Subgroups — C₂³×#88065 #16726506

Order 3328.

5758 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6127	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88065	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×C₂²⋊D₁₃	—	2	1	no	—
7	C₂⁴×Dic₁₃	—	2	1	no	—
8	C₂²×#6127	—	2	7	no	—
9	C₂²×#6127	—	2	7	no	—
10	C₂²×#6127	C₄	1	14	yes	complemented
11	C₂²×#6127	C₂²	1	14	yes	—
12	C₂³×D₅₂	C₄	1	1	yes	characteristic, complemented
13	C₂³×C₄×D₁₃	—	2	1	no	—
14	C₂³×Dic₂₆	C₄	1	1	yes	characteristic, complemented
15	C₂²×C₁₃×M₁₆	—	2	7	no	—
16	C₂²×C₁₃×M₁₆	C₂²	1	14	yes	complemented
17	C₂³×C₈×C₁₃	—	2	1	no	—
18	C₂²×C₄×Dic₁₃	—	2	7	no	—
19	C₂²×C₄×Dic₁₃	C₂²	1	14	yes	complemented
20	C₂×#88065	C₂²	1	112	yes	direct product factor
21	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
22	C₂⁴×D₁₃	—	2	1	no	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	7	yes	—
25	C₂³×C₄×C₁₃	D₄	1	7	yes	—
26	C₂³×C₄×C₁₃	D₄	1	7	yes	—
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	7	yes	—
29	C₂²×C₂²⋊D₁₃	—	2	14	no	—
30	C₂²×C₂²⋊D₁₃	—	2	14	no	—
31	C₂³×Dic₁₃	—	2	16	no	—
32	C₂³×Dic₁₃	—	2	1	no	—
33	C₂³×Dic₁₃	—	2	14	no	—
34	C₂×#6127	—	2	84	no	—
35	C₂×#6127	—	2	42	no	—
36	C₂×#6127	—	2	42	no	—
37	C₂×#6127	C₂×C₄	1	84	yes	complemented
38	C₂×#6127	C₂³	1	28	yes	—
39	C₂²×D₅₂	—	2	7	no	—
40	C₂²×D₅₂	C₂×C₄	1	14	yes	complemented
41	C₂²×C₄×D₁₃	—	2	14	no	—
42	C₂²×C₄×D₁₃	—	2	14	no	—
43	C₂²×Dic₂₆	—	2	7	no	—
44	C₂²×Dic₂₆	C₂×C₄	1	14	yes	complemented
45	C₂×C₁₃×M₁₆	—	2	42	no	—
46	C₂×C₁₃×M₁₆	C₂³	1	28	yes	complemented
47	C₂²×C₈×C₁₃	—	2	14	no	—
48	C₂×C₄×Dic₁₃	—	2	42	no	—
49	C₂×C₄×Dic₁₃	C₂³	1	28	yes	complemented
50	#88065	C₂³	1	64	yes	direct product factor
51	C₂³×#104	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	—	2	8	no	—
53	C₂⁴×C₁₃	C₂²⋊C₄	1	7	yes	—
54	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂⁴×C₁₃	C₂×D₄	1	7	yes	—
56	C₂³×D₁₃	—	2	14	no	—
57	C₂³×D₁₃	—	4	8	no	—
58	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
59	C₂²×C₄×C₁₃	C₂²⋊C₄	1	7	yes	—
60	C₂²×C₄×C₁₃	C₂²⋊C₄	1	42	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	7	yes	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	21	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×C₂²⋊D₁₃	—	2	84	no	—
68	C₂×C₂²⋊D₁₃	—	2	28	no	—
69	C₂²×Dic₁₃	—	2	112	no	—
70	C₂²×Dic₁₃	—	2	14	no	—
71	C₂²×Dic₁₃	—	2	28	no	—
72	#6127	—	2	168	no	—
73	#6127	—	2	28	no	—
74	#6127	—	2	28	no	—
75	#6127	C₂²×C₄	1	56	yes	complemented
