Subgroups — C₂³×#88063 #16726400

Order 3328.

7134 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×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×#6099	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88063	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₁₃	C₄	1	2	yes	—
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×D₅₂	—	2	1	no	—
8	C₂³×D₅₂	—	2	7	no	—
9	C₂³×D₅₂	—	2	1	no	—
10	C₂³×D₅₂	—	2	7	no	—
11	C₂³×D₅₂	C₄	1	14	yes	—
12	C₂³×D₅₂	C₂²	1	14	yes	—
13	C₂²×C₁₃×M₁₆	—	2	7	no	—
14	C₂²×C₁₃×M₁₆	C₂²	1	14	yes	complemented
15	C₂³×C₈×C₁₃	—	2	1	no	—
16	C₂²×#6099	—	2	7	no	—
17	C₂²×#6099	C₂²	1	14	yes	complemented
18	C₂³×#1055	—	2	1	no	—
19	C₂×#88063	C₂²	1	112	yes	direct product factor
20	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×D₁₃	—	2	14	no	—
22	C₂⁴×D₁₃	—	2	2	no	—
23	C₂⁴×D₁₃	—	4	16	no	—
24	C₂⁴×D₁₃	C₂×C₄	1	28	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	7	yes	—
26	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	D₄	1	7	yes	—
28	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	D₄	1	7	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
31	C₂²×D₅₂	—	2	84	no	—
32	C₂²×D₅₂	—	2	14	no	—
33	C₂²×D₅₂	—	2	14	no	—
34	C₂²×D₅₂	—	2	14	no	—
35	C₂²×D₅₂	—	2	42	no	—
36	C₂²×D₅₂	—	2	14	no	—
37	C₂²×D₅₂	—	2	42	no	—
38	C₂²×D₅₂	C₂×C₄	1	84	yes	—
39	C₂²×D₅₂	C₂³	1	28	yes	—
40	C₂×C₁₃×M₁₆	—	2	42	no	—
41	C₂×C₁₃×M₁₆	C₂³	1	28	yes	complemented
42	C₂²×C₈×C₁₃	—	2	14	no	—
43	C₂×#6099	—	2	42	no	—
44	C₂×#6099	C₂³	1	28	yes	complemented
45	C₂²×#1055	—	2	14	no	—
46	#88063	C₂³	1	64	yes	direct product factor
47	C₂³×#100	—	13	1	no	maximal, Sylow
48	C₂⁴×C₁₃	—	2	8	no	—
49	C₂⁴×C₁₃	C₂²⋊C₄	1	7	yes	—
50	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂⁴×C₁₃	C₂²×C₄	1	7	yes	—
52	C₂³×D₁₃	—	2	84	no	—
53	C₂³×D₁₃	—	2	28	no	—
54	C₂³×D₁₃	—	4	112	no	—
55	C₂³×D₁₃	—	4	16	no	—
56	C₂³×D₁₃	—	4	112	no	—
57	C₂³×D₁₃	C₂²×C₄	1	56	yes	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	42	yes	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	21	yes	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	21	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
67	C₂×D₅₂	—	2	168	no	—
68	C₂×D₅₂	—	2	84	no	—
69	C₂×D₅₂	—	2	84	no	—
70	C₂×D₅₂	—	2	28	no	—
71	C₂×D₅₂	—	2	28	no	—
72	C₂×D₅₂	—	2	28	no	—
73	C₂×D₅₂	—	2	28	no	—
74	C₂×D₅₂	C₂²×C₄	1	56	yes	—
75	C₂×D₅₂	C₂⁴	1	8	yes	—
76	C₁₃×M₁₆	—	2	28	no	—
77	C₁₃×M₁₆	C₂⁴	1	8	yes	complemented
78	C₂×C₈×C₁₃	—	2	28	no	—
79	#6099	—	2	28	no	—
80	#6099	C₂⁴	1	8	yes	complemented
81	C₂×#1055	—	2	28	no	—
82	C₂⁴×D₄	—	13	1	no	—
83	C₂³×M₁₆	—	13	1	no	—
84	C₂³×M₁₆	D₁₃	1	1	yes	characteristic, complemented
85	C₂²×#100	—	13	28	no	—
86	C₂³×C₁₃	—	2	28	no	—
87	C₂³×C₁₃	—	2	28	no	—
88	C₂³×C₁₃	#100	1	8	yes	complemented
89	C₂³×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
90	C₂³×C₁₃	C₂×C₂²⋊C₄	1	7	yes	—
91	C₂³×C₁₃	C₂³×C₄	1	7	yes	—
92	C₂²×D₁₃	—	2	56	no	—
93	C₂²×D₁₃	—	2	56	no	—
94	C₂²×D₁₃	—	4	336	no	—
95	C₂²×D₁₃	—	4	112	no	—
96	C₂²×D₁₃	—	4	112	no	—
