Subgroups — C₂²×#11719226 #16769499

Order 3328.

14590 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₅₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#11719226	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄×D₅₂	C₂	1	4	yes	maximal, complemented
4	C₂²×#88147	C₂	1	4	yes	maximal, complemented
5	C₂²×#88180	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₂²⋊C₄×D₁₃	C₂	1	4	yes	maximal, complemented
7	C₂²×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂⁵×D₁₃	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×D₅₂	C₄	1	32	yes	complemented
11	C₂³×D₅₂	C₂²	1	2	yes	—
12	C₂³×D₅₂	C₂²	1	2	yes	—
13	C₂³×D₅₂	C₂²	1	24	yes	—
14	C₂³×C₄×D₁₃	C₂²	1	8	yes	complemented
15	C₂×C₁₃×#117	C₂²	1	24	yes	complemented
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	complemented
17	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₂²×C₄²×C₁₃	C₂²	1	2	yes	complemented
19	C₂²×C₂²⋊Dic₁₃	C₂²	1	2	yes	—
20	C₂²×#6103	C₂²	1	8	yes	complemented
21	C₂×#88180	C₂²	1	24	yes	complemented
22	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	complemented
23	C₂²×C₄×Dic₁₃	C₂²	1	2	yes	complemented
24	#11719226	C₂²	1	256	yes	direct product factor
25	C₂×#88147	C₂²	1	96	yes	complemented
26	C₂×C₂²⋊C₄×D₁₃	C₂²	1	96	yes	complemented
27	C₂×C₄×D₅₂	C₂²	1	96	yes	complemented
28	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
29	C₂⁴×D₁₃	—	4	16	no	—
30	C₂⁴×D₁₃	C₂×C₄	1	32	yes	complemented
31	C₂⁴×D₁₃	C₂³	1	4	yes	—
32	C₂⁴×D₁₃	C₂³	1	24	yes	—
33	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	complemented
34	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
38	C₂³×Dic₁₃	C₂³	1	4	yes	—
39	C₂²×D₅₂	C₂×C₄	1	32	yes	complemented
40	C₂²×D₅₂	C₂×C₄	1	32	yes	complemented
41	C₂²×D₅₂	C₂×C₄	1	384	yes	complemented
42	C₂²×D₅₂	C₂³	1	24	yes	—
43	C₂²×D₅₂	C₂³	1	24	yes	—
44	C₂²×D₅₂	C₂³	1	64	yes	—
45	C₂²×C₄×D₁₃	—	4	32	no	—
46	C₂²×C₄×D₁₃	C₂³	1	96	yes	complemented
47	C₁₃×#117	C₂³	1	64	yes	complemented
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	complemented
49	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	24	yes	complemented
51	C₂×C₂²⋊Dic₁₃	C₂³	1	24	yes	—
52	C₂×#6103	C₂³	1	96	yes	complemented
53	#88180	C₂³	1	64	yes	complemented
54	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	complemented
55	C₂×C₄×Dic₁₃	C₂³	1	24	yes	complemented
56	#88147	C₂³	1	256	yes	complemented
57	C₂²⋊C₄×D₁₃	C₂³	1	256	yes	complemented
58	C₄×D₅₂	C₂³	1	256	yes	complemented
