Subgroups — C₂×#11719230 #13128147

Order 1664.

1896 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₅₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	#11719230	C₂	1	16	yes	maximal, direct product factor
5	C₂×#88149	C₂	1	4	yes	maximal, complemented
6	C₂×#88136	C₂	1	4	yes	maximal, complemented
7	C₂×#88129	C₂	1	2	yes	maximal, complemented
8	C₂×#88128	C₂	1	2	yes	maximal, complemented
9	C₂⁴×D₁₃	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₂²⋊D₁₃	C₂²	1	2	yes	—
12	C₂×#6127	—	2	8	no	—
13	C₂×#6127	C₂²	1	8	yes	complemented
14	C₂²×D₅₂	—	2	8	no	—
15	C₂²×D₅₂	C₂²	1	2	yes	—
16	C₂²×D₅₂	C₂²	1	2	yes	—
17	C₂²×D₅₂	C₂²	1	8	yes	complemented
18	C₂²×D₅₂	C₂²	1	1	yes	characteristic, complemented
19	C₂²×D₅₂	C₂²	1	8	yes	complemented
20	C₂²×C₄×D₁₃	C₂²	1	2	yes	complemented
21	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic, complemented
22	C₁₃×#117	C₂²	1	8	yes	complemented
23	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	complemented
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂²	1	2	yes	complemented
26	C₂×#6103	C₂²	1	8	yes	complemented
27	#88149	C₂²	1	32	yes	complemented
28	#88136	C₂²	1	32	yes	complemented
29	#88129	C₂²	1	16	yes	complemented
30	#88128	C₂²	1	16	yes	complemented
31	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
32	C₂³×D₁₃	—	2	8	no	—
33	C₂³×D₁₃	—	4	8	no	—
34	C₂³×D₁₃	C₂³	1	4	yes	—
35	C₂³×D₁₃	C₂³	1	2	yes	—
36	C₂³×D₁₃	C₂³	1	8	yes	—
37	C₂²×C₄×C₁₃	D₄	1	8	yes	complemented
38	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
42	C₂×C₂²⋊D₁₃	—	2	8	no	—
43	C₂×C₂²⋊D₁₃	C₂³	1	8	yes	—
44	C₂²×Dic₁₃	C₂³	1	2	yes	—
45	#6127	—	2	32	no	—
46	C₂×D₅₂	—	2	8	no	—
47	C₂×D₅₂	—	2	8	no	—
48	C₂×D₅₂	—	2	4	no	—
49	C₂×D₅₂	—	2	32	no	—
50	C₂×D₅₂	—	4	16	no	—
51	C₂×D₅₂	C₂³	1	8	yes	—
52	C₂×D₅₂	C₂³	1	8	yes	—
53	C₂×D₅₂	C₂³	1	4	yes	complemented
54	C₂×D₅₂	C₂³	1	32	yes	complemented
55	C₂×C₄×D₁₃	—	2	8	no	—
56	C₂×C₄×D₁₃	C₂³	1	8	yes	complemented
57	C₂×Dic₂₆	—	2	4	no	—
58	C₂×Dic₂₆	C₂³	1	4	yes	complemented
59	C₁₃×C₄⋊C₄	C₂³	1	8	yes	complemented
60	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
61	C₄²×C₁₃	C₂³	1	8	yes	complemented
62	#6103	C₂³	1	32	yes	complemented
63	C₂×#535	—	13	1	no	maximal, Sylow
64	C₂³×C₁₃	—	2	4	no	—
65	C₂³×C₁₃	C₂×D₄	1	4	yes	—
66	C₂³×C₁₃	C₂⁴	1	2	yes	—
67	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
68	C₂²×D₁₃	—	2	4	no	—
69	C₂²×D₁₃	—	2	16	no	—
70	C₂²×D₁₃	—	2	8	no	—
71	C₂²×D₁₃	—	2	8	no	—
72	C₂²×D₁₃	—	2	16	no	—
73	C₂²×D₁₃	—	4	16	no	—
74	C₂²×D₁₃	—	4	32	no	—
75	C₂²×D₁₃	C₂⁴	1	8	yes	—
76	C₂²×D₁₃	C₂⁴	1	4	yes	—
77	C₂×C₄×C₁₃	—	2	8	no	—
78	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
79	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	16	yes	complemented
81	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
82	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
83	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
