Subgroups — C₂³×#88135 #16769481

Order 3328.

19998 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#88135	C₂	1	56	yes	maximal, direct product factor
5	C₂³×#6101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#6103	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
10	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
11	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×C₂²⋊D₁₃	C₄	1	32	yes	complemented
14	C₂³×C₂²⋊D₁₃	C₂²	1	28	yes	—
15	C₂⁴×Dic₁₃	—	2	16	no	—
16	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
18	C₂⁴×Dic₁₃	C₂²	1	1	yes	characteristic
19	C₂⁴×Dic₁₃	C₂²	1	28	yes	complemented
20	C₂³×C₄×D₁₃	—	2	16	no	—
21	C₂³×C₄×D₁₃	C₂²	1	28	yes	complemented
22	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	28	yes	complemented
23	C₂²×#6103	C₂²	1	28	yes	complemented
24	C₂²×#6101	C₂²	1	28	yes	complemented
25	C₂²×C₄×Dic₁₃	C₂²	1	28	yes	complemented
26	C₂×#88135	C₂²	1	448	yes	direct product factor
27	C₂⁵×C₁₃	—	2	8	no	—
28	C₂⁵×C₁₃	—	2	8	no	—
29	C₂⁵×C₁₃	C₂×C₄	1	16	yes	complemented
30	C₂⁵×C₁₃	C₂³	1	14	yes	—
31	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
32	C₂⁴×D₁₃	—	2	8	no	—
33	C₂⁴×D₁₃	—	2	8	no	—
34	C₂⁴×D₁₃	C₂×C₄	1	16	yes	complemented
35	C₂⁴×D₁₃	C₂³	1	14	yes	—
36	C₂³×C₄×C₁₃	—	2	8	no	—
37	C₂³×C₄×C₁₃	—	2	8	no	—
38	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
40	C₂²×C₂²⋊D₁₃	C₂×C₄	1	448	yes	complemented
41	C₂²×C₂²⋊D₁₃	C₂³	1	112	yes	—
42	C₂³×Dic₁₃	—	2	8	no	—
43	C₂³×Dic₁₃	—	2	224	no	—
44	C₂³×Dic₁₃	C₂×C₄	1	16	yes	complemented
45	C₂³×Dic₁₃	D₄	1	16	yes	complemented
46	C₂³×Dic₁₃	C₂³	1	14	yes	—
47	C₂³×Dic₁₃	C₂³	1	14	yes	—
48	C₂³×Dic₁₃	C₂³	1	14	yes	—
49	C₂³×Dic₁₃	C₂³	1	112	yes	complemented
50	C₂²×C₄×D₁₃	—	2	224	no	—
51	C₂²×C₄×D₁₃	C₂³	1	112	yes	complemented
52	C₂×C₁₃×C₂²⋊C₄	C₂³	1	112	yes	complemented
53	C₂×#6103	C₂³	1	112	yes	complemented
54	C₂×#6101	C₂³	1	112	yes	complemented
55	C₂×C₄×Dic₁₃	C₂³	1	112	yes	complemented
56	#88135	C₂³	1	512	yes	direct product factor
57	C₂³×C₄×D₄	—	13	1	no	maximal, Sylow
58	C₂⁴×C₁₃	—	2	128	no	—
59	C₂⁴×C₁₃	—	2	56	no	—
60	C₂⁴×C₁₃	—	2	56	no	—
61	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
62	C₂⁴×C₁₃	C₂²×C₄	1	112	yes	complemented
63	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
64	C₂⁴×C₁₃	#41	1	8	yes	—
65	C₂⁴×C₁₃	C₂⁴	1	28	yes	—
