Subgroups — #72323801

Order 4096.

At least 12005 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

The subgroup lattice has 182 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72323800	C₂	1	4	yes	maximal, complemented
2	#72323799	C₂	1	4	yes	maximal, complemented
3	#72323226	C₂	1	1	yes	maximal, characteristic, complemented
4	#72310472	C₂	1	1	yes	maximal, characteristic, complemented
5	#72310469	C₂²	1	4	yes	complemented
6	#72309729	C₂	1	1	yes	maximal, characteristic, complemented
7	#72259008	C₂²	1	1	yes	characteristic
8	#72258757	C₂	1	1	yes	maximal, characteristic, complemented
9	#72258606	C₂²	1	2	yes	complemented
10	#72237864	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237816	C₂²	1	4	yes	complemented
12	#72237816	C₂²	1	4	yes	complemented
13	#72237381	C₂	1	1	yes	maximal, characteristic, complemented
14	#72237377	C₂²	1	4	yes	—
15	#72234625	C₂²	1	1	yes	characteristic
16	#72234118	C₂²	1	2	yes	complemented
17	#72233061	C₂²	1	2	yes	complemented
18	#72233061	C₂²	1	2	yes	complemented
19	#72226018	C₂	1	1	yes	maximal, characteristic, complemented
20	#72226015	C₂²	1	2	yes	complemented
21	#72226011	C₂²	1	2	yes	complemented
22	#72225870	C₂²	1	1	yes	characteristic
23	#72224136	C₂²	1	1	yes	characteristic
24	#72219314	C₂²	1	1	yes	characteristic
25	#72219311	C₂²	1	1	yes	characteristic
26	#72219257	C₂²	1	1	yes	characteristic
27	#10586404	D₄	1	1	yes	characteristic, complemented
28	C₂×#62817	D₄	1	1	yes	characteristic, complemented
29	C₂²×#3484	D₄	1	1	yes	characteristic, complemented
30	C₂×#34980	C₂³	1	1	yes	characteristic
31	#6499870	D₄	1	2	yes	complemented
32	#6499870	C₂³	1	2	yes	—
33	#6499870	C₂³	1	2	yes	—
34	#6499867	D₄	1	2	yes	complemented
35	#6374128	D₄	1	1	yes	characteristic, complemented
36	C₂×#21746	D₄	1	1	yes	characteristic, complemented
37	#4930495	D₄	1	1	yes	characteristic, complemented
38	#4930495	C₂³	1	1	yes	characteristic
39	#4930494	D₄	1	1	yes	characteristic, complemented
40	#4930494	C₂³	1	1	yes	characteristic
41	C₂²×#1438	D₄	1	1	yes	characteristic, complemented
42	#190116	D₄	1	2	yes	complemented
43	#188109	D₄	1	1	yes	characteristic, complemented
44	#188049	D₄	1	1	yes	characteristic, complemented
45	C₂×#8101	D₄	1	2	yes	complemented
46	#134037	C₂³	1	2	yes	—
47	#133826	C₂³	1	2	yes	—
48	#133824	D₄	1	2	yes	complemented
49	#115690	D₄	1	2	yes	complemented
50	#115631	D₄	1	2	yes	complemented
51	#115631	C₂³	1	2	yes	—
52	#115631	C₂³	1	2	yes	—
53	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
54	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
55	#62817	C₂×D₄	1	2	yes	—
56	#62817	C₂×D₄	1	2	yes	—
57	C₂²×#534	C₂×D₄	1	1	yes	characteristic
58	C₂²×#534	C₂×D₄	1	1	yes	characteristic
59	C₂³×#120	C₂×D₄	1	1	yes	characteristic
60	C₂²×#518	C₂×D₄	1	1	yes	characteristic, complemented
61	C₂×#2849	C₂×D₄	1	1	yes	characteristic
62	C₂×#2849	#41	1	1	yes	characteristic
63	C₂×#2846	C₂×D₄	1	1	yes	characteristic
64	C₂×#2846	C₂×D₄	1	1	yes	characteristic
65	C₂×#2613	C₂×D₄	1	1	yes	characteristic
