Subgroups — C₂²×#11719416 #16769789

Order 3328.

9606 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719416	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88128	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88130	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#88148	C₂	1	3	yes	maximal, complemented
5	C₂²×#88149	C₂	1	3	yes	maximal, complemented
6	C₂²×#88138	C₂	1	3	yes	maximal, complemented
7	C₂²×#88136	C₂	1	3	yes	maximal, complemented
8	C₂²×C₁₃×#126	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₂²⋊D₁₃	C₂²	1	3	yes	—
11	C₂³×D₅₂	C₂²	1	3	yes	—
12	C₂³×D₅₂	C₂²	1	3	yes	—
13	C₂³×D₅₂	C₂²	1	3	yes	complemented
14	C₂³×C₄×D₁₃	C₂²	1	3	yes	complemented
15	C₂×C₁₃×#126	C₂²	1	24	yes	complemented
16	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	complemented
17	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
18	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
19	C₂²×#6103	C₂²	1	3	yes	—
20	C₂²×#6103	C₂²	1	3	yes	—
21	C₂²×#6103	C₂²	1	3	yes	complemented
22	C₂²×#6101	C₂²	1	3	yes	complemented
23	#11719416	C₂²	1	256	yes	direct product factor
24	C₂×#88149	C₂²	1	72	yes	complemented
25	C₂×#88148	C₂²	1	72	yes	complemented
26	C₂×#88138	C₂²	1	72	yes	complemented
27	C₂×#88136	C₂²	1	72	yes	complemented
28	C₂×#88130	C₂²	1	24	yes	complemented
29	C₂×#88128	C₂²	1	24	yes	complemented
30	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
31	C₂⁴×D₁₃	—	4	12	no	—
32	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
33	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
34	C₂⁴×D₁₃	C₂³	1	3	yes	—
35	C₂⁴×D₁₃	C₂³	1	12	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
38	C₂²×C₂²⋊D₁₃	—	4	12	no	—
39	C₂²×C₂²⋊D₁₃	C₂³	1	36	yes	—
40	C₂³×Dic₁₃	C₂³	1	3	yes	—
41	C₂²×D₅₂	—	4	12	no	—
42	C₂²×D₅₂	—	4	12	no	—
43	C₂²×D₅₂	—	4	12	no	—
44	C₂²×D₅₂	C₂³	1	36	yes	—
45	C₂²×D₅₂	C₂³	1	36	yes	—
46	C₂²×D₅₂	C₂³	1	36	yes	complemented
47	C₂²×C₄×D₁₃	—	4	12	no	—
48	C₂²×C₄×D₁₃	C₂³	1	36	yes	complemented
49	C₁₃×#126	C₂³	1	64	yes	complemented
50	C₂×C₁₃×C₄⋊C₄	C₂³	1	36	yes	complemented
51	C₂×C₁₃×C₂²⋊C₄	C₂³	1	36	yes	—
52	C₂×C₄²×C₁₃	C₂³	1	12	yes	complemented
53	C₂×#6103	C₂³	1	36	yes	—
54	C₂×#6103	C₂³	1	36	yes	—
55	C₂×#6103	C₂³	1	36	yes	complemented
56	C₂×#6101	C₂³	1	36	yes	complemented
57	#88149	C₂³	1	192	yes	complemented
