Subgroups — #72340497

Order 4096.

At least 9781 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

The subgroup lattice has 201 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72340496	C₂	1	1	yes	maximal, characteristic, complemented
2	#72340495	C₂	1	1	yes	maximal, characteristic, complemented
3	#72340298	C₂	1	1	yes	maximal, characteristic, complemented
4	#72340294	C₂	1	1	yes	maximal, characteristic, complemented
5	#72340292	C₂²	1	1	yes	characteristic
6	#72311067	C₂	1	1	yes	maximal, characteristic, complemented
7	#72311058	C₂²	1	1	yes	characteristic
8	#72311052	C₂	1	1	yes	maximal, characteristic, complemented
9	#72311044	C₂	1	1	yes	maximal, characteristic, complemented
10	#72311038	C₂²	1	1	yes	characteristic
11	#72310948	C₂	1	1	yes	maximal, characteristic, complemented
12	#72310930	C₂²	1	1	yes	characteristic, complemented
13	#72310928	C₂	1	1	yes	maximal, characteristic, complemented
14	#72310922	C₂²	1	1	yes	characteristic, complemented
15	#72310922	C₂²	1	1	yes	characteristic, complemented
16	#72310920	C₂²	1	1	yes	characteristic
17	#72310916	C₂²	1	1	yes	characteristic, complemented
18	#72286031	C₂	1	1	yes	maximal, characteristic, complemented
19	#72285777	C₂	1	1	yes	maximal, characteristic, complemented
20	#72284988	C₂²	1	1	yes	characteristic
21	#72284657	C₂²	1	1	yes	characteristic, complemented
22	#72284221	C₂²	1	1	yes	characteristic, complemented
23	#72284136	C₂	1	1	yes	maximal, characteristic, complemented
24	#72283936	C₂²	1	1	yes	characteristic, complemented
25	#72283788	C₂²	1	1	yes	characteristic, complemented
26	#72283662	C₂	1	1	yes	maximal, characteristic, complemented
27	#72283640	C₂²	1	1	yes	characteristic, complemented
28	#72283614	C₂²	1	1	yes	characteristic, complemented
29	#72283614	C₂²	1	1	yes	characteristic, complemented
30	#72283582	C₂²	1	1	yes	characteristic, complemented
31	#72283582	C₂²	1	1	yes	characteristic, complemented
32	#72283562	C₂²	1	1	yes	characteristic, complemented
33	#72283505	C₂²	1	1	yes	characteristic, complemented
34	#72283505	C₂²	1	1	yes	characteristic, complemented
35	#72283484	C₂²	1	1	yes	characteristic
36	#72252351	C₂	1	1	yes	maximal, characteristic, complemented
37	#72247784	C₂	1	1	yes	maximal, characteristic, complemented
38	#72241392	C₂²	1	1	yes	characteristic
39	#72241392	C₂²	1	1	yes	characteristic
40	#72241386	C₂²	1	1	yes	characteristic
41	#72237008	C₂²	1	1	yes	characteristic
42	#72237008	C₂²	1	1	yes	characteristic
43	#72236934	C₂²	1	1	yes	characteristic
44	#72236934	C₂²	1	1	yes	characteristic
45	#72213808	C₂²	1	1	yes	characteristic
46	#72213804	C₂²	1	1	yes	characteristic
47	#72213791	C₂²	1	1	yes	characteristic
48	#72213786	C₂²	1	1	yes	characteristic
49	#72213523	C₂²	1	1	yes	characteristic
50	#72213513	C₂²	1	1	yes	characteristic
51	#8933820	D₄	1	1	yes	characteristic, complemented
52	#8932958	D₄	1	1	yes	characteristic
53	#8932887	D₄	1	1	yes	characteristic, complemented
54	#6628902	C₂³	1	1	yes	characteristic
55	#6628897	D₄	1	1	yes	characteristic
56	#6628896	D₄	1	1	yes	characteristic
57	#6437630	C₂³	1	1	yes	characteristic
58	#356056	C₂³	1	1	yes	characteristic
59	#346279	D₄	1	1	yes	characteristic, complemented
60	#346278	D₄	1	1	yes	characteristic, complemented
61	#346276	D₄	1	1	yes	characteristic
62	#346276	D₄	1	1	yes	characteristic
63	#346269	D₄	1	1	yes	characteristic
64	#346267	D₄	1	1	yes	characteristic
65	#346265	D₄	1	1	yes	characteristic
66	#346259	D₄	1	1	yes	characteristic, complemented
67	#346252	D₄	1	1	yes	characteristic
68	#346250	D₄	1	1	yes	characteristic, complemented
69	#346239	D₄	1	1	yes	characteristic, complemented
