Subgroups — #72262764

Order 4096.

At least 8290 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

The subgroup lattice has 188 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72262719	C₂	1	1	yes	maximal, characteristic, complemented
2	#72262581	C₂	1	1	yes	maximal, characteristic, complemented
3	#72262534	C₂	1	4	yes	maximal, complemented
4	#72262522	C₂²	1	2	yes	complemented
5	#72262493	C₂	1	4	yes	maximal, complemented
6	#72262478	C₂²	1	2	yes	complemented
7	#72262471	C₂²	1	4	yes	—
8	#72238375	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238374	C₂²	1	4	yes	complemented
10	#72238371	C₂	1	1	yes	maximal, characteristic, complemented
11	#72238368	C₂²	1	2	yes	complemented
12	#72238367	C₂²	1	2	yes	complemented
13	#72237864	C₂	1	1	yes	maximal, characteristic, complemented
14	#72237816	C₂²	1	4	yes	—
15	#72237586	C₂	1	1	yes	maximal, characteristic, complemented
16	#72237577	C₂²	1	4	yes	complemented
17	#72237368	C₂	1	1	yes	maximal, characteristic
18	#72237364	C₂²	1	2	yes	—
19	#72237362	C₄	1	2	yes	complemented
20	#72237359	C₄	1	2	yes	complemented
21	#72237354	C₄	1	2	yes	complemented
22	#72235718	C₂²	1	1	yes	characteristic
23	#72234716	C₂²	1	1	yes	characteristic
24	#72234712	C₄	1	2	yes	complemented
25	#72234711	C₂²	1	2	yes	—
26	#72234625	C₂²	1	1	yes	characteristic
27	#72225870	C₂²	1	1	yes	characteristic
28	#72219314	C₂²	1	1	yes	characteristic
29	#72219314	C₂²	1	1	yes	characteristic
30	#72219311	C₂²	1	1	yes	characteristic
31	C₂×#62817	D₄	1	1	yes	characteristic
32	C₂×#47031	C₂×C₄	1	1	yes	characteristic, complemented
33	C₂×#34980	C₂³	1	1	yes	characteristic
34	#6499870	D₄	1	2	yes	—
35	#6499870	C₂³	1	2	yes	—
36	#6499868	C₂×C₄	1	2	yes	—
37	#6499867	D₄	1	2	yes	—
38	#6451874	C₂×C₄	1	2	yes	—
39	#6451874	D₄	1	2	yes	—
40	C₂×#21746	D₄	1	1	yes	characteristic
41	#5025772	C₂³	1	1	yes	characteristic
42	#5020747	C₂×C₄	1	1	yes	characteristic, complemented
43	#4435227	C₂³	1	1	yes	characteristic
44	#686193	C₂×C₄	1	1	yes	characteristic, complemented
45	#576189	C₂×C₄	1	1	yes	characteristic, complemented
46	#485078	C₂×C₄	1	4	yes	complemented
47	#485044	C₂×C₄	1	4	yes	complemented
48	#480250	C₂×C₄	1	2	yes	complemented
49	#188109	D₄	1	1	yes	characteristic
50	#188109	D₄	1	1	yes	characteristic
51	#188049	D₄	1	1	yes	characteristic
52	#188049	D₄	1	1	yes	characteristic
53	#139625	C₂×C₄	1	4	yes	complemented
54	#139623	C₂×C₄	1	4	yes	complemented
55	#134037	C₂³	1	2	yes	—
56	#134036	C₂³	1	2	yes	—
57	#134034	C₂×C₄	1	2	yes	complemented
58	#133826	C₂³	1	2	yes	—
59	#133825	C₂³	1	2	yes	—
60	#133824	D₄	1	2	yes	—
61	#133822	D₄	1	2	yes	—
62	#115631	C₂³	1	2	yes	—
63	C₂⁵×D₄	#41	1	1	yes	characteristic
64	#62817	C₂²⋊C₄	1	2	yes	—
65	#62817	C₂×D₄	1	2	yes	—
66	C₂×#3490	C₂²⋊C₄	1	2	yes	—
67	C₂²×#534	C₂×D₄	1	1	yes	characteristic
68	C₂²×#534	C₂×D₄	1	1	yes	characteristic
69	C₂³×#120	C₂×D₄	1	1	yes	characteristic
70	C₂×#2815	C₂²×C₄	1	1	yes	characteristic, complemented
71	C₂×#2849	C₂²×C₄	1	1	yes	characteristic
72	C₂×#2849	C₂×D₄	1	1	yes	characteristic
