Subgroups — C₂³×#66873 #15617280

Order 2816.

7646 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#66873	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₁₁	—	2	1	no	—
6	C₂⁴×C₄×C₁₁	—	2	1	no	—
7	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
8	C₂³×D₄₄	—	2	8	no	—
9	C₂³×D₄₄	—	2	7	no	—
10	C₂³×D₄₄	C₄	1	16	yes	complemented
11	C₂³×D₄₄	C₂²	1	14	yes	—
12	C₂²×C₁₁×C₄⋊C₄	—	2	7	no	—
13	C₂²×C₁₁×C₄⋊C₄	C₂²	1	14	yes	complemented
14	C₂³×C₁₁⋊C₈	—	2	8	no	—
15	C₂³×C₁₁⋊C₈	C₂²	1	14	yes	complemented
16	C₂×#66873	C₂²	1	112	yes	direct product factor
17	C₂⁵×C₁₁	D₄	1	1	yes	characteristic
18	C₂⁴×D₁₁	—	2	16	no	—
19	C₂⁴×D₁₁	—	2	14	no	—
20	C₂⁴×D₁₁	—	4	8	no	—
21	C₂⁴×D₁₁	—	4	8	no	—
22	C₂³×C₄×C₁₁	—	2	14	no	—
23	C₂³×C₄×C₁₁	—	4	8	no	—
24	C₂³×C₄×C₁₁	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₁₁	D₄	1	8	yes	—
26	C₂³×C₄×C₁₁	D₄	1	7	yes	—
27	C₂³×C₄×C₁₁	C₂³	1	7	yes	—
28	C₂²×D₄₄	—	2	112	no	—
29	C₂²×D₄₄	—	2	56	no	—
30	C₂²×D₄₄	—	2	42	no	—
31	C₂²×D₄₄	C₂×C₄	1	112	yes	complemented
32	C₂²×D₄₄	C₂³	1	28	yes	—
33	C₂×C₁₁×C₄⋊C₄	—	2	42	no	—
34	C₂×C₁₁×C₄⋊C₄	C₂³	1	28	yes	complemented
35	C₂²×C₁₁⋊C₈	—	2	56	no	—
36	C₂²×C₁₁⋊C₈	C₂³	1	28	yes	complemented
37	#66873	C₂³	1	64	yes	direct product factor
38	C₂³×#102	—	11	1	no	maximal, Sylow
39	C₂⁴×C₁₁	C₂²⋊C₄	1	8	yes	—
40	C₂⁴×C₁₁	D₈	1	8	yes	—
41	C₂⁴×C₁₁	SD₁₆	1	8	yes	—
42	C₂⁴×C₁₁	C₂×D₄	1	7	yes	—
43	C₂³×D₁₁	—	2	112	no	—
44	C₂³×D₁₁	—	2	28	no	—
45	C₂³×D₁₁	—	4	128	no	—
46	C₂³×D₁₁	—	4	56	no	—
47	C₂³×D₁₁	—	4	56	no	—
48	C₂²×C₄×C₁₁	—	2	28	no	—
49	C₂²×C₄×C₁₁	—	4	56	no	—
50	C₂²×C₄×C₁₁	C₂²⋊C₄	1	56	yes	—
51	C₂²×C₄×C₁₁	C₂²×C₄	1	28	yes	—
52	C₂²×C₄×C₁₁	C₂×D₄	1	21	yes	—
53	C₂²×C₄×C₁₁	C₂×D₄	1	28	yes	—
54	C₂²×C₄×C₁₁	C₂⁴	1	7	yes	—
55	C₂×D₄₄	—	2	336	no	—
56	C₂×D₄₄	—	2	56	no	—
57	C₂×D₄₄	—	2	28	no	—
58	C₂×D₄₄	C₂²×C₄	1	112	yes	complemented
59	C₂×D₄₄	C₂⁴	1	8	yes	—
60	C₁₁×C₄⋊C₄	—	2	28	no	—
61	C₁₁×C₄⋊C₄	C₂⁴	1	8	yes	complemented
62	C₂×C₁₁⋊C₈	—	2	56	no	—
63	C₂×C₁₁⋊C₈	C₂⁴	1	8	yes	complemented
64	C₂⁴×D₄	—	11	1	no	—
65	C₂⁴×C₈	—	11	1	no	—
66	C₂³×C₄⋊C₄	D₁₁	1	1	yes	characteristic, complemented
67	C₂²×#102	—	11	28	no	—
68	C₂³×C₁₁	#102	1	64	yes	complemented
69	C₂³×C₁₁	C₂×C₂²⋊C₄	1	28	yes	—
70	C₂³×C₁₁	C₂×D₈	1	28	yes	—
71	C₂³×C₁₁	C₂×SD₁₆	1	28	yes	—
72	C₂³×C₁₁	C₂²×D₄	1	7	yes	—
73	C₂²×D₁₁	—	2	112	no	—
74	C₂²×D₁₁	—	2	8	no	—
75	C₂²×D₁₁	—	4	56	no	—
76	C₂²×D₁₁	—	4	56	no	—
77	C₂²×D₁₁	—	4	448	no	—
78	C₂×C₄×C₁₁	—	2	8	no	—
79	C₂×C₄×C₁₁	—	4	56	no	—
80	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	84	yes	—
81	C₂×C₄×C₁₁	C₂³×C₄	1	14	yes	—
82	C₂×C₄×C₁₁	C₂²×D₄	1	14	yes	—
