Subgroups — #72266256

Order 2048.

At least 7008 conjugacy classes of subgroups; only the 300 normal subgroups are shown.

The subgroup lattice has 209 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266255	C₂	1	1	yes	maximal, characteristic, complemented
2	#72259019	C₂	1	1	yes	maximal, characteristic, complemented
3	#72258744	C₂	1	2	yes	maximal, complemented
4	#72258700	C₂	1	1	yes	maximal, characteristic, complemented
5	#72258693	C₂	1	2	yes	maximal, complemented
6	#72258689	C₂	1	1	yes	maximal, characteristic, complemented
7	#72255632	C₂	1	1	yes	maximal, characteristic, complemented
8	#72240427	C₂	1	1	yes	maximal, characteristic, complemented
9	#72240426	C₂	1	2	yes	maximal, complemented
10	#72236816	C₂	1	1	yes	maximal, characteristic, complemented
11	#72236756	C₂	1	2	yes	maximal, complemented
12	C₂×#34980	C₂²	1	1	yes	characteristic
13	C₂²×#1781	C₂²	1	1	yes	characteristic
14	C₂²×#1781	C₂²	1	1	yes	characteristic
15	#4414129	C₂²	1	1	yes	characteristic
16	#759288	C₂²	1	1	yes	characteristic
17	#635984	C₂²	1	2	yes	—
18	#612424	C₂²	1	2	yes	—
19	#521311	C₂²	1	2	yes	—
20	#515285	C₂²	1	1	yes	characteristic
21	#392109	C₂²	1	2	yes	—
22	#382422	C₂²	1	2	yes	—
23	#359428	C₂²	1	2	yes	—
24	#359428	C₂²	1	2	yes	—
25	#359428	C₂²	1	2	yes	—
26	C₂×#10965	C₂²	1	1	yes	characteristic
27	#123513	C₂²	1	1	yes	characteristic
28	#123513	C₂²	1	1	yes	characteristic
29	C₂²⋊#1304	C₂²	1	2	yes	—
30	C₂²⋊#1304	C₂²	1	1	yes	characteristic
31	C₂²⋊#1304	C₂²	1	1	yes	characteristic
32	C₂²⋊#1304	C₂²	1	2	yes	—
33	C₂²⋊#1304	C₂²	1	2	yes	—
34	C₂²⋊#1304	C₂²	1	2	yes	—
35	C₂⁵×D₄	C₂³	1	1	yes	characteristic
36	C₂³×#120	C₂³	1	1	yes	characteristic
37	C₂³×#120	C₂³	1	2	yes	—
38	C₂²×#518	D₄	1	1	yes	characteristic, complemented
39	C₂×#2840	D₄	1	1	yes	characteristic, complemented
40	C₂×#2405	D₄	1	2	yes	complemented
41	C₂×#2403	D₄	1	2	yes	complemented
42	C₂×#2403	D₄	1	1	yes	characteristic, complemented
43	C₂×#2308	D₄	1	2	yes	complemented
44	C₂×#2308	D₄	1	1	yes	characteristic, complemented
45	C₂²×#379	C₂³	1	1	yes	characteristic
46	C₂²×#379	C₂³	1	1	yes	characteristic
47	C₂²×#379	C₂³	1	1	yes	characteristic
48	C₂²×#379	C₂³	1	1	yes	characteristic
49	C₂²×#379	C₂³	1	1	yes	characteristic
50	#14679	C₂³	1	2	yes	—
51	#14679	C₂³	1	2	yes	—
52	#14679	C₂³	1	2	yes	—
53	#14673	D₄	1	1	yes	characteristic, complemented
54	#14673	D₄	1	2	yes	complemented
55	#14338	D₄	1	2	yes	complemented
56	#14335	D₄	1	2	yes	complemented
57	#14127	D₄	1	1	yes	characteristic, complemented
58	#13869	D₄	1	2	yes	complemented
59	#13867	D₄	1	2	yes	complemented
60	#13867	D₄	1	2	yes	complemented
61	#13794	D₄	1	1	yes	characteristic, complemented
62	#13794	D₄	1	1	yes	characteristic, complemented
63	#13775	D₄	1	2	yes	complemented
64	C₂⁷	#41	1	1	yes	characteristic, complemented
65	C₂⁷	C₂⁴	1	1	yes	characteristic
66	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
67	C₂⁴×D₄	C₂×D₄	1	2	yes	—
68	C₂⁴×D₄	C₂×D₄	1	2	yes	—
69	C₂⁴×D₄	C₂×D₄	1	2	yes	—
70	C₂⁴×D₄	C₂×D₄	1	2	yes	—
71	C₂⁵×C₄	#41	1	1	yes	characteristic, complemented
72	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
73	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
