Subgroups — C₂³×#935 #11701833

Order 768.

15150 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#217	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#217	C₂	1	2	yes	maximal, complemented
5	C₂²×#935	C₂	1	56	yes	maximal, direct product factor
6	C₂³×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂³×#123	—	3	1	no	maximal, Sylow
8	C₂⁶×C₃	C₂²	1	1	yes	characteristic
9	C₂⁶×C₃	C₂²	1	1	yes	characteristic
10	C₂³×C₃×D₄	—	2	16	no	—
11	C₂³×C₃×D₄	C₂²	1	28	yes	—
12	C₂⁴×C₃×C₄	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₃	—	2	16	no	—
14	C₂⁴×Dic₃	C₂²	1	2	yes	—
15	C₂⁴×Dic₃	C₂²	1	2	yes	—
16	C₂⁴×Dic₃	C₂²	1	28	yes	complemented
17	C₂²×#217	C₂²	1	56	yes	complemented
18	C₂²×#217	C₂²	1	28	yes	complemented
19	C₂×#935	C₂²	1	448	yes	direct product factor
20	C₂²×Dic₃⋊C₄	C₂²	1	56	yes	complemented
21	C₂⁴×D₄	D₃	1	1	yes	characteristic
22	C₂⁵×C₄	—	3	1	no	—
23	C₂²×#123	—	3	56	no	—
24	C₂³×C₄⋊C₄	—	3	2	no	—
25	C₂³×C₂²⋊C₄	—	3	2	no	—
26	C₂³×C₂²⋊C₄	—	3	1	no	—
27	C₂⁵×C₃	—	2	16	no	—
28	C₂⁵×C₃	—	2	8	no	—
29	C₂⁵×C₃	—	2	16	no	—
30	C₂⁵×C₃	D₄	1	16	yes	—
31	C₂⁵×C₃	C₂³	1	14	yes	—
32	C₂⁵×C₃	C₂³	1	14	yes	—
33	C₂⁵×C₃	C₂³	1	1	yes	characteristic
34	C₂²×C₃×D₄	—	2	224	no	—
35	C₂²×C₃×D₄	C₂³	1	112	yes	—
36	C₂³×C₃×C₄	—	2	8	no	—
37	C₂³×C₃×C₄	C₂³	1	14	yes	—
38	C₂³×Dic₃	—	2	16	no	—
39	C₂³×Dic₃	—	2	224	no	—
40	C₂³×Dic₃	—	2	16	no	—
41	C₂³×Dic₃	C₂³	1	28	yes	—
42	C₂³×Dic₃	C₂³	1	28	yes	—
43	C₂³×Dic₃	C₂³	1	112	yes	complemented
44	C₂×#217	C₂³	1	112	yes	complemented
45	C₂×#217	C₂³	1	224	yes	complemented
46	#935	C₂³	1	512	yes	direct product factor
47	C₂×Dic₃⋊C₄	C₂³	1	224	yes	complemented
48	C₂⁶	C₂×D₃	1	1	yes	characteristic
49	C₂⁶	C₂×D₃	1	1	yes	characteristic
50	C₂³×D₄	—	2	16	no	—
51	C₂³×D₄	C₂×D₃	1	28	yes	—
52	C₂⁴×C₄	—	3	28	no	—
53	C₂⁴×C₄	—	3	2	no	—
54	C₂⁴×C₄	—	3	2	no	—
55	C₂⁴×C₄	—	6	16	no	—
56	C₂⁴×C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×#123	—	3	448	no	—
58	C₂²×C₄⋊C₄	—	3	56	no	—
59	C₂²×C₂²⋊C₄	—	3	28	no	—
60	C₂²×C₂²⋊C₄	—	3	56	no	—
61	C₂⁴×C₃	—	2	128	no	—
62	C₂⁴×C₃	—	2	112	no	—
63	C₂⁴×C₃	—	2	56	no	—
64	C₂⁴×C₃	—	2	112	no	—
65	C₂⁴×C₃	—	4	64	no	—
66	C₂⁴×C₃	C₂×D₄	1	112	yes	—
67	C₂⁴×C₃	C₂×D₄	1	8	yes	—
68	C₂⁴×C₃	#41	1	16	yes	—
69	C₂⁴×C₃	C₂⁴	1	28	yes	—
70	C₂⁴×C₃	C₂⁴	1	28	yes	—
71	C₂⁴×C₃	C₂⁴	1	7	yes	—
72	C₂×C₃×D₄	—	2	448	no	—
73	C₂×C₃×D₄	C₂⁴	1	64	yes	—
74	C₂²×C₃×C₄	—	2	56	no	—