97	C₂²×D₁₃	C₂³×C₄	1	16	yes	—
98	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	42	yes	—
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
102	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
104	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
106	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
107	D₅₂	—	2	56	no	—
108	D₅₂	—	2	56	no	—
109	D₅₂	—	2	8	no	—
110	D₅₂	—	2	8	no	—
111	C₈×C₁₃	—	2	8	no	—
112	#1055	—	2	8	no	—
113	C₂⁶	—	13	2	no	—
114	C₂³×D₄	—	13	14	no	—
115	C₂³×D₄	—	13	14	no	—
116	C₂³×D₄	—	26	7	no	—
117	C₂³×D₄	—	26	1	no	—
118	C₂³×D₄	—	26	7	no	—
119	C₂³×D₄	—	26	1	no	—
120	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
121	C₂²×M₁₆	—	2	7	no	—
122	C₂²×M₁₆	—	13	14	no	—
123	C₂²×M₁₆	—	26	7	no	—
124	C₂²×M₁₆	C₂×D₁₃	1	14	yes	complemented
125	C₂³×C₈	—	2	1	no	—
126	C₂³×C₈	—	26	1	no	—
127	C₂×#100	—	13	112	no	—
128	C₂²×C₁₃	—	2	42	no	—
129	C₂²×C₁₃	—	2	14	no	—
130	C₂²×C₁₃	C₂×#100	1	28	yes	complemented
131	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
132	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
133	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
134	C₂×D₁₃	—	2	16	no	—
135	C₂×D₁₃	—	4	112	no	—
136	C₂×D₁₃	—	4	112	no	—
137	C₂×D₁₃	—	4	16	no	—
138	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
139	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
140	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
142	C₂⁵	—	13	28	no	—
143	C₂⁵	—	26	14	no	—
144	C₂⁵	—	26	2	no	—
145	C₂⁵	—	52	16	no	—
146	C₂⁵	C₄×D₁₃	1	1	yes	characteristic
147	C₂²×D₄	—	13	84	no	—
148	C₂²×D₄	—	13	28	no	—
149	C₂²×D₄	—	26	84	no	—
150	C₂²×D₄	—	26	14	no	—
151	C₂²×D₄	—	26	14	no	—
152	C₂²×D₄	—	26	14	no	—
153	C₂²×D₄	—	26	42	no	—
154	C₂²×D₄	—	26	14	no	—
155	C₂²×D₄	—	26	42	no	—
156	C₂³×C₄	C₄×D₁₃	1	7	yes	—
157	C₂³×C₄	D₅₂	1	7	yes	—
158	C₂³×C₄	D₅₂	1	1	yes	characteristic
159	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
160	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
161	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
162	C₂×M₁₆	—	2	42	no	—
163	C₂×M₁₆	—	13	28	no	—
164	C₂×M₁₆	—	26	42	no	—
165	C₂×M₁₆	C₂²×D₁₃	1	28	yes	complemented
166	C₂²×C₈	—	2	14	no	—
167	C₂²×C₈	—	26	14	no	—
168	#100	—	13	64	no	—
169	C₂×C₁₃	—	2	7	no	—
170	C₂×C₁₃	—	2	1	no	—
171	C₂×C₁₃	C₂²×#100	1	14	yes	complemented
172	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
173	D₁₃	—	4	16	no	—
174	C₂⁴	—	2	8	no	—
175	C₂⁴	—	13	56	no	—
176	C₂⁴	—	26	84	no	—
177	C₂⁴	—	26	28	no	—
178	C₂⁴	—	52	112	no	—
179	C₂⁴	—	52	112	no	—
180	C₂⁴	—	52	16	no	—
181	C₂⁴	#6103	1	7	yes	—
182	C₂⁴	#6103	1	1	yes	characteristic
183	C₂⁴	C₂×C₄×D₁₃	1	7	yes	—
184	C₂×D₄	—	13	56	no	—
185	C₂×D₄	—	13	8	no	—
186	C₂×D₄	—	26	168	no	—
187	C₂×D₄	—	26	84	no	—
188	C₂×D₄	—	26	84	no	—
189	C₂×D₄	—	26	28	no	—
190	C₂×D₄	—	26	28	no	—
191	C₂×D₄	—	26	28	no	—
192	C₂×D₄	—	26	28	no	—
193	C₂²×C₄	#6103	1	42	yes	—
194	C₂²×C₄	#6103	1	7	yes	—
195	C₂²×C₄	#6103	1	7	yes	—
196	C₂²×C₄	C₂×C₄×D₁₃	1	21	yes	—
