Subgroups — #72240521

Order 2048.

At least 9156 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

The subgroup lattice has 295 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72240519	C₂	1	1	yes	maximal, characteristic, complemented
2	#72240518	C₂	1	1	yes	maximal, characteristic, complemented
3	#72240517	C₂	1	1	yes	maximal, characteristic, complemented
4	#72240516	C₂	1	1	yes	maximal, characteristic, complemented
5	#72240506	C₂	1	1	yes	maximal, characteristic, complemented
6	#72240494	C₂	1	1	yes	maximal, characteristic, complemented
7	#72240492	C₂	1	1	yes	maximal, characteristic, complemented
8	#72240483	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238001	C₂	1	1	yes	maximal, characteristic, complemented
10	#72234206	C₂	1	1	yes	maximal, characteristic, complemented
11	#72233775	C₂	1	1	yes	maximal, characteristic, complemented
12	#72233452	C₂	1	1	yes	maximal, characteristic, complemented
13	#72216064	C₂	1	1	yes	maximal, characteristic, complemented
14	#72214173	C₂	1	1	yes	maximal, characteristic, complemented
15	#72213729	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586066	C₂²	1	1	yes	characteristic
17	C₂²×#3484	C₂²	1	1	yes	characteristic
18	#10574840	C₂²	1	1	yes	characteristic
19	#6437628	C₂²	1	1	yes	characteristic, complemented
20	#6437623	C₂²	1	1	yes	characteristic
21	#6437623	C₂²	1	1	yes	characteristic
22	#6437616	C₂²	1	1	yes	characteristic
23	#6437616	C₂²	1	1	yes	characteristic
24	#6437616	C₂²	1	1	yes	characteristic, complemented
25	#6437463	C₂²	1	1	yes	characteristic, complemented
26	#6437449	C₂²	1	1	yes	characteristic, complemented
27	#6370693	C₂²	1	1	yes	characteristic
28	#493402	C₂²	1	1	yes	characteristic, complemented
29	#493396	C₂²	1	1	yes	characteristic, complemented
30	#493396	C₂²	1	1	yes	characteristic, complemented
31	#493390	C₂²	1	1	yes	characteristic, complemented
32	#493390	C₂²	1	1	yes	characteristic, complemented
33	#493390	C₂²	1	1	yes	characteristic, complemented
34	#392132	C₂²	1	1	yes	characteristic
35	#392132	C₂²	1	1	yes	characteristic
36	#392132	C₂²	1	1	yes	characteristic
37	#359562	C₂²	1	1	yes	characteristic, complemented
38	#359551	C₂²	1	1	yes	characteristic, complemented
39	#359513	C₂²	1	1	yes	characteristic, complemented
40	#359507	C₂²	1	1	yes	characteristic, complemented
41	#188071	C₂²	1	1	yes	characteristic, complemented
42	#188029	C₂²	1	1	yes	characteristic, complemented
43	#187756	C₂²	1	1	yes	characteristic, complemented
44	#187756	C₂²	1	1	yes	characteristic, complemented
45	#146483	C₂²	1	1	yes	characteristic, complemented
46	#146477	C₂²	1	1	yes	characteristic, complemented
47	#128042	C₂²	1	1	yes	characteristic, complemented
48	#128040	C₂²	1	1	yes	characteristic, complemented
49	#128020	C₂²	1	1	yes	characteristic, complemented
50	#128016	C₂²	1	1	yes	characteristic, complemented
51	C₂×H(7,𝔽₂)	D₄	1	1	yes	characteristic, complemented
52	C₂³×H(5,𝔽₂)	C₂³	1	1	yes	characteristic
53	C₂⁵×D₄	D₄	1	1	yes	characteristic, complemented
