Subgroups — C₂²×#11719198 #16769638

Order 3328.

15230 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	2	yes	maximal, complemented
2	C₂×#11719198	C₂	1	48	yes	maximal, direct product factor
3	C₂²×#88140	C₂	1	4	yes	maximal, complemented
4	C₂²×#88184	C₂	1	4	yes	maximal, complemented
5	C₂²×#88136	C₂	1	4	yes	maximal, complemented
6	C₂³×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
8	C₂⁵×D₁₃	C₂²	1	2	yes	complemented
9	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	complemented
10	C₂³×C₂²⋊D₁₃	—	2	32	no	—
11	C₂³×C₂²⋊D₁₃	C₂²	1	4	yes	—
12	C₂³×C₂²⋊D₁₃	C₂²	1	4	yes	complemented
13	C₂³×C₂²⋊D₁₃	C₂²	1	48	yes	complemented
14	C₂⁴×Dic₁₃	C₂²	1	2	yes	complemented
15	C₂³×D₅₂	C₂²	1	4	yes	complemented
16	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	24	yes	complemented
18	C₂×#88184	C₂²	1	96	yes	complemented
19	C₂²×#6103	C₂²	1	8	yes	complemented
20	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	complemented
21	#11719198	C₂²	1	256	yes	direct product factor
22	C₂×#88140	C₂²	1	96	yes	complemented
23	C₂×#88136	C₂²	1	96	yes	complemented
24	C₂⁵×C₁₃	—	4	8	no	—
25	C₂⁵×C₁₃	D₄	1	16	yes	complemented
26	C₂⁵×C₁₃	C₂³	1	2	yes	—
27	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
28	C₂⁵×C₁₃	C₂³	1	12	yes	—
29	C₂⁴×D₁₃	—	2	16	no	—
30	C₂⁴×D₁₃	—	4	16	no	—
31	C₂⁴×D₁₃	C₂³	1	4	yes	—
32	C₂⁴×D₁₃	C₂³	1	24	yes	complemented
33	C₂³×C₄×C₁₃	—	4	8	no	—
34	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	24	yes	complemented
36	C₂²×C₂²⋊D₁₃	—	2	32	no	—
37	C₂²×C₂²⋊D₁₃	—	2	32	no	—
38	C₂²×C₂²⋊D₁₃	—	2	384	no	—
39	C₂²×C₂²⋊D₁₃	C₂³	1	48	yes	—
40	C₂²×C₂²⋊D₁₃	C₂³	1	48	yes	complemented
41	C₂²×C₂²⋊D₁₃	C₂³	1	128	yes	complemented
42	C₂³×Dic₁₃	—	2	16	no	—
43	C₂³×Dic₁₃	C₂³	1	4	yes	—
44	C₂³×Dic₁₃	C₂³	1	24	yes	complemented
45	C₂²×D₅₂	—	4	16	no	—
46	C₂²×D₅₂	C₂³	1	48	yes	complemented
47	C₂×C₁₃×C₂²⋊C₄	C₂³	1	48	yes	complemented
48	C₂×C₁₃×C₂²⋊C₄	C₂³	1	64	yes	complemented
49	#88184	C₂³	1	256	yes	complemented
50	C₂×#6103	C₂³	1	96	yes	complemented
51	C₂×C₄⋊Dic₁₃	C₂³	1	48	yes	complemented
52	#88140	C₂³	1	256	yes	complemented
53	#88136	C₂³	1	256	yes	complemented
54	C₂²×#534	—	13	1	no	maximal, Sylow
55	C₂⁴×C₁₃	—	2	8	no	—
56	C₂⁴×C₁₃	—	2	16	no	—
57	C₂⁴×C₁₃	—	4	64	no	—
58	C₂⁴×C₁₃	—	4	48	no	—
59	C₂⁴×C₁₃	C₂×D₄	1	16	yes	—
