Subgroups — #72266300

Order 4096.

At least 11455 conjugacy classes of subgroups; only the 339 normal subgroups are shown.

The subgroup lattice has 219 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266289	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266223	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266135	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266134	C₂	1	1	yes	maximal, characteristic
5	#72266023	C₂	1	1	yes	maximal, characteristic
6	#72266021	C₄	1	1	yes	characteristic, complemented
7	#72266021	C₄	1	1	yes	characteristic, complemented
8	#72266010	C₄	1	1	yes	characteristic, complemented
9	#72236870	C₂	1	1	yes	maximal, characteristic
10	#72235937	C₂²	1	1	yes	characteristic
11	#72235793	C₄	1	4	yes	complemented
12	#72225904	C₂	1	1	yes	maximal, characteristic, complemented
13	#72225881	C₂²	1	1	yes	characteristic
14	#72225783	C₄	1	1	yes	characteristic, complemented
15	#72225621	C₂²	1	1	yes	characteristic
16	#72219377	C₂²	1	1	yes	characteristic
17	#72219307	C₂²	1	1	yes	characteristic
18	#72219307	C₂²	1	1	yes	characteristic
19	C₂×#7631286	C₂²	1	1	yes	characteristic
20	#72218467	C₄	1	4	yes	complemented
21	C₂⁴×H(5,𝔽₂)	C₂×C₄	1	1	yes	characteristic
22	C₂²×#3484	C₂×C₄	1	2	yes	—
23	C₂³×D₄²	C₂³	1	1	yes	characteristic
24	C₂×#35347	Q₈	1	2	yes	—
25	C₂×#35346	D₄	1	2	yes	—
26	C₂×#34980	D₄	1	1	yes	characteristic
27	C₂×#34980	D₄	1	1	yes	characteristic
28	C₂²×#2403	D₄	1	1	yes	characteristic
29	C₂²×#2403	D₄	1	1	yes	characteristic
30	C₂×#34256	C₂×C₄	1	2	yes	—
31	C₂×#34171	C₂×C₄	1	1	yes	characteristic
32	#517411	C₂×C₄	1	1	yes	characteristic
33	#517411	C₂×C₄	1	1	yes	characteristic
34	#517411	C₂×C₄	1	1	yes	characteristic
35	#517411	C₂×C₄	1	1	yes	characteristic
36	#392245	D₄	1	2	yes	—
37	#392109	C₂×C₄	1	2	yes	complemented
38	#392109	D₄	1	2	yes	—
39	#382422	D₄	1	2	yes	—
40	#382390	D₄	1	2	yes	—
41	#184778	C₂×C₄	1	2	yes	complemented
42	#184777	C₂×C₄	1	2	yes	complemented
43	#154094	C₂×C₄	1	2	yes	complemented
44	C₂³×H(5,𝔽₂)	C₄²	1	2	yes	—
45	C₂³×H(5,𝔽₂)	C₄⋊C₄	1	2	yes	—
46	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
47	C₂⁵×D₄	C₂²⋊C₄	1	1	yes	characteristic
48	C₂⁵×D₄	C₂²×C₄	1	1	yes	characteristic
49	C₂⁵×D₄	C₂²×C₄	1	1	yes	characteristic
50	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
51	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
52	C₂²×#534	C₄⋊C₄	1	2	yes	—
53	C₂²×#534	C₄⋊C₄	1	2	yes	—
54	C₂³×#127	#41	1	1	yes	characteristic
55	C₂³×#121	C₂×D₄	1	1	yes	characteristic
56	C₂³×#121	C₂×Q₈	1	1	yes	characteristic
57	C₂³×#121	#41	1	1	yes	characteristic
58	C₂³×#121	#41	1	1	yes	characteristic
59	C₂³×#120	C₂²⋊C₄	1	2	yes	—
60	C₂³×#120	C₂²⋊C₄	1	2	yes	—
61	C₂³×#120	C₂×D₄	1	1	yes	characteristic
62	C₂³×#120	C₂×D₄	1	1	yes	characteristic
63	C₂³×#120	C₂×D₄	1	1	yes	characteristic
64	C₂³×#120	#41	1	1	yes	characteristic
65	C₂²×#518	C₄²	1	2	yes	—
66	C₂³×C₄×D₄	C₂²×C₄	1	1	yes	characteristic
67	C₂³×C₄×D₄	C₂×D₄	1	1	yes	characteristic
68	C₂×#2308	C₂²⋊C₄	1	2	yes	—
69	C₂×#2308	C₂²⋊C₄	1	2	yes	—
70	C₂×#2308	C₄⋊C₄	1	2	yes	—
71	C₂²×#386	C₄⋊C₄	1	2	yes	—