75	C₂²×C₃×C₄	C₂⁴	1	28	yes	—
76	C₂²×Dic₃	—	2	112	no	—
77	C₂²×Dic₃	—	2	448	no	—
78	C₂²×Dic₃	—	2	112	no	—
79	C₂²×Dic₃	C₂⁴	1	56	yes	—
80	C₂²×Dic₃	C₂⁴	1	56	yes	—
81	C₂²×Dic₃	C₂⁴	1	64	yes	complemented
82	#217	C₂⁴	1	64	yes	complemented
83	#217	C₂⁴	1	128	yes	complemented
84	Dic₃⋊C₄	C₂⁴	1	128	yes	complemented
85	C₂⁵	—	2	16	no	—
86	C₂⁵	—	2	16	no	—
87	C₂⁵	—	2	8	no	—
88	C₂⁵	#67	1	16	yes	—
89	C₂⁵	C₂²×D₃	1	14	yes	—
90	C₂⁵	C₂²×D₃	1	14	yes	—
91	C₂⁵	C₂²×D₃	1	1	yes	characteristic
92	C₂²×D₄	—	2	224	no	—
93	C₂²×D₄	C₂²×D₃	1	112	yes	—
94	C₂³×C₄	—	2	8	no	—
95	C₂³×C₄	—	3	28	no	—
96	C₂³×C₄	—	3	28	no	—
97	C₂³×C₄	—	3	112	no	—
98	C₂³×C₄	—	6	16	no	—
99	C₂³×C₄	—	6	224	no	—
100	C₂³×C₄	—	6	16	no	—
101	C₂³×C₄	C₂²×D₃	1	14	yes	—
102	#123	—	3	512	no	—
103	C₂×C₄⋊C₄	—	3	224	no	—
104	C₂×C₂²⋊C₄	—	3	112	no	—
105	C₂×C₂²⋊C₄	—	3	224	no	—
106	C₂³×C₃	—	2	448	no	—
107	C₂³×C₃	—	2	112	no	—
108	C₂³×C₃	—	2	56	no	—
109	C₂³×C₃	—	2	112	no	—
110	C₂³×C₃	—	4	224	no	—
111	C₂³×C₃	#123	1	64	yes	complemented
112	C₂³×C₃	C₂²×D₄	1	112	yes	—
113	C₂³×C₃	C₂²×D₄	1	28	yes	—
114	C₂³×C₃	C₂×#41	1	56	yes	—
115	C₂³×C₃	C₂⁵	1	8	yes	—
116	C₂³×C₃	C₂⁵	1	8	yes	—
117	C₂³×C₃	C₂⁵	1	7	yes	—
118	C₃×D₄	—	2	128	no	—
119	C₂×C₃×C₄	—	2	56	no	—
120	C₂×C₃×C₄	C₂⁵	1	8	yes	—
121	C₂×Dic₃	—	2	112	no	—
122	C₂×Dic₃	—	2	128	no	—
123	C₂×Dic₃	—	2	112	no	—
124	C₂×Dic₃	C₂⁵	1	16	yes	—
125	C₂×Dic₃	C₂⁵	1	16	yes	—
126	C₂⁴	—	2	128	no	—
127	C₂⁴	—	2	112	no	—
128	C₂⁴	—	2	112	no	—
129	C₂⁴	—	2	56	no	—
130	C₂⁴	—	4	64	no	—
131	C₂⁴	#237	1	16	yes	—
132	C₂⁴	C₂×#67	1	112	yes	—
133	C₂⁴	C₂×#67	1	8	yes	—
134	C₂⁴	C₂³×D₃	1	28	yes	—
135	C₂⁴	C₂³×D₃	1	28	yes	—
136	C₂⁴	C₂³×D₃	1	7	yes	—
137	C₂×D₄	—	2	448	no	—
138	C₂×D₄	C₂³×D₃	1	64	yes	—
139	C₂²×C₄	—	2	56	no	—
140	C₂²×C₄	—	3	56	no	—
141	C₂²×C₄	—	3	56	no	—
142	C₂²×C₄	—	3	64	no	—
143	C₂²×C₄	—	6	112	no	—
144	C₂²×C₄	—	6	448	no	—
145	C₂²×C₄	—	6	112	no	—
146	C₂²×C₄	C₂³×D₃	1	28	yes	—
147	C₄⋊C₄	—	3	128	no	—
148	C₂²⋊C₄	—	3	64	no	—
149	C₂²⋊C₄	—	3	128	no	—
150	C₂²×C₃	—	2	16	no	—
151	C₂²×C₃	—	2	224	no	—
152	C₂²×C₃	—	2	8	no	—
153	C₂²×C₃	—	2	16	no	—
154	C₂²×C₃	—	4	112	no	—
155	C₂²×C₃	C₂×#123	1	112	yes	complemented
156	C₂²×C₃	C₂³×D₄	1	16	yes	—
157	C₂²×C₃	C₂³×D₄	1	14	yes	—
158	C₂²×C₃	C₂²×#41	1	28	yes	—
159	C₂²×C₃	C₂⁶	1	1	yes	characteristic
160	C₃×C₄	—	2	8	no	—
161	Dic₃	—	2	16	no	—
162	Dic₃	—	2	16	no	—
163	C₂³	—	2	112	no	—
164	C₂³	—	2	448	no	—
