Subgroups — C₂³×#66931 #15660414

Order 2816.

14078 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁵×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×#66931	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂⁶×C₁₁	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₁₁	—	2	16	no	—
13	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₁₁	C₂²	1	28	yes	—
15	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
16	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
17	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
18	C₂³×Dic₂₂	—	2	16	no	—
19	C₂³×Dic₂₂	C₂²	1	28	yes	complemented
20	C₂²×C₁₁×C₂²⋊C₄	C₂²	1	28	yes	—
21	C₂²×C₂²⋊Dic₁₁	C₂²	1	28	yes	—
22	C₂²×C₄⋊Dic₁₁	C₂²	1	28	yes	complemented
23	C₂²×Dic₁₁⋊C₄	C₂²	1	28	yes	complemented
24	C₂²×Dic₁₁⋊C₄	C₂²	1	28	yes	complemented
25	C₂×#66931	C₂²	1	448	yes	direct product factor
26	C₂⁵×C₁₁	—	2	8	no	—
27	C₂⁵×C₁₁	—	2	8	no	—
28	C₂⁵×C₁₁	Q₈	1	16	yes	complemented
29	C₂⁵×C₁₁	C₂³	1	14	yes	—
30	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₁₁	—	2	8	no	—
32	C₂³×C₄×C₁₁	—	2	8	no	—
33	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
34	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
35	C₂³×Dic₁₁	—	2	8	no	—
36	C₂³×Dic₁₁	—	2	8	no	—
37	C₂³×Dic₁₁	—	2	8	no	—
38	C₂³×Dic₁₁	—	2	224	no	—
39	C₂³×Dic₁₁	D₄	1	16	yes	—
40	C₂³×Dic₁₁	C₂³	1	112	yes	—
41	C₂³×Dic₁₁	C₂³	1	14	yes	—
42	C₂³×Dic₁₁	C₂³	1	14	yes	—
43	C₂³×Dic₁₁	C₂³	1	14	yes	—
44	C₂³×Dic₁₁	C₂³	1	14	yes	—
45	C₂²×Dic₂₂	—	2	224	no	—
46	C₂²×Dic₂₂	C₂³	1	112	yes	complemented
47	C₂×C₁₁×C₂²⋊C₄	C₂³	1	112	yes	—
48	C₂×C₂²⋊Dic₁₁	C₂³	1	112	yes	—
49	C₂×C₄⋊Dic₁₁	C₂³	1	112	yes	complemented
50	C₂×Dic₁₁⋊C₄	C₂³	1	112	yes	complemented
51	C₂×Dic₁₁⋊C₄	C₂³	1	112	yes	complemented
52	#66931	C₂³	1	512	yes	direct product factor
53	C₂³×C₂²⋊Q₈	—	11	1	no	maximal, Sylow
54	C₂⁴×C₁₁	—	2	128	no	—
55	C₂⁴×C₁₁	—	2	56	no	—
56	C₂⁴×C₁₁	—	2	56	no	—
57	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
58	C₂⁴×C₁₁	C₂×Q₈	1	8	yes	—
59	C₂⁴×C₁₁	C₂×Q₈	1	112	yes	complemented
60	C₂⁴×C₁₁	#41	1	8	yes	—
61	C₂⁴×C₁₁	C₂⁴	1	28	yes	—
62	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
63	C₂²×C₄×C₁₁	—	2	56	no	—
64	C₂²×C₄×C₁₁	—	2	56	no	—
65	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
66	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
67	C₂²×Dic₁₁	—	2	448	no	—
68	C₂²×Dic₁₁	—	2	56	no	—
69	C₂²×Dic₁₁	—	2	56	no	—
70	C₂²×Dic₁₁	—	2	56	no	—