72	C₂²×#386	C₄⋊C₄	1	2	yes	—
73	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×#379	C₂²⋊C₄	1	1	yes	characteristic
75	#14681	C₂²⋊C₄	1	2	yes	—
76	#14681	C₂²⋊C₄	1	2	yes	—
77	#14679	C₂²⋊C₄	1	2	yes	—
78	#14679	C₂²⋊C₄	1	2	yes	—
79	C₂⁷	#99	1	1	yes	characteristic
80	C₂⁷	#99	1	1	yes	characteristic
81	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂⁷	C₄×D₄	1	1	yes	characteristic
85	C₂⁴×D₄	#95	1	2	yes	—
86	C₂⁴×D₄	#95	1	2	yes	—
87	C₂⁴×D₄	#99	1	2	yes	—
88	C₂⁴×D₄	#99	1	2	yes	—
89	C₂⁴×D₄	C₂×C₄²	1	1	yes	characteristic
90	C₂⁴×D₄	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂⁴×D₄	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂⁴×D₄	#117	1	1	yes	characteristic
93	C₂⁴×D₄	#117	1	1	yes	characteristic
94	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
95	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
96	C₂⁴×D₄	#120	1	1	yes	characteristic
97	C₂⁴×D₄	#120	1	1	yes	characteristic
98	C₂⁴×D₄	#120	1	1	yes	characteristic
99	C₂⁴×D₄	#121	1	1	yes	characteristic
100	C₂⁴×D₄	#121	1	1	yes	characteristic
101	C₂⁴×D₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂⁴×D₄	#123	1	1	yes	characteristic
103	C₂⁴×D₄	#123	1	1	yes	characteristic
104	C₂⁴×D₄	#123	1	1	yes	characteristic
105	C₂⁵×C₄	#99	1	1	yes	characteristic
106	C₂⁵×C₄	#99	1	1	yes	characteristic
107	C₂⁵×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂⁵×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂²×#120	#99	1	8	yes	—
112	C₂³×C₄⋊C₄	C₄×D₄	1	1	yes	characteristic
113	C₂³×C₄⋊C₄	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂³×C₂²⋊C₄	#95	1	2	yes	—
115	C₂³×C₂²⋊C₄	#95	1	2	yes	—
116	C₂³×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
118	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
119	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
120	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
121	C₂³×C₂²⋊C₄	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂³×C₂²⋊C₄	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
124	C₂³×C₄²	C₄×D₄	1	1	yes	characteristic
125	#2308	#99	1	8	yes	—
126	C₂×#379	#99	1	4	yes	complemented
127	C₂²⋊#95	#99	1	4	yes	complemented
128	C₂⁶	#342	1	2	yes	—
129	C₂⁶	#342	1	2	yes	—
130	C₂⁶	C₂×#95	1	1	yes	characteristic
131	C₂⁶	C₄×C₂²⋊C₄	1	1	yes	characteristic
132	C₂⁶	#379	1	1	yes	characteristic
133	C₂⁶	#380	1	1	yes	characteristic
134	C₂⁶	#381	1	1	yes	characteristic
135	C₂⁶	#385	1	1	yes	characteristic
136	C₂⁶	#386	1	1	yes	characteristic
137	C₂⁶	#386	1	1	yes	characteristic
138	C₂⁶	C₂×#99	1	1	yes	characteristic
139	C₂⁶	C₂×#99	1	1	yes	characteristic
140	C₂⁶	C₂×#99	1	1	yes	characteristic
141	C₂⁶	C₂×#99	1	1	yes	characteristic
142	C₂⁶	C₂×#99	1	2	yes	—
143	C₂⁶	C₂×#99	1	4	yes	—
144	C₂⁶	#410	1	2	yes	—
145	C₂³×D₄	#342	1	4	yes	—
146	C₂³×D₄	#342	1	4	yes	—
147	C₂⁴×C₄	#342	1	2	yes	—
148	C₂⁴×C₄	#342	1	2	yes	—
149	C₂⁴×C₄	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₂⁴×C₄	#380	1	1	yes	characteristic
151	C₂⁴×C₄	#385	1	1	yes	characteristic
152	C₂⁴×C₄	#385	1	1	yes	characteristic
153	C₂⁴×C₄	#385	1	1	yes	characteristic
154	C₂⁴×C₄	#386	1	1	yes	characteristic
