Subgroups — C₂²×#11719240 #16769696

Order 3328.

8838 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11719240	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#88128	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#88148	C₂	1	4	yes	maximal, complemented
5	C₂²×#88184	C₂	1	2	yes	maximal, complemented
6	C₂²×#88138	C₂	1	4	yes	maximal, complemented
7	C₂²×C₄×D₅₂	C₂	1	2	yes	maximal, complemented
8	C₂²×C₁₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₂²⋊D₁₃	C₂²	1	4	yes	—
11	C₂³×D₅₂	C₂²	1	2	yes	—
12	C₂³×D₅₂	C₂²	1	4	yes	—
13	C₂³×C₄×D₁₃	C₂²	1	4	yes	—
14	C₂×C₁₃×#117	C₂²	1	24	yes	complemented
15	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×#88184	C₂²	1	48	yes	complemented
20	C₂²×#6103	C₂²	1	4	yes	—
21	C₂²×#6103	C₂²	1	4	yes	—
22	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
23	C₂²×#6101	C₂²	1	4	yes	—
24	#11719240	C₂²	1	256	yes	direct product factor
25	C₂×#88148	C₂²	1	96	yes	complemented
26	C₂×#88138	C₂²	1	96	yes	complemented
27	C₂×#88128	C₂²	1	24	yes	complemented
28	C₂×C₄×D₅₂	C₂²	1	48	yes	complemented
29	C₂×#88124	C₂²	1	24	yes	complemented
30	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
31	C₂⁴×D₁₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₃	—	2	8	no	—
33	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
34	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	12	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₂²⋊D₁₃	—	4	16	no	—
39	C₂²×C₂²⋊D₁₃	C₂³	1	48	yes	—
40	C₂³×Dic₁₃	C₂³	1	4	yes	—
41	C₂²×D₅₂	—	2	16	no	—
42	C₂²×D₅₂	—	4	16	no	—
43	C₂²×D₅₂	C₂³	1	24	yes	—
44	C₂²×D₅₂	C₂³	1	48	yes	—
45	C₂²×C₄×D₁₃	—	4	16	no	—
46	C₂²×C₄×D₁₃	C₂³	1	48	yes	—
47	C₁₃×#117	C₂³	1	64	yes	complemented
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	24	yes	—
49	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
52	#88184	C₂³	1	128	yes	complemented
53	C₂×#6103	C₂³	1	48	yes	—
54	C₂×#6103	C₂³	1	48	yes	—
55	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
56	C₂×#6101	C₂³	1	48	yes	—
57	#88148	C₂³	1	256	yes	complemented
58	#88138	C₂³	1	256	yes	complemented
59	#88128	C₂³	1	64	yes	complemented
60	C₄×D₅₂	C₂³	1	128	yes	complemented
61	#88124	C₂³	1	64	yes	complemented
62	C₂²×#540	—	13	1	no	maximal, Sylow
63	C₂⁴×C₁₃	—	2	4	no	—
64	C₂⁴×C₁₃	—	2	8	no	—
65	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
66	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂³×D₁₃	—	2	16	no	—
68	C₂³×D₁₃	—	2	16	no	—
69	C₂³×D₁₃	—	2	16	no	—
70	C₂³×D₁₃	C₂⁴	1	24	yes	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₃	—	2	48	no	—
75	C₂²×C₄×C₁₃	#41	1	8	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