59	C₂²×#518	—	13	1	no	maximal, Sylow
60	C₂⁴×C₁₃	—	2	8	no	—
61	C₂⁴×C₁₃	C₂²×C₄	1	8	yes	—
62	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
63	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂³×D₁₃	—	2	32	no	—
65	C₂³×D₁₃	—	4	128	no	—
66	C₂³×D₁₃	—	4	96	no	—
67	C₂³×D₁₃	C₂²×C₄	1	32	yes	—
68	C₂³×D₁₃	C₂²×C₄	1	192	yes	complemented
69	C₂³×D₁₃	C₂⁴	1	24	yes	—
70	C₂³×D₁₃	C₂⁴	1	32	yes	—
71	C₂²×C₄×C₁₃	—	2	16	no	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	96	yes	complemented
75	C₂²×C₄×C₁₃	C₂⁴	1	24	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
79	C₂²×Dic₁₃	—	2	16	no	—
80	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
81	C₂×D₅₂	C₂²×C₄	1	192	yes	complemented
82	C₂×D₅₂	C₂²×C₄	1	192	yes	complemented
83	C₂×D₅₂	C₂²×C₄	1	512	yes	complemented
84	C₂×D₅₂	C₂⁴	1	32	yes	—
85	C₂×D₅₂	C₂⁴	1	32	yes	—
86	C₂×C₄×D₁₃	—	4	192	no	—
87	C₂×C₄×D₁₃	C₂⁴	1	128	yes	complemented
88	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	complemented
89	C₁₃×C₂²⋊C₄	C₂⁴	1	32	yes	—
90	C₄²×C₁₃	C₂⁴	1	32	yes	complemented
91	C₂²⋊Dic₁₃	C₂⁴	1	32	yes	—
92	#6103	C₂⁴	1	128	yes	complemented
93	C₄⋊Dic₁₃	C₂⁴	1	32	yes	complemented
94	C₄×Dic₁₃	C₂⁴	1	32	yes	complemented
95	C₂⁴×D₄	—	13	1	no	—
96	C₂×#518	—	13	48	no	—
97	C₂²×C₄×D₄	—	13	4	no	—
98	C₂²×C₄×D₄	—	13	4	no	—
99	C₂²×#117	—	13	1	no	—
100	C₂²×#117	D₁₃	1	1	yes	characteristic, complemented
101	C₂³×C₂²⋊C₄	—	13	4	no	—
102	C₂³×C₁₃	—	2	32	no	—
103	C₂³×C₁₃	—	2	24	no	—
104	C₂³×C₁₃	C₂³×C₄	1	24	yes	—
105	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
106	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
107	C₂³×C₁₃	C₂⁵	1	4	yes	—
108	C₂³×C₁₃	C₂⁵	1	3	yes	—
109	C₂²×D₁₃	—	2	128	no	—
110	C₂²×D₁₃	—	2	96	no	—
111	C₂²×D₁₃	—	4	384	no	—
112	C₂²×D₁₃	—	4	64	no	—
113	C₂²×D₁₃	C₂³×C₄	1	96	yes	—
114	C₂²×D₁₃	C₂³×C₄	1	128	yes	complemented
115	C₂²×D₁₃	C₂⁵	1	16	yes	—
116	C₂×C₄×C₁₃	—	2	48	no	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	24	yes	—
118	C₂×C₄×C₁₃	C₂³×C₄	1	24	yes	—
119	C₂×C₄×C₁₃	C₂³×C₄	1	64	yes	complemented
120	C₂×C₄×C₁₃	C₂⁵	1	16	yes	—
121	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
122	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
123	C₂×Dic₁₃	—	2	48	no	—
124	C₂×Dic₁₃	C₂⁵	1	16	yes	—
125	D₅₂	C₂³×C₄	1	128	yes	complemented
126	D₅₂	C₂³×C₄	1	128	yes	complemented
127	C₄×D₁₃	—	4	128	no	—
128	C₂⁶	—	13	2	no	—
129	C₂³×D₄	—	13	32	no	—
130	C₂³×D₄	—	13	2	no	—
131	C₂³×D₄	—	13	2	no	—
132	C₂³×D₄	—	13	24	no	—
133	C₂⁴×C₄	—	13	8	no	—
