Subgroups — C₂³×#66999 #15660413

Order 2816.

14078 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×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×#66999	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×Dic₁₁⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂³×(C₂×D₁₁)⋊C₄	C₂	1	2	yes	maximal
6	C₂⁴×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₁	C₂²	1	1	yes	characteristic
8	C₂³×Q₈×C₁₁	—	2	16	no	—
9	C₂³×Q₈×C₁₁	C₂²	1	28	yes	complemented
10	C₂⁴×C₄×C₁₁	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
12	C₂⁴×Dic₁₁	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₁₁	C₂²	1	2	yes	—
14	C₂³×C₄×D₁₁	—	2	16	no	—
15	C₂³×C₄×D₁₁	C₂²	1	28	yes	—
16	C₂×#66999	C₂²	1	448	yes	direct product factor
17	C₂²×(C₂×D₁₁)⋊C₄	C₂²	1	56	yes	—
18	C₂²×C₄⋊Dic₁₁	C₂²	1	28	yes	complemented
19	C₂²×Dic₁₁⋊C₄	C₂²	1	56	yes	complemented
20	C₂⁵×C₁₁	C₂³	1	1	yes	characteristic
21	C₂⁴×D₁₁	—	2	8	no	—
22	C₂⁴×D₁₁	—	2	8	no	—
23	C₂⁴×D₁₁	Q₈	1	16	yes	complemented
24	C₂⁴×D₁₁	C₂³	1	14	yes	—
25	C₂²×Q₈×C₁₁	—	2	224	no	—
26	C₂²×Q₈×C₁₁	C₂³	1	112	yes	complemented
27	C₂³×C₄×C₁₁	—	2	16	no	—
28	C₂³×C₄×C₁₁	D₄	1	16	yes	—
29	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
30	C₂³×C₄×C₁₁	C₂³	1	14	yes	—
31	C₂³×Dic₁₁	—	2	8	no	—
32	C₂³×Dic₁₁	—	2	16	no	—
33	C₂³×Dic₁₁	C₂³	1	14	yes	—
34	C₂³×Dic₁₁	C₂³	1	28	yes	—
35	C₂²×C₄×D₁₁	—	2	224	no	—
36	C₂²×C₄×D₁₁	C₂³	1	112	yes	—
37	#66999	C₂³	1	512	yes	direct product factor
38	C₂×(C₂×D₁₁)⋊C₄	C₂³	1	224	yes	—
39	C₂×C₄⋊Dic₁₁	C₂³	1	112	yes	complemented
40	C₂×Dic₁₁⋊C₄	C₂³	1	224	yes	complemented
41	C₂³×C₂²⋊Q₈	—	11	1	no	maximal, Sylow
42	C₂⁴×C₁₁	C₂×D₄	1	8	yes	—
43	C₂⁴×C₁₁	C₂×Q₈	1	8	yes	—
44	C₂⁴×C₁₁	#41	1	8	yes	—
45	C₂⁴×C₁₁	C₂⁴	1	7	yes	—
46	C₂³×D₁₁	—	2	128	no	—
47	C₂³×D₁₁	—	2	56	no	—
48	C₂³×D₁₁	—	2	56	no	—
49	C₂³×D₁₁	C₂×Q₈	1	112	yes	complemented
50	C₂³×D₁₁	C₂⁴	1	28	yes	—
51	C₂×Q₈×C₁₁	—	2	448	no	—
52	C₂×Q₈×C₁₁	C₂⁴	1	64	yes	complemented
53	C₂²×C₄×C₁₁	—	2	112	no	—
