Subgroups — C₂²×#11719425 #16769785

Order 3328.

9606 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719425	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88130	C₂	1	2	yes	maximal, complemented
3	C₂²×#88189	C₂	1	3	yes	maximal, complemented
4	C₂²×#88193	C₂	1	6	yes	maximal, complemented
5	C₂²×#88191	C₂	1	3	yes	maximal, complemented
6	C₂²×C₁₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
9	C₂³×D₄×C₁₃	C₂²	1	3	yes	complemented
10	C₂³×C₂²⋊D₁₃	C₂²	1	6	yes	—
11	C₂⁴×Dic₁₃	C₂²	1	3	yes	complemented
12	C₂×C₁₃×#127	C₂²	1	24	yes	complemented
13	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₂²⋊Dic₁₃	C₂²	1	3	yes	—
15	C₂²×C₂²⋊Dic₁₃	C₂²	1	3	yes	complemented
16	C₂×#88193	C₂²	1	144	yes	complemented
17	C₂×#88191	C₂²	1	72	yes	complemented
18	C₂×#88189	C₂²	1	72	yes	complemented
19	C₂²×#6103	C₂²	1	6	yes	—
20	C₂²×#6101	C₂²	1	6	yes	complemented
21	#11719425	C₂²	1	256	yes	direct product factor
22	C₂×#88130	C₂²	1	48	yes	complemented
23	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
24	C₂⁵×C₁₃	C₂³	1	3	yes	—
25	C₂⁴×D₁₃	—	4	12	no	—
26	C₂⁴×D₁₃	C₂³	1	2	yes	—
27	C₂⁴×D₁₃	C₂³	1	12	yes	—
28	C₂²×D₄×C₁₃	—	4	12	no	—
29	C₂²×D₄×C₁₃	—	4	12	no	—
30	C₂²×D₄×C₁₃	C₂³	1	36	yes	—
31	C₂²×D₄×C₁₃	C₂³	1	36	yes	complemented
32	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
33	C₂²×C₂²⋊D₁₃	—	4	24	no	—
34	C₂²×C₂²⋊D₁₃	C₂³	1	72	yes	—
35	C₂³×Dic₁₃	—	4	12	no	—
36	C₂³×Dic₁₃	C₂³	1	6	yes	—
37	C₂³×Dic₁₃	C₂³	1	36	yes	complemented
38	C₁₃×#127	C₂³	1	64	yes	complemented
39	C₂×C₄²×C₁₃	C₂³	1	12	yes	complemented
40	C₂×C₂²⋊Dic₁₃	C₂³	1	36	yes	—
41	C₂×C₂²⋊Dic₁₃	C₂³	1	36	yes	complemented
42	#88193	C₂³	1	384	yes	complemented
43	#88191	C₂³	1	192	yes	complemented
44	#88189	C₂³	1	192	yes	complemented
45	C₂×#6103	C₂³	1	72	yes	—
46	C₂×#6101	C₂³	1	72	yes	complemented
47	#88130	C₂³	1	128	yes	complemented
48	C₂²×#560	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	—	2	12	no	—
50	C₂⁴×C₁₃	—	2	24	no	—
51	C₂⁴×C₁₃	—	2	12	no	—
52	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
53	C₂⁴×C₁₃	C₂⁴	1	18	yes	—
54	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
55	C₂³×D₁₃	—	2	24	no	—
56	C₂³×D₁₃	—	4	72	no	—
57	C₂³×D₁₃	—	4	64	no	—
58	C₂³×D₁₃	C₂⁴	1	12	yes	—
59	C₂³×D₁₃	C₂⁴	1	16	yes	—
60	C₂×D₄×C₁₃	—	4	72	no	—