66	C₂×#2613	C₂×D₄	1	1	yes	characteristic
67	C₂×#2613	C₂×D₄	1	1	yes	characteristic
68	C₂×#2613	C₂⁴	1	1	yes	characteristic
69	C₂×#2308	C₂×D₄	1	1	yes	characteristic
70	C₂×#2308	#41	1	1	yes	characteristic
71	C₂²×#379	C₂×D₄	1	1	yes	characteristic
72	#21746	C₂×D₄	1	2	yes	—
73	#21746	C₂×D₄	1	2	yes	—
74	C₂×#1438	C₂×D₄	1	1	yes	characteristic, complemented
75	C₂×#1438	C₂×D₄	1	1	yes	characteristic, complemented
76	#8548	C₂×D₄	1	2	yes	—
77	#8543	C₂×D₄	1	2	yes	—
78	#8101	C₂×D₄	1	4	yes	complemented
79	#8101	C₂×D₄	1	4	yes	complemented
80	C₂⁷	#120	1	1	yes	characteristic
81	C₂⁷	#120	1	1	yes	characteristic
82	C₂⁴×D₄	#120	1	1	yes	characteristic
83	C₂⁴×D₄	#120	1	1	yes	characteristic
84	C₂⁴×D₄	#120	1	1	yes	characteristic
85	C₂⁴×D₄	#121	1	1	yes	characteristic
86	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
87	C₂⁵×C₄	#121	1	1	yes	characteristic
88	C₂²×#123	#120	1	1	yes	characteristic
89	C₂²×#123	#121	1	1	yes	characteristic
90	C₂²×#123	#121	1	1	yes	characteristic
91	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
92	C₂²×#120	#120	1	1	yes	characteristic
93	C₂²×#120	#120	1	1	yes	characteristic
94	C₂²×#120	#120	1	1	yes	characteristic
95	C₂²×#120	#120	1	1	yes	characteristic
96	C₂²×#120	#121	1	1	yes	characteristic
97	C₂²×#120	#127	1	1	yes	characteristic
98	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
99	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
100	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
103	C₂×#381	#121	1	1	yes	characteristic
104	C₂×#381	#121	1	1	yes	characteristic
105	C₂×#381	#127	1	1	yes	characteristic
106	C₂×#381	C₂×#41	1	1	yes	characteristic
107	C₂×#379	#120	1	1	yes	characteristic
108	C₂×#379	#120	1	1	yes	characteristic
109	C₂×#379	#121	1	1	yes	characteristic
110	C₂×#379	#121	1	1	yes	characteristic
111	C₂×#379	#121	1	1	yes	characteristic
112	C₂×#379	#127	1	1	yes	characteristic
113	C₂×#379	C₂²×D₄	1	1	yes	characteristic
114	C₂×#379	C₂²×D₄	1	1	yes	characteristic
115	C₂⁶	#392	1	1	yes	characteristic
116	C₂⁶	#392	1	1	yes	characteristic
117	C₂⁶	#392	1	1	yes	characteristic
118	C₂⁶	B(4,𝔽₂)	1	2	yes	—
119	C₂⁶	B(4,𝔽₂)	1	2	yes	—
120	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
121	C₂⁶	C₂×#120	1	1	yes	characteristic
122	C₂⁶	C₂×#120	1	1	yes	characteristic
123	C₂³×D₄	#392	1	1	yes	characteristic
124	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
125	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
126	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
127	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
128	C₂³×D₄	C₂×#120	1	1	yes	characteristic
129	C₂⁴×C₄	#392	1	1	yes	characteristic
130	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
131	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
133	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
134	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
135	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
136	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
137	C₂²×C₂²⋊C₄	C₂×#127	1	1	yes	characteristic
