Subgroups — #72340839

Order 2048.

At least 6147 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

The subgroup lattice has 231 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72340837	C₂	1	1	yes	maximal, characteristic, complemented
2	#72340834	C₂	1	1	yes	maximal, characteristic, complemented
3	#72340833	C₂	1	1	yes	maximal, characteristic, complemented
4	#72338523	C₂	1	1	yes	maximal, characteristic, complemented
5	#72286706	C₂	1	1	yes	maximal, characteristic, complemented
6	#72286397	C₂	1	1	yes	maximal, characteristic, complemented
7	#72286088	C₂	1	1	yes	maximal, characteristic, complemented
8	#72284828	C₂	1	1	yes	maximal, characteristic, complemented
9	#72284524	C₂	1	1	yes	maximal, characteristic, complemented
10	#72279397	C₂	1	1	yes	maximal, characteristic, complemented
11	#72279362	C₂	1	1	yes	maximal, characteristic, complemented
12	#72279319	C₂	1	1	yes	maximal, characteristic, complemented
13	#72279311	C₂	1	1	yes	maximal, characteristic, complemented
14	#72251988	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#511819	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#63011	C₂²	1	1	yes	characteristic
17	C₂×#33570	C₂²	1	1	yes	characteristic
18	C₂×#33559	C₂²	1	1	yes	characteristic
19	C₂×#33546	C₂²	1	1	yes	characteristic
20	C₂×#33546	C₂²	1	1	yes	characteristic
21	#7504291	C₂²	1	1	yes	characteristic, complemented
22	#7504286	C₂²	1	1	yes	characteristic, complemented
23	#7503371	C₂²	1	1	yes	characteristic, complemented
24	#6446716	C₂²	1	1	yes	characteristic
25	C₂×#23872	C₂²	1	1	yes	characteristic
26	#6437452	C₂²	1	1	yes	characteristic
27	#6351567	C₂²	1	1	yes	characteristic
28	#6348078	C₂²	1	1	yes	characteristic
29	#512635	C₂²	1	1	yes	characteristic, complemented
30	#512625	C₂²	1	1	yes	characteristic, complemented
31	#512585	C₂²	1	1	yes	characteristic, complemented
32	#512578	C₂²	1	1	yes	characteristic, complemented
33	#512578	C₂²	1	1	yes	characteristic, complemented
34	#512564	C₂²	1	1	yes	characteristic, complemented
35	#493630	C₂²	1	1	yes	characteristic
36	C₂×#12778	C₂²	1	1	yes	characteristic
37	C₂×#12765	C₂²	1	1	yes	characteristic
38	#493577	C₂²	1	1	yes	characteristic
39	#493431	C₂²	1	1	yes	characteristic
40	C₂×#12718	C₂²	1	1	yes	characteristic
41	#357371	C₂²	1	1	yes	characteristic, complemented
42	#187899	C₂²	1	1	yes	characteristic
43	#187081	C₂²	1	1	yes	characteristic
44	#186576	C₂²	1	1	yes	characteristic
45	#151858	C₂²	1	1	yes	characteristic, complemented
46	#151842	C₂²	1	1	yes	characteristic, complemented
47	#151818	C₂²	1	1	yes	characteristic, complemented
48	#151816	C₂²	1	1	yes	characteristic, complemented
49	#151814	C₂²	1	1	yes	characteristic, complemented
50	#151806	C₂²	1	1	yes	characteristic, complemented
51	C₂×#3585	D₄	1	1	yes	characteristic, complemented
52	#62852	D₄	1	1	yes	characteristic, complemented
53	C₂×#3498	C₂³	1	1	yes	characteristic
54	C₂×#3549	C₂³	1	1	yes	characteristic