83	C₂×C₄×C₁₁	C₂²×D₄	1	7	yes	—
84	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
85	D₄₄	—	2	112	no	—
86	D₄₄	—	2	8	no	—
87	D₄₄	C₂³×C₄	1	16	yes	complemented
88	C₁₁⋊C₈	—	2	8	no	—
89	C₂⁶	—	22	1	no	—
90	C₂³×D₄	—	11	14	no	—
91	C₂³×D₄	—	11	16	no	—
92	C₂³×D₄	—	22	8	no	—
93	C₂³×D₄	—	22	7	no	—
94	C₂⁴×C₄	—	2	1	no	—
95	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
96	C₂³×C₈	—	11	14	no	—
97	C₂³×C₈	—	22	8	no	—
98	C₂²×C₄⋊C₄	—	2	7	no	—
99	C₂²×C₄⋊C₄	C₂×D₁₁	1	14	yes	complemented
100	C₂×#102	—	11	112	no	—
101	C₂²×C₁₁	C₂×#102	1	112	yes	complemented
102	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	14	yes	—
103	C₂²×C₁₁	C₂²×D₈	1	14	yes	—
104	C₂²×C₁₁	C₂²×SD₁₆	1	14	yes	—
105	C₂²×C₁₁	C₂³×D₄	1	1	yes	characteristic
106	C₂×D₁₁	—	2	16	no	—
107	C₂×D₁₁	—	4	224	no	—
108	C₂×D₁₁	—	4	8	no	—
109	C₂×D₁₁	—	4	8	no	—
110	C₄×C₁₁	—	4	8	no	—
111	C₄×C₁₁	C₂²×C₂²⋊C₄	1	14	yes	—
112	C₄×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
113	C₄×C₁₁	C₂³×D₄	1	1	yes	characteristic
114	C₂⁵	—	22	16	no	—
115	C₂⁵	—	22	14	no	—
116	C₂⁵	—	44	8	no	—
117	C₂⁵	—	44	8	no	—
118	C₂⁵	C₂²⋊D₁₁	1	1	yes	characteristic
119	C₂²×D₄	—	11	112	no	—
120	C₂²×D₄	—	11	28	no	—
121	C₂²×D₄	—	22	112	no	—
122	C₂²×D₄	—	22	56	no	—
123	C₂²×D₄	—	22	42	no	—
124	C₂³×C₄	—	2	14	no	—
125	C₂³×C₄	—	4	8	no	—
126	C₂³×C₄	C₄×D₁₁	1	8	yes	—
127	C₂³×C₄	D₄₄	1	8	yes	—
128	C₂³×C₄	C₂²⋊D₁₁	1	7	yes	—
129	C₂³×C₄	C₂²×D₁₁	1	7	yes	—
130	C₂²×C₈	—	11	28	no	—
131	C₂²×C₈	—	22	56	no	—
132	C₂×C₄⋊C₄	—	2	42	no	—
133	C₂×C₄⋊C₄	C₂²×D₁₁	1	28	yes	complemented
134	#102	—	11	64	no	—
135	C₂×C₁₁	C₂²×#102	1	28	yes	complemented
136	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₁₁	C₂³×D₈	1	1	yes	characteristic
138	C₂×C₁₁	C₂³×SD₁₆	1	1	yes	characteristic
139	D₁₁	—	4	16	no	—
140	C₂⁴	—	22	112	no	—
141	C₂⁴	—	22	28	no	—
142	C₂⁴	—	44	128	no	—
143	C₂⁴	—	44	56	no	—
144	C₂⁴	—	44	56	no	—
145	C₂⁴	(C₂×D₁₁)⋊C₄	1	8	yes	—
146	C₂⁴	#4323	1	8	yes	—
147	C₂⁴	#4325	1	8	yes	—
148	C₂⁴	C₂×C₂²⋊D₁₁	1	7	yes	—
149	C₂×D₄	—	11	112	no	—
150	C₂×D₄	—	11	8	no	—
151	C₂×D₄	—	22	336	no	—
152	C₂×D₄	—	22	56	no	—
153	C₂×D₄	—	22	28	no	—
154	C₂²×C₄	—	2	28	no	—
155	C₂²×C₄	—	4	56	no	—
156	C₂²×C₄	(C₂×D₁₁)⋊C₄	1	56	yes	—
157	C₂²×C₄	C₂×C₄×D₁₁	1	28	yes	—
158	C₂²×C₄	C₂×D₄₄	1	28	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₁₁	1	21	yes	—
160	C₂²×C₄	C₂³×D₁₁	1	7	yes	—
161	C₂×C₈	—	11	8	no	—
162	C₂×C₈	—	22	56	no	—
163	C₄⋊C₄	—	2	28	no	—
164	C₄⋊C₄	C₂³×D₁₁	1	8	yes	complemented
165	C₁₁	C₂³×#102	1	1	yes	characteristic, complemented, Sylow
166	C₂³	—	22	112	no	—
167	C₂³	—	22	8	no	—