134	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
135	#518	—	13	256	no	—
136	C₂×C₄×D₄	—	13	96	no	—
137	C₂×C₄×D₄	—	13	96	no	—
138	C₂×#117	—	13	24	no	—
139	C₂×#117	C₂×D₁₃	1	24	yes	complemented
140	C₂²×C₄⋊C₄	—	13	2	no	—
141	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	complemented
142	C₂²×C₂²⋊C₄	—	13	2	no	—
143	C₂²×C₂²⋊C₄	—	13	8	no	—
144	C₂²×C₂²⋊C₄	—	13	96	no	—
145	C₂²×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
146	C₂²×C₄²	—	13	2	no	—
147	C₂²×C₄²	C₂×D₁₃	1	2	yes	complemented
148	C₂²×C₁₃	—	2	8	no	—
149	C₂²×C₁₃	—	2	48	no	—
150	C₂²×C₁₃	#518	1	16	yes	complemented
151	C₂²×C₁₃	C₂⁴×C₄	1	8	yes	—
152	C₂²×C₁₃	C₂⁴×C₄	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	32	no	—
156	C₂×D₁₃	—	2	192	no	—
157	C₂×D₁₃	—	4	128	no	—
158	C₂×D₁₃	C₂⁴×C₄	1	32	yes	—
159	C₄×C₁₃	—	2	16	no	—
160	C₄×C₁₃	C₂⁴×C₄	1	8	yes	—
161	C₄×C₁₃	C₂⁴×C₄	1	8	yes	—
162	Dic₁₃	—	2	16	no	—
163	C₂⁵	—	13	4	no	—
164	C₂⁵	—	13	32	no	—
165	C₂⁵	—	13	24	no	—
166	C₂⁵	—	52	16	no	—
167	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
168	C₂²×D₄	—	13	32	no	—
169	C₂²×D₄	—	13	32	no	—
170	C₂²×D₄	—	13	384	no	—
171	C₂²×D₄	—	13	24	no	—
172	C₂²×D₄	—	13	24	no	—
173	C₂²×D₄	—	13	64	no	—
174	C₂³×C₄	—	13	4	no	—
175	C₂³×C₄	—	13	96	no	—
176	C₂³×C₄	—	52	32	no	—
177	C₂³×C₄	C₄×D₁₃	1	16	yes	complemented
178	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
179	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
180	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
181	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
182	C₄×D₄	—	13	256	no	—
183	C₄×D₄	—	13	256	no	—
184	#117	—	13	64	no	—
185	#117	C₂²×D₁₃	1	64	yes	complemented
186	C₂×C₄⋊C₄	—	13	24	no	—
187	C₂×C₄⋊C₄	C₂²×D₁₃	1	24	yes	complemented
188	C₂×C₂²⋊C₄	—	13	24	no	—
189	C₂×C₂²⋊C₄	—	13	96	no	—
190	C₂×C₂²⋊C₄	—	13	256	no	—
191	C₂×C₂²⋊C₄	C₂²×D₁₃	1	24	yes	—
192	C₂×C₄²	—	13	24	no	—
193	C₂×C₄²	C₂²×D₁₃	1	24	yes	complemented
194	C₂×C₁₃	—	2	8	no	—
195	C₂×C₁₃	C₂×#518	1	12	yes	complemented
196	C₂×C₁₃	C₂⁵×C₄	1	1	yes	characteristic
197	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	2	yes	—
198	D₁₃	—	2	32	no	—
199	C₂⁴	—	2	8	no	—
200	C₂⁴	—	13	32	no	—
201	C₂⁴	—	13	24	no	—
202	C₂⁴	—	13	192	no	—
203	C₂⁴	—	13	32	no	—
204	C₂⁴	—	26	32	no	—
205	C₂⁴	—	52	128	no	—
206	C₂⁴	—	52	96	no	—
207	C₂⁴	C₂×C₄×D₁₃	1	8	yes	—
208	C₂⁴	C₂³×D₁₃	1	6	yes	—
209	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
210	C₂×D₄	—	13	192	no	—
211	C₂×D₄	—	13	192	no	—