54	C₂²×C₄×C₁₁	C₂×D₄	1	112	yes	—
55	C₂²×C₄×C₁₁	C₂⁴	1	56	yes	—
56	C₂²×C₄×C₁₁	C₂⁴	1	28	yes	—
57	C₂²×Dic₁₁	—	2	56	no	—
58	C₂²×Dic₁₁	—	2	112	no	—
59	C₂²×Dic₁₁	C₂⁴	1	28	yes	—
60	C₂²×Dic₁₁	C₂⁴	1	56	yes	—
61	C₂×C₄×D₁₁	—	2	448	no	—
62	C₂×C₄×D₁₁	C₂⁴	1	64	yes	—
63	(C₂×D₁₁)⋊C₄	C₂⁴	1	128	yes	—
64	C₄⋊Dic₁₁	C₂⁴	1	64	yes	complemented
65	Dic₁₁⋊C₄	C₂⁴	1	128	yes	complemented
66	C₂⁴×Q₈	D₁₁	1	1	yes	characteristic, complemented
67	C₂⁵×C₄	—	11	1	no	—
68	C₂²×C₂²⋊Q₈	—	11	56	no	—
69	C₂³×C₄⋊C₄	—	11	2	no	—
70	C₂³×C₄⋊C₄	—	11	1	no	—
71	C₂³×C₂²⋊C₄	—	11	2	no	—
72	C₂³×C₁₁	C₂²⋊Q₈	1	64	yes	complemented
73	C₂³×C₁₁	C₂²×D₄	1	28	yes	—
74	C₂³×C₁₁	C₂²×Q₈	1	28	yes	—
75	C₂³×C₁₁	C₂×#41	1	28	yes	—
76	C₂³×C₁₁	C₂⁵	1	7	yes	—
77	C₂²×D₁₁	—	2	448	no	—
78	C₂²×D₁₁	—	2	56	no	—
79	C₂²×D₁₁	—	2	56	no	—
80	C₂²×D₁₁	C₂²×Q₈	1	112	yes	complemented
81	C₂²×D₁₁	C₂⁵	1	8	yes	—
82	Q₈×C₁₁	—	2	128	no	—
83	C₂×C₄×C₁₁	—	2	112	no	—
84	C₂×C₄×C₁₁	C₂²×D₄	1	112	yes	—
85	C₂×C₄×C₁₁	C₂⁵	1	16	yes	—
86	C₂×C₄×C₁₁	C₂⁵	1	8	yes	—
87	C₂×Dic₁₁	—	2	56	no	—
88	C₂×Dic₁₁	—	2	112	no	—
89	C₂×Dic₁₁	C₂⁵	1	8	yes	—
90	C₂×Dic₁₁	C₂⁵	1	16	yes	—
91	C₄×D₁₁	—	2	128	no	—
92	C₂⁶	—	11	1	no	—
93	C₂³×Q₈	—	2	16	no	—
94	C₂³×Q₈	C₂×D₁₁	1	28	yes	complemented
95	C₂⁴×C₄	—	11	1	no	—
96	C₂⁴×C₄	—	11	2	no	—
97	C₂⁴×C₄	—	11	28	no	—
98	C₂⁴×C₄	—	22	16	no	—
99	C₂⁴×C₄	C₂×D₁₁	1	2	yes	—
100	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
101	C₂×C₂²⋊Q₈	—	11	448	no	—
102	C₂²×C₄⋊C₄	—	11	56	no	—
103	C₂²×C₄⋊C₄	—	11	28	no	—
104	C₂²×C₂²⋊C₄	—	11	56	no	—
105	C₂²×C₁₁	C₂×C₂²⋊Q₈	1	112	yes	complemented
106	C₂²×C₁₁	C₂³×D₄	1	14	yes	—
107	C₂²×C₁₁	C₂³×Q₈	1	14	yes	—
108	C₂²×C₁₁	C₂²×#41	1	14	yes	—
109	C₂²×C₁₁	C₂⁶	1	1	yes	characteristic
110	C₂×D₁₁	—	2	8	no	—
111	C₂×D₁₁	—	2	8	no	—
112	C₂×D₁₁	—	2	224	no	—
113	C₂×D₁₁	C₂³×Q₈	1	16	yes	complemented
114	C₄×C₁₁	—	2	16	no	—
115	C₄×C₁₁	C₂³×D₄	1	16	yes	—
116	Dic₁₁	—	2	8	no	—
117	Dic₁₁	—	2	16	no	—
118	C₂⁵	—	11	16	no	—
119	C₂⁵	—	11	14	no	—
120	C₂⁵	—	22	8	no	—
121	C₂⁵	—	22	8	no	—
