Subgroups — #72251674

Order 2048.

At least 6930 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

The subgroup lattice has 220 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72251673	C₂	1	1	yes	maximal, characteristic, complemented
2	#72251666	C₂	1	1	yes	maximal, characteristic, complemented
3	#72251664	C₂	1	1	yes	maximal, characteristic, complemented
4	#72251662	C₂	1	1	yes	maximal, characteristic, complemented
5	#72251533	C₂	1	1	yes	maximal, characteristic, complemented
6	#72251476	C₂	1	1	yes	maximal, characteristic, complemented
7	#72251370	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238443	C₂	1	1	yes	maximal, characteristic, complemented
9	#72238342	C₂	1	1	yes	maximal, characteristic, complemented
10	#72236948	C₂	1	1	yes	maximal, characteristic, complemented
11	#72236674	C₂	1	1	yes	maximal, characteristic, complemented
12	#72236631	C₂	1	1	yes	maximal, characteristic, complemented
13	#72235250	C₂	1	1	yes	maximal, characteristic, complemented
14	#72235178	C₂	1	1	yes	maximal, characteristic, complemented
15	#72234100	C₂	1	1	yes	maximal, characteristic, complemented
16	#10586422	C₂²	1	1	yes	characteristic
17	#7626386	C₂²	1	1	yes	characteristic, complemented
18	C₂×#23895	C₂²	1	1	yes	characteristic
19	C₂×#23895	C₂²	1	1	yes	characteristic
20	C₂×#23895	C₂²	1	1	yes	characteristic, complemented
21	C₂×#23897	C₂²	1	1	yes	characteristic
22	#6437445	C₂²	1	1	yes	characteristic
23	#6437445	C₂²	1	1	yes	characteristic
24	#6431795	C₂²	1	1	yes	characteristic
25	C₂×#12705	C₂²	1	1	yes	characteristic
26	C₂×#12705	C₂²	1	1	yes	characteristic
27	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
28	C₂×#12705	C₂²	1	1	yes	characteristic, complemented
29	#493370	C₂²	1	1	yes	characteristic
30	#382544	C₂²	1	1	yes	characteristic, complemented
31	#382436	C₂²	1	1	yes	characteristic
32	#359558	C₂²	1	1	yes	characteristic
33	#359554	C₂²	1	1	yes	characteristic
34	#359554	C₂²	1	1	yes	characteristic
35	#359554	C₂²	1	1	yes	characteristic, complemented
36	#359509	C₂²	1	1	yes	characteristic
37	#359433	C₂²	1	1	yes	characteristic
38	#188050	C₂²	1	1	yes	characteristic
39	#187944	C₂²	1	1	yes	characteristic
40	#187944	C₂²	1	1	yes	characteristic
41	#187944	C₂²	1	1	yes	characteristic, complemented
42	#186835	C₂²	1	1	yes	characteristic, complemented
43	#146479	C₂²	1	1	yes	characteristic, complemented
44	#146459	C₂²	1	1	yes	characteristic, complemented
45	#128033	C₂²	1	1	yes	characteristic, complemented
46	C₂×#7316	C₂²	1	1	yes	characteristic, complemented
47	C₂×#7314	C₂²	1	1	yes	characteristic, complemented
48	C₂×#7311	C₂²	1	1	yes	characteristic, complemented
49	#108238	C₂²	1	1	yes	characteristic, complemented
50	#105817	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	#34171	C₂³	1	1	yes	characteristic
61	C₂×#2308	C₂³	1	1	yes	characteristic
62	#23907	D₄	1	2	yes	complemented
63	#23906	D₄	1	2	yes	complemented
64	#23895	D₄	1	4	yes	complemented
65	C₂×#1913	D₄	1	2	yes	—
66	C₂×#1898	D₄	1	2	yes	complemented
67	#20616	D₄	1	2	yes	complemented
68	#20605	D₄	1	2	yes	complemented
69	C₂×#1783	C₂³	1	1	yes	characteristic
70	C₂×#1783	C₂³	1	1	yes	characteristic
71	C₂×#1783	C₂³	1	1	yes	characteristic
72	C₂×#1781	C₂³	1	1	yes	characteristic
73	C₂×#1781	C₂³	1	1	yes	characteristic
74	C₂×#1781	C₂³	1	1	yes	characteristic
75	C₂×#1781	C₂³	1	1	yes	characteristic
76	#20354	D₄	1	1	yes	characteristic, complemented
77	#14335	C₂³	1	1	yes	characteristic
78	#14255	D₄	1	2	yes	complemented
79	#10740	D₄	1	2	yes	complemented