54	C₂×#3585	D₄	1	1	yes	characteristic, complemented
55	C₂×#3585	D₄	1	1	yes	characteristic, complemented
56	#62852	D₄	1	1	yes	characteristic, complemented
57	#62852	D₄	1	1	yes	characteristic, complemented
58	#62811	C₂³	1	1	yes	characteristic
59	#62811	C₂³	1	1	yes	characteristic
60	#62811	C₂³	1	1	yes	characteristic
61	C₂²×#535	D₄	1	1	yes	characteristic, complemented
62	C₂²×#535	C₂³	1	1	yes	characteristic
63	C₂²×#525	D₄	1	1	yes	characteristic, complemented
64	C₂²×#525	C₂³	1	1	yes	characteristic
65	#62652	D₄	1	1	yes	characteristic, complemented
66	C₂²×#518	D₄	1	1	yes	characteristic, complemented
67	#60087	D₄	1	1	yes	characteristic, complemented
68	#60087	D₄	1	1	yes	characteristic, complemented
69	C₂×#2883	D₄	1	2	yes	complemented
70	#60064	C₂³	1	1	yes	characteristic
71	#23904	D₄	1	4	yes	complemented
72	#23904	D₄	1	4	yes	complemented
73	#23901	D₄	1	2	yes	—
74	#23901	D₄	1	2	yes	complemented
75	#23900	D₄	1	2	yes	—
76	#23900	D₄	1	2	yes	complemented
77	C₂×#1795	D₄	1	2	yes	complemented
78	C₂×#1795	D₄	1	2	yes	complemented
79	#20605	D₄	1	2	yes	complemented
80	#20598	D₄	1	2	yes	complemented
81	#20591	C₂³	1	1	yes	characteristic
82	#20591	C₂³	1	1	yes	characteristic
83	#20591	C₂³	1	1	yes	characteristic
84	#20586	D₄	1	2	yes	complemented
85	#20584	C₂³	1	1	yes	characteristic
86	#20584	C₂³	1	1	yes	characteristic
87	#20584	C₂³	1	1	yes	characteristic
88	#20584	C₂³	1	1	yes	characteristic
89	#20584	C₂³	1	1	yes	characteristic
90	C₂×#585	C₂×D₄	1	1	yes	characteristic
91	C₂×#585	C₂×D₄	1	1	yes	characteristic
92	C₂×#585	C₂×D₄	1	1	yes	characteristic, complemented
93	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
94	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
95	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
96	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
97	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
98	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
99	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
100	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
101	C₂³×#41	C₂×D₄	1	1	yes	characteristic
102	C₂³×#41	C₂×D₄	1	1	yes	characteristic
103	C₂³×#41	#41	1	1	yes	characteristic
104	C₂³×#41	C₂⁴	1	1	yes	characteristic
105	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
106	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
107	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
108	C₂⁴×D₄	#41	1	1	yes	characteristic
109	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
110	C₂×#568	C₂×D₄	1	1	yes	characteristic, complemented
111	C₂×#568	C₂×D₄	1	1	yes	characteristic, complemented
112	#3485	C₂×D₄	1	2	yes	—
113	#3485	C₂×D₄	1	2	yes	—
114	#3484	C₂×D₄	1	2	yes	—
115	#3484	C₂×D₄	1	2	yes	—
116	C₂×#535	C₂×D₄	1	2	yes	—
117	C₂×#535	C₂×D₄	1	2	yes	—
118	C₂²×#123	C₂×D₄	1	1	yes	characteristic
119	C₂²×#123	C₂×D₄	1	1	yes	characteristic
120	C₂²×#120	C₂×D₄	1	1	yes	characteristic
121	C₂²×#120	C₂×D₄	1	1	yes	characteristic
122	#3428	C₂×D₄	1	1	yes	characteristic
123	#3428	C₂×D₄	1	1	yes	characteristic
