Subgroups — #72238387

Order 4096.

At least 10385 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

The subgroup lattice has 169 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72238386	C₂	1	4	yes	maximal, complemented
2	#72238385	C₂	1	4	yes	maximal, complemented
3	#72238383	C₂	1	1	yes	maximal, characteristic, complemented
4	#72238378	C₂	1	1	yes	maximal, characteristic, complemented
5	#72238377	C₂²	1	4	yes	complemented
6	#72238371	C₂	1	1	yes	maximal, characteristic, complemented
7	#72238368	C₂²	1	2	yes	complemented
8	#72238367	C₂²	1	2	yes	complemented
9	#72237432	C₂	1	1	yes	maximal, characteristic, complemented
10	#72237427	C₂²	1	4	yes	complemented
11	#72237427	C₂²	1	4	yes	complemented
12	#72237381	C₂	1	1	yes	maximal, characteristic, complemented
13	#72237377	C₂²	1	4	yes	—
14	#72237368	C₂	1	1	yes	maximal, characteristic, complemented
15	#72237364	C₂²	1	2	yes	complemented
16	#72235718	C₂²	1	1	yes	characteristic
17	#72234717	C₂	1	1	yes	maximal, characteristic, complemented
18	#72234716	C₂²	1	1	yes	characteristic
19	#72234711	C₂²	1	2	yes	complemented
20	#72234711	C₂²	1	2	yes	complemented
21	#72234710	C₂²	1	2	yes	complemented
22	#72234625	C₂²	1	1	yes	characteristic
23	#72225870	C₂²	1	1	yes	characteristic
24	#72225870	C₂²	1	1	yes	characteristic
25	#72224136	C₂²	1	1	yes	characteristic
26	#72219314	C₂²	1	1	yes	characteristic
27	#10586404	D₄	1	1	yes	characteristic
28	C₂×#62817	D₄	1	1	yes	characteristic
29	C₂×#47031	D₄	1	1	yes	characteristic, complemented
30	C₂×#34980	C₂³	1	1	yes	characteristic
31	#6499870	D₄	1	2	yes	—
32	#6499870	C₂³	1	2	yes	—
33	#6499870	C₂³	1	2	yes	—
34	#6499867	D₄	1	2	yes	—
35	#6374128	D₄	1	1	yes	characteristic
36	C₂×#21746	D₄	1	1	yes	characteristic
37	#5025772	C₂³	1	1	yes	characteristic
38	#5020747	D₄	1	1	yes	characteristic, complemented
39	#4435227	C₂³	1	1	yes	characteristic
40	#686193	D₄	1	1	yes	characteristic, complemented
41	#576189	D₄	1	1	yes	characteristic, complemented
42	#480250	D₄	1	2	yes	complemented
43	C₂×#11499	D₄	1	2	yes	complemented
44	#188109	D₄	1	1	yes	characteristic
45	#188049	D₄	1	1	yes	characteristic
46	#134037	D₄	1	2	yes	complemented
47	#134037	C₂³	1	2	yes	—
48	#134036	C₂³	1	2	yes	—
49	#134036	C₂³	1	2	yes	—
50	#134033	D₄	1	2	yes	complemented
51	#133826	C₂³	1	2	yes	—
52	#133824	D₄	1	2	yes	—
53	C₂⁵×D₄	#41	1	1	yes	characteristic
54	#62817	C₂×D₄	1	2	yes	—
55	#62817	C₂×D₄	1	2	yes	—
56	C₂²×#534	C₂×D₄	1	1	yes	characteristic
57	C₂²×#534	C₂×D₄	1	1	yes	characteristic
58	C₂³×#120	C₂×D₄	1	1	yes	characteristic
59	C₂×#2815	C₂×D₄	1	1	yes	characteristic, complemented
60	C₂×#2849	C₂×D₄	1	1	yes	characteristic
61	C₂×#2849	C₂×D₄	1	1	yes	characteristic
62	C₂×#2846	C₂×D₄	1	1	yes	characteristic
63	C₂×#2846	C₂×D₄	1	1	yes	characteristic
64	C₂×#2607	C₂×D₄	1	1	yes	characteristic, complemented
65	C₂×#2613	C₂×D₄	1	1	yes	characteristic
66	C₂×#2613	C₂×D₄	1	1	yes	characteristic
67	C₂×#2613	#41	1	1	yes	characteristic
68	C₂×#2613	C₂⁴	1	1	yes	characteristic
69	C₂×#2308	C₂×D₄	1	1	yes	characteristic
70	C₂×#2308	C₂×D₄	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	#13828	C₂×D₄	1	1	yes	characteristic, complemented
75	#13807	C₂×D₄	1	1	yes	characteristic, complemented
76	#11499	C₂×D₄	1	4	yes	complemented
77	#11499	C₂×D₄	1	4	yes	complemented
78	#8548	C₂×D₄	1	2	yes	—
79	#8543	C₂×D₄	1	2	yes	—
80	C₂⁷	#121	1	1	yes	characteristic
81	C₂⁷	#121	1	1	yes	characteristic
82	C₂⁴×D₄	#120	1	1	yes	characteristic
83	C₂⁴×D₄	#121	1	1	yes	characteristic
84	C₂⁴×D₄	#121	1	1	yes	characteristic
85	C₂⁴×D₄	#121	1	1	yes	characteristic
86	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
87	C₂⁵×C₄	#121	1	1	yes	characteristic
88	C₂²×#123	#120	1	1	yes	characteristic