80	#10716	D₄	1	2	yes	complemented
81	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
82	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
83	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
84	C₂²×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
85	C₂³×#41	C₂×D₄	1	1	yes	characteristic
86	C₂⁴×D₄	C₂⁴	1	1	yes	characteristic
87	#3484	C₂×D₄	1	2	yes	—
88	#3484	C₂×D₄	1	2	yes	—
89	#3484	C₂×D₄	1	2	yes	—
90	#3484	#41	1	2	yes	—
91	C₂×#534	C₂×D₄	1	2	yes	—
92	C₂²×#127	C₂×D₄	1	1	yes	characteristic
93	C₂²×#127	#41	1	1	yes	characteristic
94	C₂²×#124	C₂×D₄	1	1	yes	characteristic
95	C₂²×#124	#41	1	1	yes	characteristic
96	C₂²×#120	C₂×D₄	1	2	yes	—
97	C₂×#518	C₂×D₄	1	1	yes	characteristic
98	C₂×#518	C₂×D₄	1	1	yes	characteristic
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	C₂²×#117	C₂×D₄	1	1	yes	characteristic
104	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂²×#100	C₂×D₄	1	2	yes	—
107	C₂²×#99	C₂×D₄	1	2	yes	—
108	C₂×#407	C₂×D₄	1	2	yes	—
109	#2309	C₂×D₄	1	2	yes	—
110	#2280	C₂×D₄	1	2	yes	—
111	C₂×#379	C₂×D₄	1	2	yes	—
112	#1783	C₂×D₄	1	2	yes	—
113	#1781	C₂×D₄	1	2	yes	—
114	C₂⁶	C₂²×D₄	1	1	yes	characteristic
115	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂×H(5,𝔽₂)	#120	1	2	yes	—
117	C₂×H(5,𝔽₂)	#121	1	2	yes	complemented
118	C₂³×Q₈	#121	1	1	yes	characteristic, complemented
119	C₂³×D₄	#120	1	2	yes	—
120	C₂³×D₄	#121	1	1	yes	characteristic, complemented
121	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
122	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
123	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
124	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
125	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
126	C₂³×D₄	C₂×#41	1	1	yes	characteristic
127	C₂³×D₄	C₂×#41	1	1	yes	characteristic
128	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
129	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
130	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
134	#518	#120	1	2	yes	—
135	#518	#121	1	2	yes	complemented
136	C₂×#117	#120	1	2	yes	—
137	C₂×#117	#120	1	2	yes	—
138	C₂²×C₂²⋊C₄	#120	1	2	yes	—
139	C₂²×C₄²	#121	1	1	yes	characteristic, complemented
140	C₂²×C₄²	#121	1	1	yes	characteristic, complemented
141	C₂⁵	B(4,𝔽₂)	1	2	yes	—
142	C₂⁵	C₂×#120	1	1	yes	characteristic
143	C₂⁵	#525	1	1	yes	characteristic
144	C₂⁵	#534	1	1	yes	characteristic
145	C₂⁵	#534	1	1	yes	characteristic
146	C₂⁵	#535	1	1	yes	characteristic
147	C₂⁵	#538	1	1	yes	characteristic
148	C₂⁵	D₄²	1	1	yes	characteristic
149	C₂⁵	D₄²	1	1	yes	characteristic
150	C₂⁵	D₄²	1	1	yes	characteristic
151	C₂⁵	D₄²	1	1	yes	characteristic
152	C₂⁵	D₄²	1	1	yes	characteristic
153	C₂⁵	#546	1	1	yes	characteristic
154	C₂⁵	#560	1	1	yes	characteristic
155	C₂²×D₄	C₂×#120	1	1	yes	characteristic
156	C₂²×D₄	C₂×#121	1	1	yes	characteristic
157	C₂²×D₄	#534	1	1	yes	characteristic
158	C₂²×D₄	#534	1	1	yes	characteristic
159	C₂²×D₄	#535	1	1	yes	characteristic
160	C₂²×D₄	#538	1	1	yes	characteristic
161	C₂²×D₄	D₄²	1	1	yes	characteristic
162	C₂²×D₄	D₄²	1	1	yes	characteristic
163	C₂²×D₄	D₄²	1	1	yes	characteristic
164	C₂²×D₄	#546	1	1	yes	characteristic
165	C₂²×D₄	#546	1	1	yes	characteristic
166	C₂²×D₄	#546	1	1	yes	characteristic
167	C₂²×D₄	#546	1	1	yes	characteristic
168	C₂²×D₄	#546	1	1	yes	characteristic
169	C₂²×D₄	#546	1	1	yes	characteristic
170	C₂²×D₄	#560	1	1	yes	characteristic
171	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
172	C₂³×C₄	C₂×#120	1	1	yes	characteristic