70	#346231	D₄	1	1	yes	characteristic, complemented
71	#251855	D₄	1	1	yes	characteristic, complemented
72	#251855	C₂³	1	1	yes	characteristic
73	#251850	D₄	1	1	yes	characteristic, complemented
74	#251850	C₂³	1	1	yes	characteristic
75	#251833	D₄	1	1	yes	characteristic, complemented
76	#251833	C₂³	1	1	yes	characteristic
77	#251828	D₄	1	1	yes	characteristic, complemented
78	#251828	C₂³	1	1	yes	characteristic
79	#245243	D₄	1	1	yes	characteristic, complemented
80	#245243	D₄	1	1	yes	characteristic, complemented
81	#245236	D₄	1	1	yes	characteristic
82	#245225	D₄	1	1	yes	characteristic
83	#213863	D₄	1	1	yes	characteristic, complemented
84	#213862	D₄	1	1	yes	characteristic
85	#213862	D₄	1	1	yes	characteristic, complemented
86	#213855	C₂³	1	1	yes	characteristic
87	#213854	C₂³	1	1	yes	characteristic
88	#149742	C₂³	1	1	yes	characteristic
89	#149742	C₂³	1	1	yes	characteristic
90	#149742	C₂³	1	1	yes	characteristic
91	#149742	C₂³	1	1	yes	characteristic
92	#128062	C₂³	1	1	yes	characteristic
93	#128062	C₂³	1	1	yes	characteristic
94	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
95	#47563	C₂×D₄	1	1	yes	characteristic
96	#47195	C₂×D₄	1	1	yes	characteristic
97	#46998	C₂×D₄	1	1	yes	characteristic
98	#46384	C₂×D₄	1	1	yes	characteristic
99	#35616	C₂×D₄	1	1	yes	characteristic
100	#35616	C₂×D₄	1	1	yes	characteristic
101	#35616	C₂×D₄	1	1	yes	characteristic
102	#35446	C₂×D₄	1	1	yes	characteristic
103	#35346	C₂×D₄	1	1	yes	characteristic
104	#20586	C₂×D₄	1	1	yes	characteristic
105	#20586	C₂×D₄	1	1	yes	characteristic
106	#20586	C₂⁴	1	1	yes	characteristic
107	#20585	C₂×D₄	1	1	yes	characteristic
108	#20585	C₂×D₄	1	1	yes	characteristic
109	#10392	C₂×D₄	1	1	yes	characteristic
110	#10392	C₂×D₄	1	1	yes	characteristic
111	#10392	C₂×D₄	1	1	yes	characteristic
112	#10392	C₂×D₄	1	1	yes	characteristic
113	#10390	C₂×D₄	1	1	yes	characteristic
114	#10390	C₂×D₄	1	1	yes	characteristic
115	#10390	C₂×D₄	1	1	yes	characteristic
116	#10390	C₂×D₄	1	1	yes	characteristic
117	#10338	C₂×D₄	1	1	yes	characteristic
118	#10338	C₂×D₄	1	1	yes	characteristic
119	#10338	C₂×D₄	1	1	yes	characteristic
120	#10338	C₂×D₄	1	1	yes	characteristic
121	#10336	C₂×D₄	1	1	yes	characteristic
122	#10336	C₂×D₄	1	1	yes	characteristic
123	#10336	C₂×D₄	1	1	yes	characteristic
124	#10336	C₂×D₄	1	1	yes	characteristic
125	#8748	C₂×D₄	1	1	yes	characteristic
126	#8748	C₂×D₄	1	1	yes	characteristic
127	#8748	C₂×D₄	1	1	yes	characteristic
128	#8748	C₂×D₄	1	1	yes	characteristic
129	#8747	C₂×D₄	1	1	yes	characteristic
130	#8747	C₂×D₄	1	1	yes	characteristic
131	#8747	C₂×D₄	1	1	yes	characteristic
132	#8747	C₂×D₄	1	1	yes	characteristic
133	#8543	C₂×D₄	1	1	yes	characteristic
134	#8543	C₂×D₄	1	1	yes	characteristic
135	#8541	C₂×D₄	1	1	yes	characteristic
136	#8541	C₂×D₄	1	1	yes	characteristic
137	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
138	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
139	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
140	C₂²×#124	#120	1	1	yes	characteristic
141	C₂²×#123	#120	1	1	yes	characteristic
142	C₂²×#120	#120	1	1	yes	characteristic
143	#2842	#120	1	1	yes	characteristic
144	#2842	#120	1	1	yes	characteristic
145	#2840	#120	1	1	yes	characteristic
146	#2840	#120	1	1	yes	characteristic
147	#2483	#120	1	1	yes	characteristic
148	#2483	#120	1	1	yes	characteristic
149	#2403	#120	1	1	yes	characteristic
150	#2403	#120	1	1	yes	characteristic
151	#2309	#120	1	1	yes	characteristic
152	#2309	C₂²×D₄	1	1	yes	characteristic
153	#2309	C₂²×D₄	1	1	yes	characteristic
154	#2308	#120	1	1	yes	characteristic