89	C₂²×#123	#120	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	#121	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	#120	1	1	yes	characteristic
101	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
102	C₂³×C₂²⋊C₄	#127	1	1	yes	characteristic
103	C₂×#381	#120	1	1	yes	characteristic
104	C₂×#381	#121	1	1	yes	characteristic
105	C₂×#381	#127	1	1	yes	characteristic
106	C₂×#381	C₂²×D₄	1	1	yes	characteristic
107	C₂×#379	#120	1	1	yes	characteristic
108	C₂×#379	#120	1	1	yes	characteristic
109	C₂×#379	#120	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	1	yes	characteristic
119	C₂⁶	C₂×#121	1	1	yes	characteristic
120	C₂⁶	C₂×#121	1	1	yes	characteristic
121	C₂³×D₄	#392	1	1	yes	characteristic
122	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
123	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
124	C₂³×D₄	C₂×#120	1	1	yes	characteristic
125	C₂⁴×C₄	#392	1	1	yes	characteristic
126	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
127	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
128	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
129	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
130	C₂²×C₂²⋊C₄	#392	1	1	yes	characteristic
131	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
132	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
133	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
134	C₂²×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
135	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
136	C₂²×C₂²⋊C₄	C₂×#120	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	1	yes	characteristic
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	C₂×#392	1	1	yes	characteristic
146	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
147	C₂³×C₄	#2021	1	1	yes	characteristic
148	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
149	C₂×C₂²⋊C₄	#2021	1	2	yes	—
150	C₂×C₂²⋊C₄	#2021	1	2	yes	—
151	C₂×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
152	C₂×C₂²⋊C₄	C₂×B(4,𝔽₂)	1	2	yes	—
153	C₂⁴	#10336	1	1	yes	characteristic
154	C₂⁴	#10336	1	1	yes	characteristic
155	C₂⁴	#10336	1	1	yes	characteristic
156	C₂⁴	#10336	1	1	yes	characteristic
157	C₂⁴	C₂×#2021	1	1	yes	characteristic
158	C₂⁴	C₂×#2021	1	1	yes	characteristic
159	C₂⁴	C₂×#2021	1	1	yes	characteristic
160	C₂³	#144382	1	1	yes	characteristic
161	C₂³	#144382	1	2	yes	—
162	C₂³	#144382	1	1	yes	characteristic
163	C₂³	C₂×#10336	1	1	yes	characteristic
164	C₂²	C₂×#144382	1	1	yes	characteristic
165	C₂²	C₂×#144382	1	1	yes	characteristic
166	C₂²	#72236674	1	1	yes	characteristic
167	C₂	#72236918	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72236918
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. #72238386	C₂	semidirect
2. #72238385	C₂	semidirect
3. #72238383	C₂	semidirect
4. #72238378	C₂	semidirect
5. #72238377	C₂²	semidirect
6. #72238371	C₂	semidirect
7. #72238368	C₂²	semidirect
8. #72238367	C₂²	semidirect
9. #72237432	C₂	semidirect
10. #72237427	C₂²	semidirect
11. #72237427	C₂²	semidirect
12. #72237381	C₂	semidirect
14. #72237368	C₂	semidirect
15. #72237364	C₂²	semidirect
17. #72234717	C₂	semidirect
19. #72234711	C₂²	semidirect
20. #72234711	C₂²	semidirect
21. #72234710	C₂²	semidirect
29. C₂×#47031	D₄	semidirect
38. #5020747	D₄	semidirect
40. #686193	D₄	semidirect
41. #576189	D₄	semidirect
42. #480250	D₄	semidirect
43. C₂×#11499	D₄	semidirect
46. #134037	D₄	semidirect
50. #134033	D₄	semidirect
59. C₂×#2815	C₂×D₄	semidirect
64. C₂×#2607	C₂×D₄	semidirect
74. #13828	C₂×D₄	semidirect
75. #13807	C₂×D₄	semidirect
76. #11499	C₂×D₄	semidirect
77. #11499	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72238386	2
#72238385	2
#72238383	2
#72238378	2
#72238371	2
#72237432	2
#72237381	2
#72237368	2
#72234717	2