212	C₂×D₄	—	13	512	no	—
213	C₂×D₄	—	13	32	no	—
214	C₂×D₄	—	13	32	no	—
215	C₂²×C₄	—	2	16	no	—
216	C₂²×C₄	—	13	24	no	—
217	C₂²×C₄	—	13	128	no	—
218	C₂²×C₄	—	26	16	no	—
219	C₂²×C₄	—	52	192	no	—
220	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
221	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
222	C₂²×C₄	C₂×C₄×D₁₃	1	96	yes	complemented
223	C₂²×C₄	C₂³×D₁₃	1	24	yes	—
224	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
225	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
226	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
227	C₄⋊C₄	—	13	32	no	—
228	C₄⋊C₄	C₂³×D₁₃	1	32	yes	complemented
229	C₂²⋊C₄	—	13	32	no	—
230	C₂²⋊C₄	—	13	128	no	—
231	C₂²⋊C₄	C₂³×D₁₃	1	32	yes	—
232	C₄²	—	13	32	no	—
233	C₄²	C₂³×D₁₃	1	32	yes	complemented
234	C₁₃	C₂²×#518	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	2	32	no	—
236	C₂³	—	2	24	no	—
237	C₂³	—	13	96	no	—
238	C₂³	—	13	16	no	—
239	C₂³	—	13	128	no	—
240	C₂³	—	26	128	no	—
241	C₂³	—	26	96	no	—
242	C₂³	—	52	384	no	—
243	C₂³	—	52	64	no	—
244	C₂³	C₂²×C₄×D₁₃	1	24	yes	—
245	C₂³	C₂²×C₄×D₁₃	1	4	yes	—
246	C₂³	#88256	1	8	yes	—
247	C₂³	C₂⁴×D₁₃	1	4	yes	—
248	C₂³	C₂⁴×D₁₃	1	3	yes	—
249	D₄	—	13	128	no	—
250	D₄	—	13	128	no	—
251	C₂×C₄	—	2	48	no	—
252	C₂×C₄	—	13	16	no	—
253	C₂×C₄	—	26	48	no	—
254	C₂×C₄	—	52	128	no	—
255	C₂×C₄	C₂²×C₄×D₁₃	1	24	yes	—
256	C₂×C₄	C₂²×C₄×D₁₃	1	24	yes	—
257	C₂×C₄	C₂²×C₄×D₁₃	1	64	yes	complemented
258	C₂×C₄	C₂⁴×D₁₃	1	16	yes	—
259	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
260	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
261	C₂²	—	2	48	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	13	32	no	—
264	C₂²	—	26	192	no	—
265	C₂²	—	26	32	no	—
266	C₂²	—	52	128	no	—
267	C₂²	#11719226	1	16	yes	direct product factor
268	C₂²	C₂³×C₄×D₁₃	1	8	yes	—
269	C₂²	C₂³×C₄×D₁₃	1	6	yes	—
270	C₂²	C₂×#88256	1	12	yes	—
271	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
272	C₄	—	2	16	no	—
273	C₄	—	26	16	no	—
274	C₄	C₂³×C₄×D₁₃	1	8	yes	—
275	C₄	C₂³×C₄×D₁₃	1	8	yes	—
276	C₂	—	2	8	no	—
277	C₂	—	26	32	no	—
278	C₂	C₂×#11719226	1	12	yes	direct product factor
279	C₂	C₂⁴×C₄×D₁₃	1	1	yes	characteristic
280	C₂	C₂²×#88256	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator	C₂×C₁₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#117	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁴×D₅₂	C₂
cosocle	C₂²×C₁₃	C₂⁶
hypercenter	C₂²×#117	D₁₃
hypocenter	C₁₃	C₂²×#518