71	C₂²×Dic₁₁	C₂×D₄	1	112	yes	—
72	C₂²×Dic₁₁	C₂⁴	1	64	yes	—
73	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
74	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
75	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
76	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
77	C₂×Dic₂₂	—	2	448	no	—
78	C₂×Dic₂₂	C₂⁴	1	64	yes	complemented
79	C₁₁×C₂²⋊C₄	C₂⁴	1	64	yes	—
80	C₂²⋊Dic₁₁	C₂⁴	1	64	yes	—
81	C₄⋊Dic₁₁	C₂⁴	1	64	yes	complemented
82	Dic₁₁⋊C₄	C₂⁴	1	64	yes	complemented
83	Dic₁₁⋊C₄	C₂⁴	1	64	yes	complemented
84	C₂⁴×Q₈	—	11	1	no	—
85	C₂⁵×C₄	—	11	1	no	—
86	C₂²×C₂²⋊Q₈	—	11	56	no	—
87	C₂³×C₄⋊C₄	—	11	1	no	—
88	C₂³×C₄⋊C₄	—	11	1	no	—
89	C₂³×C₄⋊C₄	—	11	1	no	—
90	C₂³×C₂²⋊C₄	—	11	1	no	—
91	C₂³×C₂²⋊C₄	D₁₁	1	1	yes	characteristic
92	C₂³×C₁₁	—	2	448	no	—
93	C₂³×C₁₁	—	2	56	no	—
94	C₂³×C₁₁	—	2	56	no	—
95	C₂³×C₁₁	C₂²⋊Q₈	1	64	yes	complemented
96	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
97	C₂³×C₁₁	C₂²×Q₈	1	28	yes	—
98	C₂³×C₁₁	C₂²×Q₈	1	112	yes	complemented
99	C₂³×C₁₁	C₂×#41	1	28	yes	—
100	C₂³×C₁₁	C₂⁵	1	8	yes	—
101	C₂³×C₁₁	C₂⁵	1	7	yes	—
102	C₂×C₄×C₁₁	—	2	56	no	—
103	C₂×C₄×C₁₁	—	2	56	no	—
104	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
105	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
106	C₂×Dic₁₁	—	2	56	no	—
107	C₂×Dic₁₁	—	2	56	no	—
108	C₂×Dic₁₁	—	2	56	no	—
109	C₂×Dic₁₁	—	2	128	no	—
110	C₂×Dic₁₁	C₂²×D₄	1	112	yes	—
111	C₂×Dic₁₁	C₂⁵	1	8	yes	—
112	C₂×Dic₁₁	C₂⁵	1	8	yes	—
113	C₂×Dic₁₁	C₂⁵	1	8	yes	—
114	C₂×Dic₁₁	C₂⁵	1	8	yes	—
115	Dic₂₂	—	2	128	no	—
116	C₂⁶	C₂×D₁₁	1	1	yes	characteristic
117	C₂³×Q₈	—	11	28	no	—
118	C₂³×Q₈	—	22	16	no	—
119	C₂⁴×C₄	—	11	1	no	—
120	C₂⁴×C₄	—	11	28	no	—
121	C₂⁴×C₄	—	11	1	no	—
122	C₂⁴×C₄	—	11	1	no	—
123	C₂⁴×C₄	—	11	1	no	—
124	C₂⁴×C₄	—	22	16	no	—
125	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
126	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
127	C₂×C₂²⋊Q₈	—	11	448	no	—
128	C₂²×C₄⋊C₄	—	11	28	no	—
129	C₂²×C₄⋊C₄	—	11	28	no	—
130	C₂²×C₄⋊C₄	—	11	28	no	—
131	C₂²×C₂²⋊C₄	—	11	28	no	—
132	C₂²×C₂²⋊C₄	C₂×D₁₁	1	28	yes	—
133	C₂²×C₁₁	—	2	8	no	—
134	C₂²×C₁₁	—	2	8	no	—
135	C₂²×C₁₁	—	2	224	no	—
136	C₂²×C₁₁	C₂×C₂²⋊Q₈	1	112	yes	complemented
137	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
138	C₂²×C₁₁	C₂³×Q₈	1	14	yes	—
139	C₂²×C₁₁	C₂³×Q₈	1	16	yes	complemented
140	C₂²×C₁₁	C₂²×#41	1	14	yes	—
141	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
142	C₄×C₁₁	—	2	8	no	—
143	C₄×C₁₁	—	2	8	no	—
