Subgroups — #72265787

Order 2048.

At least 6844 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

The subgroup lattice has 181 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72236929	C₂	1	1	yes	maximal, characteristic, complemented
2	#72236375	C₂	1	1	yes	maximal, characteristic
3	#72219300	C₂	1	1	yes	maximal, characteristic, complemented
4	#72219245	C₂	1	1	yes	maximal, characteristic, complemented
5	#72218854	C₂	1	1	yes	maximal, characteristic, complemented
6	#72218198	C₂	1	1	yes	maximal, characteristic, complemented
7	#72210555	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#3484	C₂²	1	1	yes	characteristic
9	#7630442	C₄	1	1	yes	characteristic, complemented
10	C₂×#34980	C₂²	1	1	yes	characteristic
11	C₂×#20984	C₄	1	1	yes	characteristic, complemented
12	C₂×#20585	C₄	1	1	yes	characteristic, complemented
13	#517411	C₂²	1	1	yes	characteristic
14	#392113	C₂²	1	1	yes	characteristic
15	#392109	C₂²	1	1	yes	characteristic
16	#392109	C₂²	1	1	yes	characteristic
17	#353408	C₄	1	1	yes	characteristic, complemented
18	#154094	C₂²	1	1	yes	characteristic
19	C₂³×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic
20	C₂⁵×D₄	C₂³	1	1	yes	characteristic
21	C₂²×#535	D₄	1	2	yes	—
22	C₂²×#534	C₂×C₄	1	1	yes	characteristic
23	C₂²×#534	D₄	1	1	yes	characteristic
24	C₂²×#525	D₄	1	2	yes	—
25	C₂³×#120	D₄	1	1	yes	characteristic
26	C₂²×#518	C₂×C₄	1	1	yes	characteristic
27	C₂×#2308	C₂×C₄	1	1	yes	characteristic
28	C₂×#2308	C₂×C₄	1	1	yes	characteristic
29	C₂×#2308	C₂×C₄	1	1	yes	characteristic
30	C₂×#2308	D₄	1	1	yes	characteristic
31	C₂²×#379	D₄	1	1	yes	characteristic
32	C₂×#1783	D₄	1	2	yes	complemented
33	C₂×#1781	D₄	1	2	yes	complemented
34	C₂×#1781	D₄	1	2	yes	complemented
35	#14679	D₄	1	2	yes	—
36	#14335	D₄	1	2	yes	—
37	C₂×#1438	D₄	1	2	yes	complemented
38	C₂⁷	C₂×D₄	1	1	yes	characteristic
39	C₂⁷	C₂×D₄	1	1	yes	characteristic
40	C₂³×#41	C₂²⋊C₄	1	2	yes	—
41	C₂⁴×D₄	C₂²⋊C₄	1	1	yes	characteristic
42	C₂⁴×D₄	C₂²×C₄	1	1	yes	characteristic
43	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
44	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
45	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
46	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
47	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
48	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
49	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
50	C₂⁴×D₄	#41	1	1	yes	characteristic
51	C₂⁴×D₄	#41	1	1	yes	characteristic
52	C₂⁴×D₄	#41	1	1	yes	characteristic
53	C₂⁴×D₄	#41	1	1	yes	characteristic
54	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
55	C₂²×#117	C₂²⋊C₄	1	2	yes	—
56	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
