Subgroups — C₂²×#5720 #11702053

Order 768.

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	C₂²×C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#937	C₂	1	3	yes	maximal, complemented
3	C₂²×#935	C₂	1	3	yes	maximal, complemented
4	C₂²×#939	C₂	1	6	yes	maximal, complemented
5	C₂×#5720	C₂	1	48	yes	maximal, direct product factor
6	C₂²×#876	C₂	1	2	yes	maximal, complemented
7	C₂²×#560	—	3	1	no	maximal, Sylow
8	C₂⁵×D₃	C₂²	1	1	yes	characteristic
9	C₂³×C₃×D₄	C₂²	1	3	yes	—
10	C₂³×C₃×D₄	C₂²	1	3	yes	complemented
11	C₂³×#67	C₂²	1	6	yes	—
12	C₂⁴×Dic₃	C₂²	1	3	yes	complemented
13	C₂×C₃×#127	C₂²	1	24	yes	complemented
14	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
15	C₂²×#217	C₂²	1	3	yes	—
16	C₂²×#217	C₂²	1	3	yes	complemented
17	C₂×#939	C₂²	1	144	yes	complemented
18	C₂×#937	C₂²	1	72	yes	complemented
19	C₂×#935	C₂²	1	72	yes	complemented
20	C₂²×(C₂×D₃)⋊C₄	C₂²	1	6	yes	—
21	C₂²×Dic₃⋊C₄	C₂²	1	6	yes	complemented
22	#5720	C₂²	1	256	yes	direct product factor
23	C₂×#876	C₂²	1	48	yes	complemented
24	C₂×#560	—	3	48	no	—
25	C₂²×#127	D₃	1	1	yes	characteristic, complemented
26	C₂²×#126	—	3	2	no	—
27	C₂²×#123	—	3	3	no	—
28	C₂²×#121	—	3	6	no	—
29	C₂²×#120	—	3	3	no	—
30	C₂⁵×C₃	C₂³	1	3	yes	—
31	C₂⁵×C₃	C₂³	1	1	yes	characteristic
32	C₂⁴×D₃	—	4	12	no	—
33	C₂⁴×D₃	C₂³	1	2	yes	—
34	C₂⁴×D₃	C₂³	1	12	yes	—
35	C₂²×C₃×D₄	—	4	12	no	—
36	C₂²×C₃×D₄	—	4	12	no	—
37	C₂²×C₃×D₄	C₂³	1	36	yes	—
38	C₂²×C₃×D₄	C₂³	1	36	yes	complemented
39	C₂³×C₃×C₄	C₂³	1	3	yes	—
40	C₂²×#67	—	4	24	no	—
41	C₂²×#67	C₂³	1	72	yes	—
42	C₂³×Dic₃	—	4	12	no	—
43	C₂³×Dic₃	C₂³	1	6	yes	—
44	C₂³×Dic₃	C₂³	1	36	yes	complemented
45	C₃×#127	C₂³	1	64	yes	complemented
46	C₂×C₃×C₄²	C₂³	1	12	yes	complemented
47	C₂×#217	C₂³	1	36	yes	—
48	C₂×#217	C₂³	1	36	yes	complemented
49	#939	C₂³	1	384	yes	complemented
50	#937	C₂³	1	192	yes	complemented
51	#935	C₂³	1	192	yes	complemented
52	C₂×(C₂×D₃)⋊C₄	C₂³	1	72	yes	—
53	C₂×Dic₃⋊C₄	C₂³	1	72	yes	complemented
54	#876	C₂³	1	128	yes	complemented
55	C₂⁶	—	3	1	no	—
56	C₂³×D₄	—	3	6	no	—
57	C₂³×D₄	C₂×D₃	1	3	yes	—
58	C₂³×D₄	C₂×D₃	1	3	yes	complemented