165	C₂³	—	2	112	no	—
166	C₂³	—	2	56	no	—
167	C₂³	—	4	224	no	—
168	C₂³	#935	1	64	yes	direct product factor
169	C₂³	C₂×#237	1	56	yes	—
170	C₂³	C₂²×#67	1	112	yes	—
171	C₂³	C₂²×#67	1	28	yes	—
172	C₂³	C₂⁴×D₃	1	8	yes	—
173	C₂³	C₂⁴×D₃	1	8	yes	—
174	C₂³	C₂⁴×D₃	1	7	yes	—
175	D₄	—	2	128	no	—
176	C₂×C₄	—	2	56	no	—
177	C₂×C₄	—	3	16	no	—
178	C₂×C₄	—	3	16	no	—
179	C₂×C₄	—	6	112	no	—
180	C₂×C₄	—	6	128	no	—
181	C₂×C₄	—	6	112	no	—
182	C₂×C₄	C₂⁴×D₃	1	8	yes	—
183	C₂×C₃	—	2	16	no	—
184	C₂×C₃	—	4	8	no	—
185	C₂×C₃	C₂²×#123	1	28	yes	complemented
186	C₂×C₃	C₂⁴×D₄	1	1	yes	characteristic
187	C₂×C₃	C₂³×#41	1	2	yes	—
188	C₂²	—	2	16	no	—
189	C₂²	—	2	224	no	—
190	C₂²	—	2	16	no	—
191	C₂²	—	2	8	no	—
192	C₂²	—	4	112	no	—
193	C₂²	C₂×#935	1	112	yes	direct product factor
194	C₂²	C₂²×#237	1	28	yes	—
195	C₂²	C₂³×#67	1	16	yes	—
196	C₂²	C₂³×#67	1	14	yes	—
197	C₂²	C₂⁵×D₃	1	1	yes	characteristic
198	C₄	—	2	8	no	—
199	C₄	—	6	16	no	—
200	C₄	—	6	16	no	—
201	C₃	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
202	C₂	—	2	16	no	—
203	C₂	—	4	8	no	—
204	C₂	C₂²×#935	1	28	yes	direct product factor
205	C₂	C₂³×#237	1	2	yes	—
206	C₂	C₂⁴×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁶
Frattini	C₂²	C₂⁵×D₃
Fitting, prime radical	C₂⁴×C₃×D₄	C₂
socle	C₂⁵×C₃	C₂³
hypercenter	C₂⁴×D₄	D₃
hypocenter	C₃	C₂³×#123
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
2. C₂⁵×Dic₃	C₂	semidirect
3. C₂³×#217	C₂	semidirect
4. C₂³×#217	C₂	semidirect
5. C₂²×#935	C₂	direct (N × H)
6. C₂³×Dic₃⋊C₄	C₂	semidirect
16. C₂⁴×Dic₃	C₂²	semidirect
17. C₂²×#217	C₂²	semidirect
18. C₂²×#217	C₂²	semidirect
19. C₂×#935	C₂²	direct (N × H)
20. C₂²×Dic₃⋊C₄	C₂²	semidirect
43. C₂³×Dic₃	C₂³	semidirect
44. C₂×#217	C₂³	semidirect
45. C₂×#217	C₂³	semidirect
46. #935	C₂³	direct (N × H)
47. C₂×Dic₃⋊C₄	C₂³	semidirect
81. C₂²×Dic₃	C₂⁴	semidirect
82. #217	C₂⁴	semidirect
83. #217	C₂⁴	semidirect
84. Dic₃⋊C₄	C₂⁴	semidirect
111. C₂³×C₃	#123	semidirect
155. C₂²×C₃	C₂×#123	semidirect
168. C₂³	#935	direct (N × H)
185. C₂×C₃	C₂²×#123	semidirect
193. C₂²	C₂×#935	direct (N × H)
201. C₃	C₂³×#123	semidirect
204. C₂	C₂²×#935	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×D₄	2
C₂⁵×Dic₃	2
C₂³×#217	2
C₂²×#935	2
C₂³×Dic₃⋊C₄	2
C₂³×#123	3

Subgroups — C23×#935 #11701833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#935 #11701833