74	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
75	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
76	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
77	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
78	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
79	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
80	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
81	C₂×#379	C₂×D₄	1	2	yes	—
82	C₂×#379	C₂×D₄	1	1	yes	characteristic
83	C₂×#379	C₂×D₄	1	1	yes	characteristic
84	C₂×#379	C₂×D₄	1	2	yes	—
85	C₂×#379	C₂×D₄	1	2	yes	—
86	C₂×#379	C₂×D₄	1	2	yes	—
87	C₂×#379	C₂×D₄	1	2	yes	—
88	C₂×#379	C₂×D₄	1	1	yes	characteristic
89	C₂×#379	C₂×D₄	1	2	yes	—
90	C₂×#379	C₂×D₄	1	2	yes	—
91	C₂×#379	C₂×D₄	1	1	yes	characteristic
92	C₂×#379	C₂×D₄	1	2	yes	—
93	C₂×#379	#41	1	2	yes	complemented
94	C₂×#379	#41	1	4	yes	complemented
95	C₂⁶	#120	1	1	yes	characteristic
96	C₂⁶	#120	1	2	yes	—
97	C₂⁶	C₂²×D₄	1	1	yes	characteristic
98	C₂⁶	C₂²×D₄	1	1	yes	characteristic
99	C₂⁶	C₂²×D₄	1	2	yes	—
100	C₂⁶	C₂²×D₄	1	2	yes	—
101	C₂⁶	C₂²×D₄	1	1	yes	characteristic
102	C₂⁶	C₂×#41	1	1	yes	characteristic
103	C₂⁶	C₂×#41	1	2	yes	—
104	C₂⁶	C₂×#41	1	1	yes	characteristic
105	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂⁶	H(5,𝔽₂)	1	2	yes	—
107	C₂⁶	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂⁴×C₄	#120	1	2	yes	—
109	C₂⁴×C₄	#120	1	1	yes	characteristic
110	C₂⁴×C₄	#120	1	2	yes	—
111	C₂⁴×C₄	#120	1	2	yes	—
112	C₂⁴×C₄	#120	1	2	yes	—
113	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
114	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
115	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
116	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
117	C₂²×C₂²⋊C₄	#120	1	2	yes	—
118	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
119	C₂²×C₂²⋊C₄	#120	1	1	yes	characteristic
120	C₂²×C₂²⋊C₄	#120	1	2	yes	—
121	C₂²×C₂²⋊C₄	#120	1	2	yes	—
122	C₂²×C₂²⋊C₄	#120	1	2	yes	—
123	C₂²×C₂²⋊C₄	#120	1	2	yes	—
124	C₂⁵	B(4,𝔽₂)	1	2	yes	—
125	C₂⁵	C₂×#120	1	2	yes	—
126	C₂⁵	C₂×#120	1	2	yes	—
127	C₂⁵	C₂×#120	1	1	yes	characteristic
128	C₂⁵	C₂×#120	1	1	yes	characteristic
129	C₂⁵	C₂×#120	1	1	yes	characteristic
130	C₂⁵	#525	1	1	yes	characteristic
131	C₂⁵	#525	1	2	yes	—
132	C₂⁵	#532	1	1	yes	characteristic
133	C₂⁵	#532	1	2	yes	—
134	C₂⁵	#535	1	1	yes	characteristic
135	C₂⁵	#535	1	2	yes	—
136	C₂⁵	#537	1	1	yes	characteristic
137	C₂⁵	#537	1	2	yes	—
138	C₂⁵	#546	1	1	yes	characteristic
139	C₂⁵	#546	1	2	yes	—
140	C₂⁵	#546	1	2	yes	—
141	C₂⁵	#546	1	2	yes	—
142	C₂⁵	#546	1	2	yes	—
143	C₂⁵	#547	1	1	yes	characteristic
144	C₂⁵	#547	1	2	yes	—
145	C₂⁵	#548	1	1	yes	characteristic
146	C₂⁵	#550	1	1	yes	characteristic
147	C₂⁵	#550	1	2	yes	—
148	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
149	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
150	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	2	yes	—
151	C₂⁴	#2388	1	1	yes	characteristic
152	C₂⁴	#2404	1	1	yes	characteristic
153	C₂⁴	#2404	1	2	yes	—
154	C₂⁴	#2407	1	2	yes	—
155	C₂⁴	#2407	1	1	yes	characteristic
156	C₂⁴	#2430	1	2	yes	—
157	C₂⁴	#2430	1	1	yes	characteristic
158	C₂⁴	#2431	1	2	yes	—
