Subgroups — #72312531

Order 2048.

At least 6303 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

The subgroup lattice has 221 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72312312	C₂	1	1	yes	maximal, characteristic, complemented
2	#72310358	C₂	1	1	yes	maximal, characteristic, complemented
3	#72309794	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266834	C₂	1	1	yes	maximal, characteristic, complemented
5	#72264770	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#139742	C₂	1	1	yes	maximal, characteristic, complemented
7	#72256052	C₂	1	1	yes	maximal, characteristic, complemented
8	#72255863	C₂	1	1	yes	maximal, characteristic, complemented
9	#72251527	C₂	1	1	yes	maximal, characteristic, complemented
10	#72251485	C₂	1	1	yes	maximal, characteristic, complemented
11	#72251054	C₂	1	1	yes	maximal, characteristic, complemented
12	#72246288	C₂	1	1	yes	maximal, characteristic, complemented
13	#72238343	C₂	1	1	yes	maximal, characteristic, complemented
14	#72237634	C₂	1	1	yes	maximal, characteristic, complemented
15	#72234100	C₂	1	1	yes	maximal, characteristic
16	#10586422	C₂²	1	1	yes	characteristic
17	#6440460	C₂²	1	1	yes	characteristic
18	#6440460	C₂²	1	1	yes	characteristic
19	C₂×#23897	C₂²	1	1	yes	characteristic
20	#6437445	C₂²	1	1	yes	characteristic
21	#6431795	C₂²	1	1	yes	characteristic
22	#6374210	C₂²	1	1	yes	characteristic, complemented
23	C₂×#21776	C₂²	1	1	yes	characteristic, complemented
24	#763514	C₂²	1	1	yes	characteristic
25	#493879	C₂²	1	1	yes	characteristic, complemented
26	C₂×#12707	C₂²	1	1	yes	characteristic, complemented
27	C₂×#12705	C₂²	1	1	yes	characteristic
28	C₂×#12705	C₂²	1	1	yes	characteristic
29	#493370	C₂²	1	1	yes	characteristic
30	#382436	C₂²	1	1	yes	characteristic
31	#373637	C₂²	1	1	yes	characteristic, complemented
32	#359811	C₂²	1	1	yes	characteristic, complemented
33	#359558	C₂²	1	1	yes	characteristic
34	#359549	C₂²	1	1	yes	characteristic
35	#359549	C₂²	1	1	yes	characteristic
36	#359509	C₂²	1	1	yes	characteristic
37	#359509	C₂²	1	1	yes	characteristic
38	#190147	C₂²	1	1	yes	characteristic, complemented
39	#188050	C₂²	1	1	yes	characteristic
40	C₂×#8534	C₂²	1	1	yes	characteristic
41	C₂×#8534	C₂²	1	1	yes	characteristic
42	C₂×#8453	C₂²	1	1	yes	characteristic, complemented
43	#146475	C₂²	1	1	yes	characteristic, complemented
44	#146475	C₂²	1	1	yes	characteristic, complemented
45	C₂×#7304	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	#108222	C₂²	1	1	yes	characteristic, complemented
50	#107130	C₂²	1	1	yes	characteristic, complemented
51	#62842	D₄	1	1	yes	characteristic
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	C₂²×#535	C₂³	1	1	yes	characteristic
56	C₂×#3019	D₄	1	2	yes	—
57	#60359	D₄	1	2	yes	—
58	C₂×#2883	D₄	1	1	yes	characteristic
59	C₂×#2883	D₄	1	1	yes	characteristic
60	#34992	D₄	1	2	yes	—
61	#34171	C₂³	1	1	yes	characteristic
62	#34171	C₂³	1	1	yes	characteristic
63	C₂×#2308	C₂³	1	1	yes	characteristic
64	C₂×#1913	D₄	1	2	yes	—
65	C₂×#1911	D₄	1	2	yes	—
66	C₂×#1898	D₄	1	2	yes	—
67	#20979	D₄	1	2	yes	—