66	C₂⁴×C₁₃	C₂⁴	1	7	yes	—
67	C₂³×D₁₃	—	2	128	no	—
68	C₂³×D₁₃	—	2	56	no	—
69	C₂³×D₁₃	—	2	56	no	—
70	C₂³×D₁₃	C₂²×C₄	1	112	yes	complemented
71	C₂³×D₁₃	C₂⁴	1	28	yes	—
72	C₂²×C₄×C₁₃	—	2	56	no	—
73	C₂²×C₄×C₁₃	—	2	56	no	—
74	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
76	C₂×C₂²⋊D₁₃	C₂²×C₄	1	896	yes	complemented
77	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	—
78	C₂²×Dic₁₃	—	2	56	no	—
79	C₂²×Dic₁₃	—	2	448	no	—
80	C₂²×Dic₁₃	C₂²×C₄	1	112	yes	complemented
81	C₂²×Dic₁₃	C₂×D₄	1	112	yes	complemented
82	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
83	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
84	C₂²×Dic₁₃	C₂⁴	1	28	yes	—
85	C₂²×Dic₁₃	C₂⁴	1	64	yes	complemented
86	C₂×C₄×D₁₃	—	2	448	no	—
87	C₂×C₄×D₁₃	C₂⁴	1	64	yes	complemented
88	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
89	#6103	C₂⁴	1	64	yes	complemented
90	#6101	C₂⁴	1	64	yes	complemented
91	C₄×Dic₁₃	C₂⁴	1	64	yes	complemented
92	C₂⁴×D₄	—	13	1	no	—
93	C₂⁵×C₄	—	13	1	no	—
94	C₂⁵×C₄	—	13	1	no	—
95	C₂²×C₄×D₄	—	13	56	no	—
96	C₂³×C₄⋊C₄	—	13	1	no	—
97	C₂³×C₂²⋊C₄	—	13	1	no	—
98	C₂³×C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
99	C₂³×C₄²	—	13	1	no	—
100	C₂³×C₁₃	—	2	448	no	—
101	C₂³×C₁₃	—	2	56	no	—
102	C₂³×C₁₃	—	2	56	no	—
103	C₂³×C₁₃	C₄×D₄	1	64	yes	complemented
104	C₂³×C₁₃	C₂³×C₄	1	28	yes	—
105	C₂³×C₁₃	C₂³×C₄	1	112	yes	complemented
106	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
107	C₂³×C₁₃	C₂×#41	1	28	yes	—
108	C₂³×C₁₃	C₂⁵	1	8	yes	—
109	C₂³×C₁₃	C₂⁵	1	7	yes	—
110	C₂²×D₁₃	—	2	448	no	—
111	C₂²×D₁₃	—	2	56	no	—
112	C₂²×D₁₃	—	2	56	no	—
113	C₂²×D₁₃	C₂³×C₄	1	112	yes	complemented
114	C₂²×D₁₃	C₂⁵	1	8	yes	—
115	C₂×C₄×C₁₃	—	2	56	no	—
116	C₂×C₄×C₁₃	—	2	56	no	—
117	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
118	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
119	C₂²⋊D₁₃	C₂³×C₄	1	256	yes	complemented
120	C₂×Dic₁₃	—	2	56	no	—
121	C₂×Dic₁₃	—	2	128	no	—
122	C₂×Dic₁₃	C₂³×C₄	1	112	yes	complemented
123	C₂×Dic₁₃	C₂²×D₄	1	112	yes	complemented
124	C₂×Dic₁₃	C₂⁵	1	8	yes	—
125	C₂×Dic₁₃	C₂⁵	1	8	yes	—
126	C₂×Dic₁₃	C₂⁵	1	8	yes	—
127	C₄×D₁₃	—	2	128	no	—
128	C₂⁶	—	13	1	no	—
129	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
130	C₂³×D₄	—	13	32	no	—
131	C₂³×D₄	—	13	28	no	—
132	C₂⁴×C₄	—	13	1	no	—
133	C₂⁴×C₄	—	13	1	no	—