76	#6127	C₂⁴	1	8	yes	—
77	C₂×D₅₂	—	2	42	no	—
78	C₂×D₅₂	C₂²×C₄	1	28	yes	complemented
79	C₂×C₄×D₁₃	—	2	84	no	—
80	C₂×C₄×D₁₃	—	2	28	no	—
81	C₂×Dic₂₆	—	2	42	no	—
82	C₂×Dic₂₆	C₂²×C₄	1	28	yes	complemented
83	C₁₃×M₁₆	—	2	28	no	—
84	C₁₃×M₁₆	C₂⁴	1	8	yes	complemented
85	C₂×C₈×C₁₃	—	2	28	no	—
86	C₄×Dic₁₃	—	2	28	no	—
87	C₄×Dic₁₃	C₂⁴	1	8	yes	complemented
88	C₂³×#41	—	13	1	no	—
89	C₂³×M₁₆	D₁₃	1	1	yes	characteristic, complemented
90	C₂³×C₄²	—	13	1	no	—
91	C₂²×#104	—	13	28	no	—
92	C₂³×C₁₃	—	2	28	no	—
93	C₂³×C₁₃	—	2	28	no	—
94	C₂³×C₁₃	#104	1	8	yes	complemented
95	C₂³×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
96	C₂³×C₁₃	C₂×C₂²⋊C₄	1	7	yes	—
97	C₂³×C₁₃	C₂²×D₄	1	7	yes	—
98	C₂²×D₁₃	—	2	28	no	—
99	C₂²×D₁₃	—	4	56	no	—
100	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
101	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	42	yes	—
103	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	7	yes	—
105	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	7	yes	—
108	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
109	C₂²⋊D₁₃	—	2	56	no	—
110	C₂²⋊D₁₃	—	2	8	no	—
111	C₂×Dic₁₃	—	2	112	no	—
112	C₂×Dic₁₃	—	2	28	no	—
113	C₂×Dic₁₃	—	2	8	no	—
114	D₅₂	—	2	28	no	—
115	D₅₂	C₂³×C₄	1	8	yes	complemented
116	C₄×D₁₃	—	2	56	no	—
117	C₄×D₁₃	—	2	8	no	—
118	Dic₂₆	—	2	28	no	—
119	Dic₂₆	C₂³×C₄	1	8	yes	complemented
120	C₈×C₁₃	—	2	8	no	—
121	C₂²×#41	—	13	14	no	—
122	C₂²×#41	—	13	14	no	—
123	C₂²×#41	—	26	7	no	—
124	C₂²×#41	—	26	7	no	—
125	C₂³×Q₈	—	13	1	no	—
126	C₂³×D₄	—	13	1	no	—
127	C₂³×D₄	—	26	1	no	—
128	C₂⁴×C₄	—	26	1	no	—
129	C₂⁴×C₄	—	26	1	no	—
130	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
131	C₂²×M₁₆	—	2	7	no	—
132	C₂²×M₁₆	C₂×D₁₃	1	14	yes	complemented
133	C₂³×C₈	—	2	1	no	—
134	C₂²×C₄²	—	13	14	no	—
135	C₂²×C₄²	—	26	7	no	—
136	C₂×#104	—	13	112	no	—
137	C₂²×C₁₃	—	2	14	no	—
138	C₂²×C₁₃	—	2	42	no	—
139	C₂²×C₁₃	C₂×#104	1	28	yes	complemented
140	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
141	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
142	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
143	C₂×D₁₃	—	2	8	no	—
144	C₂×D₁₃	—	4	56	no	—
145	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
146	C₄×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
147	C₄×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
148	C₄×C₁₃	C₂³×D₄	1	1	yes	characteristic
149	Dic₁₃	—	2	16	no	—
150	Dic₁₃	—	2	8	no	—
151	C₂⁵	—	26	1	no	—
152	C₂⁵	D₅₂	1	1	yes	characteristic