68	#20960	D₄	1	2	yes	—
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
77	C₂×#1438	C₂³	1	1	yes	characteristic
78	#8546	D₄	1	2	yes	complemented
79	#8544	D₄	1	2	yes	complemented
80	#8534	D₄	1	2	yes	complemented
81	#8534	D₄	1	2	yes	complemented
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	#41	1	2	yes	—
89	C₂×#535	C₂×D₄	1	2	yes	—
90	C₂×#534	C₂×D₄	1	2	yes	—
91	C₂²×#127	C₂×D₄	1	1	yes	characteristic
92	C₂²×#127	#41	1	1	yes	characteristic
93	C₂²×#124	C₂×D₄	1	1	yes	characteristic
94	C₂²×#124	#41	1	1	yes	characteristic
95	C₂×#525	C₂×D₄	1	2	yes	—
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	C₂×#518	C₂×D₄	1	1	yes	characteristic
104	C₂²×#117	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
106	C₂³×C₂²⋊C₄	C₂×D₄	1	1	yes	characteristic
107	C₂²×#100	C₂×D₄	1	2	yes	—
108	C₂²×#99	C₂×D₄	1	2	yes	—
109	C₂×#407	C₂×D₄	1	2	yes	—
110	C₂×#379	C₂×D₄	1	2	yes	—
111	#1781	C₂×D₄	1	2	yes	—
112	#1781	C₂×D₄	1	2	yes	—
113	#1781	C₂×D₄	1	2	yes	—
114	#1438	C₂×D₄	1	1	yes	characteristic
115	#1438	C₂×D₄	1	1	yes	characteristic
116	C₂⁶	C₂²×D₄	1	1	yes	characteristic
117	C₂⁶	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂×H(5,𝔽₂)	#121	1	2	yes	—
119	C₂²×#41	#120	1	2	yes	—
120	C₂³×Q₈	#121	1	1	yes	characteristic
121	C₂³×D₄	#120	1	2	yes	—
122	C₂³×D₄	#121	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	C₂²×D₄	1	1	yes	characteristic
127	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
128	C₂³×D₄	C₂×#41	1	1	yes	characteristic
129	C₂³×D₄	C₂×#41	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	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂³×D₄	H(5,𝔽₂)	1	1	yes	characteristic
135	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
136	#518	#121	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
140	C₂²×C₄²	#121	1	1	yes	characteristic
141	#379	#120	1	4	yes	—
142	C₂⁵	B(4,𝔽₂)	1	2	yes	—
143	C₂⁵	C₂×#120	1	1	yes	characteristic
144	C₂⁵	C₂×#120	1	1	yes	characteristic
145	C₂⁵	#534	1	1	yes	characteristic
146	C₂⁵	#534	1	1	yes	characteristic
147	C₂⁵	#535	1	1	yes	characteristic
148	C₂⁵	#538	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	#546	1	1	yes	characteristic
153	C₂⁵	#546	1	1	yes	characteristic
154	C₂⁵	#546	1	1	yes	characteristic
155	C₂⁵	#560	1	1	yes	characteristic
156	C₂²×Q₈	#458	1	2	yes	complemented
157	C₂²×D₄	#458	1	2	yes	complemented
158	C₂²×D₄	C₂×#121	1	1	yes	characteristic
159	C₂²×D₄	#525	1	1	yes	characteristic
160	C₂²×D₄	#534	1	1	yes	characteristic
161	C₂²×D₄	#535	1	1	yes	characteristic
162	C₂²×D₄	#535	1	1	yes	characteristic
163	C₂²×D₄	#538	1	1	yes	characteristic
164	C₂²×D₄	D₄²	1	1	yes	characteristic
165	C₂²×D₄	D₄²	1	1	yes	characteristic
166	C₂²×D₄	D₄²	1	1	yes	characteristic
167	C₂²×D₄	D₄²	1	1	yes	characteristic
168	C₂²×D₄	D₄²	1	1	yes	characteristic
169	C₂²×D₄	#546	1	1	yes	characteristic
170	C₂²×D₄	#546	1	1	yes	characteristic
171	C₂²×D₄	#546	1	1	yes	characteristic
172	C₂²×D₄	#546	1	1	yes	characteristic
173	C₂²×D₄	#560	1	1	yes	characteristic
174	C₂³×C₄	B(4,𝔽₂)	1	2	yes	—
175	C₂³×C₄	C₂×#120	1	1	yes	characteristic
176	C₂³×C₄	C₂×#121	1	1	yes	characteristic