134	C₂⁴×C₄	—	13	28	no	—
135	C₂⁴×C₄	—	13	1	no	—
136	C₂⁴×C₄	—	13	28	no	—
137	C₂⁴×C₄	—	26	16	no	—
138	C₂⁴×C₄	—	26	16	no	—
139	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
140	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
141	C₂×C₄×D₄	—	13	448	no	—
142	C₂²×C₄⋊C₄	—	13	28	no	—
143	C₂²×C₂²⋊C₄	—	13	28	no	—
144	C₂²×C₂²⋊C₄	C₂×D₁₃	1	28	yes	complemented
145	C₂²×C₄²	—	13	28	no	—
146	C₂²×C₁₃	—	2	8	no	—
147	C₂²×C₁₃	—	2	8	no	—
148	C₂²×C₁₃	—	2	224	no	—
149	C₂²×C₁₃	C₂×C₄×D₄	1	112	yes	complemented
150	C₂²×C₁₃	C₂⁴×C₄	1	14	yes	—
151	C₂²×C₁₃	C₂⁴×C₄	1	16	yes	complemented
152	C₂²×C₁₃	C₂³×D₄	1	14	yes	—
153	C₂²×C₁₃	C₂²×#41	1	14	yes	—
154	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
155	C₂×D₁₃	—	2	8	no	—
156	C₂×D₁₃	—	2	8	no	—
157	C₂×D₁₃	—	2	224	no	—
158	C₂×D₁₃	C₂⁴×C₄	1	16	yes	complemented
159	C₄×C₁₃	—	2	8	no	—
160	C₄×C₁₃	—	2	8	no	—
161	Dic₁₃	—	2	8	no	—
162	Dic₁₃	C₂⁴×C₄	1	16	yes	complemented
163	Dic₁₃	C₂³×D₄	1	16	yes	complemented
164	C₂⁵	—	2	8	no	—
165	C₂⁵	—	2	8	no	—
166	C₂⁵	—	13	16	no	—
167	C₂⁵	—	13	14	no	—
168	C₂⁵	—	26	8	no	—
169	C₂⁵	—	26	8	no	—
170	C₂⁵	C₄×D₁₃	1	16	yes	complemented
171	C₂⁵	C₂²×D₁₃	1	14	yes	—
172	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×D₄	—	13	448	no	—
174	C₂²×D₄	—	13	112	no	—
175	C₂³×C₄	—	2	8	no	—
176	C₂³×C₄	—	2	8	no	—
177	C₂³×C₄	—	13	16	no	—
178	C₂³×C₄	—	13	16	no	—
179	C₂³×C₄	—	13	14	no	—
180	C₂³×C₄	—	13	14	no	—
181	C₂³×C₄	—	13	14	no	—
182	C₂³×C₄	—	13	112	no	—
183	C₂³×C₄	—	13	112	no	—
184	C₂³×C₄	—	26	224	no	—
185	C₂³×C₄	—	26	224	no	—
186	C₂³×C₄	—	26	8	no	—
187	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
188	C₂³×C₄	C₂²×D₁₃	1	14	yes	—
189	C₄×D₄	—	13	512	no	—
190	C₂×C₄⋊C₄	—	13	112	no	—
191	C₂×C₂²⋊C₄	—	13	112	no	—
192	C₂×C₂²⋊C₄	C₂²×D₁₃	1	112	yes	complemented
193	C₂×C₄²	—	13	112	no	—
194	C₂×C₁₃	—	2	16	no	—
195	C₂×C₁₃	C₂²×C₄×D₄	1	28	yes	complemented
196	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
197	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
198	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
199	D₁₃	—	2	16	no	—
200	C₂⁴	—	2	128	no	—
201	C₂⁴	—	2	56	no	—
202	C₂⁴	—	2	56	no	—
203	C₂⁴	—	13	112	no	—
204	C₂⁴	—	13	28	no	—
205	C₂⁴	—	26	128	no	—
206	C₂⁴	—	26	56	no	—
207	C₂⁴	—	26	56	no	—
208	C₂⁴	C₂×C₄×D₁₃	1	8	yes	—