60	C₂⁴×C₁₃	C₂×D₄	1	8	yes	—
61	C₂⁴×C₁₃	C₂×D₄	1	96	yes	complemented
62	C₂⁴×C₁₃	C₂⁴	1	12	yes	—
63	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
64	C₂⁴×C₁₃	C₂⁴	1	16	yes	—
65	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
66	C₂³×D₁₃	—	2	16	no	—
67	C₂³×D₁₃	—	2	96	no	—
68	C₂³×D₁₃	—	2	32	no	—
69	C₂³×D₁₃	—	4	96	no	—
70	C₂³×D₁₃	—	4	128	no	—
71	C₂³×D₁₃	C₂⁴	1	24	yes	—
72	C₂³×D₁₃	C₂⁴	1	32	yes	complemented
73	C₂²×C₄×C₁₃	—	2	16	no	—
74	C₂²×C₄×C₁₃	—	4	48	no	—
75	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	32	yes	complemented
77	C₂×C₂²⋊D₁₃	—	2	192	no	—
78	C₂×C₂²⋊D₁₃	—	2	192	no	—
79	C₂×C₂²⋊D₁₃	—	2	512	no	—
80	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	—
81	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	complemented
82	C₂²×Dic₁₃	—	2	16	no	—
83	C₂²×Dic₁₃	—	2	96	no	—
84	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
85	C₂²×Dic₁₃	C₂⁴	1	32	yes	complemented
86	C₂×D₅₂	—	4	96	no	—
87	C₂×D₅₂	C₂⁴	1	64	yes	complemented
88	C₁₃×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
89	#6103	C₂⁴	1	128	yes	complemented
90	C₄⋊Dic₁₃	C₂⁴	1	64	yes	complemented
91	C₂⁴×D₄	—	13	2	no	—
92	C₂×#534	—	13	48	no	—
93	C₂²×#123	—	13	4	no	—
94	C₂²×#121	—	13	4	no	—
95	C₂²×#120	—	13	4	no	—
96	C₂³×C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
97	C₂³×C₁₃	—	2	64	no	—
98	C₂³×C₁₃	—	2	32	no	—
99	C₂³×C₁₃	—	2	24	no	—
100	C₂³×C₁₃	—	2	48	no	—
101	C₂³×C₁₃	—	4	192	no	—
102	C₂³×C₁₃	—	4	32	no	—
103	C₂³×C₁₃	C₂²×D₄	1	48	yes	—
104	C₂³×C₁₃	C₂²×D₄	1	24	yes	—
105	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
106	C₂³×C₁₃	C₂²×D₄	1	64	yes	complemented
107	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
108	C₂³×C₁₃	C₂⁵	1	8	yes	—
109	C₂³×C₁₃	C₂⁵	1	4	yes	—
110	C₂³×C₁₃	C₂⁵	1	3	yes	—
111	C₂²×D₁₃	—	2	48	no	—
112	C₂²×D₁₃	—	2	64	no	—
113	C₂²×D₁₃	—	2	96	no	—
114	C₂²×D₁₃	—	4	64	no	—
115	C₂²×D₁₃	—	4	384	no	—
116	C₂²×D₁₃	—	4	64	no	—
117	C₂²×D₁₃	C₂⁵	1	16	yes	—
118	C₂×C₄×C₁₃	—	2	48	no	—
119	C₂×C₄×C₁₃	—	4	32	no	—
120	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
121	C₂²⋊D₁₃	—	2	128	no	—
122	C₂²⋊D₁₃	—	2	128	no	—
123	C₂×Dic₁₃	—	2	48	no	—
124	C₂×Dic₁₃	—	2	64	no	—
125	C₂×Dic₁₃	C₂⁵	1	16	yes	—
126	D₅₂	—	4	64	no	—
127	C₂⁶	—	13	2	no	—
128	C₂⁶	C₂×D₁₃	1	1	yes	characteristic
129	C₂³×D₄	—	13	4	no	—
130	C₂³×D₄	—	13	4	no	—
