Subgroups — C₂³×#88074 #16726380

Order 3328.

7902 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊Dic₁₃	C₂	1	2	yes	maximal, complemented
2	C₂²×#88074	C₂	1	28	yes	maximal, direct product factor
3	C₂⁴×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂⁶×C₁₃	C₄	1	1	yes	characteristic, complemented
5	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
6	C₂³×D₄×C₁₃	—	2	2	no	—
7	C₂³×D₄×C₁₃	—	2	14	no	—
8	C₂³×D₄×C₁₃	C₄	1	14	yes	complemented
9	C₂³×D₄×C₁₃	C₂²	1	14	yes	—
10	C₂⁴×C₄×C₁₃	C₄	1	1	yes	characteristic, complemented
11	C₂⁴×Dic₁₃	—	2	2	no	—
12	C₂²×C₂²⋊Dic₁₃	—	2	14	no	—
13	C₂²×C₂²⋊Dic₁₃	C₂²	1	28	yes	complemented
14	C₂×#88074	C₂²	1	112	yes	direct product factor
15	C₂⁵×C₁₃	—	2	7	no	—
16	C₂⁵×C₁₃	—	2	1	no	—
17	C₂⁵×C₁₃	—	4	8	no	—
18	C₂⁵×C₁₃	—	4	8	no	—
19	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂⁵×C₁₃	C₂×C₄	1	7	yes	—
21	C₂⁵×C₁₃	C₂×C₄	1	14	yes	complemented
22	C₂⁵×C₁₃	D₄	1	2	yes	—
23	C₂⁵×C₁₃	D₄	1	14	yes	—
24	C₂⁵×C₁₃	C₂³	1	7	yes	—
25	C₂²×D₄×C₁₃	—	2	28	no	—
26	C₂²×D₄×C₁₃	—	2	84	no	—
27	C₂²×D₄×C₁₃	—	2	28	no	—
28	C₂²×D₄×C₁₃	—	2	84	no	—
29	C₂²×D₄×C₁₃	C₂×C₄	1	84	yes	complemented
30	C₂²×D₄×C₁₃	C₂³	1	28	yes	—
31	C₂³×C₄×C₁₃	—	2	7	no	—
32	C₂³×C₄×C₁₃	—	2	1	no	—
33	C₂³×C₄×C₁₃	C₂×C₄	1	14	yes	complemented
34	C₂³×Dic₁₃	—	2	28	no	—
35	C₂³×Dic₁₃	—	4	16	no	—
36	C₂×C₂²⋊Dic₁₃	—	2	84	no	—
37	C₂×C₂²⋊Dic₁₃	C₂³	1	56	yes	complemented
38	#88074	C₂³	1	64	yes	direct product factor
39	C₂³×#99	—	13	1	no	maximal, Sylow
40	C₂⁴×C₁₃	—	2	16	no	—
41	C₂⁴×C₁₃	—	2	8	no	—
42	C₂⁴×C₁₃	—	2	42	no	—
43	C₂⁴×C₁₃	—	2	14	no	—
44	C₂⁴×C₁₃	—	4	28	no	—
45	C₂⁴×C₁₃	—	4	56	no	—
46	C₂⁴×C₁₃	—	4	28	no	—
47	C₂⁴×C₁₃	—	4	56	no	—
48	C₂⁴×C₁₃	—	4	56	no	—
49	C₂⁴×C₁₃	—	4	8	no	—
50	C₂⁴×C₁₃	C₂²⋊C₄	1	7	yes	—
51	C₂⁴×C₁₃	C₂²⋊C₄	1	14	yes	—
52	C₂⁴×C₁₃	C₂²⋊C₄	1	42	yes	—
53	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂⁴×C₁₃	C₂²×C₄	1	7	yes	—
55	C₂⁴×C₁₃	C₂²×C₄	1	21	yes	—
56	C₂⁴×C₁₃	C₂²×C₄	1	28	yes	complemented
57	C₂⁴×C₁₃	C₂×D₄	1	14	yes	—
58	C₂⁴×C₁₃	C₂×D₄	1	42	yes	—
59	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
60	C₂×D₄×C₁₃	—	2	168	no	—
61	C₂×D₄×C₁₃	—	2	168	no	—
62	C₂×D₄×C₁₃	—	2	56	no	—
63	C₂×D₄×C₁₃	—	2	56	no	—
64	C₂×D₄×C₁₃	C₂²×C₄	1	56	yes	complemented
65	C₂×D₄×C₁₃	C₂⁴	1	8	yes	—
66	C₂²×C₄×C₁₃	—	2	42	no	—
67	C₂²×C₄×C₁₃	—	2	14	no	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	28	yes	complemented
69	C₂²×Dic₁₃	—	2	56	no	—