209	C₂⁴	C₂×C₄×D₁₃	1	112	yes	complemented
210	C₂⁴	D₄×D₁₃	1	8	yes	—
211	C₂⁴	#6129	1	8	yes	—
212	C₂⁴	C₂³×D₁₃	1	28	yes	—
213	C₂⁴	C₂³×D₁₃	1	7	yes	—
214	C₂×D₄	—	13	896	no	—
215	C₂×D₄	—	13	64	no	—
216	C₂²×C₄	—	2	56	no	—
217	C₂²×C₄	—	2	56	no	—
218	C₂²×C₄	—	13	112	no	—
219	C₂²×C₄	—	13	112	no	—
220	C₂²×C₄	—	13	28	no	—
221	C₂²×C₄	—	13	28	no	—
222	C₂²×C₄	—	13	28	no	—
223	C₂²×C₄	—	13	64	no	—
224	C₂²×C₄	—	13	64	no	—
225	C₂²×C₄	—	26	448	no	—
226	C₂²×C₄	—	26	448	no	—
227	C₂²×C₄	—	26	56	no	—
228	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
229	C₂²×C₄	C₂³×D₁₃	1	28	yes	—
230	C₄⋊C₄	—	13	64	no	—
231	C₂²⋊C₄	—	13	64	no	—
232	C₂²⋊C₄	C₂³×D₁₃	1	64	yes	complemented
233	C₄²	—	13	64	no	—
234	C₁₃	C₂³×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	448	no	—
236	C₂³	—	2	56	no	—
237	C₂³	—	2	56	no	—
238	C₂³	—	13	112	no	—
239	C₂³	—	13	8	no	—
240	C₂³	—	26	448	no	—
241	C₂³	—	26	56	no	—
242	C₂³	—	26	56	no	—
243	C₂³	#88135	1	64	yes	direct product factor
244	C₂³	C₂²×C₄×D₁₃	1	28	yes	—
245	C₂³	C₂²×C₄×D₁₃	1	112	yes	complemented
246	C₂³	C₂×D₄×D₁₃	1	28	yes	—
247	C₂³	C₂×#6129	1	28	yes	—
248	C₂³	C₂⁴×D₁₃	1	8	yes	—
249	C₂³	C₂⁴×D₁₃	1	7	yes	—
250	D₄	—	13	256	no	—
251	C₂×C₄	—	2	56	no	—
252	C₂×C₄	—	2	56	no	—
253	C₂×C₄	—	13	112	no	—
254	C₂×C₄	—	13	112	no	—
255	C₂×C₄	—	13	8	no	—
256	C₂×C₄	—	13	8	no	—
257	C₂×C₄	—	13	8	no	—
258	C₂×C₄	—	26	128	no	—
259	C₂×C₄	—	26	128	no	—
260	C₂×C₄	—	26	56	no	—
261	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
262	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
263	C₂²	—	2	224	no	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	13	16	no	—
267	C₂²	—	26	224	no	—
268	C₂²	—	26	8	no	—
269	C₂²	—	26	8	no	—
270	C₂²	C₂×#88135	1	112	yes	direct product factor
271	C₂²	C₂³×C₄×D₁₃	1	14	yes	—
272	C₂²	C₂³×C₄×D₁₃	1	16	yes	complemented
273	C₂²	C₂²×D₄×D₁₃	1	14	yes	—
274	C₂²	C₂²×#6129	1	14	yes	—
275	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
276	C₄	—	2	8	no	—
277	C₄	—	2	8	no	—
278	C₄	—	13	16	no	—
279	C₄	—	13	16	no	—
280	C₄	—	26	8	no	—
281	C₂	—	2	16	no	—
282	C₂	—	26	16	no	—
283	C₂	C₂²×#88135	1	28	yes	direct product factor
284	C₂	C₂⁴×C₄×D₁₃	1	1	yes	characteristic
285	C₂	C₂³×D₄×D₁₃	1	1	yes	characteristic
286	C₂	C₂³×#6129	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂⁵×Dic₁₃	C₂	semidirect