124	#3428	C₂×D₄	1	1	yes	characteristic, complemented
125	C₂²×#117	C₂×D₄	1	1	yes	characteristic
126	C₂²×#117	C₂×D₄	1	1	yes	characteristic
127	C₂×#413	C₂×D₄	1	2	yes	complemented
128	C₂×#410	C₂×D₄	1	2	yes	—
129	C₂×#410	C₂×D₄	1	2	yes	—
130	C₂×#410	C₂×D₄	1	2	yes	complemented
131	C₂×#407	C₂×D₄	1	2	yes	complemented
132	#2280	C₂×D₄	1	2	yes	complemented
133	#1782	C₂×D₄	1	2	yes	—
134	#1782	C₂×D₄	1	2	yes	—
135	#1781	C₂×D₄	1	2	yes	—
136	#1781	C₂×D₄	1	2	yes	—
137	C₂⁶	#120	1	1	yes	characteristic
138	C₂×H(5,𝔽₂)	#120	1	2	yes	—
139	C₂×H(5,𝔽₂)	#120	1	2	yes	—
140	C₂²×#41	#120	1	1	yes	characteristic
141	C₂²×#41	#120	1	1	yes	characteristic
142	C₂²×#41	#120	1	1	yes	characteristic
143	C₂²×#41	#120	1	1	yes	characteristic
144	C₂²×#41	#120	1	1	yes	characteristic
145	C₂²×#41	#120	1	1	yes	characteristic
146	C₂²×#41	#121	1	1	yes	characteristic
147	C₂²×#41	#121	1	1	yes	characteristic
148	C₂²×#41	#121	1	1	yes	characteristic
149	C₂²×#41	#121	1	1	yes	characteristic
150	C₂²×#41	#127	1	1	yes	characteristic
151	C₂²×#41	#127	1	1	yes	characteristic
152	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
153	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
154	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
155	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
156	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
157	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
158	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂³×Q₈	#121	1	1	yes	characteristic
161	C₂³×Q₈	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂³×D₄	#120	1	1	yes	characteristic
163	C₂³×D₄	#120	1	1	yes	characteristic
164	C₂³×D₄	#120	1	1	yes	characteristic
165	C₂³×D₄	#120	1	1	yes	characteristic
166	C₂³×D₄	#120	1	1	yes	characteristic
167	C₂³×D₄	#120	1	1	yes	characteristic
168	C₂³×D₄	#120	1	2	yes	—
169	C₂³×D₄	#121	1	1	yes	characteristic
170	C₂³×D₄	#121	1	1	yes	characteristic
171	C₂³×D₄	#121	1	1	yes	characteristic
172	C₂³×D₄	#121	1	1	yes	characteristic
173	C₂³×D₄	#121	1	1	yes	characteristic
174	C₂³×D₄	#121	1	1	yes	characteristic
175	C₂³×D₄	#121	1	1	yes	characteristic
176	C₂³×D₄	#127	1	1	yes	characteristic
177	C₂³×D₄	#127	1	1	yes	characteristic
178	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
179	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
180	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂⁴×C₄	#120	1	1	yes	characteristic
182	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
183	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
184	C₂⁴×C₄	C₂×#41	1	1	yes	characteristic
185	#518	#120	1	2	yes	—
186	#518	#120	1	2	yes	—
187	C₂×#117	#120	1	2	yes	—
188	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
189	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
190	C₂×#99	#120	1	4	yes	—
191	C₂×#99	#120	1	4	yes	—
192	C₂⁵	C₂×#120	1	1	yes	characteristic
193	C₂⁵	D₄²	1	1	yes	characteristic
194	C₂⁵	#546	1	1	yes	characteristic
195	C₂×#41	B(4,𝔽₂)	1	4	yes	—
196	C₂²×Q₈	#392	1	1	yes	characteristic
197	C₂²×Q₈	#392	1	1	yes	characteristic