131	C₂³×D₄	—	13	48	no	—
132	C₂³×D₄	—	13	4	no	—
133	C₂³×D₄	—	26	32	no	—
134	C₂⁴×C₄	—	13	2	no	—
135	C₂⁴×C₄	C₂×D₁₃	1	2	yes	complemented
136	#534	—	13	256	no	—
137	C₂×#123	—	13	96	no	—
138	C₂×#121	—	13	96	no	—
139	C₂×#120	—	13	96	no	—
140	C₂²×C₄⋊C₄	—	13	4	no	—
141	C₂²×C₂²⋊C₄	—	13	8	no	—
142	C₂²×C₂²⋊C₄	C₂×D₁₃	1	4	yes	complemented
143	C₂²×C₂²⋊C₄	C₂×D₁₃	1	24	yes	complemented
144	C₂²×C₁₃	—	2	8	no	—
145	C₂²×C₁₃	—	2	16	no	—
146	C₂²×C₁₃	—	2	96	no	—
147	C₂²×C₁₃	—	2	48	no	—
148	C₂²×C₁₃	—	4	64	no	—
149	C₂²×C₁₃	#534	1	16	yes	complemented
150	C₂²×C₁₃	C₂³×D₄	1	16	yes	—
151	C₂²×C₁₃	C₂³×D₄	1	8	yes	—
152	C₂²×C₁₃	C₂³×D₄	1	6	yes	—
153	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	12	yes	—
154	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
155	C₂×D₁₃	—	2	16	no	—
156	C₂×D₁₃	—	2	32	no	—
157	C₂×D₁₃	—	4	128	no	—
158	C₂×D₁₃	—	4	96	no	—
159	C₄×C₁₃	—	2	16	no	—
160	Dic₁₃	—	2	16	no	—
161	C₂⁵	—	4	8	no	—
162	C₂⁵	—	13	4	no	—
163	C₂⁵	—	13	24	no	—
164	C₂⁵	—	26	16	no	—
165	C₂⁵	—	52	16	no	—
166	C₂⁵	D₅₂	1	16	yes	complemented
167	C₂⁵	C₂²×D₁₃	1	2	yes	—
168	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
169	C₂⁵	C₂²×D₁₃	1	12	yes	—
170	C₂²×D₄	—	13	48	no	—
171	C₂²×D₄	—	13	48	no	—
172	C₂²×D₄	—	13	128	no	—
173	C₂²×D₄	—	13	48	no	—
174	C₂²×D₄	—	26	32	no	—
175	C₂²×D₄	—	26	32	no	—
176	C₂²×D₄	—	26	384	no	—
177	C₂²×D₄	—	52	16	no	—
178	C₂³×C₄	—	4	8	no	—
179	C₂³×C₄	—	13	24	no	—
180	C₂³×C₄	—	13	4	no	—
181	C₂³×C₄	—	26	16	no	—
182	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
183	C₂³×C₄	C₂²×D₁₃	1	24	yes	complemented
184	#123	—	13	256	no	—
185	#121	—	13	256	no	—
186	#120	—	13	256	no	—
187	C₂×C₄⋊C₄	—	13	48	no	—
188	C₂×C₂²⋊C₄	—	13	96	no	—
189	C₂×C₂²⋊C₄	C₂²×D₁₃	1	48	yes	complemented
190	C₂×C₂²⋊C₄	C₂²×D₁₃	1	64	yes	complemented
191	C₂×C₁₃	—	2	16	no	—
192	C₂×C₁₃	—	2	8	no	—
193	C₂×C₁₃	C₂×#534	1	12	yes	complemented
194	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
195	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	2	yes	—
196	D₁₃	—	4	16	no	—
197	C₂⁴	—	2	8	no	—
198	C₂⁴	—	2	16	no	—
199	C₂⁴	—	4	64	no	—
200	C₂⁴	—	4	48	no	—
201	C₂⁴	—	13	24	no	—
202	C₂⁴	—	13	32	no	—
203	C₂⁴	—	26	96	no	—
204	C₂⁴	—	26	16	no	—
205	C₂⁴	—	26	32	no	—
206	C₂⁴	—	52	96	no	—
207	C₂⁴	—	52	128	no	—
208	C₂⁴	C₂×D₅₂	1	16	yes	—
209	C₂⁴	C₂×D₅₂	1	8	yes	—