177	C₂³×C₄	#534	1	1	yes	characteristic
178	C₂³×C₄	D₄²	1	1	yes	characteristic
179	C₂³×C₄	D₄²	1	1	yes	characteristic
180	C₂³×C₄	#546	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	B(4,𝔽₂)	1	4	yes	—
182	C₂⁴	#2403	1	1	yes	characteristic
183	C₂⁴	#2403	1	1	yes	characteristic
184	C₂⁴	#2404	1	1	yes	characteristic
185	C₂⁴	#2408	1	1	yes	characteristic
186	C₂⁴	#2408	1	1	yes	characteristic
187	C₂⁴	#2424	1	1	yes	characteristic
188	C₂⁴	#2668	1	1	yes	characteristic
189	C₂⁴	#2668	1	1	yes	characteristic
190	C₂⁴	#2669	1	1	yes	characteristic
191	C₂⁴	#2678	1	1	yes	characteristic
192	C₂⁴	#2678	1	1	yes	characteristic
193	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
194	C₂⁴	C₂×B(4,𝔽₂)	1	2	yes	—
195	C₂²×C₄	#2410	1	1	yes	characteristic
196	C₂²×C₄	#2410	1	1	yes	characteristic
197	C₂²×C₄	#2424	1	1	yes	characteristic
198	C₂²×C₄	#2669	1	1	yes	characteristic
199	C₂²×C₄	C₂×#458	1	2	yes	—
200	C₂²×C₄	#3026	1	2	yes	—
201	C₂³	#15257	1	1	yes	characteristic
202	C₂³	#23895	1	1	yes	characteristic
203	C₂³	#23896	1	1	yes	characteristic
204	C₂³	#23900	1	2	yes	—
205	C₂³	#23900	1	1	yes	characteristic
206	C₂³	#23900	1	1	yes	characteristic
207	C₂³	#23901	1	1	yes	characteristic
208	C₂³	#23903	1	1	yes	characteristic
209	C₂³	#23984	1	2	yes	—
210	C₂×C₄	#24405	1	2	yes	—
211	C₂×C₄	#24405	1	2	yes	—
212	C₂²	#245226	1	1	yes	characteristic
213	C₂²	#245230	1	1	yes	characteristic
214	C₂²	#245254	1	1	yes	characteristic
215	C₂²	#245255	1	1	yes	characteristic
216	C₂²	#417493	1	1	yes	characteristic
217	C₂²	#417499	1	1	yes	characteristic
218	C₂²	#417510	1	1	yes	characteristic
219	C₂	#72265994	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72265994
commutator, Frattini, cosocle	C₂⁴×D₄	C₂⁴
prime radical	#72234100	C₂
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. #72312312	C₂	semidirect
2. #72310358	C₂	semidirect
3. #72309794	C₂	semidirect
4. #72266834	C₂	semidirect
5. #72264770	C₂	semidirect
6. C₂×#139742	C₂	semidirect
7. #72256052	C₂	semidirect
8. #72255863	C₂	semidirect
9. #72251527	C₂	semidirect
10. #72251485	C₂	semidirect
11. #72251054	C₂	semidirect
12. #72246288	C₂	semidirect
13. #72238343	C₂	semidirect
14. #72237634	C₂	semidirect
22. #6374210	C₂²	semidirect
23. C₂×#21776	C₂²	semidirect
25. #493879	C₂²	semidirect
26. C₂×#12707	C₂²	semidirect
31. #373637	C₂²	semidirect
32. #359811	C₂²	semidirect
38. #190147	C₂²	semidirect
42. C₂×#8453	C₂²	semidirect
43. #146475	C₂²	semidirect
44. #146475	C₂²	semidirect
45. C₂×#7304	C₂²	semidirect
46. C₂×#7316	C₂²	semidirect
47. C₂×#7314	C₂²	semidirect
48. C₂×#7311	C₂²	semidirect
49. #108222	C₂²	semidirect
50. #107130	C₂²	semidirect
78. #8546	D₄	semidirect
79. #8544	D₄	semidirect
80. #8534	D₄	semidirect
81. #8534	D₄	semidirect
156. C₂²×Q₈	#458	semidirect
157. C₂²×D₄	#458	semidirect

Maximal subgroups

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

subgroup	index
#72312312	2
#72310358	2
#72309794	2
#72266834	2
#72264770	2
C₂×#139742	2
#72256052	2
#72255863	2
#72251527	2
#72251485	2
#72251054	2
#72246288	2
#72238343	2
#72237634	2
#72234100	2