159	C₂⁴	#2431	1	1	yes	characteristic
160	C₂⁴	#2615	1	1	yes	characteristic
161	C₂⁴	#2846	1	1	yes	characteristic
162	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
163	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
164	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
165	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
166	C₂⁴	#3025	1	1	yes	characteristic
167	C₂⁴	#3025	1	1	yes	characteristic
168	C₂⁴	#3026	1	1	yes	characteristic
169	C₂⁴	#3026	1	1	yes	characteristic
170	C₂⁴	#3026	1	1	yes	characteristic
171	C₂⁴	#3026	1	1	yes	characteristic
172	C₂⁴	#3027	1	1	yes	characteristic
173	C₂⁴	#3027	1	1	yes	characteristic
174	C₂³	#14885	1	1	yes	characteristic
175	C₂³	#23595	1	1	yes	characteristic
176	C₂³	#23597	1	1	yes	characteristic
177	C₂³	#23984	1	1	yes	characteristic
178	C₂³	#23987	1	1	yes	characteristic
179	C₂³	#23996	1	4	yes	—
180	C₂³	#24140	1	1	yes	characteristic
181	C₂³	#24142	1	1	yes	characteristic
182	C₂³	#25563	1	1	yes	characteristic
183	C₂³	#25563	1	1	yes	characteristic
184	C₂³	#25585	1	1	yes	characteristic
185	C₂³	#25587	1	1	yes	characteristic
186	C₂³	#29684	1	1	yes	characteristic
187	C₂³	#29686	1	1	yes	characteristic
188	C₂³	#32898	1	1	yes	characteristic
189	C₂³	#32898	1	1	yes	characteristic
190	C₂³	#32898	1	1	yes	characteristic
191	C₂³	#32898	1	1	yes	characteristic
192	C₂³	#32899	1	1	yes	characteristic
193	C₂³	#32899	1	1	yes	characteristic
194	C₂²	#347042	1	1	yes	characteristic
195	C₂²	#347043	1	1	yes	characteristic
196	C₂²	#347043	1	1	yes	characteristic
197	C₂²	#347044	1	1	yes	characteristic
198	C₂²	#403939	1	1	yes	characteristic
199	C₂²	#403941	1	1	yes	characteristic
200	C₂²	#403943	1	2	yes	—
201	C₂²	#403952	1	1	yes	characteristic
202	C₂²	#404015	1	2	yes	—
203	C₂	#72232805	1	1	yes	characteristic
204	C₂	#72234956	1	2	yes	—
205	C₂	#72234958	1	2	yes	—
206	C₂	#72258395	1	1	yes	characteristic
207	C₂	#72265945	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#14885
commutator, Frattini, cosocle	C₂⁷	C₂⁴
LCS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266255	C₂	semidirect
2. #72259019	C₂	semidirect
3. #72258744	C₂	semidirect
4. #72258700	C₂	semidirect
5. #72258693	C₂	semidirect
6. #72258689	C₂	semidirect
7. #72255632	C₂	semidirect
8. #72240427	C₂	semidirect
9. #72240426	C₂	semidirect
10. #72236816	C₂	semidirect
11. #72236756	C₂	semidirect
38. C₂²×#518	D₄	semidirect
39. C₂×#2840	D₄	semidirect
40. C₂×#2405	D₄	semidirect
41. C₂×#2403	D₄	semidirect
42. C₂×#2403	D₄	semidirect
43. C₂×#2308	D₄	semidirect
44. C₂×#2308	D₄	semidirect
53. #14673	D₄	semidirect
54. #14673	D₄	semidirect
55. #14338	D₄	semidirect
56. #14335	D₄	semidirect
57. #14127	D₄	semidirect
58. #13869	D₄	semidirect
59. #13867	D₄	semidirect
60. #13867	D₄	semidirect
61. #13794	D₄	semidirect
62. #13794	D₄	semidirect
63. #13775	D₄	semidirect
64. C₂⁷	#41	semidirect
71. C₂⁵×C₄	#41	semidirect
93. C₂×#379	#41	semidirect
94. C₂×#379	#41	semidirect

Maximal subgroups

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

subgroup	index
#72266255	2
#72259019	2
#72258744	2
#72258700	2
#72258693	2
#72258689	2
#72255632	2
#72240427	2
#72240426	2
#72236816	2
#72236756	2