59	C₂⁴×C₄	—	3	3	no	—
60	#560	—	3	256	no	—
61	C₂×#127	C₂×D₃	1	24	yes	complemented
62	C₂×#126	—	3	48	no	—
63	C₂×#123	—	3	72	no	—
64	C₂×#121	—	3	144	no	—
65	C₂×#120	—	3	72	no	—
66	C₂²×C₄⋊C₄	—	3	6	no	—
67	C₂²×C₂²⋊C₄	—	3	3	no	—
68	C₂²×C₂²⋊C₄	—	3	3	no	—
69	C₂²×C₂²⋊C₄	—	3	6	no	—
70	C₂²×C₄²	C₂×D₃	1	1	yes	characteristic, complemented
71	C₂⁴×C₃	—	2	12	no	—
72	C₂⁴×C₃	—	2	24	no	—
73	C₂⁴×C₃	—	2	12	no	—
74	C₂⁴×C₃	C₂⁴	1	6	yes	—
75	C₂⁴×C₃	C₂⁴	1	18	yes	—
76	C₂⁴×C₃	C₂⁴	1	1	yes	characteristic
77	C₂³×D₃	—	2	24	no	—
78	C₂³×D₃	—	4	72	no	—
79	C₂³×D₃	—	4	64	no	—
80	C₂³×D₃	C₂⁴	1	12	yes	—
81	C₂³×D₃	C₂⁴	1	16	yes	—
82	C₂×C₃×D₄	—	4	72	no	—
83	C₂×C₃×D₄	—	4	72	no	—
84	C₂×C₃×D₄	C₂⁴	1	48	yes	—
85	C₂×C₃×D₄	C₂⁴	1	48	yes	complemented
86	C₂²×C₃×C₄	—	2	12	no	—
87	C₂²×C₃×C₄	C₂⁴	1	18	yes	—
88	C₂×#67	—	4	144	no	—
89	C₂×#67	C₂⁴	1	96	yes	—
90	C₂²×Dic₃	—	2	24	no	—
91	C₂²×Dic₃	—	4	72	no	—
92	C₂²×Dic₃	C₂⁴	1	36	yes	—
93	C₂²×Dic₃	C₂⁴	1	48	yes	complemented
94	C₃×C₄²	C₂⁴	1	16	yes	complemented
95	#217	C₂⁴	1	48	yes	—
96	#217	C₂⁴	1	48	yes	complemented
97	(C₂×D₃)⋊C₄	C₂⁴	1	96	yes	—
98	Dic₃⋊C₄	C₂⁴	1	96	yes	complemented
99	C₂⁵	—	3	2	no	—
100	C₂⁵	—	3	12	no	—
101	C₂⁵	—	12	12	no	—
102	C₂⁵	C₂²×D₃	1	1	yes	characteristic
103	C₂⁵	C₂²×D₃	1	3	yes	—
104	C₂²×D₄	—	3	72	no	—
105	C₂²×D₄	—	4	12	no	—
106	C₂²×D₄	—	4	12	no	—
107	C₂²×D₄	—	12	24	no	—
108	C₂²×D₄	C₂²×D₃	1	36	yes	—
109	C₂²×D₄	C₂²×D₃	1	36	yes	complemented
110	C₂³×C₄	—	3	6	no	—
111	C₂³×C₄	—	3	36	no	—
112	C₂³×C₄	—	12	12	no	—
113	C₂³×C₄	C₂²×D₃	1	3	yes	—
114	#127	C₂²×D₃	1	64	yes	complemented
115	#126	—	3	128	no	—
116	#123	—	3	192	no	—
117	#121	—	3	384	no	—
118	#120	—	3	192	no	—
119	C₂×C₄⋊C₄	—	3	72	no	—
120	C₂×C₂²⋊C₄	—	3	36	no	—
121	C₂×C₂²⋊C₄	—	3	36	no	—
122	C₂×C₂²⋊C₄	—	3	72	no	—
123	C₂×C₄²	C₂²×D₃	1	12	yes	complemented
124	C₂³×C₃	—	2	36	no	—
125	C₂³×C₃	—	2	72	no	—
126	C₂³×C₃	—	2	36	no	—
127	C₂³×C₃	—	4	16	no	—
128	C₂³×C₃	—	4	48	no	—
129	C₂³×C₃	H(5,𝔽₂)	1	12	yes	—
130	C₂³×C₃	C₂⁵	1	4	yes	—
131	C₂³×C₃	C₂⁵	1	12	yes	—
132	C₂³×C₃	C₂⁵	1	3	yes	—
133	C₂²×D₃	—	2	72	no	—
134	C₂²×D₃	—	4	48	no	—
135	C₂²×D₃	—	4	192	no	—