70	C₂²×Dic₁₃	—	4	112	no	—
71	C₂²⋊Dic₁₃	—	2	56	no	—
72	C₂²⋊Dic₁₃	C₂⁴	1	16	yes	complemented
73	C₂⁴×D₄	D₁₃	1	1	yes	characteristic
74	C₂³×C₂²⋊C₄	—	13	2	no	—
75	C₂²×#99	—	13	28	no	—
76	C₂³×C₁₃	—	2	56	no	—
77	C₂³×C₁₃	—	2	28	no	—
78	C₂³×C₁₃	—	2	28	no	—
79	C₂³×C₁₃	—	2	56	no	—
80	C₂³×C₁₃	—	2	28	no	—
81	C₂³×C₁₃	—	2	28	no	—
82	C₂³×C₁₃	—	4	84	no	—
83	C₂³×C₁₃	—	4	42	no	—
84	C₂³×C₁₃	—	4	84	no	—
85	C₂³×C₁₃	—	4	14	no	—
86	C₂³×C₁₃	—	4	168	no	—
87	C₂³×C₁₃	—	4	56	no	—
88	C₂³×C₁₃	—	4	56	no	—
89	C₂³×C₁₃	#99	1	8	yes	complemented
90	C₂³×C₁₃	C₂×C₂²⋊C₄	1	21	yes	—
91	C₂³×C₁₃	C₂×C₂²⋊C₄	1	42	yes	—
92	C₂³×C₁₃	C₂×C₂²⋊C₄	1	42	yes	—
93	C₂³×C₁₃	C₂×C₂²⋊C₄	1	7	yes	—
94	C₂³×C₁₃	C₂³×C₄	1	7	yes	—
95	C₂³×C₁₃	C₂³×C₄	1	7	yes	—
96	C₂³×C₁₃	C₂³×C₄	1	8	yes	complemented
97	C₂³×C₁₃	C₂²×D₄	1	14	yes	—
98	C₂³×C₁₃	C₂²×D₄	1	14	yes	—
99	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
100	D₄×C₁₃	—	2	112	no	—
101	D₄×C₁₃	—	2	16	no	—
102	C₂×C₄×C₁₃	—	2	28	no	—
103	C₂×C₄×C₁₃	—	2	28	no	—
104	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	complemented
105	C₂×Dic₁₃	—	2	16	no	—
106	C₂×Dic₁₃	—	4	112	no	—
107	C₂⁶	Dic₁₃	1	1	yes	characteristic, complemented
108	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
109	C₂³×D₄	—	2	14	no	—
110	C₂³×D₄	—	2	2	no	—
111	C₂³×D₄	Dic₁₃	1	14	yes	complemented
112	C₂³×D₄	C₂×D₁₃	1	14	yes	—
113	C₂⁴×C₄	—	26	2	no	—
114	C₂⁴×C₄	Dic₁₃	1	1	yes	characteristic, complemented
115	C₂²×C₂²⋊C₄	—	13	28	no	—
116	C₂²×C₂²⋊C₄	—	26	14	no	—
117	C₂×#99	—	13	112	no	—
118	C₂²×C₁₃	—	2	28	no	—
119	C₂²×C₁₃	—	2	42	no	—
120	C₂²×C₁₃	—	2	14	no	—
121	C₂²×C₁₃	—	2	84	no	—
122	C₂²×C₁₃	—	2	8	no	—
123	C₂²×C₁₃	—	4	42	no	—
124	C₂²×C₁₃	—	4	7	no	—
125	C₂²×C₁₃	—	4	14	no	—
126	C₂²×C₁₃	—	4	1	no	—
127	C₂²×C₁₃	—	4	8	no	—
128	C₂²×C₁₃	—	4	56	no	—
129	C₂²×C₁₃	—	4	56	no	—
130	C₂²×C₁₃	C₂×#99	1	28	yes	complemented
131	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
132	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	14	yes	—
133	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	7	yes	—
134	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₂³×D₄	1	2	yes	—
136	C₄×C₁₃	—	2	8	no	—
137	Dic₁₃	—	4	16	no	—
138	C₂⁵	—	2	7	no	—
139	C₂⁵	—	2	1	no	—
140	C₂⁵	—	4	8	no	—
141	C₂⁵	—	4	8	no	—
142	C₂⁵	C₂×Dic₁₃	1	1	yes	characteristic
143	C₂⁵	C₂×Dic₁₃	1	7	yes	—
144	C₂⁵	C₂×Dic₁₃	1	14	yes	complemented
145	C₂⁵	C₂²⋊D₁₃	1	2	yes	—
146	C₂⁵	C₂²⋊D₁₃	1	14	yes	—