57	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
58	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×#379	C₂²⋊C₄	1	2	yes	—
60	C₂×#379	C₂²⋊C₄	1	2	yes	—
61	C₂⁶	#99	1	1	yes	characteristic
62	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂⁶	#120	1	1	yes	characteristic
65	C₂⁶	#120	1	1	yes	characteristic
66	C₂⁶	#120	1	1	yes	characteristic
67	C₂⁶	#120	1	1	yes	characteristic
68	C₂⁶	#121	1	1	yes	characteristic
69	C₂⁶	#121	1	1	yes	characteristic
70	C₂⁶	#121	1	1	yes	characteristic
71	C₂⁶	#121	1	1	yes	characteristic
72	C₂⁶	#123	1	1	yes	characteristic
73	C₂⁶	#124	1	1	yes	characteristic
74	C₂⁶	#127	1	1	yes	characteristic
75	C₂³×Q₈	#99	1	2	yes	complemented
76	C₂³×D₄	#99	1	2	yes	complemented
77	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
78	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
79	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
80	C₂³×D₄	C₄×D₄	1	1	yes	characteristic
81	C₂³×D₄	#120	1	1	yes	characteristic
82	C₂³×D₄	#120	1	1	yes	characteristic
83	C₂³×D₄	#121	1	1	yes	characteristic
84	C₂³×D₄	#121	1	1	yes	characteristic
85	C₂³×D₄	#121	1	1	yes	characteristic
86	C₂³×D₄	#121	1	1	yes	characteristic
87	C₂³×D₄	#121	1	1	yes	characteristic
88	C₂³×D₄	#121	1	1	yes	characteristic
89	C₂³×D₄	#123	1	1	yes	characteristic
90	C₂³×D₄	#123	1	1	yes	characteristic
91	C₂³×D₄	#124	1	1	yes	characteristic
92	C₂³×D₄	#127	1	1	yes	characteristic
93	C₂⁴×C₄	#99	1	1	yes	characteristic
94	C₂⁴×C₄	#99	1	1	yes	characteristic
95	C₂⁴×C₄	#99	1	1	yes	characteristic
96	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂⁴×C₄	#120	1	1	yes	characteristic
98	C₂⁴×C₄	#120	1	1	yes	characteristic
99	C₂⁴×C₄	#121	1	1	yes	characteristic
100	C₂⁴×C₄	#121	1	1	yes	characteristic
101	C₂⁴×C₄	#123	1	1	yes	characteristic
102	C₂×#117	#99	1	4	yes	complemented
103	C₂²×C₂²⋊C₄	#99	1	2	yes	—
104	C₂²×C₂²⋊C₄	#99	1	2	yes	—
105	C₂²×C₂²⋊C₄	#99	1	4	yes	complemented
106	C₂⁵	#379	1	1	yes	characteristic
107	C₂⁵	#386	1	1	yes	characteristic
108	C₂⁵	#386	1	1	yes	characteristic
109	C₂⁵	#390	1	1	yes	characteristic
110	C₂⁵	#390	1	1	yes	characteristic
111	C₂⁵	#392	1	1	yes	characteristic
112	C₂⁵	#392	1	1	yes	characteristic
113	C₂⁵	#392	1	1	yes	characteristic
114	C₂⁵	#394	1	1	yes	characteristic
115	C₂⁵	#394	1	1	yes	characteristic
116	C₂⁵	#394	1	1	yes	characteristic
117	C₂⁵	C₂×#99	1	1	yes	characteristic
118	C₂⁵	C₂×#99	1	1	yes	characteristic
119	C₂⁵	C₂×#99	1	1	yes	characteristic
120	C₂⁵	C₂×#99	1	1	yes	characteristic
121	C₂⁵	C₂×#99	1	2	yes	—
122	C₂⁵	#410	1	2	yes	—
123	C₂⁵	B(4,𝔽₂)	1	4	yes	—
124	C₂²×D₄	B(4,𝔽₂)	1	4	yes	—
125	C₂³×C₄	#386	1	1	yes	characteristic
126	C₂³×C₄	#386	1	1	yes	characteristic
127	C₂³×C₄	#392	1	1	yes	characteristic