198	C₂²×Q₈	#534	1	1	yes	characteristic
199	C₂²×Q₈	#535	1	1	yes	characteristic
200	C₂²×Q₈	D₄²	1	1	yes	characteristic
201	C₂²×Q₈	D₄²	1	1	yes	characteristic
202	C₂²×D₄	#392	1	1	yes	characteristic
203	C₂²×D₄	#392	1	1	yes	characteristic
204	C₂²×D₄	#392	1	1	yes	characteristic
205	C₂²×D₄	#392	1	1	yes	characteristic
206	C₂²×D₄	#392	1	1	yes	characteristic
207	C₂²×D₄	#392	1	1	yes	characteristic
208	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
209	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
210	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
211	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
212	C₂²×D₄	C₂×#120	1	1	yes	characteristic
213	C₂²×D₄	C₂×#120	1	1	yes	characteristic
214	C₂²×D₄	C₂×#120	1	1	yes	characteristic
215	C₂²×D₄	C₂×#120	1	1	yes	characteristic
216	C₂²×D₄	#534	1	1	yes	characteristic
217	C₂²×D₄	#535	1	1	yes	characteristic
218	C₂²×D₄	D₄²	1	1	yes	characteristic
219	C₂²×D₄	D₄²	1	1	yes	characteristic
220	C₂²×D₄	D₄²	1	1	yes	characteristic
221	C₂²×D₄	D₄²	1	1	yes	characteristic
222	C₂²×D₄	D₄²	1	1	yes	characteristic
223	C₂²×D₄	D₄²	1	1	yes	characteristic
224	C₂³×C₄	C₂×#120	1	1	yes	characteristic
225	C₂³×C₄	C₂×#120	1	1	yes	characteristic
226	C₂³×C₄	C₂×#120	1	1	yes	characteristic
227	C₂³×C₄	C₂×#120	1	1	yes	characteristic
228	C₂³×C₄	C₂×#121	1	1	yes	characteristic
229	C₂³×C₄	C₂×#121	1	1	yes	characteristic
230	C₂³×C₄	C₂×#121	1	1	yes	characteristic
231	C₂³×C₄	C₂×#127	1	1	yes	characteristic
232	C₂³×C₄	#534	1	1	yes	characteristic
233	C₂³×C₄	#534	1	1	yes	characteristic
234	C₂³×C₄	D₄²	1	1	yes	characteristic
235	C₂³×C₄	D₄²	1	1	yes	characteristic
236	C₂³×C₄	D₄²	1	1	yes	characteristic
237	C₂³×C₄	#546	1	1	yes	characteristic
238	C₂³×C₄	#546	1	1	yes	characteristic
239	C₂³×C₄	#546	1	1	yes	characteristic
240	C₂⁴	#2403	1	1	yes	characteristic
241	C₂⁴	#2403	1	1	yes	characteristic
242	C₂⁴	#2408	1	1	yes	characteristic
243	C₂⁴	#2410	1	1	yes	characteristic
244	C₂⁴	#2410	1	1	yes	characteristic
245	C₂⁴	#2433	1	1	yes	characteristic
246	C₂⁴	#2846	1	1	yes	characteristic
247	C₂×Q₈	#2021	1	2	yes	—
248	C₂×D₄	#2021	1	2	yes	—
249	C₂×D₄	#2021	1	2	yes	—
250	C₂×D₄	#2021	1	2	yes	—
251	C₂×D₄	#2021	1	2	yes	—
252	C₂×D₄	#2021	1	2	yes	—
253	C₂×D₄	#2021	1	2	yes	—
254	C₂×D₄	#2021	1	2	yes	—
255	C₂×D₄	#2021	1	2	yes	—
256	C₂×D₄	#2021	1	2	yes	—
257	C₂²×C₄	C₂×#392	1	1	yes	characteristic
258	C₂²×C₄	#2403	1	1	yes	characteristic
259	C₂²×C₄	#2403	1	1	yes	characteristic
260	C₂²×C₄	#2408	1	1	yes	characteristic
261	C₂²×C₄	#2410	1	1	yes	characteristic
262	C₂²×C₄	#2410	1	1	yes	characteristic
263	C₂²×C₄	#2433	1	1	yes	characteristic
264	C₂²×C₄	#2613	1	1	yes	characteristic
265	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
266	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
267	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
268	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
269	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
270	C₂²×C₄	#3026	1	2	yes	—
271	C₂³	#14715	1	1	yes	characteristic
272	C₂³	#23900	1	2	yes	—
