Subgroups — C₂³×#13063300 #26003235

Order 11008.

19998 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₁₇₂	C₂	1	2	yes	maximal, complemented
2	C₂⁴×D₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₄×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#13063300	C₂	1	56	yes	maximal, direct product factor
5	C₂³×(C₂×D₄₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₄₃	C₂²	1	2	yes	—
8	C₂⁵×D₄₃	C₂²	1	1	yes	characteristic
9	C₂³×D₁₇₂	—	2	32	no	—
10	C₂³×D₁₇₂	—	2	16	no	—
11	C₂³×D₁₇₂	C₂²	1	56	yes	complemented
12	C₂³×D₁₇₂	C₂²	1	28	yes	complemented
13	C₂⁴×Dic₄₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×D₄₃	—	2	16	no	—
15	C₂³×C₄×D₄₃	C₂²	1	28	yes	complemented
16	C₂×#13063300	C₂²	1	448	yes	direct product factor
17	C₂²×(C₂×D₄₃)⋊C₄	C₂²	1	56	yes	complemented
18	C₂⁴×C₄×C₄₃	C₂²	1	2	yes	—
19	C₂⁴×C₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₄₃	C₂²	1	28	yes	complemented
21	C₂⁴×D₄₃	—	2	8	no	—
22	C₂⁴×D₄₃	—	2	8	no	—
23	C₂⁴×D₄₃	—	2	16	no	—
24	C₂⁴×D₄₃	—	2	16	no	—
25	C₂⁴×D₄₃	—	2	16	no	—
26	C₂⁴×D₄₃	D₄	1	16	yes	complemented
27	C₂⁴×D₄₃	C₂³	1	28	yes	—
28	C₂⁴×D₄₃	C₂³	1	14	yes	—
29	C₂²×D₁₇₂	—	2	448	no	—
30	C₂²×D₁₇₂	—	2	224	no	—
31	C₂²×D₁₇₂	C₂³	1	112	yes	complemented
32	C₂²×D₁₇₂	C₂³	1	224	yes	complemented
33	C₂³×Dic₄₃	—	2	8	no	—
34	C₂³×Dic₄₃	C₂³	1	14	yes	—
35	C₂²×C₄×D₄₃	—	2	224	no	—
36	C₂²×C₄×D₄₃	C₂³	1	112	yes	complemented
37	#13063300	C₂³	1	512	yes	direct product factor
38	C₂×(C₂×D₄₃)⋊C₄	C₂³	1	224	yes	complemented
39	C₂⁵×C₄₃	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₄₃	—	2	16	no	—
41	C₂³×C₄×C₄₃	D₄	1	16	yes	complemented
42	C₂³×C₄×C₄₃	C₂³	1	28	yes	—
43	C₂³×C₄×C₄₃	C₂³	1	14	yes	—
44	C₂×C₄⋊C₄×C₄₃	C₂³	1	112	yes	complemented
45	C₂⁴×C₄₃	C₂×D₄	1	8	yes	—
46	C₂⁴×C₄₃	C₂×D₄	1	8	yes	—
47	C₂⁴×C₄₃	#41	1	8	yes	—
48	C₂⁴×C₄₃	C₂⁴	1	7	yes	—
49	C₂³×D₄₃	—	2	128	no	—
50	C₂³×D₄₃	—	2	56	no	—
51	C₂³×D₄₃	—	2	56	no	—
52	C₂³×D₄₃	—	2	112	no	—
53	C₂³×D₄₃	—	2	112	no	—
54	C₂³×D₄₃	—	2	112	no	—
55	C₂³×D₄₃	—	4	128	no	—
56	C₂³×D₄₃	C₂×D₄	1	112	yes	complemented
57	C₂³×D₄₃	C₂⁴	1	56	yes	—
58	C₂³×D₄₃	C₂⁴	1	28	yes	—
59	C₂²×C₄×C₄₃	—	2	112	no	—
60	C₂²×C₄×C₄₃	C₂×D₄	1	112	yes	complemented
61	C₂²×C₄×C₄₃	C₂⁴	1	56	yes	—
62	C₂²×C₄×C₄₃	C₂⁴	1	28	yes	—
63	C₂²×Dic₄₃	—	2	56	no	—
64	C₂²×Dic₄₃	C₂⁴	1	28	yes	—
65	C₂×D₁₇₂	—	2	448	no	—