84	C₂²⋊D₁₃	—	2	16	no	—
85	C₂×Dic₁₃	—	2	4	no	—
86	C₂×Dic₁₃	C₂⁴	1	4	yes	—
87	D₅₂	—	2	16	no	—
88	D₅₂	—	2	16	no	—
89	D₅₂	—	2	8	no	—
90	D₅₂	—	4	32	no	—
91	C₄×D₁₃	—	2	16	no	—
92	Dic₂₆	—	2	8	no	—
93	C₂²×#41	—	13	1	no	—
94	C₂³×D₄	—	13	1	no	—
95	#535	—	13	16	no	—
96	C₂×#127	—	13	2	no	—
97	C₂×#124	—	13	2	no	—
98	C₂×#121	—	13	4	no	—
99	C₂×#120	—	13	4	no	—
100	C₂×#117	D₁₃	1	1	yes	characteristic, complemented
101	C₂²×C₁₃	—	2	8	no	—
102	C₂²×C₁₃	—	2	4	no	—
103	C₂²×C₁₃	C₂²×D₄	1	4	yes	—
104	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
105	C₂²×C₁₃	H(5,𝔽₂)	1	4	yes	—
106	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
107	C₂×D₁₃	—	2	4	no	—
108	C₂×D₁₃	—	2	8	no	—
109	C₂×D₁₃	—	2	8	no	—
110	C₂×D₁₃	—	2	16	no	—
111	C₂×D₁₃	—	4	32	no	—
112	C₂×D₁₃	—	4	8	no	—
113	C₂×D₁₃	—	4	16	no	—
114	C₄×C₁₃	—	2	8	no	—
115	C₄×C₁₃	C₂²×D₄	1	4	yes	—
116	C₄×C₁₃	C₂²×D₄	1	4	yes	—
117	Dic₁₃	—	2	4	no	—
118	C₂⁵	—	13	2	no	—
119	C₂×#41	—	13	8	no	—
120	C₂×#41	—	26	8	no	—
121	C₂²×Q₈	—	13	1	no	—
122	C₂²×D₄	—	13	2	no	—
123	C₂²×D₄	—	13	2	no	—
124	C₂²×D₄	—	13	2	no	—
125	C₂²×D₄	—	13	1	no	—
126	C₂²×D₄	—	13	8	no	—
127	C₂²×D₄	—	13	8	no	—
128	C₂²×D₄	—	26	8	no	—
129	C₂³×C₄	—	13	2	no	—
130	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
131	#127	—	13	16	no	—
132	#124	—	13	16	no	—
133	#121	—	13	32	no	—
134	#120	—	13	32	no	—
135	#117	C₂×D₁₃	1	8	yes	complemented
136	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	complemented
137	C₂×C₂²⋊C₄	—	13	8	no	—
138	C₂×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
139	C₂×C₄²	C₂×D₁₃	1	2	yes	complemented
140	C₂×C₁₃	—	2	4	no	—
141	C₂×C₁₃	#535	1	4	yes	complemented
142	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
143	C₂×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
144	D₁₃	—	4	4	no	—
145	D₁₃	—	4	8	no	—
146	C₂⁴	—	13	4	no	—
147	C₂⁴	—	13	2	no	—
148	C₂⁴	—	13	8	no	—
149	C₂⁴	—	26	8	no	—
150	C₂⁴	—	52	8	no	—
151	C₂⁴	C₂²×D₁₃	1	1	yes	characteristic
152	#41	—	26	32	no	—
153	C₂×Q₈	—	13	4	no	—
154	C₂×Q₈	—	26	4	no	—
155	C₂×D₄	—	13	8	no	—
156	C₂×D₄	—	13	8	no	—
157	C₂×D₄	—	13	8	no	—
158	C₂×D₄	—	13	4	no	—
159	C₂×D₄	—	13	32	no	—
160	C₂×D₄	—	26	8	no	—
161	C₂×D₄	—	26	8	no	—
162	C₂×D₄	—	26	8	no	—
163	C₂×D₄	—	26	4	no	—
164	C₂×D₄	—	26	32	no	—
165	C₂×D₄	—	52	16	no	—
166	C₂²×C₄	—	13	2	no	—
167	C₂²×C₄	—	13	8	no	—
168	C₂²×C₄	—	26	8	no	—
169	C₂²×C₄	D₅₂	1	8	yes	complemented
170	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
171	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
174	C₄⋊C₄	C₂²×D₁₃	1	8	yes	complemented
175	C₂²⋊C₄	—	13	32	no	—
176	C₂²⋊C₄	C₂²×D₁₃	1	8	yes	—
177	C₄²	C₂²×D₁₃	1	8	yes	complemented
178	C₁₃	C₂×#535	1	1	yes	characteristic, complemented, Sylow
179	C₂³	—	2	4	no	—
180	C₂³	—	13	8	no	—
181	C₂³	—	13	4	no	—
182	C₂³	—	26	4	no	—
183	C₂³	—	26	16	no	—
184	C₂³	—	26	8	no	—
185	C₂³	—	26	8	no	—