61	C₂×D₄×C₁₃	—	4	72	no	—
62	C₂×D₄×C₁₃	C₂⁴	1	48	yes	—
63	C₂×D₄×C₁₃	C₂⁴	1	48	yes	complemented
64	C₂²×C₄×C₁₃	—	2	12	no	—
65	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
66	C₂×C₂²⋊D₁₃	—	4	144	no	—
67	C₂×C₂²⋊D₁₃	C₂⁴	1	96	yes	—
68	C₂²×Dic₁₃	—	2	24	no	—
69	C₂²×Dic₁₃	—	4	72	no	—
70	C₂²×Dic₁₃	C₂⁴	1	36	yes	—
71	C₂²×Dic₁₃	C₂⁴	1	48	yes	complemented
72	C₄²×C₁₃	C₂⁴	1	16	yes	complemented
73	C₂²⋊Dic₁₃	C₂⁴	1	48	yes	—
74	C₂²⋊Dic₁₃	C₂⁴	1	48	yes	complemented
75	#6103	C₂⁴	1	96	yes	—
76	#6101	C₂⁴	1	96	yes	complemented
77	C₂×#560	—	13	48	no	—
78	C₂²×#127	D₁₃	1	1	yes	characteristic, complemented
79	C₂²×#126	—	13	2	no	—
80	C₂²×#123	—	13	3	no	—
81	C₂²×#121	—	13	6	no	—
82	C₂²×#120	—	13	3	no	—
83	C₂³×C₁₃	—	2	36	no	—
84	C₂³×C₁₃	—	2	72	no	—
85	C₂³×C₁₃	—	2	36	no	—
86	C₂³×C₁₃	—	4	16	no	—
87	C₂³×C₁₃	—	4	48	no	—
88	C₂³×C₁₃	H(5,𝔽₂)	1	12	yes	—
89	C₂³×C₁₃	C₂⁵	1	4	yes	—
90	C₂³×C₁₃	C₂⁵	1	12	yes	—
91	C₂³×C₁₃	C₂⁵	1	3	yes	—
92	C₂²×D₁₃	—	2	72	no	—
93	C₂²×D₁₃	—	4	48	no	—
94	C₂²×D₁₃	—	4	192	no	—
95	C₂²×D₁₃	—	4	32	no	—
96	C₂²×D₁₃	C₂⁵	1	8	yes	—
97	D₄×C₁₃	—	4	48	no	—
98	D₄×C₁₃	—	4	48	no	—
99	C₂×C₄×C₁₃	—	2	36	no	—
100	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
101	C₂²⋊D₁₃	—	4	96	no	—
102	C₂×Dic₁₃	—	2	72	no	—
103	C₂×Dic₁₃	—	4	48	no	—
104	C₂×Dic₁₃	C₂⁵	1	24	yes	—
105	C₂⁶	—	13	1	no	—
106	C₂³×D₄	—	13	6	no	—
107	C₂³×D₄	C₂×D₁₃	1	3	yes	—
108	C₂³×D₄	C₂×D₁₃	1	3	yes	complemented
109	C₂⁴×C₄	—	13	3	no	—
110	#560	—	13	256	no	—
111	C₂×#127	C₂×D₁₃	1	24	yes	complemented
112	C₂×#126	—	13	48	no	—
113	C₂×#123	—	13	72	no	—
114	C₂×#121	—	13	144	no	—
115	C₂×#120	—	13	72	no	—
116	C₂²×C₄⋊C₄	—	13	6	no	—
117	C₂²×C₂²⋊C₄	—	13	3	no	—
118	C₂²×C₂²⋊C₄	—	13	3	no	—
119	C₂²×C₂²⋊C₄	—	13	6	no	—
120	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
121	C₂²×C₁₃	—	2	12	no	—
122	C₂²×C₁₃	—	2	24	no	—
123	C₂²×C₁₃	—	2	12	no	—
124	C₂²×C₁₃	—	4	24	no	—
125	C₂²×C₁₃	—	4	72	no	—
126	C₂²×C₁₃	#560	1	16	yes	complemented
127	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	18	yes	—
128	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
129	C₂×D₁₃	—	2	24	no	—
130	C₂×D₁₃	—	4	64	no	—
131	C₂×D₁₃	—	4	48	no	—
132	C₄×C₁₃	—	2	12	no	—
133	Dic₁₃	—	2	24	no	—
134	C₂⁵	—	13	2	no	—
135	C₂⁵	—	13	12	no	—
136	C₂⁵	—	52	12	no	—
