Subgroups — C₂³×#88152 #16769591

Order 3328.

9486 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88152	C₂	1	56	yes	maximal, direct product factor
2	C₂³×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#6101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#6103	C₂	1	1	yes	maximal, characteristic
5	C₂³×#6103	C₂	1	1	yes	maximal, characteristic
6	C₂³×#6103	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
10	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	28	yes	complemented
17	C₂²×#6103	C₂²	1	28	yes	—
18	C₂²×#6103	C₂²	1	28	yes	—
19	C₂²×#6103	C₂²	1	28	yes	—
20	C₂²×C₄⋊Dic₁₃	C₂²	1	28	yes	complemented
21	C₂²×#6101	C₂²	1	28	yes	complemented
22	C₂²×C₄×Dic₁₃	C₂²	1	28	yes	complemented
23	C₂×#88152	C₂²	1	448	yes	direct product factor
24	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
25	C₂⁴×D₁₃	—	2	8	no	—
26	C₂⁴×D₁₃	—	2	8	no	—
27	C₂⁴×D₁₃	—	2	8	no	—
28	C₂⁴×D₁₃	C₂³	1	14	yes	—
29	C₂³×C₄×C₁₃	—	2	8	no	—
30	C₂³×C₄×C₁₃	—	2	8	no	—
31	C₂³×C₄×C₁₃	—	2	8	no	—
32	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
35	C₂³×Dic₁₃	—	2	8	no	—
36	C₂³×Dic₁₃	—	2	8	no	—
37	C₂³×Dic₁₃	—	2	8	no	—
38	C₂³×Dic₁₃	C₂³	1	14	yes	—
39	C₂³×Dic₁₃	C₂³	1	14	yes	—
40	C₂³×Dic₁₃	C₂³	1	14	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	112	yes	complemented
42	C₂×#6103	C₂³	1	112	yes	—
43	C₂×#6103	C₂³	1	112	yes	—
44	C₂×#6103	C₂³	1	112	yes	—
45	C₂×C₄⋊Dic₁₃	C₂³	1	112	yes	complemented
46	C₂×#6101	C₂³	1	112	yes	complemented
47	C₂×C₄×Dic₁₃	C₂³	1	112	yes	complemented
48	#88152	C₂³	1	512	yes	direct product factor
49	C₂³×#126	—	13	1	no	maximal, Sylow
50	C₂⁴×C₁₃	#41	1	8	yes	—
51	C₂⁴×C₁₃	#41	1	8	yes	—
52	C₂⁴×C₁₃	#41	1	8	yes	—
53	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
54	C₂³×D₁₃	—	2	56	no	—
55	C₂³×D₁₃	—	2	56	no	—
56	C₂³×D₁₃	—	2	56	no	—
57	C₂³×D₁₃	—	4	64	no	—
58	C₂³×D₁₃	C₂⁴	1	28	yes	—
59	C₂²×C₄×C₁₃	—	2	56	no	—
60	C₂²×C₄×C₁₃	—	2	56	no	—
61	C₂²×C₄×C₁₃	—	2	56	no	—
62	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
65	C₂²×Dic₁₃	—	2	56	no	—
66	C₂²×Dic₁₃	—	2	56	no	—
67	C₂²×Dic₁₃	—	2	56	no	—
68	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
69	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
70	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
71	C₁₃×C₄⋊C₄	C₂⁴	1	64	yes	complemented
72	#6103	C₂⁴	1	64	yes	—
73	#6103	C₂⁴	1	64	yes	—
74	#6103	C₂⁴	1	64	yes	—
75	C₄⋊Dic₁₃	C₂⁴	1	64	yes	complemented
76	#6101	C₂⁴	1	64	yes	complemented
77	C₄×Dic₁₃	C₂⁴	1	64	yes	complemented
78	C₂²×#126	—	13	56	no	—
79	C₂³×C₄⋊C₄	—	13	1	no	—
80	C₂³×C₄⋊C₄	—	13	1	no	—
81	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
82	C₂³×C₂²⋊C₄	—	13	1	no	—
83	C₂³×C₂²⋊C₄	—	13	1	no	—
84	C₂³×C₂²⋊C₄	—	13	1	no	—
85	C₂³×C₄²	—	13	1	no	—
86	C₂³×C₁₃	#126	1	64	yes	complemented
87	C₂³×C₁₃	C₂×#41	1	28	yes	—
88	C₂³×C₁₃	C₂×#41	1	28	yes	—
89	C₂³×C₁₃	C₂×#41	1	28	yes	—
90	C₂³×C₁₃	C₂⁵	1	7	yes	—
91	C₂²×D₁₃	—	2	56	no	—
92	C₂²×D₁₃	—	2	56	no	—
93	C₂²×D₁₃	—	2	56	no	—
94	C₂²×D₁₃	—	4	224	no	—
95	C₂²×D₁₃	C₂⁵	1	8	yes	—