186	C₂³	—	26	16	no	—
187	C₂³	—	52	16	no	—
188	C₂³	—	52	32	no	—
189	C₂³	C₂×D₅₂	1	4	yes	—
190	C₂³	C₂³×D₁₃	1	2	yes	—
191	C₂³	C₂³×D₁₃	1	1	yes	characteristic
192	Q₈	—	26	8	no	—
193	D₄	—	26	16	no	—
194	D₄	—	26	16	no	—
195	D₄	—	26	16	no	—
196	D₄	—	26	8	no	—
197	D₄	—	52	32	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	13	4	no	—
200	C₂×C₄	—	26	4	no	—
201	C₂×C₄	—	26	16	no	—
202	C₂×C₄	C₂×D₅₂	1	4	yes	—
203	C₂×C₄	C₂×D₅₂	1	4	yes	—
204	C₂×C₄	C₂×D₅₂	1	16	yes	complemented
205	C₂×C₄	C₂³×D₁₃	1	8	yes	—
206	C₂×C₄	C₂³×D₁₃	1	2	yes	—
207	C₂×C₄	C₂³×D₁₃	1	2	yes	—
208	C₂²	—	2	8	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	26	4	no	—
211	C₂²	—	26	8	no	—
212	C₂²	—	26	8	no	—
213	C₂²	—	26	16	no	—
214	C₂²	—	52	32	no	—
215	C₂²	—	52	8	no	—
216	C₂²	—	52	16	no	—
217	C₂²	C₂²×D₅₂	1	4	yes	—
218	C₂²	C₂²×D₅₂	1	2	yes	—
219	C₂²	#88256	1	4	yes	—
220	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
221	C₄	—	2	8	no	—
222	C₄	—	26	4	no	—
223	C₄	C₂²×D₅₂	1	4	yes	—
224	C₄	C₂²×D₅₂	1	4	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	52	8	no	—
227	C₂	—	52	4	no	—
228	C₂	#11719230	1	4	yes	direct product factor
229	C₂	C₂³×D₅₂	1	1	yes	characteristic
230	C₂	C₂×#88256	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#117	C₂
socle	C₂³×C₁₃	C₂⁴
hypercenter	C₂×#117	D₁₃
hypocenter	C₁₃	C₂×#535
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. C₂²×#6127	C₂	semidirect
2. C₂³×D₅₂	C₂	semidirect
3. C₂×C₁₃×#117	C₂	semidirect
4. #11719230	C₂	direct (N × H)
5. C₂×#88149	C₂	semidirect
6. C₂×#88136	C₂	semidirect
7. C₂×#88129	C₂	semidirect
8. C₂×#88128	C₂	semidirect
13. C₂×#6127	C₂²	semidirect
17. C₂²×D₅₂	C₂²	semidirect
18. C₂²×D₅₂	C₂²	semidirect
19. C₂²×D₅₂	C₂²	semidirect
20. C₂²×C₄×D₁₃	C₂²	semidirect
21. C₂²×Dic₂₆	C₂²	semidirect
22. C₁₃×#117	C₂²	semidirect
23. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
25. C₂×C₄²×C₁₃	C₂²	semidirect
26. C₂×#6103	C₂²	semidirect
27. #88149	C₂²	semidirect
28. #88136	C₂²	semidirect
29. #88129	C₂²	semidirect
30. #88128	C₂²	semidirect
37. C₂²×C₄×C₁₃	D₄	semidirect
53. C₂×D₅₂	C₂³	semidirect
54. C₂×D₅₂	C₂³	semidirect
56. C₂×C₄×D₁₃	C₂³	semidirect
58. C₂×Dic₂₆	C₂³	semidirect
59. C₁₃×C₄⋊C₄	C₂³	semidirect
61. C₄²×C₁₃	C₂³	semidirect
62. #6103	C₂³	semidirect
80. C₂×C₄×C₁₃	C₂×D₄	semidirect
100. C₂×#117	D₁₃	semidirect
135. #117	C₂×D₁₃	semidirect
136. C₂×C₄⋊C₄	C₂×D₁₃	semidirect
139. C₂×C₄²	C₂×D₁₃	semidirect
141. C₂×C₁₃	#535	semidirect
169. C₂²×C₄	D₅₂	semidirect
174. C₄⋊C₄	C₂²×D₁₃	semidirect
177. C₄²	C₂²×D₁₃	semidirect
178. C₁₃	C₂×#535	semidirect
204. C₂×C₄	C₂×D₅₂	semidirect
228. C₂	#11719230	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×#6127	2
C₂³×D₅₂	2
C₂×C₁₃×#117	2
#11719230	2
C₂×#88149	2
C₂×#88136	2
C₂×#88129	2
C₂×#88128	2
C₂×#535	13

Subgroups — C2×#11719230 #13128147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719230 #13128147