137	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
138	C₂⁵	C₂²×D₁₃	1	3	yes	—
139	C₂²×D₄	—	4	12	no	—
140	C₂²×D₄	—	4	12	no	—
141	C₂²×D₄	—	13	72	no	—
142	C₂²×D₄	—	52	24	no	—
143	C₂²×D₄	C₂²×D₁₃	1	36	yes	—
144	C₂²×D₄	C₂²×D₁₃	1	36	yes	complemented
145	C₂³×C₄	—	13	6	no	—
146	C₂³×C₄	—	13	36	no	—
147	C₂³×C₄	—	52	12	no	—
148	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
149	#127	C₂²×D₁₃	1	64	yes	complemented
150	#126	—	13	128	no	—
151	#123	—	13	192	no	—
152	#121	—	13	384	no	—
153	#120	—	13	192	no	—
154	C₂×C₄⋊C₄	—	13	72	no	—
155	C₂×C₂²⋊C₄	—	13	36	no	—
156	C₂×C₂²⋊C₄	—	13	36	no	—
157	C₂×C₂²⋊C₄	—	13	72	no	—
158	C₂×C₄²	C₂²×D₁₃	1	12	yes	complemented
159	C₂×C₁₃	—	4	4	no	—
160	C₂×C₁₃	—	4	12	no	—
161	C₂×C₁₃	C₂×#560	1	12	yes	complemented
162	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	3	yes	—
163	D₁₃	—	4	8	no	—
164	C₂⁴	—	2	12	no	—
165	C₂⁴	—	2	24	no	—
166	C₂⁴	—	2	12	no	—
167	C₂⁴	—	13	12	no	—
168	C₂⁴	—	13	16	no	—
169	C₂⁴	—	26	24	no	—
170	C₂⁴	—	52	72	no	—
171	C₂⁴	—	52	64	no	—
172	C₂⁴	C₂³×D₁₃	1	6	yes	—
173	C₂⁴	C₂³×D₁₃	1	18	yes	—
174	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
175	C₂×D₄	—	4	72	no	—
176	C₂×D₄	—	4	72	no	—
177	C₂×D₄	—	13	96	no	—
178	C₂×D₄	—	52	144	no	—
179	C₂×D₄	C₂³×D₁₃	1	48	yes	—
180	C₂×D₄	C₂³×D₁₃	1	48	yes	complemented
181	C₂²×C₄	—	2	12	no	—
182	C₂²×C₄	—	13	36	no	—
183	C₂²×C₄	—	13	48	no	—
184	C₂²×C₄	—	26	24	no	—
185	C₂²×C₄	—	52	72	no	—
186	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
187	C₄⋊C₄	—	13	96	no	—
188	C₂²⋊C₄	—	13	48	no	—
189	C₂²⋊C₄	—	13	48	no	—
190	C₂²⋊C₄	—	13	96	no	—
191	C₄²	C₂³×D₁₃	1	16	yes	complemented
192	C₁₃	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	36	no	—
194	C₂³	—	2	72	no	—
195	C₂³	—	2	36	no	—
196	C₂³	—	4	16	no	—
197	C₂³	—	4	48	no	—
198	C₂³	—	13	8	no	—
199	C₂³	—	26	72	no	—
200	C₂³	—	52	48	no	—
201	C₂³	—	52	192	no	—
202	C₂³	—	52	32	no	—
203	C₂³	#88251	1	12	yes	—
204	C₂³	C₂⁴×D₁₃	1	4	yes	—
205	C₂³	C₂⁴×D₁₃	1	12	yes	—
206	C₂³	C₂⁴×D₁₃	1	3	yes	—
207	D₄	—	4	48	no	—
208	D₄	—	4	48	no	—
209	D₄	—	52	96	no	—
210	C₂×C₄	—	2	36	no	—
211	C₂×C₄	—	13	24	no	—
212	C₂×C₄	—	26	72	no	—
213	C₂×C₄	—	52	48	no	—
214	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
215	C₂²	—	2	12	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	12	no	—
218	C₂²	—	4	72	no	—