147	C₂⁵	C₂²×D₁₃	1	7	yes	—
148	C₂²×D₄	—	2	28	no	—
149	C₂²×D₄	—	2	84	no	—
150	C₂²×D₄	—	2	28	no	—
151	C₂²×D₄	—	2	84	no	—
152	C₂²×D₄	C₂×Dic₁₃	1	84	yes	complemented
153	C₂²×D₄	C₂²×D₁₃	1	28	yes	—
154	C₂³×C₄	—	2	7	no	—
155	C₂³×C₄	—	2	1	no	—
156	C₂³×C₄	—	26	28	no	—
157	C₂³×C₄	—	52	16	no	—
158	C₂³×C₄	C₂×Dic₁₃	1	14	yes	complemented
159	C₂×C₂²⋊C₄	—	13	56	no	—
160	C₂×C₂²⋊C₄	—	26	84	no	—
161	#99	—	13	64	no	—
162	C₂×C₁₃	—	2	2	no	—
163	C₂×C₁₃	—	2	7	no	—
164	C₂×C₁₃	—	2	1	no	—
165	C₂×C₁₃	—	2	14	no	—
166	C₂×C₁₃	—	4	8	no	—
167	C₂×C₁₃	C₂²×#99	1	14	yes	complemented
168	C₂×C₁₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic
169	C₂⁴	—	2	16	no	—
170	C₂⁴	—	2	8	no	—
171	C₂⁴	—	2	42	no	—
172	C₂⁴	—	2	14	no	—
173	C₂⁴	—	4	28	no	—
174	C₂⁴	—	4	56	no	—
175	C₂⁴	—	4	28	no	—
176	C₂⁴	—	4	56	no	—
177	C₂⁴	—	4	56	no	—
178	C₂⁴	—	4	8	no	—
179	C₂⁴	C₂²⋊Dic₁₃	1	7	yes	—
180	C₂⁴	C₂²⋊Dic₁₃	1	14	yes	—
181	C₂⁴	C₂²⋊Dic₁₃	1	42	yes	—
182	C₂⁴	C₂²⋊Dic₁₃	1	1	yes	characteristic
183	C₂⁴	C₂²×Dic₁₃	1	7	yes	—
184	C₂⁴	C₂²×Dic₁₃	1	21	yes	—
185	C₂⁴	C₂²×Dic₁₃	1	28	yes	complemented
186	C₂⁴	C₂×C₂²⋊D₁₃	1	14	yes	—
187	C₂⁴	C₂×C₂²⋊D₁₃	1	42	yes	—
188	C₂⁴	C₂³×D₁₃	1	7	yes	—
189	C₂×D₄	—	2	168	no	—
190	C₂×D₄	—	2	168	no	—
191	C₂×D₄	—	2	56	no	—
192	C₂×D₄	—	2	56	no	—
193	C₂×D₄	C₂²×Dic₁₃	1	56	yes	complemented
194	C₂×D₄	C₂³×D₁₃	1	8	yes	—
195	C₂²×C₄	—	2	42	no	—
196	C₂²×C₄	—	2	14	no	—
197	C₂²×C₄	—	26	56	no	—
198	C₂²×C₄	—	52	112	no	—
199	C₂²×C₄	C₂²×Dic₁₃	1	28	yes	complemented
200	C₂²⋊C₄	—	13	16	no	—
201	C₂²⋊C₄	—	26	56	no	—
202	C₁₃	C₂³×#99	1	1	yes	characteristic, complemented, Sylow
203	C₂³	—	2	56	no	—
204	C₂³	—	2	28	no	—
205	C₂³	—	2	28	no	—
206	C₂³	—	2	56	no	—
207	C₂³	—	2	28	no	—
208	C₂³	—	2	28	no	—
209	C₂³	—	4	84	no	—
210	C₂³	—	4	42	no	—
211	C₂³	—	4	84	no	—
212	C₂³	—	4	14	no	—
213	C₂³	—	4	56	no	—
214	C₂³	—	4	168	no	—
215	C₂³	—	4	56	no	—
216	C₂³	#88074	1	8	yes	direct product factor
217	C₂³	C₂×C₂²⋊Dic₁₃	1	21	yes	—
218	C₂³	C₂×C₂²⋊Dic₁₃	1	42	yes	—
219	C₂³	C₂×C₂²⋊Dic₁₃	1	42	yes	—
220	C₂³	C₂×C₂²⋊Dic₁₃	1	7	yes	—
221	C₂³	C₂³×Dic₁₃	1	7	yes	—
222	C₂³	C₂³×Dic₁₃	1	7	yes	—
223	C₂³	C₂³×Dic₁₃	1	8	yes	complemented
224	C₂³	C₂²×C₂²⋊D₁₃	1	14	yes	—
225	C₂³	C₂²×C₂²⋊D₁₃	1	14	yes	—
226	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
227	D₄	—	2	112	no	—
228	D₄	—	2	16	no	—
229	C₂×C₄	—	2	28	no	—
230	C₂×C₄	—	2	28	no	—
231	C₂×C₄	—	26	16	no	—
232	C₂×C₄	—	52	112	no	—