128	C₂³×C₄	#394	1	1	yes	characteristic
129	C₂³×C₄	C₂×#99	1	2	yes	—
130	C₂³×C₄	C₂×#99	1	2	yes	—
131	C₂⁴	#1438	1	1	yes	characteristic
132	C₂⁴	#1781	1	1	yes	characteristic
133	C₂⁴	#1781	1	1	yes	characteristic
134	C₂⁴	#1781	1	1	yes	characteristic
135	C₂⁴	#1781	1	1	yes	characteristic
136	C₂⁴	#1781	1	1	yes	characteristic
137	C₂⁴	#1782	1	1	yes	characteristic
138	C₂⁴	#1889	1	2	yes	—
139	C₂⁴	#1889	1	2	yes	—
140	C₂⁴	#1898	1	1	yes	characteristic
141	C₂⁴	#1898	1	1	yes	characteristic
142	C₂⁴	#1898	1	1	yes	characteristic
143	C₂⁴	#1898	1	1	yes	characteristic
144	C₂⁴	#1913	1	2	yes	—
145	C₂⁴	#1913	1	2	yes	—
146	C₂⁴	#1964	1	2	yes	—
147	C₂⁴	#1964	1	2	yes	—
148	C₂⁴	#2021	1	2	yes	—
149	C₂⁴	#2021	1	2	yes	—
150	C₂⁴	#2021	1	2	yes	—
151	C₂²×C₄	#1898	1	2	yes	—
152	C₂²×C₄	#1898	1	2	yes	—
153	C₂²×C₄	#2021	1	2	yes	—
154	C₂³	#8541	1	1	yes	characteristic
155	C₂³	#8542	1	1	yes	characteristic
156	C₂³	#8542	1	1	yes	characteristic
157	C₂³	#8546	1	1	yes	characteristic
158	C₂³	#9495	1	2	yes	—
159	C₂³	#9495	1	2	yes	—
160	C₂³	#9497	1	2	yes	—
161	C₂³	#9497	1	2	yes	—
162	C₂³	#10965	1	1	yes	characteristic
163	C₂³	#10965	1	1	yes	characteristic
164	C₂³	#10972	1	1	yes	characteristic
165	C₂³	#10972	1	1	yes	characteristic
166	C₂³	#10980	1	1	yes	characteristic
167	C₂³	#10991	1	1	yes	characteristic
168	C₂³	#10991	1	1	yes	characteristic
169	C₂²	#111516	1	1	yes	characteristic
170	C₂²	#115133	1	1	yes	characteristic
171	C₂²	#115133	1	1	yes	characteristic
172	C₂²	#115163	1	1	yes	characteristic
173	C₂²	#115163	1	1	yes	characteristic
174	C₂²	#115631	1	1	yes	characteristic
175	C₂²	#115690	1	1	yes	characteristic
176	C₂²	#133839	1	2	yes	—
177	C₂	#72219443	1	1	yes	characteristic
178	C₂	#72251552	1	1	yes	characteristic
179	C₂	#72264770	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#111516
commutator	C₂⁴×D₄	C₂²×C₄
Frattini, cosocle	C₂⁵×D₄	C₂³
prime radical	#72236375	C₂
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁵×D₄ ◁ G
LECS	G ▷ C₂⁵×D₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72236929	C₂	semidirect
3. #72219300	C₂	semidirect
4. #72219245	C₂	semidirect
5. #72218854	C₂	semidirect
6. #72218198	C₂	semidirect
7. #72210555	C₂	semidirect
9. #7630442	C₄	semidirect
11. C₂×#20984	C₄	semidirect
12. C₂×#20585	C₄	semidirect
17. #353408	C₄	semidirect
32. C₂×#1783	D₄	semidirect
33. C₂×#1781	D₄	semidirect
34. C₂×#1781	D₄	semidirect
37. C₂×#1438	D₄	semidirect
75. C₂³×Q₈	#99	semidirect
76. C₂³×D₄	#99	semidirect
102. C₂×#117	#99	semidirect
105. C₂²×C₂²⋊C₄	#99	semidirect

Maximal subgroups

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

subgroup	index
#72236929	2
#72236375	2
#72219300	2
#72219245	2
#72218854	2
#72218198	2
#72210555	2