Subgroups — #72265043

Order 2048.

At least 7253 conjugacy classes of subgroups; only the 380 normal subgroups are shown.

The subgroup lattice has 201 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72264992	C₂	1	1	yes	maximal, characteristic, complemented
2	#72264957	C₂	1	2	yes	maximal, complemented
3	#72264916	C₂	1	1	yes	maximal, characteristic, complemented
4	#72219423	C₂	1	1	yes	maximal, characteristic, complemented
5	#72218846	C₂	1	2	yes	maximal, complemented
6	#72218387	C₂	1	4	yes	maximal, complemented
7	#72214832	C₂	1	2	yes	maximal, complemented
8	#72214549	C₂	1	2	yes	maximal, complemented
9	C₂²×#3484	C₂²	1	1	yes	characteristic
10	C₂²×#2403	C₂²	1	2	yes	—
11	C₂²×#2403	C₂²	1	1	yes	characteristic, complemented
12	#612424	C₂²	1	4	yes	—
13	#517411	C₂²	1	1	yes	characteristic
14	C₂×#13867	C₂²	1	1	yes	characteristic
15	#392109	C₂²	1	2	yes	—
16	#392109	C₂²	1	4	yes	complemented
17	#359428	C₂²	1	2	yes	—
18	#359428	C₂²	1	4	yes	—
19	#359428	C₂²	1	2	yes	—
20	#155632	C₂²	1	1	yes	characteristic, complemented
21	#154094	C₂²	1	2	yes	complemented
22	C₂²⋊#1304	C₂²	1	4	yes	complemented
23	C₂²⋊#1304	C₂²	1	2	yes	complemented
24	C₂²⋊#1304	C₂²	1	2	yes	complemented
25	C₂³×H(5,𝔽₂)	D₄	1	1	yes	characteristic, complemented
26	C₂⁵×D₄	C₂³	1	2	yes	—
27	C₂⁵×D₄	C₂³	1	1	yes	characteristic
28	C₂²×#534	D₄	1	2	yes	complemented
29	C₂³×#120	D₄	1	2	yes	complemented
30	C₂³×#120	D₄	1	2	yes	complemented
31	C₂³×#120	C₂³	1	2	yes	—
32	C₂²×#518	D₄	1	1	yes	characteristic, complemented
33	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic, complemented
34	C₂²×#386	D₄	1	2	yes	complemented
35	C₂²×#386	D₄	1	2	yes	complemented
36	C₂²×#379	D₄	1	1	yes	characteristic, complemented
37	C₂²×#379	C₂³	1	2	yes	—
38	C₂²×#379	C₂³	1	1	yes	characteristic
39	C₂²×#379	C₂³	1	1	yes	characteristic
40	C₂×#1781	D₄	1	4	yes	complemented
41	C₂×#1781	D₄	1	4	yes	complemented
42	C₂×#1781	D₄	1	4	yes	complemented
43	#14679	C₂³	1	2	yes	—
44	#14679	C₂³	1	4	yes	—
45	C₂×C₂²⋊#95	D₄	1	1	yes	characteristic, complemented
46	C₂×C₂²⋊#95	D₄	1	1	yes	characteristic, complemented
47	#10725	D₄	1	4	yes	complemented
48	C₂⁷	C₂×D₄	1	1	yes	characteristic
49	C₂⁷	C₂×D₄	1	2	yes	—
50	C₂⁷	C₂⁴	1	1	yes	characteristic
51	C₂⁴×D₄	C₂×D₄	1	2	yes	—
52	C₂⁴×D₄	C₂×D₄	1	2	yes	—
53	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
54	C₂⁴×D₄	C₂×D₄	1	1	yes	characteristic
55	C₂⁴×D₄	C₂×D₄	1	2	yes	—
56	C₂⁴×D₄	C₂×D₄	1	2	yes	—
57	C₂⁴×D₄	C₂×D₄	1	2	yes	—
58	C₂⁴×D₄	C₂×D₄	1	2	yes	—
59	C₂⁴×D₄	#41	1	1	yes	characteristic
60	C₂⁴×D₄	#41	1	1	yes	characteristic
61	C₂⁴×D₄	#41	1	2	yes	—
62	C₂⁵×C₄	C₂×D₄	1	2	yes	—
63	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
65	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
66	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
67	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