78	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
79	C₂²×C₄×C₁₃	C₂⁴	1	16	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
81	C₂×C₂²⋊D₁₃	—	4	96	no	—
82	C₂×C₂²⋊D₁₃	C₂⁴	1	64	yes	—
83	C₂²×Dic₁₃	—	2	16	no	—
84	C₂²×Dic₁₃	C₂⁴	1	24	yes	—
85	C₂×D₅₂	—	2	96	no	—
86	C₂×D₅₂	—	4	96	no	—
87	C₂×D₅₂	C₂⁴	1	32	yes	—
88	C₂×D₅₂	C₂⁴	1	64	yes	—
89	C₂×C₄×D₁₃	—	4	96	no	—
90	C₂×C₄×D₁₃	C₂⁴	1	64	yes	—
91	C₁₃×C₄⋊C₄	C₂⁴	1	32	yes	—
92	C₁₃×C₂²⋊C₄	C₂⁴	1	32	yes	—
93	C₄²×C₁₃	C₂⁴	1	16	yes	—
94	C₄²×C₁₃	C₂⁴	1	16	yes	—
95	#6103	C₂⁴	1	64	yes	—
96	#6103	C₂⁴	1	64	yes	—
97	C₄⋊Dic₁₃	C₂⁴	1	32	yes	—
98	#6101	C₂⁴	1	64	yes	—
99	C₂×#540	—	13	48	no	—
100	C₂²×#127	—	13	1	no	—
101	C₂²×#125	—	13	1	no	—
102	C₂²×#123	—	13	4	no	—
103	C₂²×#121	—	13	2	no	—
104	C₂²×#121	—	13	4	no	—
105	C₂²×C₄×D₄	—	13	2	no	—
106	C₂²×#117	D₁₃	1	1	yes	characteristic, complemented
107	C₂³×C₁₃	—	2	12	no	—
108	C₂³×C₁₃	—	2	24	no	—
109	C₂³×C₁₃	—	4	16	no	—
110	C₂³×C₁₃	C₂×#41	1	4	yes	—
111	C₂³×C₁₃	H(5,𝔽₂)	1	8	yes	—
112	C₂³×C₁₃	C₂⁵	1	4	yes	—
113	C₂³×C₁₃	C₂⁵	1	3	yes	—
114	C₂²×D₁₃	—	2	48	no	—
115	C₂²×D₁₃	—	2	48	no	—
116	C₂²×D₁₃	—	2	48	no	—
117	C₂²×D₁₃	—	4	64	no	—
118	C₂²×D₁₃	C₂⁵	1	16	yes	—
119	C₂×C₄×C₁₃	—	2	12	no	—
120	C₂×C₄×C₁₃	—	2	24	no	—
121	C₂×C₄×C₁₃	—	2	24	no	—
122	C₂×C₄×C₁₃	—	2	32	no	—
123	C₂×C₄×C₁₃	C₂×#41	1	24	yes	—
124	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
125	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
126	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
127	C₂×C₄×C₁₃	C₂⁵	1	4	yes	—
128	C₂²⋊D₁₃	—	4	64	no	—
129	C₂×Dic₁₃	—	2	48	no	—
130	C₂×Dic₁₃	C₂⁵	1	16	yes	—
131	D₅₂	—	2	64	no	—
132	D₅₂	—	4	64	no	—
133	C₄×D₁₃	—	4	64	no	—
134	C₂³×D₄	—	13	4	no	—
135	C₂³×D₄	—	13	2	no	—
136	C₂³×D₄	—	13	4	no	—
137	C₂⁴×C₄	—	13	4	no	—
138	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
139	#540	—	13	256	no	—
140	C₂×#127	—	13	24	no	—
141	C₂×#125	—	13	24	no	—
142	C₂×#123	—	13	96	no	—
143	C₂×#121	—	13	48	no	—
144	C₂×#121	—	13	96	no	—
145	C₂×C₄×D₄	—	13	48	no	—
146	C₂×#117	C₂×D₁₃	1	24	yes	complemented
147	C₂²×C₄⋊C₄	—	13	4	no	—
148	C₂²×C₄⋊C₄	—	13	2	no	—
149	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
150	C₂²×C₂²⋊C₄	—	13	4	no	—
151	C₂²×C₂²⋊C₄	—	13	4	no	—
152	C₂²×C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
153	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
154	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
155	C₂²×C₁₃	—	2	4	no	—
156	C₂²×C₁₃	—	2	8	no	—
157	C₂²×C₁₃	—	4	24	no	—
158	C₂²×C₁₃	#540	1	16	yes	complemented
159	C₂²×C₁₃	C₂²×#41	1	6	yes	—
160	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	12	yes	—