144	Dic₁₁	—	2	8	no	—
145	Dic₁₁	—	2	8	no	—
146	Dic₁₁	—	2	8	no	—
147	Dic₁₁	C₂³×D₄	1	16	yes	—
148	C₂⁵	—	2	8	no	—
149	C₂⁵	—	2	8	no	—
150	C₂⁵	Dic₂₂	1	16	yes	complemented
151	C₂⁵	C₂²×D₁₁	1	14	yes	—
152	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
153	C₂²×Q₈	—	11	112	no	—
154	C₂²×Q₈	—	22	224	no	—
155	C₂³×C₄	—	2	8	no	—
156	C₂³×C₄	—	2	8	no	—
157	C₂³×C₄	—	11	16	no	—
158	C₂³×C₄	—	11	14	no	—
159	C₂³×C₄	—	11	14	no	—
160	C₂³×C₄	—	11	14	no	—
161	C₂³×C₄	—	11	14	no	—
162	C₂³×C₄	—	11	112	no	—
163	C₂³×C₄	—	22	224	no	—
164	C₂³×C₄	—	22	8	no	—
165	C₂³×C₄	—	22	8	no	—
166	C₂³×C₄	—	22	8	no	—
167	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
168	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
169	C₂²⋊Q₈	—	11	512	no	—
170	C₂×C₄⋊C₄	—	11	112	no	—
171	C₂×C₄⋊C₄	—	11	112	no	—
172	C₂×C₄⋊C₄	—	11	112	no	—
173	C₂×C₂²⋊C₄	—	11	112	no	—
174	C₂×C₂²⋊C₄	C₂²×D₁₁	1	112	yes	—
175	C₂×C₁₁	—	2	16	no	—
176	C₂×C₁₁	C₂²×C₂²⋊Q₈	1	28	yes	complemented
177	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
178	C₂×C₁₁	C₂⁴×Q₈	1	1	yes	characteristic
179	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
180	C₂⁴	—	2	128	no	—
181	C₂⁴	—	2	56	no	—
182	C₂⁴	—	2	56	no	—
183	C₂⁴	C₂×Dic₂₂	1	8	yes	—
184	C₂⁴	C₂×Dic₂₂	1	112	yes	complemented
185	C₂⁴	D₄×D₁₁	1	8	yes	—
186	C₂⁴	#4341	1	8	yes	—
187	C₂⁴	C₂³×D₁₁	1	28	yes	—
188	C₂⁴	C₂³×D₁₁	1	7	yes	—
189	C₂×Q₈	—	11	64	no	—
190	C₂×Q₈	—	22	448	no	—
191	C₂²×C₄	—	2	56	no	—
192	C₂²×C₄	—	2	56	no	—
193	C₂²×C₄	—	11	112	no	—
194	C₂²×C₄	—	11	28	no	—
195	C₂²×C₄	—	11	28	no	—
196	C₂²×C₄	—	11	28	no	—
197	C₂²×C₄	—	11	28	no	—
198	C₂²×C₄	—	11	64	no	—
199	C₂²×C₄	—	22	448	no	—
200	C₂²×C₄	—	22	56	no	—
201	C₂²×C₄	—	22	56	no	—
202	C₂²×C₄	—	22	56	no	—
203	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
204	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
205	C₄⋊C₄	—	11	64	no	—
206	C₄⋊C₄	—	11	64	no	—
207	C₄⋊C₄	—	11	64	no	—
208	C₂²⋊C₄	—	11	64	no	—
209	C₂²⋊C₄	C₂³×D₁₁	1	64	yes	—
210	C₁₁	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	2	448	no	—
212	C₂³	—	2	56	no	—
213	C₂³	—	2	56	no	—
214	C₂³	#66931	1	64	yes	direct product factor
215	C₂³	C₂²×Dic₂₂	1	28	yes	—
216	C₂³	C₂²×Dic₂₂	1	112	yes	complemented
217	C₂³	C₂×D₄×D₁₁	1	28	yes	—
218	C₂³	C₂×#4341	1	28	yes	—
219	C₂³	C₂⁴×D₁₁	1	8	yes	—
220	C₂³	C₂⁴×D₁₁	1	7	yes	—
221	Q₈	—	22	128	no	—
222	C₂×C₄	—	2	56	no	—
223	C₂×C₄	—	2	56	no	—
224	C₂×C₄	—	11	112	no	—
225	C₂×C₄	—	11	8	no	—