153	C₂×#41	—	13	84	no	—
154	C₂×#41	—	13	28	no	—
155	C₂×#41	—	26	84	no	—
156	C₂×#41	—	26	42	no	—
157	C₂×#41	—	26	42	no	—
158	C₂²×Q₈	—	13	14	no	—
159	C₂²×Q₈	—	26	7	no	—
160	C₂²×D₄	—	13	14	no	—
161	C₂²×D₄	—	26	14	no	—
162	C₂²×D₄	—	26	14	no	—
163	C₂²×D₄	—	26	7	no	—
164	C₂³×C₄	—	26	16	no	—
165	C₂³×C₄	—	26	1	no	—
166	C₂³×C₄	—	26	14	no	—
167	C₂³×C₄	—	26	14	no	—
168	C₂³×C₄	—	26	14	no	—
169	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
170	C₂³×C₄	C₄×D₁₃	1	7	yes	—
171	C₂³×C₄	D₅₂	1	7	yes	—
172	C₂³×C₄	C₂²⋊D₁₃	1	7	yes	—
173	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₁₃	1	7	yes	—
175	C₂×M₁₆	—	2	42	no	—
176	C₂×M₁₆	C₂²×D₁₃	1	28	yes	complemented
177	C₂²×C₈	—	2	14	no	—
178	C₂×C₄²	—	13	28	no	—
179	C₂×C₄²	—	26	42	no	—
180	#104	—	13	64	no	—
181	C₂×C₁₃	—	2	1	no	—
182	C₂×C₁₃	—	2	7	no	—
183	C₂×C₁₃	C₂²×#104	1	14	yes	complemented
184	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
185	D₁₃	—	4	8	no	—
186	C₂⁴	—	2	8	no	—
187	C₂⁴	—	26	14	no	—
188	C₂⁴	—	52	8	no	—
189	C₂⁴	#6103	1	7	yes	—
190	C₂⁴	#6103	1	1	yes	characteristic
191	C₂⁴	C₂×D₅₂	1	7	yes	—
192	#41	—	13	56	no	—
193	#41	—	13	8	no	—
194	#41	—	26	168	no	—
195	#41	—	26	28	no	—
196	#41	—	26	28	no	—
197	C₂×Q₈	—	13	28	no	—
198	C₂×Q₈	—	26	42	no	—
199	C₂×D₄	—	13	28	no	—
200	C₂×D₄	—	26	84	no	—
201	C₂×D₄	—	26	28	no	—
202	C₂×D₄	—	26	42	no	—
203	C₂²×C₄	—	26	112	no	—
204	C₂²×C₄	—	26	14	no	—
205	C₂²×C₄	—	26	28	no	—
206	C₂²×C₄	—	26	84	no	—
207	C₂²×C₄	—	26	28	no	—
208	C₂²×C₄	#6103	1	7	yes	—
209	C₂²×C₄	#6103	1	7	yes	—
210	C₂²×C₄	#6103	1	42	yes	—
211	C₂²×C₄	C₂×C₄×D₁₃	1	7	yes	—
212	C₂²×C₄	C₂×C₄×D₁₃	1	21	yes	—
213	C₂²×C₄	C₂×D₅₂	1	21	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₃	1	7	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₁₃	1	21	yes	—
216	C₂²×C₄	C₂³×D₁₃	1	7	yes	—
217	M₁₆	—	2	28	no	—
218	M₁₆	C₂³×D₁₃	1	8	yes	complemented
219	C₂×C₈	—	2	28	no	—
220	C₄²	—	13	8	no	—
221	C₄²	—	26	28	no	—
222	C₁₃	C₂³×#104	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	28	no	—
224	C₂³	—	2	28	no	—
225	C₂³	—	26	28	no	—
226	C₂³	—	52	56	no	—
227	C₂³	#88065	1	8	yes	direct product factor
228	C₂³	C₂×#6103	1	21	yes	—
229	C₂³	C₂×#6103	1	7	yes	—
230	C₂³	C₂²×D₅₂	1	7	yes	—
231	Q₈	—	13	8	no	—
232	Q₈	—	26	28	no	—
233	D₄	—	13	8	no	—
234	D₄	—	26	56	no	—
235	D₄	—	26	8	no	—
236	D₄	—	26	28	no	—
237	C₂×C₄	—	26	112	no	—
238	C₂×C₄	—	26	28	no	—
239	C₂×C₄	—	26	8	no	—
240	C₂×C₄	—	26	56	no	—
241	C₂×C₄	—	26	8	no	—