161	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
162	C₂×D₁₃	—	2	16	no	—
163	C₂×D₁₃	—	2	16	no	—
164	C₂×D₁₃	—	2	16	no	—
165	C₂×D₁₃	—	4	96	no	—
166	C₄×C₁₃	—	2	4	no	—
167	C₄×C₁₃	—	2	8	no	—
168	C₄×C₁₃	—	2	8	no	—
169	C₄×C₁₃	C₂²×#41	1	8	yes	—
170	Dic₁₃	—	2	16	no	—
171	C₂⁵	—	13	4	no	—
172	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
173	C₂²×D₄	—	13	48	no	—
174	C₂²×D₄	—	13	24	no	—
175	C₂²×D₄	—	13	48	no	—
176	C₂²×D₄	—	26	16	no	—
177	C₂²×D₄	—	52	16	no	—
178	C₂²×D₄	—	52	16	no	—
179	C₂³×C₄	—	2	8	no	—
180	C₂³×C₄	—	13	4	no	—
181	C₂³×C₄	—	13	48	no	—
182	C₂³×C₄	—	52	16	no	—
183	C₂³×C₄	C₂²×D₁₃	1	12	yes	—
184	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
186	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
187	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
188	#127	—	13	64	no	—
189	#125	—	13	64	no	—
190	#123	—	13	256	no	—
191	#121	—	13	128	no	—
192	#121	—	13	256	no	—
193	C₄×D₄	—	13	128	no	—
194	#117	C₂²×D₁₃	1	64	yes	complemented
195	C₂×C₄⋊C₄	—	13	24	no	—
196	C₂×C₄⋊C₄	—	13	48	no	—
197	C₂×C₄⋊C₄	C₂²×D₁₃	1	24	yes	—
198	C₂×C₂²⋊C₄	—	13	48	no	—
199	C₂×C₂²⋊C₄	—	13	48	no	—
200	C₂×C₂²⋊C₄	C₂²×D₁₃	1	24	yes	—
201	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
202	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
203	C₂×C₁₃	—	4	4	no	—
204	C₂×C₁₃	C₂×#540	1	12	yes	complemented
205	C₂×C₁₃	C₂³×#41	1	1	yes	characteristic
206	C₂×C₁₃	C₂²×H(5,𝔽₂)	1	2	yes	—
207	D₁₃	—	4	16	no	—
208	C₂⁴	—	2	4	no	—
209	C₂⁴	—	2	8	no	—
210	C₂⁴	—	13	24	no	—
211	C₂⁴	—	26	16	no	—
212	C₂⁴	—	26	16	no	—
213	C₂⁴	—	26	16	no	—
214	C₂⁴	C₂³×D₁₃	1	6	yes	—
215	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
216	C₂×D₄	—	13	64	no	—
217	C₂×D₄	—	13	32	no	—
218	C₂×D₄	—	13	64	no	—
219	C₂×D₄	—	26	96	no	—
220	C₂×D₄	—	52	96	no	—
221	C₂×D₄	—	52	96	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	48	no	—
226	C₂²×C₄	—	13	24	no	—
227	C₂²×C₄	—	13	64	no	—
228	C₂²×C₄	—	26	16	no	—
229	C₂²×C₄	—	52	96	no	—
230	C₂²×C₄	#6127	1	8	yes	—
231	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
232	C₂²×C₄	C₂³×D₁₃	1	12	yes	—
233	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
234	C₂²×C₄	C₂³×D₁₃	1	16	yes	—
235	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
236	C₄⋊C₄	—	13	32	no	—
237	C₄⋊C₄	—	13	64	no	—
238	C₄⋊C₄	C₂³×D₁₃	1	32	yes	—
239	C₂²⋊C₄	—	13	64	no	—
240	C₂²⋊C₄	—	13	64	no	—
241	C₂²⋊C₄	C₂³×D₁₃	1	32	yes	—
242	C₄²	C₂³×D₁₃	1	16	yes	—
243	C₄²	C₂³×D₁₃	1	16	yes	—
244	C₁₃	C₂²×#540	1	1	yes	characteristic, complemented, Sylow
245	C₂³	—	2	12	no	—
246	C₂³	—	2	24	no	—
247	C₂³	—	4	16	no	—
248	C₂³	—	13	16	no	—