55	C₂×#3481	C₂³	1	1	yes	characteristic
56	#60618	C₂³	1	1	yes	characteristic
57	#60484	D₄	1	1	yes	characteristic, complemented
58	C₂×#3027	D₄	1	1	yes	characteristic, complemented
59	C₂×#3021	D₄	1	1	yes	characteristic, complemented
60	C₂×#3018	C₂³	1	1	yes	characteristic
61	C₂×#3018	C₂³	1	1	yes	characteristic
62	#35285	D₄	1	1	yes	characteristic, complemented
63	C₂×#2132	C₂³	1	1	yes	characteristic
64	C₂×#2126	C₂³	1	1	yes	characteristic
65	C₂×#2126	C₂³	1	1	yes	characteristic
66	C₂×#2126	C₂³	1	1	yes	characteristic
67	C₂×#1965	D₄	1	1	yes	characteristic, complemented
68	#21209	C₂³	1	1	yes	characteristic
69	C₂×#1914	D₄	1	1	yes	characteristic, complemented
70	#20956	D₄	1	1	yes	characteristic, complemented
71	#20952	C₂³	1	1	yes	characteristic
72	C₂×#1884	C₂³	1	1	yes	characteristic
73	C₂×#1795	D₄	1	1	yes	characteristic, complemented
74	#20592	D₄	1	1	yes	characteristic, complemented
75	C₄×#421	D₄	1	1	yes	characteristic, complemented
76	#17738	C₂³	1	1	yes	characteristic
77	#17663	C₂³	1	1	yes	characteristic
78	#12778	D₄	1	4	yes	complemented
79	#12765	D₄	1	4	yes	complemented
80	#12763	D₄	1	4	yes	complemented
81	#12760	D₄	1	2	yes	complemented
82	#12759	D₄	1	2	yes	complemented
83	#8530	D₄	1	4	yes	complemented
84	#8440	D₄	1	4	yes	complemented
85	C₂×#585	C₂×D₄	1	1	yes	characteristic
86	C₂×#585	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
89	C₂×#568	C₂×D₄	1	1	yes	characteristic
90	#3549	C₂×D₄	1	2	yes	—
91	#3549	C₂×D₄	1	2	yes	—
92	C₂×H(3,𝔽₄)	C₂×D₄	1	2	yes	—
93	#3481	C₂×D₄	1	2	yes	—
94	#3481	C₂×D₄	1	2	yes	—
95	C₂×#559	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄²	C₂×D₄	1	1	yes	characteristic
97	C₂×#535	C₂×D₄	1	1	yes	characteristic
98	C₂×#535	C₂×D₄	1	1	yes	characteristic
99	C₂×#533	C₂×D₄	1	1	yes	characteristic
100	C₂×#532	C₂⁴	1	1	yes	characteristic
101	C₂²×#127	C₂×D₄	1	1	yes	characteristic
102	#3428	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×#41	C₂×D₄	1	1	yes	characteristic
104	C₂×#423	C₂×D₄	1	1	yes	characteristic
105	C₂×#413	C₂×D₄	1	1	yes	characteristic
106	C₂×#413	C₂×D₄	1	1	yes	characteristic
107	C₂×#413	C₂×D₄	1	1	yes	characteristic
108	C₂×#410	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×M₁₆	C₂×D₄	1	1	yes	characteristic
110	C₄×#117	C₂×D₄	1	1	yes	characteristic
111	#2132	C₂×D₄	1	2	yes	—
112	#2126	C₂×D₄	1	2	yes	—
113	C₂×#354	C₂×D₄	1	2	yes	—
114	C₂×#353	C₂×D₄	1	2	yes	—
115	#1962	C₂×D₄	1	2	yes	—
116	#1962	C₂×D₄	1	2	yes	—
117	#1887	C₂×D₄	1	2	yes	—
118	#1885	C₂×D₄	1	2	yes	—
119	#1884	C₂×D₄	1	2	yes	—
120	#1884	C₂×D₄	1	2	yes	—
121	#1560	C₂×D₄	1	2	yes	—
122	#1525	C₂×D₄	1	2	yes	—
123	C₂²×#41	#120	1	1	yes	characteristic
124	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
125	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
126	C₂³×D₄	#120	1	1	yes	characteristic
127	C₂³×D₄	#120	1	1	yes	characteristic
128	#568	#120	1	2	yes	complemented
129	H(3,𝔽₄)	#127	1	4	yes	—
130	#559	#127	1	2	yes	—
131	D₄²	#127	1	2	yes	—