210	C₂⁴	C₂×D₅₂	1	96	yes	complemented
211	C₂⁴	C₂³×D₁₃	1	12	yes	—
212	C₂⁴	C₂³×D₁₃	1	6	yes	—
213	C₂⁴	C₂³×D₁₃	1	16	yes	—
214	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
215	C₂×D₄	—	13	64	no	—
216	C₂×D₄	—	13	64	no	—
217	C₂×D₄	—	13	64	no	—
218	C₂×D₄	—	26	192	no	—
219	C₂×D₄	—	26	192	no	—
220	C₂×D₄	—	26	512	no	—
221	C₂×D₄	—	52	96	no	—
222	C₂²×C₄	—	2	16	no	—
223	C₂²×C₄	—	4	48	no	—
224	C₂²×C₄	—	13	24	no	—
225	C₂²×C₄	—	13	32	no	—
226	C₂²×C₄	—	26	96	no	—
227	C₂²×C₄	—	26	16	no	—
228	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
229	C₂²×C₄	C₂³×D₁₃	1	32	yes	complemented
230	C₄⋊C₄	—	13	64	no	—
231	C₂²⋊C₄	—	13	128	no	—
232	C₂²⋊C₄	C₂³×D₁₃	1	64	yes	complemented
233	C₁₃	C₂²×#534	1	1	yes	characteristic, complemented, Sylow
234	C₂³	—	2	64	no	—
235	C₂³	—	2	32	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	2	48	no	—
238	C₂³	—	4	192	no	—
239	C₂³	—	4	32	no	—
240	C₂³	—	13	16	no	—
241	C₂³	—	26	64	no	—
242	C₂³	—	26	48	no	—
243	C₂³	—	26	96	no	—
244	C₂³	—	52	64	no	—
245	C₂³	—	52	384	no	—
246	C₂³	—	52	64	no	—
247	C₂³	C₂²×D₅₂	1	48	yes	—
248	C₂³	C₂²×D₅₂	1	24	yes	—
249	C₂³	C₂²×D₅₂	1	4	yes	—
250	C₂³	C₂²×D₅₂	1	64	yes	complemented
251	C₂³	#88251	1	8	yes	—
252	C₂³	C₂⁴×D₁₃	1	8	yes	—
253	C₂³	C₂⁴×D₁₃	1	4	yes	—
254	C₂³	C₂⁴×D₁₃	1	3	yes	—
255	D₄	—	26	128	no	—
256	D₄	—	26	128	no	—
257	D₄	—	52	64	no	—
258	C₂×C₄	—	2	48	no	—
259	C₂×C₄	—	4	32	no	—
260	C₂×C₄	—	13	16	no	—
261	C₂×C₄	—	26	64	no	—
262	C₂×C₄	—	26	48	no	—
263	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	16	no	—
266	C₂²	—	2	96	no	—
267	C₂²	—	2	48	no	—
268	C₂²	—	4	64	no	—
269	C₂²	—	26	16	no	—
270	C₂²	—	26	32	no	—
271	C₂²	—	52	128	no	—
272	C₂²	—	52	96	no	—
273	C₂²	#11719198	1	16	yes	direct product factor
274	C₂²	C₂³×D₅₂	1	16	yes	—
275	C₂²	C₂³×D₅₂	1	8	yes	—
276	C₂²	C₂³×D₅₂	1	6	yes	—
277	C₂²	C₂×#88251	1	12	yes	—
278	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
279	C₄	—	2	16	no	—
280	C₄	—	26	16	no	—
281	C₂	—	2	16	no	—
282	C₂	—	2	8	no	—
283	C₂	—	52	16	no	—
284	C₂	C₂×#11719198	1	12	yes	direct product factor
285	C₂	C₂⁴×D₅₂	1	1	yes	characteristic
286	C₂	C₂²×#88251	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂³×C₁₃×C₂²⋊C₄	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂³×C₂²⋊C₄	D₁₃
hypocenter	C₁₃	C₂²×#534
