Subgroups — #72312555

Order 2048.

At least 6929 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

The subgroup lattice has 227 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72312216	C₂	1	1	yes	maximal, characteristic, complemented
2	#72309794	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266834	C₂	1	1	yes	maximal, characteristic, complemented
4	#72260584	C₂	1	1	yes	maximal, characteristic, complemented
5	#72258368	C₂	1	1	yes	maximal, characteristic, complemented
6	#72256052	C₂	1	1	yes	maximal, characteristic, complemented
7	#72256025	C₂	1	1	yes	maximal, characteristic, complemented
8	#72251528	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251073	C₂	1	1	yes	maximal, characteristic, complemented
10	#72238342	C₂	1	1	yes	maximal, characteristic, complemented
11	#72237635	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236674	C₂	1	1	yes	maximal, characteristic, complemented
13	#72234100	C₂	1	1	yes	maximal, characteristic, complemented
14	#72219159	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#144364	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#6440460	C₂²	1	1	yes	characteristic
18	#6440460	C₂²	1	1	yes	characteristic
19	#6440460	C₂²	1	1	yes	characteristic, complemented
20	C₂×#23895	C₂²	1	1	yes	characteristic
21	C₂×#23895	C₂²	1	1	yes	characteristic
22	C₂×#23895	C₂²	1	1	yes	characteristic, complemented
23	C₂×#23897	C₂²	1	1	yes	characteristic
24	#6437445	C₂²	1	1	yes	characteristic
25	#6437445	C₂²	1	1	yes	characteristic
26	#6431795	C₂²	1	1	yes	characteristic
27	#6374210	C₂²	1	1	yes	characteristic, complemented
28	#763514	C₂²	1	1	yes	characteristic
29	#493879	C₂²	1	1	yes	characteristic, complemented
30	C₂×#12705	C₂²	1	1	yes	characteristic
31	C₂×#12705	C₂²	1	1	yes	characteristic
32	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
33	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
34	#493370	C₂²	1	1	yes	characteristic
35	#382436	C₂²	1	1	yes	characteristic
36	#359558	C₂²	1	1	yes	characteristic
37	#359509	C₂²	1	1	yes	characteristic
38	#355096	C₂²	1	1	yes	characteristic, complemented
39	#190147	C₂²	1	1	yes	characteristic, complemented
40	#188050	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8534	C₂²	1	1	yes	characteristic
43	C₂×#8534	C₂²	1	1	yes	characteristic, complemented
44	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
45	#146479	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7304	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
49	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
50	#107130	C₂²	1	1	yes	characteristic, complemented
51	#62842	D₄	1	1	yes	characteristic, complemented
52	C₂×#3484	C₂³	1	1	yes	characteristic
53	C₂×#3484	C₂³	1	1	yes	characteristic
54	C₂×#3484	C₂³	1	1	yes	characteristic
55	C₂×#3484	C₂³	1	1	yes	characteristic
56	C₂²×#535	C₂³	1	1	yes	characteristic
57	C₂×#3026	D₄	1	2	yes	—
58	C₂×#2883	D₄	1	1	yes	characteristic, complemented
59	C₂×#2883	D₄	1	1	yes	characteristic, complemented
60	#34992	D₄	1	2	yes	complemented
61	#34171	C₂³	1	1	yes	characteristic
62	C₂×#2308	C₂³	1	1	yes	characteristic
63	C₂×#1913	D₄	1	2	yes	—
64	C₂×#1898	D₄	1	2	yes	complemented
65	#20616	D₄	1	2	yes	complemented
66	#20605	D₄	1	2	yes	complemented
67	C₂×#1783	C₂³	1	1	yes	characteristic
68	C₂×#1783	C₂³	1	1	yes	characteristic
69	C₂×#1783	C₂³	1	1	yes	characteristic
70	C₂×#1781	C₂³	1	1	yes	characteristic
71	C₂×#1781	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	#20354	D₄	1	1	yes	characteristic, complemented
75	C₂×#1438	C₂³	1	1	yes	characteristic
76	#12705	D₄	1	4	yes	complemented
77	#12694	D₄	1	2	yes	complemented