168	C₂³	—	44	56	no	—
169	C₂³	—	44	56	no	—
170	C₂³	—	44	448	no	—
171	C₂³	#66873	1	64	yes	direct product factor
172	C₂³	C₂×(C₂×D₁₁)⋊C₄	1	28	yes	—
173	C₂³	C₂×#4323	1	28	yes	—
174	C₂³	C₂×#4325	1	28	yes	—
175	C₂³	C₂²×C₂²⋊D₁₁	1	7	yes	—
176	D₄	—	11	16	no	—
177	D₄	—	22	112	no	—
178	D₄	—	22	8	no	—
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	—	4	56	no	—
181	C₂×C₄	C₂×(C₂×D₁₁)⋊C₄	1	84	yes	—
182	C₂×C₄	C₂²×C₄×D₁₁	1	14	yes	—
183	C₂×C₄	C₂²×D₄₄	1	14	yes	—
184	C₂×C₄	C₂²×C₂²⋊D₁₁	1	7	yes	—
185	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
186	C₈	—	22	8	no	—
187	C₂²	—	22	16	no	—
188	C₂²	—	44	224	no	—
189	C₂²	—	44	8	no	—
190	C₂²	—	44	8	no	—
191	C₂²	C₂×#66873	1	112	yes	direct product factor
192	C₂²	C₂²×(C₂×D₁₁)⋊C₄	1	14	yes	—
193	C₂²	C₂²×#4323	1	14	yes	—
194	C₂²	C₂²×#4325	1	14	yes	—
195	C₂²	C₂³×C₂²⋊D₁₁	1	1	yes	characteristic
196	C₄	—	4	8	no	—
197	C₄	C₂²×(C₂×D₁₁)⋊C₄	1	14	yes	—
198	C₄	C₂³×C₄×D₁₁	1	1	yes	characteristic
199	C₄	C₂³×D₄₄	1	1	yes	characteristic
200	C₂	—	44	16	no	—
201	C₂	C₂²×#66873	1	28	yes	direct product factor
202	C₂	C₂³×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
203	C₂	C₂³×#4323	1	1	yes	characteristic
204	C₂	C₂³×#4325	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²⋊D₁₁
commutator	C₄×C₁₁	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂³×C₁₁×C₄⋊C₄	C₂
socle	C₂⁵×C₁₁	D₄
prime radical	C₂⁴×D₄₄	C₂
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₁₁
hypocenter	C₁₁	C₂³×#102
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₁₁⋊C₈	C₂	semidirect
3. C₂²×#66873	C₂	direct (N × H)
4. C₂³×C₁₁×C₄⋊C₄	C₂	semidirect
10. C₂³×D₄₄	C₄	semidirect
13. C₂²×C₁₁×C₄⋊C₄	C₂²	semidirect
15. C₂³×C₁₁⋊C₈	C₂²	semidirect
16. C₂×#66873	C₂²	direct (N × H)
31. C₂²×D₄₄	C₂×C₄	semidirect
34. C₂×C₁₁×C₄⋊C₄	C₂³	semidirect
36. C₂²×C₁₁⋊C₈	C₂³	semidirect
37. #66873	C₂³	direct (N × H)
58. C₂×D₄₄	C₂²×C₄	semidirect
61. C₁₁×C₄⋊C₄	C₂⁴	semidirect
63. C₂×C₁₁⋊C₈	C₂⁴	semidirect
66. C₂³×C₄⋊C₄	D₁₁	semidirect
68. C₂³×C₁₁	#102	semidirect
87. D₄₄	C₂³×C₄	semidirect
99. C₂²×C₄⋊C₄	C₂×D₁₁	semidirect
101. C₂²×C₁₁	C₂×#102	semidirect
133. C₂×C₄⋊C₄	C₂²×D₁₁	semidirect
135. C₂×C₁₁	C₂²×#102	semidirect
164. C₄⋊C₄	C₂³×D₁₁	semidirect
165. C₁₁	C₂³×#102	semidirect
171. C₂³	#66873	direct (N × H)
191. C₂²	C₂×#66873	direct (N × H)
201. C₂	C₂²×#66873	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×D₄₄	2
C₂⁴×C₁₁⋊C₈	2
C₂²×#66873	2
C₂³×C₁₁×C₄⋊C₄	2
C₂³×#102	11

Subgroups — C23×#66873 #15617280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#66873 #15617280