Subgroups — #72284294

Order 2048.

At least 6194 conjugacy classes of subgroups; only the 428 normal subgroups are shown.

The subgroup lattice has 222 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72284249	C₂	1	1	yes	maximal, characteristic, complemented
2	#72284187	C₂	1	4	yes	maximal, complemented
3	#72284157	C₂	1	2	yes	maximal, complemented
4	#72284143	C₂	1	2	yes	maximal, complemented
5	C₂²×#23984	C₂	1	1	yes	maximal, characteristic
6	#72258104	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#150393	C₂	1	4	yes	maximal, complemented
8	C₂²×#3490	C₄	1	1	yes	characteristic, complemented
9	C₂³×#379	C₂²	1	1	yes	characteristic
10	C₂×#23984	C₄	1	2	yes	complemented
11	C₂×#23984	C₄	1	2	yes	complemented
12	C₂×#23984	C₂²	1	2	yes	—
13	C₂×#23984	C₂²	1	2	yes	—
14	C₂²×#2029	C₄	1	1	yes	characteristic, complemented
15	C₂²×#2029	C₂²	1	1	yes	characteristic
16	C₂²×#2021	C₄	1	1	yes	characteristic, complemented
17	C₂²×#2021	C₂²	1	1	yes	characteristic
18	C₂²×#1819	C₄	1	1	yes	characteristic, complemented
19	C₂×#11499	C₂²	1	2	yes	complemented
20	C₂×#8101	C₂²	1	2	yes	complemented
21	C₂×#7426	C₂²	1	8	yes	complemented
22	C₂×#7394	C₂²	1	8	yes	complemented
23	#129209	C₂²	1	2	yes	—
24	#129209	C₂²	1	2	yes	complemented
25	C₂²⋊#1304	C₂²	1	2	yes	—
26	C₂²⋊#1304	C₂²	1	2	yes	complemented
27	C₂⁸	D₄	1	1	yes	characteristic
28	C₂×#3490	C₂×C₄	1	2	yes	—
29	C₂×#3490	C₂×C₄	1	2	yes	complemented
30	C₂²×#543	C₂×C₄	1	1	yes	characteristic, complemented
31	C₂²×#534	C₂×C₄	1	1	yes	characteristic, complemented
32	C₂⁴×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
33	C₂⁴×C₂²⋊C₄	D₄	1	1	yes	characteristic
34	C₂²×#379	C₂×C₄	1	1	yes	characteristic
35	C₂²×#379	C₂×C₄	1	1	yes	characteristic
36	C₂²×#379	C₂×C₄	1	1	yes	characteristic
37	C₂²×#379	D₄	1	1	yes	characteristic
38	C₂²×#379	D₄	1	1	yes	characteristic
39	C₂²×#379	D₄	1	1	yes	characteristic
40	C₂²×#379	D₄	1	1	yes	characteristic
41	C₂²×#379	D₄	1	1	yes	characteristic
42	C₂²×#379	D₄	1	1	yes	characteristic
43	C₂²×#379	C₂³	1	1	yes	characteristic
44	C₂²×#379	C₂³	1	1	yes	characteristic
45	C₂²×#379	C₂³	1	1	yes	characteristic
46	#23984	D₄	1	16	yes	—
47	C₂×#2029	C₂×C₄	1	2	yes	—
48	C₂×#2029	C₂×C₄	1	2	yes	complemented
49	C₂×#2029	C₂×C₄	1	2	yes	complemented
50	C₂×#2029	C₂³	1	2	yes	—
51	C₂×#2021	C₂×C₄	1	2	yes	—
52	C₂×#2021	C₂×C₄	1	2	yes	complemented
53	C₂×#2021	C₂×C₄	1	2	yes	complemented
54	C₂×#2021	C₂³	1	2	yes	—
55	C₂×#1819	C₂×C₄	1	2	yes	—
56	C₂×#1819	C₂×C₄	1	2	yes	complemented
57	C₂²×#342	C₂×C₄	1	1	yes	characteristic, complemented
58	C₂²×#342	C₂×C₄	1	1	yes	characteristic, complemented
59	C₂×#1316	C₂³	1	4	yes	—
60	C₂×#1304	C₂³	1	4	yes	—
61	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
62	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
63	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
64	C₂⁷	C₂²⋊C₄	1	1	yes	characteristic
65	C₂⁷	C₂×D₄	1	1	yes	characteristic
66	C₂⁷	C₂×D₄	1	1	yes	characteristic
67	C₂⁷	C₂×D₄	1	1	yes	characteristic
68	#3490	C₂²⋊C₄	1	8	yes	complemented
69	C₂×#543	C₂²×C₄	1	2	yes	—
70	C₂×#534	C₂²×C₄	1	2	yes	—
71	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
73	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