78	#12693	D₄	1	2	yes	complemented
79	#10740	D₄	1	2	yes	complemented
80	#10716	D₄	1	2	yes	complemented
81	#8546	D₄	1	2	yes	complemented
82	#8544	D₄	1	2	yes	complemented
83	#8534	D₄	1	2	yes	complemented
84	#8534	D₄	1	2	yes	complemented
85	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
86	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
87	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
88	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
89	C₂³×#41	C₂×D₄	1	1	yes	characteristic
90	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
91	#3484	C₂×D₄	1	2	yes	—
92	#3484	C₂×D₄	1	2	yes	—
93	C₂×#534	C₂×D₄	1	2	yes	—
94	C₂²×#127	C₂×D₄	1	1	yes	characteristic
95	C₂²×#127	C₂×D₄	1	1	yes	characteristic
96	C₂²×#124	C₂×D₄	1	1	yes	characteristic
97	C₂²×#124	C₂×D₄	1	1	yes	characteristic
98	C₂²×#120	C₂×D₄	1	2	yes	—
99	C₂×#518	C₂×D₄	1	1	yes	characteristic
100	C₂×#518	C₂×D₄	1	1	yes	characteristic
101	C₂×#518	C₂×D₄	1	1	yes	characteristic
102	C₂×#518	C₂×D₄	1	1	yes	characteristic
103	C₂×#518	C₂×D₄	1	1	yes	characteristic
104	C₂×#518	C₂×D₄	1	1	yes	characteristic
105	C₂²×#117	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
108	C₂²×#100	C₂×D₄	1	2	yes	—
109	C₂²×#99	C₂×D₄	1	2	yes	—
110	C₂×#407	C₂×D₄	1	2	yes	—
111	#2309	C₂×D₄	1	2	yes	—
112	#2280	C₂×D₄	1	2	yes	—
113	C₂×#379	C₂×D₄	1	2	yes	—
114	#1783	C₂×D₄	1	2	yes	—
115	#1781	C₂×D₄	1	2	yes	—
116	#1781	C₂×D₄	1	2	yes	—
117	#1781	C₂×D₄	1	2	yes	—
118	#1781	C₂×D₄	1	2	yes	—
119	#1781	C₂×D₄	1	2	yes	—
120	#1781	C₂×D₄	1	2	yes	—
121	#1438	C₂×D₄	1	1	yes	characteristic
122	#1438	C₂×D₄	1	1	yes	characteristic
123	C₂⁶	C₂²×D₄	1	1	yes	characteristic
124	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
125	C₂×H(5,𝔽₂)	#127	1	2	yes	—
126	C₂³×Q₈	#127	1	1	yes	characteristic
127	C₂³×D₄	#120	1	2	yes	—
128	C₂³×D₄	#127	1	1	yes	characteristic
129	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
130	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
131	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
132	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
133	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
134	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
135	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
136	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
138	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
142	#518	#127	1	2	yes	—
143	C₂×#117	#120	1	2	yes	—
144	C₂×#117	#120	1	2	yes	—
145	C₂²×C₂²⋊C₄	#120	1	2	yes	—
146	C₂²×C₄²	#127	1	1	yes	characteristic
147	C₂²×C₄²	#127	1	1	yes	characteristic
148	C₂×#99	#120	1	4	yes	—
149	#379	#120	1	4	yes	—
150	C₂⁵	B(4,𝔽₂)	1	2	yes	—
151	C₂⁵	C₂×#120	1	1	yes	characteristic
152	C₂⁵	C₂×#120	1	1	yes	characteristic
153	C₂⁵	#534	1	1	yes	characteristic
154	C₂⁵	#534	1	1	yes	characteristic
155	C₂⁵	#535	1	1	yes	characteristic
156	C₂⁵	#535	1	1	yes	characteristic
157	C₂⁵	D₄²	1	1	yes	characteristic
158	C₂⁵	D₄²	1	1	yes	characteristic
159	C₂⁵	D₄²	1	1	yes	characteristic
160	C₂⁵	D₄²	1	1	yes	characteristic
161	C₂⁵	D₄²	1	1	yes	characteristic
162	C₂⁵	D₄²	1	1	yes	characteristic
163	C₂⁵	#560	1	1	yes	characteristic
164	C₂²×D₄	C₂×#120	1	1	yes	characteristic
165	C₂²×D₄	C₂×#127	1	1	yes	characteristic
166	C₂²×D₄	#534	1	1	yes	characteristic
167	C₂²×D₄	#535	1	1	yes	characteristic
168	C₂²×D₄	#535	1	1	yes	characteristic
169	C₂²×D₄	#535	1	1	yes	characteristic
170	C₂²×D₄	D₄²	1	1	yes	characteristic
171	C₂²×D₄	D₄²	1	1	yes	characteristic
172	C₂²×D₄	D₄²	1	1	yes	characteristic
173	C₂²×D₄	D₄²	1	1	yes	characteristic
174	C₂²×D₄	D₄²	1	1	yes	characteristic
175	C₂²×D₄	D₄²	1	1	yes	characteristic
176	C₂²×D₄	D₄²	1	1	yes	characteristic