96	C₂×C₄×C₁₃	—	2	56	no	—
97	C₂×C₄×C₁₃	—	2	56	no	—
98	C₂×C₄×C₁₃	—	2	56	no	—
99	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
100	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
101	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
102	C₂×Dic₁₃	—	2	56	no	—
103	C₂×Dic₁₃	—	2	56	no	—
104	C₂×Dic₁₃	—	2	56	no	—
105	C₂×Dic₁₃	C₂⁵	1	8	yes	—
106	C₂×Dic₁₃	C₂⁵	1	8	yes	—
107	C₂×Dic₁₃	C₂⁵	1	8	yes	—
108	C₂⁶	—	13	1	no	—
109	C₂⁴×C₄	—	13	1	no	—
110	C₂⁴×C₄	—	13	1	no	—
111	C₂⁴×C₄	—	13	1	no	—
112	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
113	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
114	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
115	C₂×#126	—	13	448	no	—
116	C₂²×C₄⋊C₄	—	13	28	no	—
117	C₂²×C₄⋊C₄	—	13	28	no	—
118	C₂²×C₄⋊C₄	C₂×D₁₃	1	28	yes	complemented
119	C₂²×C₂²⋊C₄	—	13	28	no	—
120	C₂²×C₂²⋊C₄	—	13	28	no	—
121	C₂²×C₂²⋊C₄	—	13	28	no	—
122	C₂²×C₄²	—	13	28	no	—
123	C₂²×C₁₃	C₂×#126	1	112	yes	complemented
124	C₂²×C₁₃	C₂²×#41	1	14	yes	—
125	C₂²×C₁₃	C₂²×#41	1	14	yes	—
126	C₂²×C₁₃	C₂²×#41	1	14	yes	—
127	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
128	C₂×D₁₃	—	2	8	no	—
129	C₂×D₁₃	—	2	8	no	—
130	C₂×D₁₃	—	2	8	no	—
131	C₂×D₁₃	—	4	112	no	—
132	C₄×C₁₃	—	2	8	no	—
133	C₄×C₁₃	—	2	8	no	—
134	C₄×C₁₃	—	2	8	no	—
135	Dic₁₃	—	2	8	no	—
136	Dic₁₃	—	2	8	no	—
137	Dic₁₃	—	2	8	no	—
138	C₂⁵	—	13	14	no	—
139	C₂⁵	—	26	8	no	—
140	C₂⁵	—	26	8	no	—
141	C₂⁵	—	26	8	no	—
142	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
143	C₂³×C₄	—	2	8	no	—
144	C₂³×C₄	—	2	8	no	—
145	C₂³×C₄	—	2	8	no	—
146	C₂³×C₄	—	13	14	no	—
147	C₂³×C₄	—	13	14	no	—
148	C₂³×C₄	—	13	14	no	—
149	C₂³×C₄	—	26	8	no	—
150	C₂³×C₄	—	26	8	no	—
151	C₂³×C₄	—	26	8	no	—
152	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
153	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
154	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
155	#126	—	13	512	no	—
156	C₂×C₄⋊C₄	—	13	112	no	—
157	C₂×C₄⋊C₄	—	13	112	no	—
158	C₂×C₄⋊C₄	C₂²×D₁₃	1	112	yes	complemented
159	C₂×C₂²⋊C₄	—	13	112	no	—
160	C₂×C₂²⋊C₄	—	13	112	no	—
161	C₂×C₂²⋊C₄	—	13	112	no	—
162	C₂×C₄²	—	13	112	no	—
163	C₂×C₁₃	C₂²×#126	1	28	yes	complemented
164	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
165	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
166	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
167	D₁₃	—	4	8	no	—
168	C₂⁴	—	13	28	no	—
169	C₂⁴	—	26	56	no	—
170	C₂⁴	—	26	56	no	—
171	C₂⁴	—	26	56	no	—
172	C₂⁴	—	52	64	no	—
173	C₂⁴	#6127	1	8	yes	—
174	C₂⁴	#6129	1	8	yes	—
175	C₂⁴	#6131	1	8	yes	—
176	C₂⁴	C₂³×D₁₃	1	7	yes	—
177	C₂²×C₄	—	2	56	no	—
178	C₂²×C₄	—	2	56	no	—
179	C₂²×C₄	—	2	56	no	—
180	C₂²×C₄	—	13	28	no	—
181	C₂²×C₄	—	13	28	no	—
182	C₂²×C₄	—	13	28	no	—
183	C₂²×C₄	—	26	56	no	—
184	C₂²×C₄	—	26	56	no	—
185	C₂²×C₄	—	26	56	no	—
186	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
187	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
188	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