233	C₂×C₄	C₂³×Dic₁₃	1	8	yes	complemented
234	C₂²	—	2	28	no	—
235	C₂²	—	2	42	no	—
236	C₂²	—	2	14	no	—
237	C₂²	—	2	84	no	—
238	C₂²	—	2	8	no	—
239	C₂²	—	4	42	no	—
240	C₂²	—	4	7	no	—
241	C₂²	—	4	14	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	8	no	—
244	C₂²	—	4	56	no	—
245	C₂²	—	4	56	no	—
246	C₂²	C₂×#88074	1	28	yes	direct product factor
247	C₂²	C₂²×C₂²⋊Dic₁₃	1	7	yes	—
248	C₂²	C₂²×C₂²⋊Dic₁₃	1	14	yes	—
249	C₂²	C₂²×C₂²⋊Dic₁₃	1	7	yes	—
250	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
251	C₂²	C₂³×C₂²⋊D₁₃	1	2	yes	—
252	C₄	—	2	8	no	—
253	C₄	—	52	16	no	—
254	C₂	—	2	2	no	—
255	C₂	—	2	7	no	—
256	C₂	—	2	1	no	—
257	C₂	—	2	14	no	—
258	C₂	—	4	8	no	—
259	C₂	C₂²×#88074	1	14	yes	direct product factor
260	C₂	C₂³×C₂²⋊Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₁₃
Fitting, prime radical	C₂⁴×D₄×C₁₃	C₂
socle	C₂⁴×C₁₃	C₂²⋊C₄
cosocle	C₂³×C₁₃	C₂⁵
hypercenter	C₂⁴×D₄	D₁₃
hypocenter	C₁₃	C₂³×#99
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁶ ◁ C₂⁴×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₂²⋊Dic₁₃	C₂	semidirect
2. C₂²×#88074	C₂	direct (N × H)
4. C₂⁶×C₁₃	C₄	semidirect
8. C₂³×D₄×C₁₃	C₄	semidirect
10. C₂⁴×C₄×C₁₃	C₄	semidirect
13. C₂²×C₂²⋊Dic₁₃	C₂²	semidirect
14. C₂×#88074	C₂²	direct (N × H)
21. C₂⁵×C₁₃	C₂×C₄	semidirect
29. C₂²×D₄×C₁₃	C₂×C₄	semidirect
33. C₂³×C₄×C₁₃	C₂×C₄	semidirect
37. C₂×C₂²⋊Dic₁₃	C₂³	semidirect
38. #88074	C₂³	direct (N × H)
56. C₂⁴×C₁₃	C₂²×C₄	semidirect
64. C₂×D₄×C₁₃	C₂²×C₄	semidirect
68. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
72. C₂²⋊Dic₁₃	C₂⁴	semidirect
89. C₂³×C₁₃	#99	semidirect
96. C₂³×C₁₃	C₂³×C₄	semidirect
104. C₂×C₄×C₁₃	C₂³×C₄	semidirect
107. C₂⁶	Dic₁₃	semidirect
111. C₂³×D₄	Dic₁₃	semidirect
114. C₂⁴×C₄	Dic₁₃	semidirect
130. C₂²×C₁₃	C₂×#99	semidirect
144. C₂⁵	C₂×Dic₁₃	semidirect
152. C₂²×D₄	C₂×Dic₁₃	semidirect
158. C₂³×C₄	C₂×Dic₁₃	semidirect
167. C₂×C₁₃	C₂²×#99	semidirect
185. C₂⁴	C₂²×Dic₁₃	semidirect
193. C₂×D₄	C₂²×Dic₁₃	semidirect
199. C₂²×C₄	C₂²×Dic₁₃	semidirect
202. C₁₃	C₂³×#99	semidirect
216. C₂³	#88074	direct (N × H)
223. C₂³	C₂³×Dic₁₃	semidirect
233. C₂×C₄	C₂³×Dic₁₃	semidirect
246. C₂²	C₂×#88074	direct (N × H)
259. C₂	C₂²×#88074	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×C₂²⋊Dic₁₃	2
C₂²×#88074	2
C₂⁴×D₄×C₁₃	2
C₂³×#99	13

Subgroups — C23×#88074 #16726380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88074 #16726380