177	C₂²×D₄	D₄²	1	1	yes	characteristic
178	C₂²×D₄	D₄²	1	1	yes	characteristic
179	C₂²×D₄	#560	1	1	yes	characteristic
180	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
181	C₂³×C₄	C₂×#120	1	1	yes	characteristic
182	C₂³×C₄	C₂×#127	1	1	yes	characteristic
183	C₂³×C₄	#535	1	1	yes	characteristic
184	C₂³×C₄	D₄²	1	1	yes	characteristic
185	C₂³×C₄	D₄²	1	1	yes	characteristic
186	C₂³×C₄	D₄²	1	1	yes	characteristic
187	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
188	C₂⁴	#2403	1	1	yes	characteristic
189	C₂⁴	#2403	1	1	yes	characteristic
190	C₂⁴	#2403	1	1	yes	characteristic
191	C₂⁴	#2433	1	1	yes	characteristic
192	C₂⁴	#2433	1	1	yes	characteristic
193	C₂⁴	#2433	1	1	yes	characteristic
194	C₂⁴	#2669	1	1	yes	characteristic
195	C₂⁴	#2669	1	1	yes	characteristic
196	C₂⁴	#2669	1	1	yes	characteristic
197	C₂⁴	#2669	1	1	yes	characteristic
198	C₂⁴	#2669	1	1	yes	characteristic
199	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
200	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
201	C₂²×C₄	#2433	1	1	yes	characteristic
202	C₂²×C₄	#2433	1	1	yes	characteristic
203	C₂²×C₄	#2433	1	1	yes	characteristic
204	C₂²×C₄	#2669	1	1	yes	characteristic
205	C₂²×C₄	#3025	1	2	yes	—
206	C₂²×C₄	#3026	1	2	yes	—
207	C₂³	#15258	1	1	yes	characteristic
208	C₂³	#23895	1	1	yes	characteristic
209	C₂³	#23895	1	1	yes	characteristic
210	C₂³	#23900	1	2	yes	—
211	C₂³	#23900	1	2	yes	—
212	C₂³	#23900	1	1	yes	characteristic
213	C₂³	#23900	1	1	yes	characteristic
214	C₂³	#23902	1	1	yes	characteristic
215	C₂³	#23903	1	1	yes	characteristic
216	C₂×C₄	#25773	1	2	yes	—
217	C₂×C₄	#25773	1	2	yes	—
218	C₂²	#245228	1	1	yes	characteristic
219	C₂²	#245231	1	1	yes	characteristic
220	C₂²	#245246	1	1	yes	characteristic
221	C₂²	#245255	1	1	yes	characteristic
222	C₂²	#417744	1	1	yes	characteristic
223	C₂²	#417823	1	1	yes	characteristic
224	C₂²	#417824	1	1	yes	characteristic
225	C₂	#72241200	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72241200
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
LCS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72312216	C₂	semidirect
2. #72309794	C₂	semidirect
3. #72266834	C₂	semidirect
4. #72260584	C₂	semidirect
5. #72258368	C₂	semidirect
6. #72256052	C₂	semidirect
7. #72256025	C₂	semidirect
8. #72251528	C₂	semidirect
9. #72251073	C₂	semidirect
10. #72238342	C₂	semidirect
11. #72237635	C₂	semidirect
12. #72236674	C₂	semidirect
13. #72234100	C₂	semidirect
14. #72219159	C₂	semidirect
15. C₂×#144364	C₂	semidirect
19. #6440460	C₂²	semidirect
22. C₂×#23895	C₂²	semidirect
27. #6374210	C₂²	semidirect
29. #493879	C₂²	semidirect
32. C₂×#12705	C₂²	semidirect
33. C₂×#12705	C₂²	semidirect
38. #355096	C₂²	semidirect
39. #190147	C₂²	semidirect
43. C₂×#8534	C₂²	semidirect
44. C₂×#8453	C₂²	semidirect
45. #146479	C₂²	semidirect
46. C₂×#7304	C₂²	semidirect
47. C₂×#7316	C₂²	semidirect
48. C₂×#7314	C₂²	semidirect
49. C₂×#7311	C₂²	semidirect
50. #107130	C₂²	semidirect
51. #62842	D₄	semidirect
58. C₂×#2883	D₄	semidirect
59. C₂×#2883	D₄	semidirect
60. #34992	D₄	semidirect
64. C₂×#1898	D₄	semidirect
65. #20616	D₄	semidirect
66. #20605	D₄	semidirect
74. #20354	D₄	semidirect
76. #12705	D₄	semidirect
77. #12694	D₄	semidirect
78. #12693	D₄	semidirect
79. #10740	D₄	semidirect
80. #10716	D₄	semidirect
81. #8546	D₄	semidirect
82. #8544	D₄	semidirect
83. #8534	D₄	semidirect
84. #8534	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72312216	2
#72309794	2
#72266834	2
#72260584	2
#72258368	2
#72256052	2
#72256025	2
#72251528	2
#72251073	2
#72238342	2
#72237635	2
#72236674	2
#72234100	2
#72219159	2
C₂×#144364	2