249	C₂³	—	26	48	no	—
250	C₂³	—	26	48	no	—
251	C₂³	—	26	48	no	—
252	C₂³	—	52	64	no	—
253	C₂³	C₂×#6127	1	4	yes	—
254	C₂³	#88256	1	8	yes	—
255	C₂³	C₂⁴×D₁₃	1	4	yes	—
256	C₂³	C₂⁴×D₁₃	1	3	yes	—
257	D₄	—	26	64	no	—
258	D₄	—	52	64	no	—
259	D₄	—	52	64	no	—
260	C₂×C₄	—	2	12	no	—
261	C₂×C₄	—	2	24	no	—
262	C₂×C₄	—	2	24	no	—
263	C₂×C₄	—	2	32	no	—
264	C₂×C₄	—	13	16	no	—
265	C₂×C₄	—	26	48	no	—
266	C₂×C₄	—	52	64	no	—
267	C₂×C₄	C₂×#6127	1	24	yes	—
268	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
269	C₂×C₄	C₂⁴×D₁₃	1	8	yes	—
270	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
271	C₂×C₄	C₂⁴×D₁₃	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	4	24	no	—
275	C₂²	—	26	16	no	—
276	C₂²	—	26	16	no	—
277	C₂²	—	26	16	no	—
278	C₂²	—	52	96	no	—
279	C₂²	#11719240	1	16	yes	direct product factor
280	C₂²	C₂²×#6127	1	6	yes	—
281	C₂²	C₂×#88256	1	12	yes	—
282	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
283	C₄	—	2	4	no	—
284	C₄	—	2	8	no	—
285	C₄	—	2	8	no	—
286	C₄	—	26	16	no	—
287	C₄	C₂²×#6127	1	8	yes	—
288	C₂	—	4	4	no	—
289	C₂	—	52	16	no	—
290	C₂	C₂×#11719240	1	12	yes	direct product factor
291	C₂	C₂³×#6127	1	1	yes	characteristic
292	C₂	C₂²×#88256	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#117	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×#117	D₁₃
hypocenter	C₁₃	C₂²×#540
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. C₂×#11719240	C₂	direct (N × H)
2. C₂²×#88128	C₂	semidirect
3. C₂²×#88124	C₂	semidirect
4. C₂²×#88148	C₂	semidirect
5. C₂²×#88184	C₂	semidirect
6. C₂²×#88138	C₂	semidirect
7. C₂²×C₄×D₅₂	C₂	semidirect
8. C₂²×C₁₃×#117	C₂	semidirect
14. C₂×C₁₃×#117	C₂²	semidirect
19. C₂×#88184	C₂²	semidirect
24. #11719240	C₂²	direct (N × H)
25. C₂×#88148	C₂²	semidirect
26. C₂×#88138	C₂²	semidirect
27. C₂×#88128	C₂²	semidirect
28. C₂×C₄×D₅₂	C₂²	semidirect
29. C₂×#88124	C₂²	semidirect
47. C₁₃×#117	C₂³	semidirect
52. #88184	C₂³	semidirect
57. #88148	C₂³	semidirect
58. #88138	C₂³	semidirect
59. #88128	C₂³	semidirect
60. C₄×D₅₂	C₂³	semidirect
61. #88124	C₂³	semidirect
106. C₂²×#117	D₁₃	semidirect
146. C₂×#117	C₂×D₁₃	semidirect
158. C₂²×C₁₃	#540	semidirect
194. #117	C₂²×D₁₃	semidirect
204. C₂×C₁₃	C₂×#540	semidirect
244. C₁₃	C₂²×#540	semidirect
279. C₂²	#11719240	direct (N × H)
290. C₂	C₂×#11719240	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×#11719240	2
C₂²×#88128	2
C₂²×#88124	2
C₂²×#88148	2
C₂²×#88184	2
C₂²×#88138	2
C₂²×C₄×D₅₂	2
C₂²×C₁₃×#117	2
C₂²×#540	13

Subgroups — C22×#11719240 #16769696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11719240 #16769696