58	#88148	C₂³	1	192	yes	complemented
59	#88138	C₂³	1	192	yes	complemented
60	#88136	C₂³	1	192	yes	complemented
61	#88130	C₂³	1	64	yes	complemented
62	#88128	C₂³	1	64	yes	complemented
63	C₂²×#560	—	13	1	no	maximal, Sylow
64	C₂⁴×C₁₃	—	2	12	no	—
65	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
66	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂³×D₁₃	—	2	12	no	—
68	C₂³×D₁₃	—	2	12	no	—
69	C₂³×D₁₃	—	2	12	no	—
70	C₂³×D₁₃	—	2	12	no	—
71	C₂³×D₁₃	—	2	12	no	—
72	C₂³×D₁₃	—	4	72	no	—
73	C₂³×D₁₃	—	4	64	no	—
74	C₂³×D₁₃	C₂⁴	1	6	yes	—
75	C₂³×D₁₃	C₂⁴	1	6	yes	—
76	C₂³×D₁₃	C₂⁴	1	18	yes	—
77	C₂³×D₁₃	C₂⁴	1	16	yes	—
78	C₂²×C₄×C₁₃	—	2	12	no	—
79	C₂²×C₄×C₁₃	—	2	12	no	—
80	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
82	C₂×C₂²⋊D₁₃	—	4	72	no	—
83	C₂×C₂²⋊D₁₃	C₂⁴	1	48	yes	—
84	C₂²×Dic₁₃	—	2	12	no	—
85	C₂²×Dic₁₃	C₂⁴	1	18	yes	—
86	C₂×D₅₂	—	4	72	no	—
87	C₂×D₅₂	—	4	72	no	—
88	C₂×D₅₂	—	4	72	no	—
89	C₂×D₅₂	C₂⁴	1	48	yes	—
90	C₂×D₅₂	C₂⁴	1	48	yes	—
91	C₂×D₅₂	C₂⁴	1	48	yes	complemented
92	C₂×C₄×D₁₃	—	4	72	no	—
93	C₂×C₄×D₁₃	C₂⁴	1	48	yes	complemented
94	C₁₃×C₄⋊C₄	C₂⁴	1	48	yes	complemented
95	C₁₃×C₂²⋊C₄	C₂⁴	1	48	yes	—
96	C₄²×C₁₃	C₂⁴	1	16	yes	complemented
97	#6103	C₂⁴	1	48	yes	—
98	#6103	C₂⁴	1	48	yes	—
99	#6103	C₂⁴	1	48	yes	complemented
100	#6101	C₂⁴	1	48	yes	complemented
101	C₂×#560	—	13	48	no	—
102	C₂²×#127	—	13	1	no	—
103	C₂²×#126	—	13	1	no	—
104	C₂²×#126	D₁₃	1	1	yes	characteristic, complemented
105	C₂²×#123	—	13	3	no	—
106	C₂²×#121	—	13	3	no	—
107	C₂²×#121	—	13	3	no	—
108	C₂²×#120	—	13	3	no	—
109	C₂³×C₁₃	—	2	36	no	—
110	C₂³×C₁₃	—	4	16	no	—
111	C₂³×C₁₃	H(5,𝔽₂)	1	12	yes	—
112	C₂³×C₁₃	C₂⁵	1	4	yes	—
113	C₂³×C₁₃	C₂⁵	1	3	yes	—
114	C₂²×D₁₃	—	2	36	no	—
115	C₂²×D₁₃	—	2	36	no	—
116	C₂²×D₁₃	—	2	36	no	—
117	C₂²×D₁₃	—	2	36	no	—
118	C₂²×D₁₃	—	2	36	no	—
119	C₂²×D₁₃	—	4	48	no	—
120	C₂²×D₁₃	—	4	192	no	—
121	C₂²×D₁₃	—	4	48	no	—
122	C₂²×D₁₃	—	4	16	no	—
123	C₂²×D₁₃	—	4	16	no	—
124	C₂²×D₁₃	C₂⁵	1	4	yes	—
125	C₂²×D₁₃	C₂⁵	1	4	yes	—
126	C₂²×D₁₃	C₂⁵	1	12	yes	—
127	C₂×C₄×C₁₃	—	2	36	no	—
128	C₂×C₄×C₁₃	—	2	36	no	—
129	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
130	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
131	C₂²⋊D₁₃	—	4	48	no	—
132	C₂×Dic₁₃	—	2	36	no	—
133	C₂×Dic₁₃	C₂⁵	1	12	yes	—
134	D₅₂	—	4	48	no	—
135	D₅₂	—	4	48	no	—