189	C₄⋊C₄	—	13	64	no	—
190	C₄⋊C₄	—	13	64	no	—
191	C₄⋊C₄	C₂³×D₁₃	1	64	yes	complemented
192	C₂²⋊C₄	—	13	64	no	—
193	C₂²⋊C₄	—	13	64	no	—
194	C₂²⋊C₄	—	13	64	no	—
195	C₄²	—	13	64	no	—
196	C₁₃	C₂³×#126	1	1	yes	characteristic, complemented, Sylow
197	C₂³	—	13	8	no	—
198	C₂³	—	26	56	no	—
199	C₂³	—	26	56	no	—
200	C₂³	—	26	56	no	—
201	C₂³	—	52	224	no	—
202	C₂³	#88152	1	64	yes	direct product factor
203	C₂³	C₂×#6127	1	28	yes	—
204	C₂³	C₂×#6129	1	28	yes	—
205	C₂³	C₂×#6131	1	28	yes	—
206	C₂³	C₂⁴×D₁₃	1	7	yes	—
207	C₂×C₄	—	2	56	no	—
208	C₂×C₄	—	2	56	no	—
209	C₂×C₄	—	2	56	no	—
210	C₂×C₄	—	13	8	no	—
211	C₂×C₄	—	13	8	no	—
212	C₂×C₄	—	13	8	no	—
213	C₂×C₄	—	26	56	no	—
214	C₂×C₄	—	26	56	no	—
215	C₂×C₄	—	26	56	no	—
216	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
217	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
218	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
219	C₂²	—	26	8	no	—
220	C₂²	—	26	8	no	—
221	C₂²	—	26	8	no	—
222	C₂²	—	52	112	no	—
223	C₂²	C₂×#88152	1	112	yes	direct product factor
224	C₂²	C₂²×#6127	1	14	yes	—
225	C₂²	C₂²×#6129	1	14	yes	—
226	C₂²	C₂²×#6131	1	14	yes	—
227	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
228	C₄	—	2	8	no	—
229	C₄	—	2	8	no	—
230	C₄	—	2	8	no	—
231	C₄	—	26	8	no	—
232	C₄	—	26	8	no	—
233	C₄	—	26	8	no	—
234	C₂	—	52	8	no	—
235	C₂	C₂²×#88152	1	28	yes	direct product factor
236	C₂	C₂³×#6129	1	1	yes	characteristic
237	C₂	C₂³×#6127	1	1	yes	characteristic
238	C₂	C₂³×#6131	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle	C₂⁵×C₁₃	C₂³
prime radical	C₂⁵×D₁₃	C₂²
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂³×#126
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#88152	C₂	direct (N × H)
2. C₂³×C₄⋊Dic₁₃	C₂	semidirect
3. C₂³×#6101	C₂	semidirect
7. C₂³×C₄×Dic₁₃	C₂	semidirect
8. C₂³×C₁₃×C₄⋊C₄	C₂	semidirect
16. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
20. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
21. C₂²×#6101	C₂²	semidirect
22. C₂²×C₄×Dic₁₃	C₂²	semidirect
23. C₂×#88152	C₂²	direct (N × H)
41. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
45. C₂×C₄⋊Dic₁₃	C₂³	semidirect
46. C₂×#6101	C₂³	semidirect
47. C₂×C₄×Dic₁₃	C₂³	semidirect
48. #88152	C₂³	direct (N × H)
71. C₁₃×C₄⋊C₄	C₂⁴	semidirect
75. C₄⋊Dic₁₃	C₂⁴	semidirect
76. #6101	C₂⁴	semidirect
77. C₄×Dic₁₃	C₂⁴	semidirect
81. C₂³×C₄⋊C₄	D₁₃	semidirect
86. C₂³×C₁₃	#126	semidirect
118. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
123. C₂²×C₁₃	C₂×#126	semidirect
158. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
163. C₂×C₁₃	C₂²×#126	semidirect
191. C₄⋊C₄	C₂³×D₁₃	semidirect
196. C₁₃	C₂³×#126	semidirect
202. C₂³	#88152	direct (N × H)
223. C₂²	C₂×#88152	direct (N × H)
235. C₂	C₂²×#88152	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88152	2
C₂³×C₄⋊Dic₁₃	2
C₂³×#6101	2
C₂³×#6103	2
C₂³×C₄×Dic₁₃	2
C₂³×C₁₃×C₄⋊C₄	2
C₂³×#126	13

Subgroups — C23×#88152 #16769591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88152 #16769591