197	C₂²×C₄	C₂×D₅₂	1	7	yes	—
198	C₂²×C₄	C₂×D₅₂	1	21	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₁₃	1	7	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
201	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
202	M₁₆	—	2	28	no	—
203	M₁₆	—	13	8	no	—
204	M₁₆	—	26	28	no	—
205	M₁₆	C₂³×D₁₃	1	8	yes	complemented
206	C₂×C₈	—	2	28	no	—
207	C₂×C₈	—	26	28	no	—
208	C₁₃	C₂³×#100	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	2	28	no	—
210	C₂³	—	2	28	no	—
211	C₂³	—	13	16	no	—
212	C₂³	—	26	56	no	—
213	C₂³	—	26	56	no	—
214	C₂³	—	52	112	no	—
215	C₂³	—	52	336	no	—
216	C₂³	—	52	112	no	—
217	C₂³	#88063	1	8	yes	direct product factor
218	C₂³	C₂×#6103	1	21	yes	—
219	C₂³	C₂×#6103	1	7	yes	—
220	C₂³	C₂²×C₄×D₁₃	1	7	yes	—
221	D₄	—	26	56	no	—
222	D₄	—	26	56	no	—
223	D₄	—	26	8	no	—
224	D₄	—	26	8	no	—
225	C₂×C₄	C₂×#6103	1	42	yes	—
226	C₂×C₄	C₂×#6103	1	21	yes	—
227	C₂×C₄	C₂×#6103	1	21	yes	—
228	C₂×C₄	C₂²×C₄×D₁₃	1	7	yes	—
229	C₂×C₄	C₂²×D₅₂	1	7	yes	—
230	C₂×C₄	C₂²×D₅₂	1	7	yes	—
231	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
232	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
233	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
234	C₈	—	2	8	no	—
235	C₈	—	26	8	no	—
236	C₂²	—	2	42	no	—
237	C₂²	—	2	14	no	—
238	C₂²	—	26	16	no	—
239	C₂²	—	52	16	no	—
240	C₂²	—	52	112	no	—
241	C₂²	—	52	112	no	—
242	C₂²	C₂×#88063	1	28	yes	direct product factor
243	C₂²	C₂²×#6103	1	7	yes	—
244	C₂²	C₂²×#6103	1	7	yes	—
245	C₂²	C₂³×C₄×D₁₃	1	1	yes	characteristic
246	C₄	C₂²×#6103	1	7	yes	—
247	C₄	C₂²×#6103	1	7	yes	—
248	C₄	C₂³×D₅₂	1	1	yes	characteristic
249	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
250	C₂	—	2	7	no	—
251	C₂	—	2	1	no	—
252	C₂	—	52	16	no	—
253	C₂	C₂²×#88063	1	14	yes	direct product factor
254	C₂	C₂³×#6103	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	#6103
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×M₁₆	C₂
socle	C₂⁴×C₁₃	C₂²⋊C₄
prime radical	C₂⁴×D₅₂	C₂
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×M₁₆	D₁₃
hypocenter	C₁₃	C₂³×#100
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×#6099	C₂	semidirect
3. C₂²×#88063	C₂	direct (N × H)
4. C₂³×C₁₃×M₁₆	C₂	semidirect
14. C₂²×C₁₃×M₁₆	C₂²	semidirect
17. C₂²×#6099	C₂²	semidirect
19. C₂×#88063	C₂²	direct (N × H)
41. C₂×C₁₃×M₁₆	C₂³	semidirect
44. C₂×#6099	C₂³	semidirect
46. #88063	C₂³	direct (N × H)
77. C₁₃×M₁₆	C₂⁴	semidirect
80. #6099	C₂⁴	semidirect
84. C₂³×M₁₆	D₁₃	semidirect
88. C₂³×C₁₃	#100	semidirect
124. C₂²×M₁₆	C₂×D₁₃	semidirect
130. C₂²×C₁₃	C₂×#100	semidirect
165. C₂×M₁₆	C₂²×D₁₃	semidirect
171. C₂×C₁₃	C₂²×#100	semidirect
205. M₁₆	C₂³×D₁₃	semidirect
208. C₁₃	C₂³×#100	semidirect
217. C₂³	#88063	direct (N × H)
242. C₂²	C₂×#88063	direct (N × H)
253. C₂	C₂²×#88063	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×D₅₂	2
C₂³×#6099	2
C₂²×#88063	2
C₂³×C₁₃×M₁₆	2
C₂³×#100	13

Subgroups — C23×#88063 #16726400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88063 #16726400