3. C₂⁴×C₄×D₁₃	C₂	semidirect
4. C₂²×#88135	C₂	direct (N × H)
5. C₂³×#6101	C₂	semidirect
6. C₂³×#6103	C₂	semidirect
7. C₂³×C₄×Dic₁₃	C₂	semidirect
8. C₂³×C₁₃×C₂²⋊C₄	C₂	semidirect
13. C₂³×C₂²⋊D₁₃	C₄	semidirect
19. C₂⁴×Dic₁₃	C₂²	semidirect
21. C₂³×C₄×D₁₃	C₂²	semidirect
22. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
23. C₂²×#6103	C₂²	semidirect
24. C₂²×#6101	C₂²	semidirect
25. C₂²×C₄×Dic₁₃	C₂²	semidirect
26. C₂×#88135	C₂²	direct (N × H)
29. C₂⁵×C₁₃	C₂×C₄	semidirect
34. C₂⁴×D₁₃	C₂×C₄	semidirect
40. C₂²×C₂²⋊D₁₃	C₂×C₄	semidirect
44. C₂³×Dic₁₃	C₂×C₄	semidirect
45. C₂³×Dic₁₃	D₄	semidirect
49. C₂³×Dic₁₃	C₂³	semidirect
51. C₂²×C₄×D₁₃	C₂³	semidirect
52. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
53. C₂×#6103	C₂³	semidirect
54. C₂×#6101	C₂³	semidirect
55. C₂×C₄×Dic₁₃	C₂³	semidirect
56. #88135	C₂³	direct (N × H)
62. C₂⁴×C₁₃	C₂²×C₄	semidirect
70. C₂³×D₁₃	C₂²×C₄	semidirect
76. C₂×C₂²⋊D₁₃	C₂²×C₄	semidirect
80. C₂²×Dic₁₃	C₂²×C₄	semidirect
81. C₂²×Dic₁₃	C₂×D₄	semidirect
85. C₂²×Dic₁₃	C₂⁴	semidirect
87. C₂×C₄×D₁₃	C₂⁴	semidirect
88. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
89. #6103	C₂⁴	semidirect
90. #6101	C₂⁴	semidirect
91. C₄×Dic₁₃	C₂⁴	semidirect
98. C₂³×C₂²⋊C₄	D₁₃	semidirect
103. C₂³×C₁₃	C₄×D₄	semidirect
105. C₂³×C₁₃	C₂³×C₄	semidirect
113. C₂²×D₁₃	C₂³×C₄	semidirect
119. C₂²⋊D₁₃	C₂³×C₄	semidirect
122. C₂×Dic₁₃	C₂³×C₄	semidirect
123. C₂×Dic₁₃	C₂²×D₄	semidirect
144. C₂²×C₂²⋊C₄	C₂×D₁₃	semidirect
149. C₂²×C₁₃	C₂×C₄×D₄	semidirect
151. C₂²×C₁₃	C₂⁴×C₄	semidirect
158. C₂×D₁₃	C₂⁴×C₄	semidirect
162. Dic₁₃	C₂⁴×C₄	semidirect
163. Dic₁₃	C₂³×D₄	semidirect
170. C₂⁵	C₄×D₁₃	semidirect
192. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
195. C₂×C₁₃	C₂²×C₄×D₄	semidirect
209. C₂⁴	C₂×C₄×D₁₃	semidirect
232. C₂²⋊C₄	C₂³×D₁₃	semidirect
234. C₁₃	C₂³×C₄×D₄	semidirect
243. C₂³	#88135	direct (N × H)
245. C₂³	C₂²×C₄×D₁₃	semidirect
270. C₂²	C₂×#88135	direct (N × H)
272. C₂²	C₂³×C₄×D₁₃	semidirect
283. C₂	C₂²×#88135	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×C₂²⋊D₁₃	2
C₂⁵×Dic₁₃	2
C₂⁴×C₄×D₁₃	2
C₂²×#88135	2
C₂³×#6101	2
C₂³×#6103	2
C₂³×C₄×Dic₁₃	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂³×C₄×D₄	13

Subgroups — C23×#88135 #16769481

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88135 #16769481