122	C₂⁵	C₂²×D₁₁	1	1	yes	characteristic
123	C₂²×Q₈	—	2	224	no	—
124	C₂²×Q₈	C₂²×D₁₁	1	112	yes	complemented
125	C₂³×C₄	—	2	16	no	—
126	C₂³×C₄	—	11	28	no	—
127	C₂³×C₄	—	11	14	no	—
128	C₂³×C₄	—	11	112	no	—
129	C₂³×C₄	—	22	16	no	—
130	C₂³×C₄	—	22	8	no	—
131	C₂³×C₄	—	22	224	no	—
132	C₂³×C₄	C₂²⋊D₁₁	1	16	yes	—
133	C₂³×C₄	C₂²×D₁₁	1	28	yes	—
134	C₂³×C₄	C₂²×D₁₁	1	14	yes	—
135	C₂²⋊Q₈	—	11	512	no	—
136	C₂×C₄⋊C₄	—	11	224	no	—
137	C₂×C₄⋊C₄	—	11	112	no	—
138	C₂×C₂²⋊C₄	—	11	224	no	—
139	C₂×C₁₁	C₂²×C₂²⋊Q₈	1	28	yes	complemented
140	C₂×C₁₁	C₂⁴×D₄	1	1	yes	characteristic
141	C₂×C₁₁	C₂⁴×Q₈	1	1	yes	characteristic
142	C₂×C₁₁	C₂³×#41	1	1	yes	characteristic
143	D₁₁	—	2	16	no	—
144	C₂⁴	—	11	112	no	—
145	C₂⁴	—	11	28	no	—
146	C₂⁴	—	22	128	no	—
147	C₂⁴	—	22	56	no	—
148	C₂⁴	—	22	56	no	—
149	C₂⁴	Q₈×D₁₁	1	8	yes	—
150	C₂⁴	#4343	1	8	yes	—
151	C₂⁴	C₂×C₂²⋊D₁₁	1	8	yes	—
152	C₂⁴	C₂³×D₁₁	1	7	yes	—
153	C₂×Q₈	—	2	448	no	—
154	C₂×Q₈	C₂³×D₁₁	1	64	yes	complemented
155	C₂²×C₄	—	2	112	no	—
156	C₂²×C₄	—	11	56	no	—
157	C₂²×C₄	—	11	28	no	—
158	C₂²×C₄	—	11	64	no	—
159	C₂²×C₄	—	22	112	no	—
160	C₂²×C₄	—	22	56	no	—
161	C₂²×C₄	—	22	448	no	—
162	C₂²×C₄	C₂×C₂²⋊D₁₁	1	112	yes	—
163	C₂²×C₄	C₂³×D₁₁	1	56	yes	—
164	C₂²×C₄	C₂³×D₁₁	1	28	yes	—
165	C₄⋊C₄	—	11	128	no	—
166	C₄⋊C₄	—	11	64	no	—
167	C₂²⋊C₄	—	11	128	no	—
168	C₁₁	C₂³×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
169	C₂³	—	11	112	no	—
170	C₂³	—	11	8	no	—
171	C₂³	—	22	448	no	—
172	C₂³	—	22	56	no	—
173	C₂³	—	22	56	no	—
174	C₂³	#66999	1	64	yes	direct product factor
175	C₂³	C₂×Q₈×D₁₁	1	28	yes	—
176	C₂³	C₂×#4343	1	28	yes	—
177	C₂³	C₂²×C₂²⋊D₁₁	1	28	yes	—
178	C₂³	C₂⁴×D₁₁	1	7	yes	—
179	Q₈	—	2	128	no	—
180	C₂×C₄	—	2	112	no	—
181	C₂×C₄	—	11	16	no	—
182	C₂×C₄	—	11	8	no	—
183	C₂×C₄	—	22	112	no	—
184	C₂×C₄	—	22	56	no	—
185	C₂×C₄	—	22	128	no	—
186	C₂×C₄	C₂²×C₂²⋊D₁₁	1	112	yes	—
187	C₂×C₄	C₂⁴×D₁₁	1	16	yes	—
188	C₂×C₄	C₂⁴×D₁₁	1	8	yes	—
189	C₂²	—	11	16	no	—
190	C₂²	—	22	8	no	—