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. C₂×#11719226	C₂	direct (N × H)
3. C₂²×C₄×D₅₂	C₂	semidirect
4. C₂²×#88147	C₂	semidirect
5. C₂²×#88180	C₂	semidirect
6. C₂²×C₂²⋊C₄×D₁₃	C₂	semidirect
7. C₂²×C₁₃×#117	C₂	semidirect
10. C₂³×D₅₂	C₄	semidirect
14. C₂³×C₄×D₁₃	C₂²	semidirect
15. C₂×C₁₃×#117	C₂²	semidirect
16. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₄²×C₁₃	C₂²	semidirect
20. C₂²×#6103	C₂²	semidirect
21. C₂×#88180	C₂²	semidirect
22. C₂²×C₄⋊Dic₁₃	C₂²	semidirect
23. C₂²×C₄×Dic₁₃	C₂²	semidirect
24. #11719226	C₂²	direct (N × H)
25. C₂×#88147	C₂²	semidirect
26. C₂×C₂²⋊C₄×D₁₃	C₂²	semidirect
27. C₂×C₄×D₅₂	C₂²	semidirect
30. C₂⁴×D₁₃	C₂×C₄	semidirect
33. C₂³×C₄×C₁₃	C₂×C₄	semidirect
39. C₂²×D₅₂	C₂×C₄	semidirect
40. C₂²×D₅₂	C₂×C₄	semidirect
41. C₂²×D₅₂	C₂×C₄	semidirect
46. C₂²×C₄×D₁₃	C₂³	semidirect
47. C₁₃×#117	C₂³	semidirect
48. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
50. C₂×C₄²×C₁₃	C₂³	semidirect
52. C₂×#6103	C₂³	semidirect
53. #88180	C₂³	semidirect
54. C₂×C₄⋊Dic₁₃	C₂³	semidirect
55. C₂×C₄×Dic₁₃	C₂³	semidirect
56. #88147	C₂³	semidirect
57. C₂²⋊C₄×D₁₃	C₂³	semidirect
58. C₄×D₅₂	C₂³	semidirect
68. C₂³×D₁₃	C₂²×C₄	semidirect
74. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
81. C₂×D₅₂	C₂²×C₄	semidirect
82. C₂×D₅₂	C₂²×C₄	semidirect
83. C₂×D₅₂	C₂²×C₄	semidirect
87. C₂×C₄×D₁₃	C₂⁴	semidirect
88. C₁₃×C₄⋊C₄	C₂⁴	semidirect
90. C₄²×C₁₃	C₂⁴	semidirect
92. #6103	C₂⁴	semidirect
93. C₄⋊Dic₁₃	C₂⁴	semidirect
94. C₄×Dic₁₃	C₂⁴	semidirect
100. C₂²×#117	D₁₃	semidirect
114. C₂²×D₁₃	C₂³×C₄	semidirect
119. C₂×C₄×C₁₃	C₂³×C₄	semidirect
125. D₅₂	C₂³×C₄	semidirect
126. D₅₂	C₂³×C₄	semidirect
139. C₂×#117	C₂×D₁₃	semidirect
141. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
147. C₂²×C₄²	C₂×D₁₃	semidirect
150. C₂²×C₁₃	#518	semidirect
177. C₂³×C₄	C₄×D₁₃	semidirect
185. #117	C₂²×D₁₃	semidirect
187. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
193. C₂×C₄²	C₂²×D₁₃	semidirect
195. C₂×C₁₃	C₂×#518	semidirect
222. C₂²×C₄	C₂×C₄×D₁₃	semidirect
228. C₄⋊C₄	C₂³×D₁₃	semidirect
233. C₄²	C₂³×D₁₃	semidirect
234. C₁₃	C₂²×#518	semidirect
257. C₂×C₄	C₂²×C₄×D₁₃	semidirect
267. C₂²	#11719226	direct (N × H)
278. C₂	C₂×#11719226	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×D₅₂	2
C₂×#11719226	2
C₂²×C₄×D₅₂	2
C₂²×#88147	2
C₂²×#88180	2
C₂²×C₂²⋊C₄×D₁₃	2
C₂²×C₁₃×#117	2
C₂²×#518	13

Subgroups — C22×#11719226 #16769499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719226 #16769499