219	C₂²	—	4	24	no	—
220	C₂²	—	26	24	no	—
221	C₂²	—	52	64	no	—
222	C₂²	—	52	48	no	—
223	C₂²	#11719425	1	16	yes	direct product factor
224	C₂²	C₂×#88251	1	18	yes	—
225	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
226	C₄	—	2	12	no	—
227	C₄	—	26	24	no	—
228	C₂	—	4	4	no	—
229	C₂	—	4	12	no	—
230	C₂	—	52	8	no	—
231	C₂	C₂×#11719425	1	12	yes	direct product factor
232	C₂	C₂²×#88251	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#127	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×#127	D₁₃
hypocenter	C₁₃	C₂²×#560
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂×#11719425	C₂	direct (N × H)
2. C₂²×#88130	C₂	semidirect
3. C₂²×#88189	C₂	semidirect
4. C₂²×#88193	C₂	semidirect
5. C₂²×#88191	C₂	semidirect
6. C₂²×C₁₃×#127	C₂	semidirect
9. C₂³×D₄×C₁₃	C₂²	semidirect
11. C₂⁴×Dic₁₃	C₂²	semidirect
12. C₂×C₁₃×#127	C₂²	semidirect
13. C₂²×C₄²×C₁₃	C₂²	semidirect
15. C₂²×C₂²⋊Dic₁₃	C₂²	semidirect
16. C₂×#88193	C₂²	semidirect
17. C₂×#88191	C₂²	semidirect
18. C₂×#88189	C₂²	semidirect
20. C₂²×#6101	C₂²	semidirect
21. #11719425	C₂²	direct (N × H)
22. C₂×#88130	C₂²	semidirect
31. C₂²×D₄×C₁₃	C₂³	semidirect
37. C₂³×Dic₁₃	C₂³	semidirect
38. C₁₃×#127	C₂³	semidirect
39. C₂×C₄²×C₁₃	C₂³	semidirect
41. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
42. #88193	C₂³	semidirect
43. #88191	C₂³	semidirect
44. #88189	C₂³	semidirect
46. C₂×#6101	C₂³	semidirect
47. #88130	C₂³	semidirect
63. C₂×D₄×C₁₃	C₂⁴	semidirect
71. C₂²×Dic₁₃	C₂⁴	semidirect
72. C₄²×C₁₃	C₂⁴	semidirect
74. C₂²⋊Dic₁₃	C₂⁴	semidirect
76. #6101	C₂⁴	semidirect
78. C₂²×#127	D₁₃	semidirect
108. C₂³×D₄	C₂×D₁₃	semidirect
111. C₂×#127	C₂×D₁₃	semidirect
120. C₂²×C₄²	C₂×D₁₃	semidirect
126. C₂²×C₁₃	#560	semidirect
144. C₂²×D₄	C₂²×D₁₃	semidirect
149. #127	C₂²×D₁₃	semidirect
158. C₂×C₄²	C₂²×D₁₃	semidirect
161. C₂×C₁₃	C₂×#560	semidirect
180. C₂×D₄	C₂³×D₁₃	semidirect
191. C₄²	C₂³×D₁₃	semidirect
192. C₁₃	C₂²×#560	semidirect
223. C₂²	#11719425	direct (N × H)
231. C₂	C₂×#11719425	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×#11719425	2
C₂²×#88130	2
C₂²×#88189	2
C₂²×#88193	2
C₂²×#88191	2
C₂²×C₁₃×#127	2
C₂²×#560	13

Subgroups — C22×#11719425 #16769785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719425 #16769785