73	C₂×#2846	C₂×D₄	1	1	yes	characteristic
74	C₂×#2846	C₂×D₄	1	1	yes	characteristic
75	C₂×#2607	C₂²×C₄	1	1	yes	characteristic, complemented
76	C₂×#2613	C₂×D₄	1	1	yes	characteristic
77	C₂×#2613	#41	1	1	yes	characteristic
78	C₂×#2613	#41	1	1	yes	characteristic
79	C₂×#2613	C₂⁴	1	1	yes	characteristic
80	C₂×#2308	C₂²×C₄	1	1	yes	characteristic
81	C₂×#2308	C₂×D₄	1	1	yes	characteristic
82	C₂²×#379	#41	1	1	yes	characteristic
83	#23985	C₂²⋊C₄	1	4	yes	—
84	#23985	C₂²×C₄	1	4	yes	—
85	#23984	C₂²⋊C₄	1	4	yes	—
86	#23984	C₂×D₄	1	4	yes	—
87	#21746	C₂²⋊C₄	1	2	yes	—
88	#21746	C₂×D₄	1	2	yes	—
89	C₂×#1819	C₂²⋊C₄	1	2	yes	—
90	#13828	C₂²×C₄	1	1	yes	characteristic, complemented
91	#13807	C₂²×C₄	1	1	yes	characteristic, complemented
92	#11500	C₂²×C₄	1	4	yes	complemented
93	#8548	C₂×D₄	1	2	yes	—
94	#8547	C₂×D₄	1	2	yes	—
95	#8543	C₂×D₄	1	2	yes	—
96	#8541	C₂×D₄	1	2	yes	—
97	C₂⁷	#121	1	1	yes	characteristic
98	C₂⁷	#123	1	1	yes	characteristic
99	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
100	C₂⁴×D₄	#120	1	1	yes	characteristic
101	C₂⁴×D₄	#121	1	1	yes	characteristic
102	C₂⁴×D₄	#123	1	1	yes	characteristic
103	C₂⁴×D₄	C₂×#41	1	1	yes	characteristic
104	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
105	#3490	#99	1	4	yes	—
106	#3490	C₂×C₂²⋊C₄	1	4	yes	—
107	C₂×#543	#99	1	2	yes	—
108	C₂×#534	#99	1	2	yes	—
109	C₂²×#123	C₄×D₄	1	1	yes	characteristic
110	C₂²×#123	C₄×D₄	1	1	yes	characteristic
111	C₂²×#123	#120	1	1	yes	characteristic
112	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
113	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×#120	#120	1	1	yes	characteristic
115	C₂²×#120	#120	1	1	yes	characteristic
116	C₂²×#120	#121	1	1	yes	characteristic
117	C₂²×#120	#121	1	1	yes	characteristic
118	C₂²×#120	#123	1	1	yes	characteristic
119	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
120	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
123	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
124	C₂×#381	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×#381	C₄×D₄	1	1	yes	characteristic
126	C₂×#381	#121	1	1	yes	characteristic
127	C₂×#381	C₂³×C₄	1	1	yes	characteristic
128	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂×#379	C₄×D₄	1	1	yes	characteristic
130	C₂×#379	C₄×D₄	1	1	yes	characteristic
131	C₂×#379	#121	1	1	yes	characteristic
132	C₂×#379	#121	1	1	yes	characteristic
133	C₂×#379	#123	1	1	yes	characteristic
134	C₂×#379	C₂²×D₄	1	1	yes	characteristic
135	C₂×#379	C₂×#41	1	1	yes	characteristic
136	#1819	C₂×C₂²⋊C₄	1	4	yes	—
137	C₂⁶	#386	1	1	yes	characteristic
138	C₂⁶	#386	1	1	yes	characteristic
139	C₂⁶	#386	1	1	yes	characteristic
140	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
141	C₂⁶	C₂×#121	1	1	yes	characteristic
142	C₂⁶	C₂×#123	1	1	yes	characteristic
143	C₂³×D₄	#386	1	1	yes	characteristic
144	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
145	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
146	C₂³×D₄	C₂×#120	1	1	yes	characteristic