136	C₂²×D₃	—	4	32	no	—
137	C₂²×D₃	C₂⁵	1	8	yes	—
138	C₃×D₄	—	4	48	no	—
139	C₃×D₄	—	4	48	no	—
140	C₂×C₃×C₄	—	2	36	no	—
141	C₂×C₃×C₄	C₂⁵	1	12	yes	—
142	#67	—	4	96	no	—
143	C₂×Dic₃	—	2	72	no	—
144	C₂×Dic₃	—	4	48	no	—
145	C₂×Dic₃	C₂⁵	1	24	yes	—
146	C₂⁴	—	2	12	no	—
147	C₂⁴	—	2	24	no	—
148	C₂⁴	—	2	12	no	—
149	C₂⁴	—	3	12	no	—
150	C₂⁴	—	3	16	no	—
151	C₂⁴	—	6	24	no	—
152	C₂⁴	—	12	72	no	—
153	C₂⁴	—	12	64	no	—
154	C₂⁴	C₂³×D₃	1	6	yes	—
155	C₂⁴	C₂³×D₃	1	18	yes	—
156	C₂⁴	C₂³×D₃	1	1	yes	characteristic
157	C₂×D₄	—	3	96	no	—
158	C₂×D₄	—	4	72	no	—
159	C₂×D₄	—	4	72	no	—
160	C₂×D₄	—	12	144	no	—
161	C₂×D₄	C₂³×D₃	1	48	yes	—
162	C₂×D₄	C₂³×D₃	1	48	yes	complemented
163	C₂²×C₄	—	2	12	no	—
164	C₂²×C₄	—	3	36	no	—
165	C₂²×C₄	—	3	48	no	—
166	C₂²×C₄	—	6	24	no	—
167	C₂²×C₄	—	12	72	no	—
168	C₂²×C₄	C₂³×D₃	1	18	yes	—
169	C₄⋊C₄	—	3	96	no	—
170	C₂²⋊C₄	—	3	48	no	—
171	C₂²⋊C₄	—	3	48	no	—
172	C₂²⋊C₄	—	3	96	no	—
173	C₄²	C₂³×D₃	1	16	yes	complemented
174	C₂²×C₃	—	2	12	no	—
175	C₂²×C₃	—	2	24	no	—
176	C₂²×C₃	—	2	12	no	—
177	C₂²×C₃	—	4	24	no	—
178	C₂²×C₃	—	4	72	no	—
179	C₂²×C₃	#560	1	16	yes	complemented
180	C₂²×C₃	C₂×H(5,𝔽₂)	1	18	yes	—
181	C₂²×C₃	C₂⁶	1	1	yes	characteristic
182	C₂×D₃	—	2	24	no	—
183	C₂×D₃	—	4	64	no	—
184	C₂×D₃	—	4	48	no	—
185	C₃×C₄	—	2	12	no	—
186	Dic₃	—	2	24	no	—
187	C₂³	—	2	36	no	—
188	C₂³	—	2	72	no	—
189	C₂³	—	2	36	no	—
190	C₂³	—	3	8	no	—
191	C₂³	—	4	16	no	—
192	C₂³	—	4	48	no	—
193	C₂³	—	6	72	no	—
194	C₂³	—	12	48	no	—
195	C₂³	—	12	192	no	—
196	C₂³	—	12	32	no	—
197	C₂³	#1004	1	12	yes	—
198	C₂³	C₂⁴×D₃	1	4	yes	—
199	C₂³	C₂⁴×D₃	1	12	yes	—
200	C₂³	C₂⁴×D₃	1	3	yes	—
201	D₄	—	4	48	no	—
202	D₄	—	4	48	no	—
203	D₄	—	12	96	no	—
204	C₂×C₄	—	2	36	no	—
205	C₂×C₄	—	3	24	no	—
206	C₂×C₄	—	6	72	no	—
207	C₂×C₄	—	12	48	no	—
208	C₂×C₄	C₂⁴×D₃	1	12	yes	—
209	C₂×C₃	—	4	4	no	—
210	C₂×C₃	—	4	12	no	—
211	C₂×C₃	C₂×#560	1	12	yes	complemented
212	C₂×C₃	C₂²×H(5,𝔽₂)	1	3	yes	—
213	D₃	—	4	8	no	—
214	C₂²	—	2	12	no	—
215	C₂²	—	2	24	no	—
216	C₂²	—	2	12	no	—
217	C₂²	—	4	24	no	—
218	C₂²	—	4	72	no	—
219	C₂²	—	6	24	no	—
220	C₂²	—	12	64	no	—
221	C₂²	—	12	48	no	—