226	C₂×C₄	—	11	8	no	—
227	C₂×C₄	—	11	8	no	—
228	C₂×C₄	—	11	8	no	—
229	C₂×C₄	—	22	128	no	—
230	C₂×C₄	—	22	56	no	—
231	C₂×C₄	—	22	56	no	—
232	C₂×C₄	—	22	56	no	—
233	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
234	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
235	C₂²	—	2	8	no	—
236	C₂²	—	2	8	no	—
237	C₂²	—	2	224	no	—
238	C₂²	C₂×#66931	1	112	yes	direct product factor
239	C₂²	C₂³×Dic₂₂	1	14	yes	—
240	C₂²	C₂³×Dic₂₂	1	16	yes	complemented
241	C₂²	C₂²×D₄×D₁₁	1	14	yes	—
242	C₂²	C₂²×#4341	1	14	yes	—
243	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
244	C₄	—	2	8	no	—
245	C₄	—	2	8	no	—
246	C₄	—	11	16	no	—
247	C₄	—	22	8	no	—
248	C₄	—	22	8	no	—
249	C₄	—	22	8	no	—
250	C₂	—	2	16	no	—
251	C₂	C₂²×#66931	1	28	yes	direct product factor
252	C₂	C₂³×D₄×D₁₁	1	1	yes	characteristic
253	C₂	C₂⁴×Dic₂₂	1	1	yes	characteristic
254	C₂	C₂³×#4341	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂³×C₁₁×C₂²⋊C₄	C₂
socle	C₂⁵×C₁₁	C₂³
prime radical	C₂⁶×C₁₁	C₂²
hypercenter	C₂³×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×Dic₂₂	C₂	semidirect
3. C₂²×#66931	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₁₁	C₂	semidirect
5. C₂³×Dic₁₁⋊C₄	C₂	semidirect
6. C₂³×Dic₁₁⋊C₄	C₂	semidirect
19. C₂³×Dic₂₂	C₂²	semidirect
22. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
23. C₂²×Dic₁₁⋊C₄	C₂²	semidirect
24. C₂²×Dic₁₁⋊C₄	C₂²	semidirect
25. C₂×#66931	C₂²	direct (N × H)
28. C₂⁵×C₁₁	Q₈	semidirect
46. C₂²×Dic₂₂	C₂³	semidirect
49. C₂×C₄⋊Dic₁₁	C₂³	semidirect
50. C₂×Dic₁₁⋊C₄	C₂³	semidirect
51. C₂×Dic₁₁⋊C₄	C₂³	semidirect
52. #66931	C₂³	direct (N × H)
59. C₂⁴×C₁₁	C₂×Q₈	semidirect
78. C₂×Dic₂₂	C₂⁴	semidirect
81. C₄⋊Dic₁₁	C₂⁴	semidirect
82. Dic₁₁⋊C₄	C₂⁴	semidirect
83. Dic₁₁⋊C₄	C₂⁴	semidirect
95. C₂³×C₁₁	C₂²⋊Q₈	semidirect
98. C₂³×C₁₁	C₂²×Q₈	semidirect
136. C₂²×C₁₁	C₂×C₂²⋊Q₈	semidirect
139. C₂²×C₁₁	C₂³×Q₈	semidirect
150. C₂⁵	Dic₂₂	semidirect
176. C₂×C₁₁	C₂²×C₂²⋊Q₈	semidirect
184. C₂⁴	C₂×Dic₂₂	semidirect
210. C₁₁	C₂³×C₂²⋊Q₈	semidirect
214. C₂³	#66931	direct (N × H)
216. C₂³	C₂²×Dic₂₂	semidirect
238. C₂²	C₂×#66931	direct (N × H)
240. C₂²	C₂³×Dic₂₂	semidirect
251. C₂	C₂²×#66931	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×Dic₂₂	2
C₂⁵×Dic₁₁	2
C₂²×#66931	2
C₂³×C₄⋊Dic₁₁	2
C₂³×Dic₁₁⋊C₄	2
C₂³×C₂²⋊Dic₁₁	2
C₂³×C₁₁×C₂²⋊C₄	2
C₂³×C₂²⋊Q₈	11

Subgroups — C23×#66931 #15660414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66931 #15660414