136	D₅₂	—	4	48	no	—
137	C₄×D₁₃	—	4	48	no	—
138	C₂⁶	—	13	1	no	—
139	C₂³×D₄	—	13	3	no	—
140	C₂³×D₄	—	13	3	no	—
141	C₂³×D₄	—	13	3	no	—
142	C₂³×D₄	—	13	3	no	—
143	C₂⁴×C₄	—	13	3	no	—
144	#560	—	13	256	no	—
145	C₂×#127	—	13	24	no	—
146	C₂×#126	—	13	24	no	—
147	C₂×#126	C₂×D₁₃	1	24	yes	complemented
148	C₂×#123	—	13	72	no	—
149	C₂×#121	—	13	72	no	—
150	C₂×#121	—	13	72	no	—
151	C₂×#120	—	13	72	no	—
152	C₂²×C₄⋊C₄	—	13	3	no	—
153	C₂²×C₄⋊C₄	C₂×D₁₃	1	3	yes	complemented
154	C₂²×C₂²⋊C₄	—	13	3	no	—
155	C₂²×C₂²⋊C₄	—	13	3	no	—
156	C₂²×C₂²⋊C₄	—	13	3	no	—
157	C₂²×C₂²⋊C₄	C₂×D₁₃	1	3	yes	—
158	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
159	C₂²×C₁₃	—	2	12	no	—
160	C₂²×C₁₃	—	4	24	no	—
161	C₂²×C₁₃	#560	1	16	yes	complemented
162	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	18	yes	—
163	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
164	C₂×D₁₃	—	2	12	no	—
165	C₂×D₁₃	—	2	12	no	—
166	C₂×D₁₃	—	2	12	no	—
167	C₂×D₁₃	—	2	12	no	—
168	C₂×D₁₃	—	2	12	no	—
169	C₂×D₁₃	—	4	64	no	—
170	C₂×D₁₃	—	4	72	no	—
171	C₂×D₁₃	—	4	24	no	—
172	C₂×D₁₃	—	4	24	no	—
173	C₄×C₁₃	—	2	12	no	—
174	C₄×C₁₃	—	2	12	no	—
175	Dic₁₃	—	2	12	no	—
176	C₂⁵	—	13	1	no	—
177	C₂⁵	—	13	1	no	—
178	C₂⁵	—	13	3	no	—
179	C₂⁵	—	13	12	no	—
180	C₂⁵	—	52	12	no	—
181	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
182	C₂²×D₄	—	13	36	no	—
183	C₂²×D₄	—	13	36	no	—
184	C₂²×D₄	—	13	36	no	—
185	C₂²×D₄	—	13	36	no	—
186	C₂²×D₄	—	52	12	no	—
187	C₂²×D₄	—	52	12	no	—
188	C₂²×D₄	—	52	12	no	—
189	C₂²×D₄	—	52	12	no	—
190	C₂³×C₄	—	13	3	no	—
191	C₂³×C₄	—	13	36	no	—
192	C₂³×C₄	—	52	12	no	—
193	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
194	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
195	#127	—	13	64	no	—
196	#126	—	13	64	no	—
197	#126	C₂²×D₁₃	1	64	yes	complemented
198	#123	—	13	192	no	—
199	#121	—	13	192	no	—
200	#121	—	13	192	no	—
201	#120	—	13	192	no	—
202	C₂×C₄⋊C₄	—	13	36	no	—
203	C₂×C₄⋊C₄	C₂²×D₁₃	1	36	yes	complemented
204	C₂×C₂²⋊C₄	—	13	36	no	—
205	C₂×C₂²⋊C₄	—	13	36	no	—
206	C₂×C₂²⋊C₄	—	13	36	no	—
207	C₂×C₂²⋊C₄	C₂²×D₁₃	1	36	yes	—
208	C₂×C₄²	C₂²×D₁₃	1	12	yes	complemented
209	C₂×C₁₃	—	4	4	no	—
210	C₂×C₁₃	C₂×#560	1	12	yes	complemented
211	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	3	yes	—
212	D₁₃	—	4	12	no	—
213	D₁₃	—	4	4	no	—