155	#2308	#120	1	1	yes	characteristic
156	#1781	#120	1	1	yes	characteristic
157	#1781	#120	1	1	yes	characteristic
158	#1781	#120	1	1	yes	characteristic
159	#1781	#120	1	1	yes	characteristic
160	#1781	C₂²×D₄	1	1	yes	characteristic
161	#1781	C₂²×D₄	1	1	yes	characteristic
162	#1781	C₂²×D₄	1	1	yes	characteristic
163	#1781	C₂²×D₄	1	1	yes	characteristic
164	#1304	#120	1	1	yes	characteristic
165	#1304	#120	1	1	yes	characteristic
166	#1304	#120	1	1	yes	characteristic
167	#1304	#120	1	1	yes	characteristic
168	#1304	#120	1	1	yes	characteristic
169	#1304	#120	1	1	yes	characteristic
170	#1304	#120	1	1	yes	characteristic
171	#1304	#120	1	1	yes	characteristic
172	C₂⁶	#453	1	1	yes	characteristic
173	C₂³×Q₈	B(4,𝔽₂)	1	1	yes	characteristic
174	C₂³×D₄	C₂×#120	1	1	yes	characteristic
175	C₂³×D₄	C₂×#120	1	1	yes	characteristic
176	C₂³×D₄	C₂×#120	1	1	yes	characteristic
177	C₂⁴×C₄	#453	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	#453	1	1	yes	characteristic
179	C₂²×C₂²⋊C₄	#453	1	1	yes	characteristic
180	C₂²×C₄²	B(4,𝔽₂)	1	1	yes	characteristic
181	#379	C₂×#120	1	1	yes	characteristic
182	#379	C₂×#120	1	1	yes	characteristic
183	#379	C₂×#120	1	1	yes	characteristic
184	#379	C₂×#120	1	1	yes	characteristic
185	C₂⁵	#2846	1	1	yes	characteristic
186	C₂⁵	C₂×#453	1	1	yes	characteristic
187	C₂⁵	C₂×#453	1	1	yes	characteristic
188	C₂⁵	#3026	1	1	yes	characteristic
189	C₂³×C₄	#3019	1	1	yes	characteristic
190	C₂³×C₄	#3019	1	1	yes	characteristic
191	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
192	C₂⁴	#32888	1	1	yes	characteristic
193	C₂⁴	#32888	1	1	yes	characteristic
194	C₂⁴	#32898	1	1	yes	characteristic
195	C₂³	#346446	1	1	yes	characteristic
196	C₂³	#511790	1	1	yes	characteristic
197	C₂³	#511790	1	1	yes	characteristic
198	C₂²	#72216464	1	1	yes	characteristic
199	C₂	#72216466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72216466
commutator, Frattini, cosocle	#20586	C₂⁴
LCS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ #6628902 ◁ G
LECS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72340496	C₂	semidirect
2. #72340495	C₂	semidirect
3. #72340298	C₂	semidirect
4. #72340294	C₂	semidirect
6. #72311067	C₂	semidirect
8. #72311052	C₂	semidirect
9. #72311044	C₂	semidirect
11. #72310948	C₂	semidirect
12. #72310930	C₂²	semidirect
13. #72310928	C₂	semidirect
14. #72310922	C₂²	semidirect
15. #72310922	C₂²	semidirect
17. #72310916	C₂²	semidirect
18. #72286031	C₂	semidirect
19. #72285777	C₂	semidirect
21. #72284657	C₂²	semidirect
22. #72284221	C₂²	semidirect
23. #72284136	C₂	semidirect
24. #72283936	C₂²	semidirect
25. #72283788	C₂²	semidirect
26. #72283662	C₂	semidirect
27. #72283640	C₂²	semidirect
28. #72283614	C₂²	semidirect
29. #72283614	C₂²	semidirect
30. #72283582	C₂²	semidirect
31. #72283582	C₂²	semidirect
32. #72283562	C₂²	semidirect
33. #72283505	C₂²	semidirect
34. #72283505	C₂²	semidirect
36. #72252351	C₂	semidirect
37. #72247784	C₂	semidirect
51. #8933820	D₄	semidirect
53. #8932887	D₄	semidirect
59. #346279	D₄	semidirect
60. #346278	D₄	semidirect
66. #346259	D₄	semidirect
68. #346250	D₄	semidirect
69. #346239	D₄	semidirect
70. #346231	D₄	semidirect
71. #251855	D₄	semidirect
73. #251850	D₄	semidirect
75. #251833	D₄	semidirect
77. #251828	D₄	semidirect
79. #245243	D₄	semidirect
80. #245243	D₄	semidirect
83. #213863	D₄	semidirect
85. #213862	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72340496	2
#72340495	2
#72340298	2
#72340294	2
#72311067	2
#72311052	2
#72311044	2
#72310948	2
#72310928	2
#72286031	2
#72285777	2
#72284136	2
#72283662	2
#72252351	2
#72247784	2