191	C₂²	—	22	8	no	—
192	C₂²	—	22	224	no	—
193	C₂²	C₂×#66999	1	112	yes	direct product factor
194	C₂²	C₂²×Q₈×D₁₁	1	14	yes	—
195	C₂²	C₂²×#4343	1	14	yes	—
196	C₂²	C₂³×C₂²⋊D₁₁	1	14	yes	—
197	C₂²	C₂⁵×D₁₁	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	22	8	no	—
200	C₄	—	22	16	no	—
201	C₄	C₂³×C₂²⋊D₁₁	1	16	yes	—
202	C₂	—	22	16	no	—
203	C₂	C₂²×#66999	1	28	yes	direct product factor
204	C₂	C₂⁴×C₂²⋊D₁₁	1	1	yes	characteristic
205	C₂	C₂³×Q₈×D₁₁	1	1	yes	characteristic
206	C₂	C₂³×#4343	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₁
commutator, cosocle	C₂²×C₁₁	C₂⁶
Frattini	C₂²	C₂⁵×D₁₁
Fitting	C₂⁴×Q₈×C₁₁	C₂
socle	C₂⁵×C₁₁	C₂³
prime radical	C₂⁵×D₁₁	C₂²
hypercenter	C₂⁴×Q₈	D₁₁
hypocenter	C₁₁	C₂³×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#66999	C₂	direct (N × H)
3. C₂³×C₄⋊Dic₁₁	C₂	semidirect
4. C₂³×Dic₁₁⋊C₄	C₂	semidirect
6. C₂⁴×Q₈×C₁₁	C₂	semidirect
9. C₂³×Q₈×C₁₁	C₂²	semidirect
16. C₂×#66999	C₂²	direct (N × H)
18. C₂²×C₄⋊Dic₁₁	C₂²	semidirect
19. C₂²×Dic₁₁⋊C₄	C₂²	semidirect
23. C₂⁴×D₁₁	Q₈	semidirect
26. C₂²×Q₈×C₁₁	C₂³	semidirect
37. #66999	C₂³	direct (N × H)
39. C₂×C₄⋊Dic₁₁	C₂³	semidirect
40. C₂×Dic₁₁⋊C₄	C₂³	semidirect
49. C₂³×D₁₁	C₂×Q₈	semidirect
52. C₂×Q₈×C₁₁	C₂⁴	semidirect
64. C₄⋊Dic₁₁	C₂⁴	semidirect
65. Dic₁₁⋊C₄	C₂⁴	semidirect
66. C₂⁴×Q₈	D₁₁	semidirect
72. C₂³×C₁₁	C₂²⋊Q₈	semidirect
80. C₂²×D₁₁	C₂²×Q₈	semidirect
94. C₂³×Q₈	C₂×D₁₁	semidirect
105. C₂²×C₁₁	C₂×C₂²⋊Q₈	semidirect
113. C₂×D₁₁	C₂³×Q₈	semidirect
124. C₂²×Q₈	C₂²×D₁₁	semidirect
139. C₂×C₁₁	C₂²×C₂²⋊Q₈	semidirect
154. C₂×Q₈	C₂³×D₁₁	semidirect
168. C₁₁	C₂³×C₂²⋊Q₈	semidirect
174. C₂³	#66999	direct (N × H)
193. C₂²	C₂×#66999	direct (N × H)
203. C₂	C₂²×#66999	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×C₄×D₁₁	2
C₂²×#66999	2
C₂³×C₄⋊Dic₁₁	2
C₂³×Dic₁₁⋊C₄	2
C₂³×(C₂×D₁₁)⋊C₄	2
C₂⁴×Q₈×C₁₁	2
C₂³×C₂²⋊Q₈	11

Subgroups — C23×#66999 #15660413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66999 #15660413