68	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
69	C₂³×C₂²⋊C₄	C₂×D₄	1	2	yes	—
70	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
71	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₂²⋊C₄	#41	1	2	yes	—
73	C₂×#379	C₂×D₄	1	4	yes	—
74	C₂×#379	C₂×D₄	1	8	yes	—
75	C₂×#379	C₂×D₄	1	4	yes	—
76	C₂⁶	#120	1	1	yes	characteristic
77	C₂⁶	#120	1	1	yes	characteristic
78	C₂⁶	#121	1	1	yes	characteristic
79	C₂⁶	#121	1	1	yes	characteristic
80	C₂⁶	#121	1	2	yes	complemented
81	C₂⁶	#127	1	1	yes	characteristic
82	C₂⁶	#127	1	1	yes	characteristic
83	C₂⁶	C₂²×D₄	1	1	yes	characteristic
84	C₂⁶	C₂²×D₄	1	2	yes	—
85	C₂⁶	C₂²×D₄	1	1	yes	characteristic
86	C₂⁶	C₂²×D₄	1	2	yes	—
87	C₂⁶	C₂²×D₄	1	2	yes	—
88	C₂⁶	C₂×#41	1	2	yes	—
89	C₂⁶	C₂×#41	1	1	yes	characteristic
90	C₂⁶	H(5,𝔽₂)	1	2	yes	—
91	C₂⁶	H(5,𝔽₂)	1	2	yes	—
92	C₂³×D₄	#120	1	2	yes	—
93	C₂³×D₄	#120	1	2	yes	—
94	C₂³×D₄	#120	1	2	yes	—
95	C₂³×D₄	#120	1	2	yes	—
96	C₂³×D₄	#121	1	2	yes	—
97	C₂³×D₄	#121	1	2	yes	—
98	C₂³×D₄	#121	1	2	yes	—
99	C₂³×D₄	#121	1	2	yes	—
100	C₂⁴×C₄	#120	1	2	yes	—
101	C₂⁴×C₄	#121	1	1	yes	characteristic
102	C₂⁴×C₄	#121	1	1	yes	characteristic
103	C₂⁴×C₄	#121	1	2	yes	complemented
104	C₂⁴×C₄	#121	1	2	yes	complemented
105	C₂⁴×C₄	#121	1	2	yes	complemented
106	C₂⁴×C₄	#127	1	2	yes	—
107	C₂²×C₂²⋊C₄	#120	1	2	yes	—
108	C₂²×C₂²⋊C₄	#120	1	2	yes	—
109	C₂⁵	#392	1	1	yes	characteristic
110	C₂⁵	#392	1	1	yes	characteristic
111	C₂⁵	#392	1	1	yes	characteristic
112	C₂⁵	#392	1	1	yes	characteristic
113	C₂⁵	B(4,𝔽₂)	1	2	yes	—
114	C₂⁵	B(4,𝔽₂)	1	2	yes	—
115	C₂⁵	B(4,𝔽₂)	1	2	yes	—
116	C₂⁵	B(4,𝔽₂)	1	2	yes	—
117	C₂⁵	C₂×#120	1	1	yes	characteristic
118	C₂⁵	C₂×#120	1	1	yes	characteristic
119	C₂⁵	C₂×#120	1	2	yes	—
120	C₂⁵	C₂×#121	1	1	yes	characteristic
121	C₂⁵	C₂×#121	1	2	yes	—
122	C₂⁵	C₂×#121	1	2	yes	—
123	C₂⁵	#525	1	1	yes	characteristic
124	C₂⁵	#525	1	2	yes	—
125	C₂⁵	C₂×#127	1	1	yes	characteristic
126	C₂⁵	#534	1	1	yes	characteristic
127	C₂⁵	#534	1	2	yes	—
128	C₂⁵	#535	1	1	yes	characteristic
129	C₂⁵	#538	1	2	yes	—
130	C₂⁵	D₄²	1	4	yes	—
131	C₂⁵	D₄²	1	4	yes	—
132	C₂⁵	#546	1	4	yes	—
133	C₂⁵	#546	1	4	yes	—
134	C₂²×D₄	B(4,𝔽₂)	1	8	yes	—
135	C₂³×C₄	#392	1	2	yes	—
136	C₂³×C₄	#392	1	2	yes	—
137	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	complemented
138	C₂⁴	#2021	1	2	yes	—
139	C₂⁴	#2021	1	2	yes	—
140	C₂⁴	#2021	1	2	yes	—
141	C₂⁴	#2021	1	2	yes	—
142	C₂⁴	#2021	1	4	yes	—
143	C₂⁴	C₂×#392	1	1	yes	characteristic
144	C₂⁴	#2403	1	1	yes	characteristic
145	C₂⁴	#2404	1	1	yes	characteristic
146	C₂⁴	#2410	1	4	yes	—
147	C₂⁴	#2424	1	4	yes	—
148	C₂⁴	#2433	1	1	yes	characteristic
149	C₂⁴	#2539	1	1	yes	characteristic
150	C₂⁴	#2617	1	1	yes	characteristic
151	C₂⁴	#2847	1	1	yes	characteristic
152	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
153	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
154	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