76	C₂³×C₂²⋊C₄	C₂²⋊C₄	1	1	yes	characteristic
77	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₂²⋊C₄	C₂²×C₄	1	1	yes	characteristic
80	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
81	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
82	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×#379	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×#379	C₂²×C₄	1	1	yes	characteristic
90	C₂×#379	C₂²×C₄	1	1	yes	characteristic
91	C₂×#379	C₂²×C₄	1	1	yes	characteristic
92	C₂×#379	C₂×D₄	1	1	yes	characteristic
93	C₂×#379	C₂×D₄	1	1	yes	characteristic
94	C₂×#379	C₂×D₄	1	1	yes	characteristic
95	C₂×#379	C₂×D₄	1	8	yes	—
96	C₂×#379	C₂×D₄	1	1	yes	characteristic
97	C₂×#379	C₂×D₄	1	1	yes	characteristic
98	C₂×#379	C₂×D₄	1	1	yes	characteristic
99	C₂×#379	C₂⁴	1	1	yes	characteristic
100	#2029	C₂²⋊C₄	1	8	yes	complemented
101	#2029	C₂×D₄	1	8	yes	—
102	#2021	C₂²⋊C₄	1	8	yes	complemented
103	#2021	C₂×D₄	1	8	yes	—
104	#1819	C₂²⋊C₄	1	8	yes	complemented
105	C₂×#342	C₂²×C₄	1	2	yes	—
106	C₂×#342	C₂²×C₄	1	2	yes	—
107	C₂⁶	#99	1	1	yes	characteristic
108	C₂⁶	#99	1	1	yes	characteristic
109	C₂⁶	#99	1	1	yes	characteristic
110	C₂⁶	#99	1	1	yes	characteristic
111	C₂⁶	#104	1	2	yes	—
112	C₂⁶	#104	1	2	yes	—
113	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂⁶	#120	1	4	yes	—
120	C₂⁶	AGL(1,ℤ/8)	1	2	yes	—
121	C₂⁶	#137	1	2	yes	—
122	C₂⁶	C₂²×D₄	1	1	yes	characteristic
123	#543	C₂×C₂²⋊C₄	1	4	yes	complemented
124	#534	C₂×C₂²⋊C₄	1	4	yes	complemented
125	C₂²×C₂²⋊C₄	#99	1	4	yes	—
126	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₂²⋊C₄	#120	1	4	yes	—
134	C₂²×C₂²⋊C₄	C₂³×C₄	1	1	yes	characteristic
135	C₂²×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
136	#379	C₂×C₂²⋊C₄	1	4	yes	—
137	#379	#120	1	4	yes	—
138	#379	#120	1	4	yes	—
139	#379	C₂²×D₄	1	4	yes	—
140	#342	C₂×C₂²⋊C₄	1	4	yes	complemented
141	#342	C₂×C₂²⋊C₄	1	4	yes	complemented
142	C₂⁵	AΣL(1,𝔽₁₆)	1	2	yes	—
143	C₂⁵	AΣL(1,𝔽₁₆)	1	4	yes	—
144	C₂⁵	AΣL(1,𝔽₁₆)	1	2	yes	—
145	C₂⁵	#379	1	2	yes	—
146	C₂⁵	#379	1	2	yes	—
147	C₂⁵	C₂×#99	1	1	yes	characteristic
148	C₂⁵	C₂×#99	1	1	yes	characteristic
149	C₂⁵	C₂×#99	1	1	yes	characteristic
150	C₂⁵	C₂×#99	1	2	yes	—
151	C₂⁵	C₂×#99	1	1	yes	characteristic
152	C₂⁵	C₂×#99	1	1	yes	characteristic
153	C₂⁵	C₂×#99	1	1	yes	characteristic
154	C₂⁵	#417	1	2	yes	—
155	C₂⁵	#417	1	2	yes	—
156	C₂⁵	C₂×#104	1	2	yes	—
157	C₂⁵	C₂×#104	1	2	yes	—
158	C₂⁵	C₂×#104	1	1	yes	characteristic
159	C₂⁵	C₂×#104	1	1	yes	characteristic
160	C₂⁵	#421	1	2	yes	—
161	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
162	C₂⁵	C₂×#120	1	2	yes	—
163	C₂⁵	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂⁵	C₂×#137	1	1	yes	characteristic
165	C₂⁵	#575	1	2	yes	—
166	C₂×C₂²⋊C₄	#379	1	2	yes	—
167	C₂×C₂²⋊C₄	#379	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂×#99	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂×#99	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂×#120	1	2	yes	—
172	C₂⁴	#1504	1	2	yes	—
173	C₂⁴	#1504	1	2	yes	—
174	C₂⁴	#1781	1	2	yes	—
175	C₂⁴	#1781	1	2	yes	—
176	C₂⁴	#1789	1	2	yes	—
177	C₂⁴	#1789	1	2	yes	—
178	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	1	yes	characteristic
179	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