214	D₁₃	—	4	4	no	—
215	C₂⁴	—	2	12	no	—
216	C₂⁴	—	13	6	no	—
217	C₂⁴	—	13	6	no	—
218	C₂⁴	—	13	18	no	—
219	C₂⁴	—	13	16	no	—
220	C₂⁴	—	26	12	no	—
221	C₂⁴	—	26	12	no	—
222	C₂⁴	—	26	12	no	—
223	C₂⁴	—	26	12	no	—
224	C₂⁴	—	26	12	no	—
225	C₂⁴	—	52	72	no	—
226	C₂⁴	—	52	64	no	—
227	C₂⁴	C₂³×D₁₃	1	6	yes	—
228	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
229	C₂×D₄	—	13	48	no	—
230	C₂×D₄	—	13	48	no	—
231	C₂×D₄	—	13	48	no	—
232	C₂×D₄	—	13	48	no	—
233	C₂×D₄	—	52	72	no	—
234	C₂×D₄	—	52	72	no	—
235	C₂×D₄	—	52	72	no	—
236	C₂×D₄	—	52	72	no	—
237	C₂²×C₄	—	2	12	no	—
238	C₂²×C₄	—	2	12	no	—
239	C₂²×C₄	—	13	18	no	—
240	C₂²×C₄	—	13	48	no	—
241	C₂²×C₄	—	26	12	no	—
242	C₂²×C₄	—	52	72	no	—
243	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
244	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
245	C₄⋊C₄	—	13	48	no	—
246	C₄⋊C₄	C₂³×D₁₃	1	48	yes	complemented
247	C₂²⋊C₄	—	13	48	no	—
248	C₂²⋊C₄	—	13	48	no	—
249	C₂²⋊C₄	—	13	48	no	—
250	C₂²⋊C₄	C₂³×D₁₃	1	48	yes	—
251	C₄²	C₂³×D₁₃	1	16	yes	complemented
252	C₁₃	C₂²×#560	1	1	yes	characteristic, complemented, Sylow
253	C₂³	—	2	36	no	—
254	C₂³	—	4	16	no	—
255	C₂³	—	13	4	no	—
256	C₂³	—	13	4	no	—
257	C₂³	—	13	12	no	—
258	C₂³	—	26	36	no	—
259	C₂³	—	26	36	no	—
260	C₂³	—	26	36	no	—
261	C₂³	—	26	36	no	—
262	C₂³	—	26	36	no	—
263	C₂³	—	52	48	no	—
264	C₂³	—	52	192	no	—
265	C₂³	—	52	16	no	—
266	C₂³	—	52	16	no	—
267	C₂³	—	52	48	no	—
268	C₂³	#88256	1	12	yes	—
269	C₂³	C₂⁴×D₁₃	1	4	yes	—
270	C₂³	C₂⁴×D₁₃	1	3	yes	—
271	D₄	—	52	48	no	—
272	D₄	—	52	48	no	—
273	D₄	—	52	48	no	—
274	D₄	—	52	48	no	—
275	C₂×C₄	—	2	36	no	—
276	C₂×C₄	—	2	36	no	—
277	C₂×C₄	—	13	12	no	—
278	C₂×C₄	—	26	36	no	—
279	C₂×C₄	—	52	48	no	—
280	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
281	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
282	C₂²	—	2	12	no	—
283	C₂²	—	4	24	no	—
284	C₂²	—	26	12	no	—
285	C₂²	—	26	12	no	—
286	C₂²	—	26	12	no	—
287	C₂²	—	26	12	no	—
288	C₂²	—	26	12	no	—
289	C₂²	—	52	64	no	—
290	C₂²	—	52	72	no	—
291	C₂²	—	52	24	no	—
292	C₂²	—	52	24	no	—
293	C₂²	#11719416	1	16	yes	direct product factor
294	C₂²	C₂×#88256	1	18	yes	—
295	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
296	C₄	—	2	12	no	—
297	C₄	—	2	12	no	—
298	C₄	—	26	12	no	—
299	C₂	—	4	4	no	—
300	C₂	—	52	4	no	—
301	C₂	—	52	4	no	—