222	C₂²	#5720	1	16	yes	direct product factor
223	C₂²	C₂×#1004	1	18	yes	—
224	C₂²	C₂⁵×D₃	1	1	yes	characteristic
225	C₄	—	2	12	no	—
226	C₄	—	6	24	no	—
227	C₃	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
228	C₂	—	4	4	no	—
229	C₂	—	4	12	no	—
230	C₂	—	12	8	no	—
231	C₂	C₂×#5720	1	12	yes	direct product factor
232	C₂	C₂²×#1004	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁶
Frattini	C₂²	C₂⁵×D₃
Fitting	C₂²×C₃×#127	C₂
socle	C₂⁴×C₃	C₂⁴
hypercenter	C₂²×#127	D₃
hypocenter	C₃	C₂²×#560
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×#127	C₂	semidirect
2. C₂²×#937	C₂	semidirect
3. C₂²×#935	C₂	semidirect
4. C₂²×#939	C₂	semidirect
5. C₂×#5720	C₂	direct (N × H)
6. C₂²×#876	C₂	semidirect
10. C₂³×C₃×D₄	C₂²	semidirect
12. C₂⁴×Dic₃	C₂²	semidirect
13. C₂×C₃×#127	C₂²	semidirect
14. C₂²×C₃×C₄²	C₂²	semidirect
16. C₂²×#217	C₂²	semidirect
17. C₂×#939	C₂²	semidirect
18. C₂×#937	C₂²	semidirect
19. C₂×#935	C₂²	semidirect
21. C₂²×Dic₃⋊C₄	C₂²	semidirect
22. #5720	C₂²	direct (N × H)
23. C₂×#876	C₂²	semidirect
25. C₂²×#127	D₃	semidirect
38. C₂²×C₃×D₄	C₂³	semidirect
44. C₂³×Dic₃	C₂³	semidirect
45. C₃×#127	C₂³	semidirect
46. C₂×C₃×C₄²	C₂³	semidirect
48. C₂×#217	C₂³	semidirect
49. #939	C₂³	semidirect
50. #937	C₂³	semidirect
51. #935	C₂³	semidirect
53. C₂×Dic₃⋊C₄	C₂³	semidirect
54. #876	C₂³	semidirect
58. C₂³×D₄	C₂×D₃	semidirect
61. C₂×#127	C₂×D₃	semidirect
70. C₂²×C₄²	C₂×D₃	semidirect
85. C₂×C₃×D₄	C₂⁴	semidirect
93. C₂²×Dic₃	C₂⁴	semidirect
94. C₃×C₄²	C₂⁴	semidirect
96. #217	C₂⁴	semidirect
98. Dic₃⋊C₄	C₂⁴	semidirect
109. C₂²×D₄	C₂²×D₃	semidirect
114. #127	C₂²×D₃	semidirect
123. C₂×C₄²	C₂²×D₃	semidirect
162. C₂×D₄	C₂³×D₃	semidirect
173. C₄²	C₂³×D₃	semidirect
179. C₂²×C₃	#560	semidirect
211. C₂×C₃	C₂×#560	semidirect
222. C₂²	#5720	direct (N × H)
227. C₃	C₂²×#560	semidirect
231. C₂	C₂×#5720	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×C₃×#127	2
C₂²×#937	2
C₂²×#935	2
C₂²×#939	2
C₂×#5720	2
C₂²×#876	2
C₂²×#560	3

Subgroups — C22×#5720 #11702053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#5720 #11702053