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₂²⋊D₁₃	C₂	semidirect
2. C₂×#11719198	C₂	direct (N × H)
3. C₂²×#88140	C₂	semidirect
4. C₂²×#88184	C₂	semidirect
5. C₂²×#88136	C₂	semidirect
6. C₂³×C₁₃×C₂²⋊C₄	C₂	semidirect
8. C₂⁵×D₁₃	C₂²	semidirect
9. C₂⁴×C₄×C₁₃	C₂²	semidirect
12. C₂³×C₂²⋊D₁₃	C₂²	semidirect
13. C₂³×C₂²⋊D₁₃	C₂²	semidirect
14. C₂⁴×Dic₁₃	C₂²	semidirect
15. C₂³×D₅₂	C₂²	semidirect
16. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
17. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
18. C₂×#88184	C₂²	semidirect
19. C₂²×#6103	C₂²	semidirect
20. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
21. #11719198	C₂²	direct (N × H)
22. C₂×#88140	C₂²	semidirect
23. C₂×#88136	C₂²	semidirect
25. C₂⁵×C₁₃	D₄	semidirect
32. C₂⁴×D₁₃	C₂³	semidirect
35. C₂³×C₄×C₁₃	C₂³	semidirect
40. C₂²×C₂²⋊D₁₃	C₂³	semidirect
41. C₂²×C₂²⋊D₁₃	C₂³	semidirect
44. C₂³×Dic₁₃	C₂³	semidirect
46. C₂²×D₅₂	C₂³	semidirect
47. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
48. C₂×C₁₃×C₂²⋊C₄	C₂³	semidirect
49. #88184	C₂³	semidirect
50. C₂×#6103	C₂³	semidirect
51. C₂×C₄⋊Dic₁₃	C₂³	semidirect
52. #88140	C₂³	semidirect
53. #88136	C₂³	semidirect
61. C₂⁴×C₁₃	C₂×D₄	semidirect
72. C₂³×D₁₃	C₂⁴	semidirect
76. C₂²×C₄×C₁₃	C₂⁴	semidirect
81. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
85. C₂²×Dic₁₃	C₂⁴	semidirect
87. C₂×D₅₂	C₂⁴	semidirect
88. C₁₃×C₂²⋊C₄	C₂⁴	semidirect
89. #6103	C₂⁴	semidirect
90. C₄⋊Dic₁₃	C₂⁴	semidirect
96. C₂³×C₂²⋊C₄	D₁₃	semidirect
106. C₂³×C₁₃	C₂²×D₄	semidirect
135. C₂⁴×C₄	C₂×D₁₃	semidirect
142. C₂²×C₂²⋊C₄	C₂×D₁₃	semidirect
143. C₂²×C₂²⋊C₄	C₂×D₁₃	semidirect
149. C₂²×C₁₃	#534	semidirect
166. C₂⁵	D₅₂	semidirect
183. C₂³×C₄	C₂²×D₁₃	semidirect
189. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
190. C₂×C₂²⋊C₄	C₂²×D₁₃	semidirect
193. C₂×C₁₃	C₂×#534	semidirect
210. C₂⁴	C₂×D₅₂	semidirect
229. C₂²×C₄	C₂³×D₁₃	semidirect
232. C₂²⋊C₄	C₂³×D₁₃	semidirect
233. C₁₃	C₂²×#534	semidirect
250. C₂³	C₂²×D₅₂	semidirect
273. C₂²	#11719198	direct (N × H)
284. C₂	C₂×#11719198	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
C₂×#11719198	2
C₂²×#88140	2
C₂²×#88184	2
C₂²×#88136	2
C₂³×C₁₃×C₂²⋊C₄	2
C₂²×#534	13

Subgroups — C22×#11719198 #16769638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719198 #16769638