138	C₂⁵	#2021	1	1	yes	characteristic
139	C₂⁵	#2021	1	1	yes	characteristic
140	C₂⁵	#2021	1	2	yes	—
141	C₂⁵	#2021	1	1	yes	characteristic
142	C₂⁵	#2021	1	1	yes	characteristic
143	C₂⁵	#2021	1	1	yes	characteristic
144	C₂⁵	#2021	1	1	yes	characteristic
145	C₂⁵	#2021	1	1	yes	characteristic
146	C₂⁵	#2196	1	2	yes	—
147	C₂⁵	#2196	1	2	yes	—
148	C₂⁵	#2199	1	2	yes	—
149	C₂⁵	#2199	1	2	yes	—
150	C₂⁵	C₂×#392	1	1	yes	characteristic
151	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
152	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
153	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
154	C₂³×C₄	#2021	1	2	yes	—
155	C₂³×C₄	#2021	1	1	yes	characteristic
156	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
157	C₂⁴	#10336	1	1	yes	characteristic
158	C₂⁴	#10336	1	1	yes	characteristic
159	C₂⁴	#10336	1	1	yes	characteristic
160	C₂⁴	#10336	1	1	yes	characteristic
161	C₂⁴	#13343	1	2	yes	—
162	C₂⁴	#13343	1	2	yes	—
163	C₂⁴	#13343	1	2	yes	—
164	C₂⁴	#13343	1	2	yes	—
165	C₂⁴	C₂×#2021	1	1	yes	characteristic
166	C₂⁴	C₂×#2021	1	1	yes	characteristic
167	C₂⁴	C₂×#2021	1	1	yes	characteristic
168	C₂⁴	C₂×#2196	1	2	yes	—
169	C₂⁴	C₂×#2199	1	2	yes	—
170	C₂⁴	#33551	1	4	yes	—
171	C₂³	#144366	1	2	yes	—
172	C₂³	#144366	1	1	yes	characteristic
173	C₂³	#144366	1	1	yes	characteristic
174	C₂³	C₂×#10336	1	1	yes	characteristic
175	C₂³	C₂×#13343	1	2	yes	—
176	C₂³	#498886	1	2	yes	—
177	C₂²	C₂×#144366	1	1	yes	characteristic
178	C₂²	#72218838	1	1	yes	characteristic
179	C₂²	#72236508	1	1	yes	characteristic
180	C₂	#72266315	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266315
commutator, Frattini, cosocle	C₂×#2613	C₂⁴
LCS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72323800	C₂	semidirect
2. #72323799	C₂	semidirect
3. #72323226	C₂	semidirect
4. #72310472	C₂	semidirect
5. #72310469	C₂²	semidirect
6. #72309729	C₂	semidirect
8. #72258757	C₂	semidirect
9. #72258606	C₂²	semidirect
10. #72237864	C₂	semidirect
11. #72237816	C₂²	semidirect
12. #72237816	C₂²	semidirect
13. #72237381	C₂	semidirect
16. #72234118	C₂²	semidirect
17. #72233061	C₂²	semidirect
18. #72233061	C₂²	semidirect
19. #72226018	C₂	semidirect
20. #72226015	C₂²	semidirect
21. #72226011	C₂²	semidirect
27. #10586404	D₄	semidirect
28. C₂×#62817	D₄	semidirect
29. C₂²×#3484	D₄	semidirect
31. #6499870	D₄	semidirect
34. #6499867	D₄	semidirect
35. #6374128	D₄	semidirect
36. C₂×#21746	D₄	semidirect
37. #4930495	D₄	semidirect
39. #4930494	D₄	semidirect
41. C₂²×#1438	D₄	semidirect
42. #190116	D₄	semidirect
43. #188109	D₄	semidirect
44. #188049	D₄	semidirect
45. C₂×#8101	D₄	semidirect
48. #133824	D₄	semidirect
49. #115690	D₄	semidirect
50. #115631	D₄	semidirect
53. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
60. C₂²×#518	C₂×D₄	semidirect
74. C₂×#1438	C₂×D₄	semidirect
75. C₂×#1438	C₂×D₄	semidirect
78. #8101	C₂×D₄	semidirect
79. #8101	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72323800	2
#72323799	2
#72323226	2
#72310472	2
#72309729	2
#72258757	2
#72237864	2
#72237381	2
#72226018	2