302	C₂	—	52	12	no	—
303	C₂	C₂×#11719416	1	12	yes	direct product factor
304	C₂	C₂²×#88256	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#126	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×#126	D₁₃
hypocenter	C₁₃	C₂²×#560
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#126

Complements

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

N	H ≅ G/N	type
1. C₂×#11719416	C₂	direct (N × H)
2. C₂²×#88128	C₂	semidirect
3. C₂²×#88130	C₂	semidirect
4. C₂²×#88148	C₂	semidirect
5. C₂²×#88149	C₂	semidirect
6. C₂²×#88138	C₂	semidirect
7. C₂²×#88136	C₂	semidirect
8. C₂²×C₁₃×#126	C₂	semidirect
13. C₂³×D₅₂	C₂²	semidirect
14. C₂³×C₄×D₁₃	C₂²	semidirect
15. C₂×C₁₃×#126	C₂²	semidirect
16. C₂²×C₁₃×C₄⋊C₄	C₂²	semidirect
18. C₂²×C₄²×C₁₃	C₂²	semidirect
21. C₂²×#6103	C₂²	semidirect
22. C₂²×#6101	C₂²	semidirect
23. #11719416	C₂²	direct (N × H)
24. C₂×#88149	C₂²	semidirect
25. C₂×#88148	C₂²	semidirect
26. C₂×#88138	C₂²	semidirect
27. C₂×#88136	C₂²	semidirect
28. C₂×#88130	C₂²	semidirect
29. C₂×#88128	C₂²	semidirect
46. C₂²×D₅₂	C₂³	semidirect
48. C₂²×C₄×D₁₃	C₂³	semidirect
49. C₁₃×#126	C₂³	semidirect
50. C₂×C₁₃×C₄⋊C₄	C₂³	semidirect
52. C₂×C₄²×C₁₃	C₂³	semidirect
55. C₂×#6103	C₂³	semidirect
56. C₂×#6101	C₂³	semidirect
57. #88149	C₂³	semidirect
58. #88148	C₂³	semidirect
59. #88138	C₂³	semidirect
60. #88136	C₂³	semidirect
61. #88130	C₂³	semidirect
62. #88128	C₂³	semidirect
91. C₂×D₅₂	C₂⁴	semidirect
93. C₂×C₄×D₁₃	C₂⁴	semidirect
94. C₁₃×C₄⋊C₄	C₂⁴	semidirect
96. C₄²×C₁₃	C₂⁴	semidirect
99. #6103	C₂⁴	semidirect
100. #6101	C₂⁴	semidirect
104. C₂²×#126	D₁₃	semidirect
147. C₂×#126	C₂×D₁₃	semidirect
153. C₂²×C₄⋊C₄	C₂×D₁₃	semidirect
158. C₂²×C₄²	C₂×D₁₃	semidirect
161. C₂²×C₁₃	#560	semidirect
197. #126	C₂²×D₁₃	semidirect
203. C₂×C₄⋊C₄	C₂²×D₁₃	semidirect
208. C₂×C₄²	C₂²×D₁₃	semidirect
210. C₂×C₁₃	C₂×#560	semidirect
246. C₄⋊C₄	C₂³×D₁₃	semidirect
251. C₄²	C₂³×D₁₃	semidirect
252. C₁₃	C₂²×#560	semidirect
293. C₂²	#11719416	direct (N × H)
303. C₂	C₂×#11719416	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#11719416	2
C₂²×#88128	2
C₂²×#88130	2
C₂²×#88148	2
C₂²×#88149	2
C₂²×#88138	2
C₂²×#88136	2
C₂²×C₁₃×#126	2
C₂²×#560	13

Subgroups — C22×#11719416 #16769789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719416 #16769789