173	C₂³×C₄	C₂×#121	1	1	yes	characteristic
174	C₂³×C₄	#535	1	1	yes	characteristic
175	C₂³×C₄	D₄²	1	1	yes	characteristic
176	C₂³×C₄	D₄²	1	1	yes	characteristic
177	C₂³×C₄	#546	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
179	C₂×C₄²	B(4,𝔽₂)	1	2	yes	—
180	C₂×C₄²	B(4,𝔽₂)	1	2	yes	—
181	C₂⁴	#2403	1	1	yes	characteristic
182	C₂⁴	#2403	1	1	yes	characteristic
183	C₂⁴	#2404	1	1	yes	characteristic
184	C₂⁴	#2410	1	1	yes	characteristic
185	C₂⁴	#2424	1	1	yes	characteristic
186	C₂⁴	#2424	1	1	yes	characteristic
187	C₂⁴	#2668	1	1	yes	characteristic
188	C₂⁴	#2668	1	1	yes	characteristic
189	C₂⁴	#2669	1	1	yes	characteristic
190	C₂⁴	#2669	1	1	yes	characteristic
191	C₂⁴	#2678	1	1	yes	characteristic
192	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
193	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
194	C₂²×C₄	#2408	1	1	yes	characteristic
195	C₂²×C₄	#2408	1	1	yes	characteristic
196	C₂²×C₄	#2410	1	1	yes	characteristic
197	C₂²×C₄	#2678	1	1	yes	characteristic
198	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
199	C₂²×C₄	#3025	1	2	yes	—
200	C₂³	#15257	1	1	yes	characteristic
201	C₂³	#23895	1	1	yes	characteristic
202	C₂³	#23895	1	1	yes	characteristic
203	C₂³	#23900	1	2	yes	—
204	C₂³	#23900	1	1	yes	characteristic
205	C₂³	#23900	1	1	yes	characteristic
206	C₂³	#23900	1	1	yes	characteristic
207	C₂³	#23902	1	1	yes	characteristic
208	C₂³	#23984	1	2	yes	—
209	C₂×C₄	#24260	1	2	yes	—
210	C₂×C₄	#24260	1	2	yes	—
211	C₂²	#245225	1	1	yes	characteristic
212	C₂²	#245228	1	1	yes	characteristic
213	C₂²	#245231	1	1	yes	characteristic
214	C₂²	#245233	1	1	yes	characteristic
215	C₂²	#417493	1	1	yes	characteristic
216	C₂²	#417526	1	1	yes	characteristic
217	C₂²	#417532	1	1	yes	characteristic
218	C₂	#72237498	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72237498
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. #72251673	C₂	semidirect
2. #72251666	C₂	semidirect
3. #72251664	C₂	semidirect
4. #72251662	C₂	semidirect
5. #72251533	C₂	semidirect
6. #72251476	C₂	semidirect
7. #72251370	C₂	semidirect
8. #72238443	C₂	semidirect
9. #72238342	C₂	semidirect
10. #72236948	C₂	semidirect
11. #72236674	C₂	semidirect
12. #72236631	C₂	semidirect
13. #72235250	C₂	semidirect
14. #72235178	C₂	semidirect
15. #72234100	C₂	semidirect
17. #7626386	C₂²	semidirect
20. C₂×#23895	C₂²	semidirect
27. C₂×#12705	C₂²	semidirect
28. C₂×#12705	C₂²	semidirect
30. #382544	C₂²	semidirect
35. #359554	C₂²	semidirect
41. #187944	C₂²	semidirect
42. #186835	C₂²	semidirect
43. #146479	C₂²	semidirect
44. #146459	C₂²	semidirect
45. #128033	C₂²	semidirect
46. C₂×#7316	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	C₂²	semidirect
49. #108238	C₂²	semidirect
50. #105817	C₂²	semidirect
51. #62842	D₄	semidirect
58. C₂×#2883	D₄	semidirect
59. C₂×#2883	D₄	semidirect
62. #23907	D₄	semidirect
63. #23906	D₄	semidirect
64. #23895	D₄	semidirect
66. C₂×#1898	D₄	semidirect
67. #20616	D₄	semidirect
68. #20605	D₄	semidirect
76. #20354	D₄	semidirect
78. #14255	D₄	semidirect
79. #10740	D₄	semidirect
80. #10716	D₄	semidirect
117. C₂×H(5,𝔽₂)	#121	semidirect
118. C₂³×Q₈	#121	semidirect
120. C₂³×D₄	#121	semidirect
135. #518	#121	semidirect
139. C₂²×C₄²	#121	semidirect
140. C₂²×C₄²	#121	semidirect

Maximal subgroups

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

subgroup	index
#72251673	2
#72251666	2
#72251664	2
#72251662	2
#72251533	2
#72251476	2
#72251370	2
#72238443	2
#72238342	2
#72236948	2
#72236674	2
#72236631	2
#72235250	2
#72235178	2
#72234100	2