132	#532	C₂²×D₄	1	1	yes	characteristic
133	#532	C₂²×D₄	1	1	yes	characteristic
134	#532	C₂²×D₄	1	1	yes	characteristic
135	#532	C₂²×D₄	1	1	yes	characteristic
136	#532	H(5,𝔽₂)	1	1	yes	characteristic
137	#532	H(5,𝔽₂)	1	1	yes	characteristic
138	#532	H(5,𝔽₂)	1	1	yes	characteristic
139	#532	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄⋊Q₈	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×#127	C₂²×D₄	1	1	yes	characteristic
142	C₂×#124	C₂²×D₄	1	1	yes	characteristic
143	C₂×#124	H(5,𝔽₂)	1	1	yes	characteristic
144	#518	#120	1	2	yes	—
145	C₂×#117	#120	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×D₄	1	1	yes	characteristic
147	#423	#120	1	2	yes	complemented
148	C₂×#100	#120	1	2	yes	—
149	#410	#120	1	2	yes	complemented
150	C₄³	#120	1	2	yes	complemented
151	C₂⁵	B(4,𝔽₂)	1	2	yes	—
152	C₂×#41	C₂×#120	1	1	yes	characteristic
153	C₂×#41	#534	1	1	yes	characteristic
154	C₂×#41	D₄²	1	1	yes	characteristic
155	C₂×#41	D₄²	1	1	yes	characteristic
156	C₂×#41	D₄²	1	1	yes	characteristic
157	C₂×#41	D₄²	1	1	yes	characteristic
158	C₂×#41	D₄²	1	1	yes	characteristic
159	C₂×#41	D₄²	1	1	yes	characteristic
160	C₂²×Q₈	B(4,𝔽₂)	1	1	yes	characteristic
161	C₂²×Q₈	#534	1	1	yes	characteristic
162	C₂²×Q₈	D₄²	1	1	yes	characteristic
163	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
164	C₂²×D₄	B(4,𝔽₂)	1	1	yes	characteristic
165	C₂²×D₄	C₂×#120	1	1	yes	characteristic
166	C₂²×D₄	D₄²	1	1	yes	characteristic
167	C₂³×C₄	C₂×#120	1	1	yes	characteristic
168	C₄⋊Q₈	#535	1	1	yes	characteristic
169	C₄⋊Q₈	#535	1	1	yes	characteristic
170	C₄⋊Q₈	D₄²	1	1	yes	characteristic
171	C₄⋊Q₈	#560	1	1	yes	characteristic
172	#127	C₂×#127	1	1	yes	characteristic
173	#127	#535	1	1	yes	characteristic
174	#127	D₄²	1	1	yes	characteristic
175	#127	D₄²	1	1	yes	characteristic
176	#124	C₂×#127	1	1	yes	characteristic
177	#124	#535	1	1	yes	characteristic
178	#124	#535	1	1	yes	characteristic
179	#124	#535	1	1	yes	characteristic
180	#124	D₄²	1	1	yes	characteristic
181	#124	D₄²	1	1	yes	characteristic
182	#124	D₄²	1	1	yes	characteristic
183	#124	#560	1	1	yes	characteristic
184	#117	B(4,𝔽₂)	1	2	yes	—
185	C₂×C₄²	C₂×#120	1	1	yes	characteristic
186	C₂×C₄²	#534	1	1	yes	characteristic
187	C₂×C₄²	D₄²	1	1	yes	characteristic
188	C₂×C₄²	D₄²	1	1	yes	characteristic
189	C₂×C₄²	D₄²	1	1	yes	characteristic
190	C₂×C₄²	D₄²	1	1	yes	characteristic
191	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
192	C₂×Q₈	#2433	1	1	yes	characteristic
193	C₂×Q₈	#2669	1	1	yes	characteristic
194	C₂×Q₈	#2669	1	1	yes	characteristic
195	C₂×Q₈	#2669	1	1	yes	characteristic
196	C₂×D₄	#2199	1	2	yes	—
197	C₂×D₄	#2433	1	1	yes	characteristic
198	C₂×D₄	#2433	1	1	yes	characteristic
199	C₂×D₄	#2433	1	1	yes	characteristic
200	C₂×D₄	#2669	1	1	yes	characteristic
201	C₂²×C₄	#2403	1	1	yes	characteristic
202	C₂²×C₄	#2403	1	1	yes	characteristic
203	C₂²×C₄	#2403	1	1	yes	characteristic
204	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
