₂×C₄⋊Q₈	—	13	1	no	—
88	C₂×#126	—	13	4	no	—
89	C₂×#124	—	13	2	no	—
90	C₂×C₂²⋊Q₈	—	13	2	no	—
91	C₂×C₄×Q₈	—	13	2	no	—
92	C₂×C₄×Q₈	—	13	1	no	—
93	C₂×C₄×D₄	D₁₃	1	1	yes	characteristic
94	C₂×#117	—	13	2	no	—
95	C₂²×C₁₃	—	2	4	no	—
96	C₂²×C₁₃	—	2	4	no	—
97	C₂²×C₁₃	—	2	4	no	—
98	C₂²×C₁₃	—	4	8	no	—
99	C₂²×C₁₃	C₂×#41	1	2	yes	—
100	C₂²×C₁₃	C₂×#41	1	2	yes	—
101	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
102	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
103	C₄×C₁₃	—	2	4	no	—
104	C₄×C₁₃	—	2	2	no	—
105	C₄×C₁₃	C₂×#41	1	4	yes	—
106	C₄×C₁₃	C₂×#41	1	4	yes	—
107	Dic₁₃	—	2	8	no	—
108	Dic₁₃	—	2	4	no	—
109	Dic₁₃	—	2	4	no	—
110	C₂²×Q₈	—	13	1	no	—
111	C₂²×Q₈	—	13	2	no	—
112	C₂²×D₄	C₂×D₁₃	1	1	yes	characteristic
113	C₂³×C₄	C₂×D₁₃	1	2	yes	—
114	C₄⋊Q₈	—	13	8	no	—
115	#126	—	13	32	no	—
116	#124	—	13	16	no	—
117	C₂²⋊Q₈	—	13	16	no	—
118	C₄×Q₈	—	13	16	no	—
119	C₄×Q₈	—	13	8	no	—
120	C₄×D₄	C₂×D₁₃	1	8	yes	—
121	#117	—	13	16	no	—
122	C₂×C₄⋊C₄	—	13	1	no	—
123	C₂×C₄⋊C₄	—	13	2	no	—
124	C₂×C₄⋊C₄	—	13	4	no	—
125	C₂×C₄⋊C₄	—	13	2	no	—
126	C₂×C₄⋊C₄	—	13	2	no	—
127	C₂×C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
128	C₂×C₂²⋊C₄	—	13	4	no	—
129	C₂×C₂²⋊C₄	—	13	4	no	—
130	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
131	C₂×C₄²	—	13	4	no	—
132	C₂×C₄²	—	13	2	no	—
133	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
134	C₂×C₁₃	—	4	4	no	—
135	C₂×C₁₃	#556	1	4	yes	complemented
136	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
137	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
138	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
139	C₂⁴	C₂²×D₁₃	1	2	yes	—
140	C₂×Q₈	—	13	4	no	—
141	C₂×Q₈	—	13	8	no	—
142	C₂×Q₈	—	26	8	no	—
143	C₂×Q₈	—	26	4	no	—
144	C₂×D₄	—	2	4	no	—
145	C₂×D₄	C₂²×D₁₃	1	4	yes	—
146	C₂²×C₄	—	2	8	no	—
147	C₂²×C₄	—	13	2	no	—
148	C₂²×C₄	—	13	4	no	—
149	C₂²×C₄	—	13	2	no	—
150	C₂²×C₄	C₂²×D₁₃	1	8	yes	—
151	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
152	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
154	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
155	C₄⋊C₄	—	13	4	no	—
156	C₄⋊C₄	—	13	8	no	—
157	C₄⋊C₄	—	13	16	no	—
158	C₄⋊C₄	—	13	8	no	—
159	C₄⋊C₄	—	13	8	no	—
160	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
161	C₂²⋊C₄	—	13	16	no	—
162	C₂²⋊C₄	—	13	16	no	—
163	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
164	C₄²	—	13	16	no	—
165	C₄²	—	13	8	no	—
166	C₄²	C₂²×D₁₃	1	4	yes	—
167	C₁₃	C₂×#556	1	1	yes	characteristic, complemented, Sylow
168	C₂³	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	—	2	4	no	—
171	C₂³	C₂³×D₁₃	1	4	yes	—
172	C₂³	C₂³×D₁₃	1	1	yes	characteristic
173	Q₈	—	26	16	no	—
174	Q₈	—	26	8	no	—
175	D₄	—	2	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	16	no	—
179	C₂×C₄	—	13	4	no	—
180	C₂×C₄	—	13	8	no	—
181	C₂×C₄	—	13	4	no	—
182	C₂×C₄	—	26	8	no	—
183	C₂×C₄	—	26	4	no	—
184	C₂×C₄	—	26	4	no	—
185	C₂×C₄	#6127	1	4	yes	—
186	C₂×C₄	#6129	1	4	yes	—
187	C₂×C₄	C₂³×D₁₃	1	4	yes	—
188	C₂×C₄	C₂³×D₁₃	1	2	yes	—
189	C₂×C₄	C₂³×D₁₃	1	2	yes	—
190	C₂×C₄	C₂³×D₁₃	1	2	yes	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	4	8	no	—
195	C₂²	C₂×#6127	1	2	yes	—
196	C₂²	C₂×#6129	1	2	yes	—
197	C₂²	#88257	1	2	yes	—
198	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	26	8	no	—
202	C₄	—	26	4	no	—
203	C₄	—	26	4	no	—
204	C₄	C₂×#6127	1	4	yes	—
205	C₄	C₂×#6129	1	4	yes	—
206	C₂	—	4	4	no	—
207	C₂	#11719247	1	4	yes	direct product factor
208	C₂	C₂²×#6127	1	1	yes	characteristic
209	C₂	C₂²×#6129	1	1	yes	characteristic
210	C₂	C₂×#88257	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₄×D₄×C₁₃	C₂
socle	C₂³×C₁₃	C₂⁴
prime radical	C₂²×D₄×C₁₃	C₂²
hypercenter	C₂×C₄×D₄	D₁₃
hypocenter	C₁₃	C₂×#556
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
3. C₂×#88180	C₂	semidirect
4. C₂×#88178	C₂	semidirect
5. #11719247	C₂	direct (N × H)
6. C₂×#88141	C₂	semidirect
7. C₂×#88133	C₂	semidirect
8. C₂×#88123	C₂	semidirect
9. C₂×C₄×Dic₂₆	C₂	semidirect
21. #88180	C₂²	semidirect
25. #88178	C₂²	semidirect
30. #88141	C₂²	semidirect
31. #88133	C₂²	semidirect
32. #88123	C₂²	semidirect
33. C₄×Dic₂₆	C₂²	semidirect
135. C₂×C₁₃	#556	semidirect
167. C₁₃	C₂×#556	semidirect
207. C₂	#11719247	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×D₄×C₁₃	2
C₂×#88190	2
C₂×#88180	2
C₂×#88178	2
#11719247	2
C₂×#88141	2
C₂×#88133	2
C₂×#88123	2
C₂×C₄×Dic₂₆	2
C₂×#556	13

Subgroups — C2×#11719247 #13128382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719247 #13128382