147	C₂⁴×C₄	#386	1	1	yes	characteristic
148	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
149	C₂⁴×C₄	C₂×C₄×D₄	1	1	yes	characteristic
150	#543	C₂×#99	1	4	yes	—
151	#534	C₂×#99	1	4	yes	—
152	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
153	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
154	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
155	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
156	C₂²×C₂²⋊C₄	C₂×#99	1	2	yes	—
157	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
160	C₂⁵	#1889	1	1	yes	characteristic
161	C₂⁵	#1889	1	1	yes	characteristic
162	C₂⁵	#1889	1	1	yes	characteristic
163	C₂⁵	#1898	1	1	yes	characteristic
164	C₂⁵	#1898	1	1	yes	characteristic
165	C₂⁵	#1913	1	1	yes	characteristic
166	C₂⁵	#1913	1	1	yes	characteristic
167	C₂⁵	C₂×#386	1	1	yes	characteristic
168	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
169	C₂³×C₄	#1889	1	1	yes	characteristic
170	C₂³×C₄	C₂×B(4,𝔽₂)	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	#1898	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×#99	1	2	yes	—
173	C₂⁴	#9495	1	1	yes	characteristic
174	C₂⁴	#9495	1	1	yes	characteristic
175	C₂⁴	#9495	1	1	yes	characteristic
176	C₂⁴	#9495	1	1	yes	characteristic
177	C₂⁴	C₂×#1889	1	1	yes	characteristic
178	C₂⁴	C₂×#1898	1	1	yes	characteristic
179	C₂⁴	C₂×#1913	1	1	yes	characteristic
180	C₂³	#139731	1	1	yes	characteristic
181	C₂³	#139731	1	1	yes	characteristic
182	C₂³	C₂×#9495	1	1	yes	characteristic
183	C₂²	C₂×#139731	1	1	yes	characteristic
184	C₂²	#72251471	1	1	yes	characteristic
185	C₂²	#72251527	1	1	yes	characteristic
186	C₂	#72262368	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72262368
commutator	C₂×#381	C₂³×C₄
Frattini, cosocle	C₂×#2613	C₂⁴
prime radical	#72237368	C₂
LCS	G ▷ C₂×#381 ▷ 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. #72262719	C₂	semidirect
2. #72262581	C₂	semidirect
3. #72262534	C₂	semidirect
4. #72262522	C₂²	semidirect
5. #72262493	C₂	semidirect
6. #72262478	C₂²	semidirect
8. #72238375	C₂	semidirect
9. #72238374	C₂²	semidirect
10. #72238371	C₂	semidirect
11. #72238368	C₂²	semidirect
12. #72238367	C₂²	semidirect
13. #72237864	C₂	semidirect
15. #72237586	C₂	semidirect
16. #72237577	C₂²	semidirect
19. #72237362	C₄	semidirect
20. #72237359	C₄	semidirect
21. #72237354	C₄	semidirect
24. #72234712	C₄	semidirect
32. C₂×#47031	C₂×C₄	semidirect
42. #5020747	C₂×C₄	semidirect
44. #686193	C₂×C₄	semidirect
45. #576189	C₂×C₄	semidirect
46. #485078	C₂×C₄	semidirect
47. #485044	C₂×C₄	semidirect
48. #480250	C₂×C₄	semidirect
53. #139625	C₂×C₄	semidirect
54. #139623	C₂×C₄	semidirect
57. #134034	C₂×C₄	semidirect
70. C₂×#2815	C₂²×C₄	semidirect
75. C₂×#2607	C₂²×C₄	semidirect
90. #13828	C₂²×C₄	semidirect
91. #13807	C₂²×C₄	semidirect
92. #11500	C₂²×C₄	semidirect

Maximal subgroups

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

subgroup	index
#72262719	2
#72262581	2
#72262534	2
#72262493	2
#72238375	2
#72238371	2
#72237864	2
#72237586	2
#72237368	2