155	C₂⁴	C₂×B(4,𝔽₂)	1	1	yes	characteristic
156	C₂⁴	C₂×B(4,𝔽₂)	1	4	yes	—
157	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
158	C₂⁴	#3026	1	4	yes	—
159	C₂⁴	#3027	1	2	yes	—
160	C₂²×C₄	#2021	1	4	yes	—
161	C₂³	#10336	1	4	yes	complemented
162	C₂³	#14716	1	1	yes	characteristic
163	C₂³	C₂×#2021	1	2	yes	—
164	C₂³	C₂×#2021	1	1	yes	characteristic
165	C₂³	C₂×#2021	1	1	yes	characteristic
166	C₂³	#23444	1	2	yes	—
167	C₂³	#23895	1	1	yes	characteristic
168	C₂³	#23896	1	1	yes	characteristic
169	C₂³	#23900	1	1	yes	characteristic
170	C₂³	#23900	1	1	yes	characteristic
171	C₂³	#23900	1	1	yes	characteristic
172	C₂³	#23904	1	1	yes	characteristic
173	C₂³	#23984	1	1	yes	characteristic
174	C₂³	#23984	1	1	yes	characteristic
175	C₂³	#23996	1	2	yes	—
176	C₂³	#23996	1	2	yes	—
177	C₂³	#24404	1	8	yes	—
178	C₂³	#25771	1	2	yes	—
179	C₂³	#32926	1	2	yes	—
180	C₂²	C₂×#10336	1	1	yes	characteristic
181	C₂²	#233541	1	2	yes	—
182	C₂²	#233549	1	1	yes	characteristic
183	C₂²	#245225	1	1	yes	characteristic
184	C₂²	#245229	1	1	yes	characteristic
185	C₂²	#245247	1	1	yes	characteristic
186	C₂²	#245252	1	1	yes	characteristic
187	C₂²	#251827	1	1	yes	characteristic
188	C₂²	#251854	1	1	yes	characteristic
189	C₂²	#251859	1	2	yes	—
190	C₂²	#394167	1	1	yes	characteristic
191	C₂²	#394173	1	1	yes	characteristic
192	C₂²	#394174	1	1	yes	characteristic
193	C₂²	#394267	1	2	yes	—
194	C₂²	#394273	1	2	yes	—
195	C₂	#72216587	1	2	yes	—
196	C₂	#72233056	1	2	yes	—
197	C₂	#72237500	1	1	yes	characteristic
198	C₂	#72240427	1	1	yes	characteristic
199	C₂	#72241287	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	#14716
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. #72264992	C₂	semidirect
2. #72264957	C₂	semidirect
3. #72264916	C₂	semidirect
4. #72219423	C₂	semidirect
5. #72218846	C₂	semidirect
6. #72218387	C₂	semidirect
7. #72214832	C₂	semidirect
8. #72214549	C₂	semidirect
11. C₂²×#2403	C₂²	semidirect
16. #392109	C₂²	semidirect
20. #155632	C₂²	semidirect
21. #154094	C₂²	semidirect
22. C₂²⋊#1304	C₂²	semidirect
23. C₂²⋊#1304	C₂²	semidirect
24. C₂²⋊#1304	C₂²	semidirect
25. C₂³×H(5,𝔽₂)	D₄	semidirect
28. C₂²×#534	D₄	semidirect
29. C₂³×#120	D₄	semidirect
30. C₂³×#120	D₄	semidirect
32. C₂²×#518	D₄	semidirect
33. C₂⁴×C₂²⋊C₄	D₄	semidirect
34. C₂²×#386	D₄	semidirect
35. C₂²×#386	D₄	semidirect
36. C₂²×#379	D₄	semidirect
40. C₂×#1781	D₄	semidirect
41. C₂×#1781	D₄	semidirect
42. C₂×#1781	D₄	semidirect
45. C₂×C₂²⋊#95	D₄	semidirect
46. C₂×C₂²⋊#95	D₄	semidirect
47. #10725	D₄	semidirect
80. C₂⁶	#121	semidirect
103. C₂⁴×C₄	#121	semidirect
104. C₂⁴×C₄	#121	semidirect
105. C₂⁴×C₄	#121	semidirect
137. C₂×C₂²⋊C₄	B(4,𝔽₂)	semidirect
161. C₂³	#10336	semidirect

Maximal subgroups

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

subgroup	index
#72264992	2
#72264957	2
#72264916	2
#72219423	2
#72218846	2
#72218387	2
#72214832	2
#72214549	2