66	C₂×D₁₇₂	—	2	896	no	—
67	C₂×D₁₇₂	C₂⁴	1	64	yes	complemented
68	C₂×D₁₇₂	C₂⁴	1	128	yes	complemented
69	C₂×C₄×D₄₃	—	2	448	no	—
70	C₂×C₄×D₄₃	C₂⁴	1	64	yes	complemented
71	C₄⋊C₄×C₄₃	C₂⁴	1	64	yes	complemented
72	(C₂×D₄₃)⋊C₄	C₂⁴	1	128	yes	complemented
73	C₂³×C₄₃	#121	1	64	yes	complemented
74	C₂³×C₄₃	C₂²×D₄	1	28	yes	—
75	C₂³×C₄₃	C₂²×D₄	1	28	yes	—
76	C₂³×C₄₃	C₂×#41	1	28	yes	—
77	C₂³×C₄₃	C₂⁵	1	7	yes	—
78	C₂²×D₄₃	—	2	448	no	—
79	C₂²×D₄₃	—	2	56	no	—
80	C₂²×D₄₃	—	2	56	no	—
81	C₂²×D₄₃	—	2	112	no	—
82	C₂²×D₄₃	—	2	112	no	—
83	C₂²×D₄₃	—	2	112	no	—
84	C₂²×D₄₃	—	4	448	no	—
85	C₂²×D₄₃	C₂²×D₄	1	112	yes	complemented
86	C₂²×D₄₃	C₂⁵	1	16	yes	—
87	C₂²×D₄₃	C₂⁵	1	8	yes	—
88	C₂×C₄×C₄₃	—	2	112	no	—
89	C₂×C₄×C₄₃	C₂²×D₄	1	112	yes	complemented
90	C₂×C₄×C₄₃	C₂⁵	1	16	yes	—
91	C₂×C₄×C₄₃	C₂⁵	1	8	yes	—
92	C₂×Dic₄₃	—	2	56	no	—
93	C₂×Dic₄₃	C₂⁵	1	8	yes	—
94	D₁₇₂	—	2	256	no	—
95	D₁₇₂	—	2	128	no	—
96	C₄×D₄₃	—	2	128	no	—
97	C₂³×#121	—	43	1	no	maximal, Sylow
98	C₂²×C₄₃	C₂×#121	1	112	yes	complemented
99	C₂²×C₄₃	C₂³×D₄	1	14	yes	—
100	C₂²×C₄₃	C₂³×D₄	1	14	yes	—
101	C₂²×C₄₃	C₂²×#41	1	14	yes	—
102	C₂²×C₄₃	C₂⁶	1	1	yes	characteristic
103	C₂×D₄₃	—	2	8	no	—
104	C₂×D₄₃	—	2	8	no	—
105	C₂×D₄₃	—	2	224	no	—
106	C₂×D₄₃	—	2	16	no	—
107	C₂×D₄₃	—	2	16	no	—
108	C₂×D₄₃	—	2	16	no	—
109	C₂×D₄₃	—	4	224	no	—
110	C₂×D₄₃	C₂³×D₄	1	16	yes	complemented
111	C₄×C₄₃	—	2	16	no	—
112	C₄×C₄₃	C₂³×D₄	1	16	yes	complemented
113	Dic₄₃	—	2	8	no	—
114	C₂⁴×D₄	—	43	2	no	—
115	C₂⁴×D₄	—	43	1	no	—
116	C₂⁵×C₄	—	43	1	no	—
117	C₂²×#121	—	43	56	no	—
118	C₂³×C₄⋊C₄	D₄₃	1	1	yes	characteristic, complemented
119	C₂³×C₂²⋊C₄	—	43	2	no	—
120	C₂×C₄₃	C₂²×#121	1	28	yes	complemented
121	C₂×C₄₃	C₂⁴×D₄	1	1	yes	characteristic
122	C₂×C₄₃	C₂⁴×D₄	1	1	yes	characteristic
123	C₂×C₄₃	C₂³×#41	1	1	yes	characteristic
124	D₄₃	—	2	16	no	—
125	D₄₃	—	4	16	no	—
126	C₂⁶	—	43	1	no	—
127	C₂⁶	—	43	2	no	—
128	C₂³×D₄	—	43	56	no	—
129	C₂³×D₄	—	43	28	no	—
130	C₂³×D₄	—	86	32	no	—
131	C₂³×D₄	—	86	16	no	—
132	C₂⁴×C₄	—	43	1	no	—
133	C₂⁴×C₄	—	43	28	no	—
134	C₂⁴×C₄	—	86	16	no	—
135	C₂⁴×C₄	C₂×D₄₃	1	2	yes	—
136	C₂⁴×C₄	C₂×D₄₃	1	1	yes	characteristic
137	C₂×#121	—	43	448	no	—
138	C₂²×C₄⋊C₄	C₂×D₄₃	1	28	yes	complemented
139	C₂²×C₂²⋊C₄	—	43	56	no	—
140	C₄₃	C₂³×#121	1	1	yes	characteristic, complemented, Sylow