242	C₂×C₄	C₂×#6103	1	21	yes	—
243	C₂×C₄	C₂×#6103	1	21	yes	—
244	C₂×C₄	C₂×#6103	1	42	yes	—
245	C₂×C₄	C₂²×C₄×D₁₃	1	7	yes	—
246	C₂×C₄	C₂²×C₄×D₁₃	1	7	yes	—
247	C₂×C₄	C₂²×D₅₂	1	7	yes	—
248	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
249	C₂×C₄	C₂²×C₂²⋊D₁₃	1	7	yes	—
250	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
251	C₈	—	2	8	no	—
252	C₂²	—	2	14	no	—
253	C₂²	—	2	42	no	—
254	C₂²	—	26	8	no	—
255	C₂²	—	52	56	no	—
256	C₂²	C₂×#88065	1	28	yes	direct product factor
257	C₂²	C₂²×#6103	1	7	yes	—
258	C₂²	C₂²×#6103	1	7	yes	—
259	C₂²	C₂³×D₅₂	1	1	yes	characteristic
260	C₄	—	26	16	no	—
261	C₄	—	26	8	no	—
262	C₄	C₂²×#6103	1	7	yes	—
263	C₄	C₂²×#6103	1	7	yes	—
264	C₄	C₂³×C₄×D₁₃	1	1	yes	characteristic
265	C₄	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	7	no	—
268	C₂	—	52	8	no	—
269	C₂	C₂²×#88065	1	14	yes	direct product factor
270	C₂	C₂³×#6103	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₁₃
commutator	C₄×C₁₃	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×M₁₆	C₂
socle	C₂⁴×C₁₃	C₂²⋊C₄
prime radical	C₂³×#6127	C₂
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂³×M₁₆	D₁₃
hypocenter	C₁₃	C₂³×#104
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄×Dic₁₃	C₂	semidirect
3. C₂²×#88065	C₂	direct (N × H)
4. C₂³×C₁₃×M₁₆	C₂	semidirect
10. C₂²×#6127	C₄	semidirect
12. C₂³×D₅₂	C₄	semidirect
14. C₂³×Dic₂₆	C₄	semidirect
16. C₂²×C₁₃×M₁₆	C₂²	semidirect
19. C₂²×C₄×Dic₁₃	C₂²	semidirect
20. C₂×#88065	C₂²	direct (N × H)
37. C₂×#6127	C₂×C₄	semidirect
40. C₂²×D₅₂	C₂×C₄	semidirect
44. C₂²×Dic₂₆	C₂×C₄	semidirect
46. C₂×C₁₃×M₁₆	C₂³	semidirect
49. C₂×C₄×Dic₁₃	C₂³	semidirect
50. #88065	C₂³	direct (N × H)
75. #6127	C₂²×C₄	semidirect
78. C₂×D₅₂	C₂²×C₄	semidirect
82. C₂×Dic₂₆	C₂²×C₄	semidirect
84. C₁₃×M₁₆	C₂⁴	semidirect
87. C₄×Dic₁₃	C₂⁴	semidirect
89. C₂³×M₁₆	D₁₃	semidirect
94. C₂³×C₁₃	#104	semidirect
115. D₅₂	C₂³×C₄	semidirect
119. Dic₂₆	C₂³×C₄	semidirect
132. C₂²×M₁₆	C₂×D₁₃	semidirect
139. C₂²×C₁₃	C₂×#104	semidirect
176. C₂×M₁₆	C₂²×D₁₃	semidirect
183. C₂×C₁₃	C₂²×#104	semidirect
218. M₁₆	C₂³×D₁₃	semidirect
222. C₁₃	C₂³×#104	semidirect
227. C₂³	#88065	direct (N × H)
256. C₂²	C₂×#88065	direct (N × H)
269. C₂	C₂²×#88065	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#6127	2
C₂³×C₄×Dic₁₃	2
C₂²×#88065	2
C₂³×C₁₃×M₁₆	2
C₂³×#104	13

Subgroups — C23×#88065 #16726506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88065 #16726506