205	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
206	C₂²×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
207	C₄²	#2433	1	1	yes	characteristic
208	C₄²	#2433	1	1	yes	characteristic
209	C₄²	#2669	1	1	yes	characteristic
210	C₄²	#2669	1	1	yes	characteristic
211	C₂³	#23900	1	1	yes	characteristic
212	C₂³	#23900	1	1	yes	characteristic
213	C₂³	#23900	1	1	yes	characteristic
214	C₂×C₄	#15258	1	1	yes	characteristic
215	C₂×C₄	#23900	1	1	yes	characteristic
216	C₂×C₄	#23900	1	1	yes	characteristic
217	C₂×C₄	#23900	1	1	yes	characteristic
218	C₂×C₄	C₂×#2199	1	1	yes	characteristic
219	C₂×C₄	#33551	1	1	yes	characteristic
220	C₂×C₄	#33551	1	1	yes	characteristic
221	C₂×C₄	#33558	1	1	yes	characteristic
222	C₂²	#417744	1	1	yes	characteristic
223	C₂²	#499795	1	1	yes	characteristic
224	C₂²	#499807	1	1	yes	characteristic
225	C₄	#499795	1	1	yes	characteristic
226	C₄	#499807	1	1	yes	characteristic
227	C₄	#499832	1	1	yes	characteristic
228	C₄	#499852	1	1	yes	characteristic
229	C₂	#72318321	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72318321
commutator, Frattini, cosocle	C₂×#532	C₂⁴
LCS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72340837	C₂	semidirect
2. #72340834	C₂	semidirect
3. #72340833	C₂	semidirect
4. #72338523	C₂	semidirect
5. #72286706	C₂	semidirect
6. #72286397	C₂	semidirect
7. #72286088	C₂	semidirect
8. #72284828	C₂	semidirect
9. #72284524	C₂	semidirect
10. #72279397	C₂	semidirect
11. #72279362	C₂	semidirect
12. #72279319	C₂	semidirect
13. #72279311	C₂	semidirect
14. #72251988	C₂	semidirect
15. C₂×#511819	C₂	semidirect
21. #7504291	C₂²	semidirect
22. #7504286	C₂²	semidirect
23. #7503371	C₂²	semidirect
29. #512635	C₂²	semidirect
30. #512625	C₂²	semidirect
31. #512585	C₂²	semidirect
32. #512578	C₂²	semidirect
33. #512578	C₂²	semidirect
34. #512564	C₂²	semidirect
41. #357371	C₂²	semidirect
45. #151858	C₂²	semidirect
46. #151842	C₂²	semidirect
47. #151818	C₂²	semidirect
48. #151816	C₂²	semidirect
49. #151814	C₂²	semidirect
50. #151806	C₂²	semidirect
51. C₂×#3585	D₄	semidirect
52. #62852	D₄	semidirect
57. #60484	D₄	semidirect
58. C₂×#3027	D₄	semidirect
59. C₂×#3021	D₄	semidirect
62. #35285	D₄	semidirect
67. C₂×#1965	D₄	semidirect
69. C₂×#1914	D₄	semidirect
70. #20956	D₄	semidirect
73. C₂×#1795	D₄	semidirect
74. #20592	D₄	semidirect
75. C₄×#421	D₄	semidirect
78. #12778	D₄	semidirect
79. #12765	D₄	semidirect
80. #12763	D₄	semidirect
81. #12760	D₄	semidirect
82. #12759	D₄	semidirect
83. #8530	D₄	semidirect
84. #8440	D₄	semidirect
128. #568	#120	semidirect
147. #423	#120	semidirect
149. #410	#120	semidirect
150. C₄³	#120	semidirect

Maximal subgroups

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

subgroup	index
#72340837	2
#72340834	2
#72340833	2
#72338523	2
#72286706	2
#72286397	2
#72286088	2
#72284828	2
#72284524	2
#72279397	2
#72279362	2
#72279319	2
#72279311	2
#72251988	2
C₂×#511819	2