141	C₂⁵	—	43	16	no	—
142	C₂⁵	—	43	14	no	—
143	C₂⁵	—	43	28	no	—
144	C₂⁵	—	86	8	no	—
145	C₂⁵	—	86	8	no	—
146	C₂⁵	—	86	16	no	—
147	C₂⁵	—	86	16	no	—
148	C₂⁵	—	86	16	no	—
149	C₂⁵	C₂²×D₄₃	1	1	yes	characteristic
150	C₂²×D₄	—	43	224	no	—
151	C₂²×D₄	—	43	112	no	—
152	C₂²×D₄	—	86	448	no	—
153	C₂²×D₄	—	86	224	no	—
154	C₂³×C₄	—	2	16	no	—
155	C₂³×C₄	—	43	14	no	—
156	C₂³×C₄	—	43	112	no	—
157	C₂³×C₄	—	86	8	no	—
158	C₂³×C₄	—	86	224	no	—
159	C₂³×C₄	D₁₇₂	1	16	yes	complemented
160	C₂³×C₄	C₂²×D₄₃	1	28	yes	—
161	C₂³×C₄	C₂²×D₄₃	1	14	yes	—
162	#121	—	43	512	no	—
163	C₂×C₄⋊C₄	C₂²×D₄₃	1	112	yes	complemented
164	C₂×C₂²⋊C₄	—	43	224	no	—
165	C₂⁴	—	43	112	no	—
166	C₂⁴	—	43	28	no	—
167	C₂⁴	—	43	56	no	—
168	C₂⁴	—	86	128	no	—
169	C₂⁴	—	86	56	no	—
170	C₂⁴	—	86	56	no	—
171	C₂⁴	—	86	112	no	—
172	C₂⁴	—	86	112	no	—
173	C₂⁴	—	86	112	no	—
174	C₂⁴	—	172	128	no	—
175	C₂⁴	C₂×D₁₇₂	1	8	yes	—
176	C₂⁴	D₄×D₄₃	1	8	yes	—
177	C₂⁴	#10621902	1	8	yes	—
178	C₂⁴	C₂³×D₄₃	1	7	yes	—
179	C₂×D₄	—	43	128	no	—
180	C₂×D₄	—	43	64	no	—
181	C₂×D₄	—	86	896	no	—
182	C₂×D₄	—	86	448	no	—
183	C₂²×C₄	—	2	112	no	—
184	C₂²×C₄	—	43	28	no	—
185	C₂²×C₄	—	43	64	no	—
186	C₂²×C₄	—	86	56	no	—
187	C₂²×C₄	—	86	448	no	—
188	C₂²×C₄	C₂×D₁₇₂	1	112	yes	complemented
189	C₂²×C₄	C₂³×D₄₃	1	56	yes	—
190	C₂²×C₄	C₂³×D₄₃	1	28	yes	—
191	C₄⋊C₄	C₂³×D₄₃	1	64	yes	complemented
192	C₂²⋊C₄	—	43	128	no	—
193	C₂³	—	43	112	no	—
194	C₂³	—	43	8	no	—
195	C₂³	—	43	16	no	—
196	C₂³	—	86	448	no	—
197	C₂³	—	86	56	no	—
198	C₂³	—	86	56	no	—
199	C₂³	—	86	112	no	—
200	C₂³	—	86	112	no	—
201	C₂³	—	86	112	no	—
202	C₂³	—	172	448	no	—
203	C₂³	#13063300	1	64	yes	direct product factor
204	C₂³	C₂×D₄×D₄₃	1	28	yes	—
205	C₂³	C₂²×D₁₇₂	1	28	yes	—
206	C₂³	C₂×#10621902	1	28	yes	—
207	C₂³	C₂⁴×D₄₃	1	7	yes	—
208	D₄	—	86	256	no	—
209	D₄	—	86	128	no	—
210	C₂×C₄	—	2	112	no	—
211	C₂×C₄	—	43	8	no	—
212	C₂×C₄	—	86	56	no	—
213	C₂×C₄	—	86	128	no	—
214	C₂×C₄	C₂²×D₁₇₂	1	112	yes	complemented
215	C₂×C₄	C₂⁴×D₄₃	1	16	yes	—
216	C₂×C₄	C₂⁴×D₄₃	1	8	yes	—
217	C₂²	—	43	16	no	—
218	C₂²	—	86	224	no	—
219	C₂²	—	86	8	no	—
220	C₂²	—	86	8	no	—
221	C₂²	—	86	16	no	—
222	C₂²	—	86	16	no	—
223	C₂²	—	86	16	no	—
224	C₂²	—	172	224	no	—
225	C₂²	C₂×#13063300	1	112	yes	direct product factor
226	C₂²	C₂²×D₄×D₄₃	1	14	yes	—
227	C₂²	C₂³×D₁₇₂	1	14	yes	—