273	C₂³	#23900	1	2	yes	—
274	C₂³	#32898	1	2	yes	—
275	D₄	#10336	1	4	yes	—
276	D₄	#10336	1	4	yes	—
277	C₂×C₄	C₂×#2021	1	1	yes	characteristic
278	C₂×C₄	C₂×#2021	1	1	yes	characteristic
279	C₂×C₄	C₂×#2021	1	1	yes	characteristic
280	C₂×C₄	C₂×#2021	1	1	yes	characteristic
281	C₂×C₄	C₂×#2021	1	1	yes	characteristic
282	C₂×C₄	#23444	1	1	yes	characteristic
283	C₂×C₄	#23900	1	2	yes	—
284	C₂×C₄	#23900	1	2	yes	—
285	C₂×C₄	#29634	1	2	yes	—
286	C₂²	#393929	1	1	yes	characteristic
287	C₂²	#394058	1	1	yes	characteristic
288	C₂²	#394058	1	1	yes	characteristic
289	C₄	C₂×#10336	1	1	yes	characteristic
290	C₄	C₂×#10336	1	1	yes	characteristic
291	C₄	#233541	1	1	yes	characteristic
292	C₄	#233541	1	1	yes	characteristic
293	C₂	#72214549	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72214549
commutator, Frattini, cosocle	C₂³×#41	C₂⁴
LCS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72240519	C₂	semidirect
2. #72240518	C₂	semidirect
3. #72240517	C₂	semidirect
4. #72240516	C₂	semidirect
5. #72240506	C₂	semidirect
6. #72240494	C₂	semidirect
7. #72240492	C₂	semidirect
8. #72240483	C₂	semidirect
9. #72238001	C₂	semidirect
10. #72234206	C₂	semidirect
11. #72233775	C₂	semidirect
12. #72233452	C₂	semidirect
13. #72216064	C₂	semidirect
14. #72214173	C₂	semidirect
15. #72213729	C₂	semidirect
19. #6437628	C₂²	semidirect
24. #6437616	C₂²	semidirect
25. #6437463	C₂²	semidirect
26. #6437449	C₂²	semidirect
28. #493402	C₂²	semidirect
29. #493396	C₂²	semidirect
30. #493396	C₂²	semidirect
31. #493390	C₂²	semidirect
32. #493390	C₂²	semidirect
33. #493390	C₂²	semidirect
37. #359562	C₂²	semidirect
38. #359551	C₂²	semidirect
39. #359513	C₂²	semidirect
40. #359507	C₂²	semidirect
41. #188071	C₂²	semidirect
42. #188029	C₂²	semidirect
43. #187756	C₂²	semidirect
44. #187756	C₂²	semidirect
45. #146483	C₂²	semidirect
46. #146477	C₂²	semidirect
47. #128042	C₂²	semidirect
48. #128040	C₂²	semidirect
49. #128020	C₂²	semidirect
50. #128016	C₂²	semidirect
51. C₂×H(7,𝔽₂)	D₄	semidirect
53. C₂⁵×D₄	D₄	semidirect
54. C₂×#3585	D₄	semidirect
55. C₂×#3585	D₄	semidirect
56. #62852	D₄	semidirect
57. #62852	D₄	semidirect
61. C₂²×#535	D₄	semidirect
63. C₂²×#525	D₄	semidirect
65. #62652	D₄	semidirect
66. C₂²×#518	D₄	semidirect
67. #60087	D₄	semidirect
68. #60087	D₄	semidirect
69. C₂×#2883	D₄	semidirect
71. #23904	D₄	semidirect
72. #23904	D₄	semidirect
74. #23901	D₄	semidirect
76. #23900	D₄	semidirect
77. C₂×#1795	D₄	semidirect
78. C₂×#1795	D₄	semidirect
79. #20605	D₄	semidirect
80. #20598	D₄	semidirect
84. #20586	D₄	semidirect
92. C₂×#585	C₂×D₄	semidirect
110. C₂×#568	C₂×D₄	semidirect
111. C₂×#568	C₂×D₄	semidirect
124. #3428	C₂×D₄	semidirect
127. C₂×#413	C₂×D₄	semidirect
130. C₂×#410	C₂×D₄	semidirect
131. C₂×#407	C₂×D₄	semidirect
132. #2280	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72240519	2
#72240518	2
#72240517	2
#72240516	2
#72240506	2
#72240494	2
#72240492	2
#72240483	2
#72238001	2
#72234206	2
#72233775	2
#72233452	2
#72216064	2
#72214173	2
#72213729	2