180	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
181	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
182	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	1	yes	characteristic
183	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
184	C₂⁴	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
185	C₂⁴	#2127	1	2	yes	—
186	C₂⁴	#2127	1	2	yes	—
187	C₂⁴	C₂×#379	1	1	yes	characteristic
188	C₂⁴	C₂²×#99	1	1	yes	characteristic
189	C₂⁴	C₂²×#99	1	1	yes	characteristic
190	C₂⁴	C₂×#417	1	1	yes	characteristic
191	C₂⁴	#2896	1	1	yes	characteristic
192	C₂⁴	C₂²×#104	1	1	yes	characteristic
193	C₂⁴	C₂×#421	1	1	yes	characteristic
194	C₂⁴	#3006	1	2	yes	—
195	C₂⁴	#3007	1	2	yes	—
196	C₂³	#11753	1	2	yes	—
197	C₂³	#11753	1	2	yes	—
198	C₂³	#11753	1	4	yes	—
199	C₂³	#12687	1	4	yes	—
200	C₂³	C₂×#1504	1	1	yes	characteristic
201	C₂³	#17542	1	1	yes	characteristic
202	C₂³	#17587	1	1	yes	characteristic
203	C₂³	#17592	1	1	yes	characteristic
204	C₂³	C₂×#1781	1	1	yes	characteristic
205	C₂³	#20601	1	1	yes	characteristic
206	C₂³	C₂×#1789	1	1	yes	characteristic
207	C₂³	C₂²×AΣL(1,𝔽₁₆)	1	1	yes	characteristic
208	C₂³	C₂²×AΣL(1,𝔽₁₆)	1	1	yes	characteristic
209	C₂³	C₂×#2127	1	1	yes	characteristic
210	C₂³	C₂×#2127	1	1	yes	characteristic
211	C₂²	#150393	1	4	yes	complemented
212	C₂²	#185022	1	1	yes	characteristic
213	C₂²	C₂×#11753	1	1	yes	characteristic
214	C₂²	C₂×#11753	1	1	yes	characteristic
215	C₂²	#482549	1	1	yes	characteristic
216	C₂²	#482549	1	1	yes	characteristic
217	C₂²	C₂×#12687	1	1	yes	characteristic
218	C₂²	#493370	1	1	yes	characteristic
219	C₂	C₂×#150393	1	2	yes	—
220	C₂	#72283946	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#11753
commutator	C₂²×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×#379	C₂⁴
prime radical	C₂²×#23984	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁷ ◁ C₂³×#379 ◁ G
LECS	G ▷ C₂×#379 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72284249	C₂	semidirect
2. #72284187	C₂	semidirect
3. #72284157	C₂	semidirect
4. #72284143	C₂	semidirect
6. #72258104	C₂	semidirect
7. C₂×#150393	C₂	semidirect
8. C₂²×#3490	C₄	semidirect
10. C₂×#23984	C₄	semidirect
11. C₂×#23984	C₄	semidirect
14. C₂²×#2029	C₄	semidirect
16. C₂²×#2021	C₄	semidirect
18. C₂²×#1819	C₄	semidirect
19. C₂×#11499	C₂²	semidirect
20. C₂×#8101	C₂²	semidirect
21. C₂×#7426	C₂²	semidirect
22. C₂×#7394	C₂²	semidirect
24. #129209	C₂²	semidirect
26. C₂²⋊#1304	C₂²	semidirect
29. C₂×#3490	C₂×C₄	semidirect
30. C₂²×#543	C₂×C₄	semidirect
31. C₂²×#534	C₂×C₄	semidirect
48. C₂×#2029	C₂×C₄	semidirect
49. C₂×#2029	C₂×C₄	semidirect
52. C₂×#2021	C₂×C₄	semidirect
53. C₂×#2021	C₂×C₄	semidirect
56. C₂×#1819	C₂×C₄	semidirect
57. C₂²×#342	C₂×C₄	semidirect
58. C₂²×#342	C₂×C₄	semidirect
68. #3490	C₂²⋊C₄	semidirect
100. #2029	C₂²⋊C₄	semidirect
102. #2021	C₂²⋊C₄	semidirect
104. #1819	C₂²⋊C₄	semidirect
123. #543	C₂×C₂²⋊C₄	semidirect
124. #534	C₂×C₂²⋊C₄	semidirect
140. #342	C₂×C₂²⋊C₄	semidirect
141. #342	C₂×C₂²⋊C₄	semidirect
211. C₂²	#150393	semidirect

Maximal subgroups

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

subgroup	index
#72284249	2
#72284187	2
#72284157	2
#72284143	2
C₂²×#23984	2
#72258104	2
C₂×#150393	2