228	C₂²	C₂²×#10621902	1	14	yes	—
229	C₂²	C₂⁵×D₄₃	1	1	yes	characteristic
230	C₄	—	2	16	no	—
231	C₄	—	86	8	no	—
232	C₄	C₂³×D₁₇₂	1	16	yes	complemented
233	C₂	—	86	16	no	—
234	C₂	—	172	16	no	—
235	C₂	C₂²×#13063300	1	28	yes	direct product factor
236	C₂	C₂³×D₄×D₄₃	1	1	yes	characteristic
237	C₂	C₂⁴×D₁₇₂	1	1	yes	characteristic
238	C₂	C₂³×#10621902	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₄₃
commutator, cosocle	C₂²×C₄₃	C₂⁶
Frattini	C₂²	C₂⁵×D₄₃
Fitting	C₂³×C₄⋊C₄×C₄₃	C₂
socle	C₂⁵×C₄₃	C₂³
hypercenter	C₂³×C₄⋊C₄	D₄₃
hypocenter	C₄₃	C₂³×#121
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₁₇₂	C₂	semidirect
2. C₂⁴×D₁₇₂	C₂	semidirect
3. C₂⁴×C₄×D₄₃	C₂	semidirect
4. C₂²×#13063300	C₂	direct (N × H)
5. C₂³×(C₂×D₄₃)⋊C₄	C₂	semidirect
6. C₂³×C₄⋊C₄×C₄₃	C₂	semidirect
11. C₂³×D₁₇₂	C₂²	semidirect
12. C₂³×D₁₇₂	C₂²	semidirect
15. C₂³×C₄×D₄₃	C₂²	semidirect
16. C₂×#13063300	C₂²	direct (N × H)
17. C₂²×(C₂×D₄₃)⋊C₄	C₂²	semidirect
20. C₂²×C₄⋊C₄×C₄₃	C₂²	semidirect
26. C₂⁴×D₄₃	D₄	semidirect
31. C₂²×D₁₇₂	C₂³	semidirect
32. C₂²×D₁₇₂	C₂³	semidirect
36. C₂²×C₄×D₄₃	C₂³	semidirect
37. #13063300	C₂³	direct (N × H)
38. C₂×(C₂×D₄₃)⋊C₄	C₂³	semidirect
41. C₂³×C₄×C₄₃	D₄	semidirect
44. C₂×C₄⋊C₄×C₄₃	C₂³	semidirect
56. C₂³×D₄₃	C₂×D₄	semidirect
60. C₂²×C₄×C₄₃	C₂×D₄	semidirect
67. C₂×D₁₇₂	C₂⁴	semidirect
68. C₂×D₁₇₂	C₂⁴	semidirect
70. C₂×C₄×D₄₃	C₂⁴	semidirect
71. C₄⋊C₄×C₄₃	C₂⁴	semidirect
72. (C₂×D₄₃)⋊C₄	C₂⁴	semidirect
73. C₂³×C₄₃	#121	semidirect
85. C₂²×D₄₃	C₂²×D₄	semidirect
89. C₂×C₄×C₄₃	C₂²×D₄	semidirect
98. C₂²×C₄₃	C₂×#121	semidirect
110. C₂×D₄₃	C₂³×D₄	semidirect
112. C₄×C₄₃	C₂³×D₄	semidirect
118. C₂³×C₄⋊C₄	D₄₃	semidirect
120. C₂×C₄₃	C₂²×#121	semidirect
138. C₂²×C₄⋊C₄	C₂×D₄₃	semidirect
140. C₄₃	C₂³×#121	semidirect
159. C₂³×C₄	D₁₇₂	semidirect
163. C₂×C₄⋊C₄	C₂²×D₄₃	semidirect
188. C₂²×C₄	C₂×D₁₇₂	semidirect
191. C₄⋊C₄	C₂³×D₄₃	semidirect
203. C₂³	#13063300	direct (N × H)
214. C₂×C₄	C₂²×D₁₇₂	semidirect
225. C₂²	C₂×#13063300	direct (N × H)
232. C₄	C₂³×D₁₇₂	semidirect
235. C₂	C₂²×#13063300	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×D₁₇₂	2
C₂⁴×C₄×D₄₃	2
C₂²×#13063300	2
C₂³×(C₂×D₄₃)⋊C₄	2
C₂³×C₄⋊C₄×C₄₃	2
C₂³×#121	43

Subgroups — C23×#13063300 #26003235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#13063300 #26003235