155	C₂⁴×C₄	#386	1	1	yes	characteristic
156	C₂⁴×C₄	C₂×#99	1	2	yes	—
157	C₂⁴×C₄	#410	1	2	yes	—
158	C₂²×C₂²⋊C₄	#342	1	4	yes	—
159	C₂²×C₂²⋊C₄	#342	1	4	yes	—
160	C₂²×C₂²⋊C₄	C₂×#99	1	4	yes	—
161	C₂⁵	#1304	1	4	yes	—
162	C₂⁵	C₂²⋊#95	1	1	yes	characteristic
163	C₂⁵	C₂×#342	1	1	yes	characteristic
164	C₂⁵	C₂×#342	1	2	yes	—
165	C₂⁵	C₂×#342	1	2	yes	—
166	C₂⁵	C₂×#342	1	1	yes	characteristic
167	C₂⁵	C₄×#99	1	2	yes	—
168	C₂⁵	#1781	1	1	yes	characteristic
169	C₂⁵	#1781	1	1	yes	characteristic
170	C₂⁵	#1781	1	1	yes	characteristic
171	C₂⁵	#1781	1	1	yes	characteristic
172	C₂⁵	#1782	1	1	yes	characteristic
173	C₂⁵	#1782	1	1	yes	characteristic
174	C₂⁵	#1799	1	2	yes	—
175	C₂⁵	#1819	1	1	yes	characteristic
176	C₂⁵	#1819	1	1	yes	characteristic
177	C₂⁵	#1828	1	2	yes	—
178	C₂⁵	#1828	1	2	yes	—
179	C₂⁵	#1898	1	2	yes	—
180	C₂⁵	#1898	1	2	yes	—
181	C₂³×C₄	#1304	1	4	yes	—
182	C₂³×C₄	C₄×#99	1	2	yes	—
183	C₂³×C₄	#1799	1	2	yes	—
184	C₂³×C₄	#1898	1	2	yes	—
185	C₂³×C₄	#1898	1	2	yes	—
186	C₂⁴	#7316	1	8	yes	complemented
187	C₂⁴	C₂×#1304	1	2	yes	—
188	C₂⁴	#8101	1	2	yes	—
189	C₂⁴	#8541	1	1	yes	characteristic
190	C₂⁴	#8542	1	1	yes	characteristic
191	C₂⁴	#8542	1	1	yes	characteristic
192	C₂⁴	#8543	1	1	yes	characteristic
193	C₂⁴	#8746	1	2	yes	—
194	C₂⁴	#8747	1	2	yes	—
195	C₂⁴	#10725	1	1	yes	characteristic
196	C₂⁴	#10735	1	1	yes	characteristic
197	C₂⁴	#10735	1	1	yes	characteristic
198	C₂⁴	#10761	1	1	yes	characteristic
199	C₂⁴	#10761	1	1	yes	characteristic
200	C₂⁴	#10768	1	1	yes	characteristic
201	C₂⁴	#10768	1	1	yes	characteristic
202	C₂³	C₂²⋊#1304	1	1	yes	characteristic
203	C₂³	#105808	1	1	yes	characteristic
204	C₂³	#106132	1	1	yes	characteristic
205	C₂³	#106654	1	1	yes	characteristic
206	C₂³	#106654	1	1	yes	characteristic
207	C₂³	#106654	1	1	yes	characteristic
208	C₂³	#106654	1	1	yes	characteristic
209	C₂³	C₂×#7316	1	4	yes	—
210	C₂³	#133839	1	2	yes	—
211	C₂²	#72219414	1	2	yes	—
212	C₂²	#72219423	1	1	yes	characteristic
213	C₂²	#72246292	1	1	yes	characteristic
214	C₂²	#72251550	1	1	yes	characteristic
215	C₂	#72265763	1	1	yes	characteristic
216	C₂	#72265788	1	1	yes	characteristic
217	C₂	#72265897	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72219423
commutator	C₂⁴×D₄	C₂×C₄²
Frattini, cosocle	C₂³×D₄²	C₂³
prime radical	C₂×#7631286	C₂²
LCS	G ▷ C₂⁴×D₄ ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×D₄² ◁ G
LECS	G ▷ C₂³×D₄² ▷ C₂⁵ ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72266289	C₂	semidirect
2. #72266223	C₂	semidirect
3. #72266135	C₂	semidirect
6. #72266021	C₄	semidirect
7. #72266021	C₄	semidirect
8. #72266010	C₄	semidirect
11. #72235793	C₄	semidirect
12. #72225904	C₂	semidirect
14. #72225783	C₄	semidirect
20. #72218467	C₄	semidirect
37. #392109	C₂×C₄	semidirect
41. #184778	C₂×C₄	semidirect
42. #184777	C₂×C₄	semidirect
43. #154094	C₂×C₄	semidirect
126. C₂×#379	#99	semidirect
127. C₂²⋊#95	#99	semidirect
186. C₂⁴	#7316	semidirect

Maximal subgroups

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

subgroup	index
#72266289	2
#72266223	2
#72266135	2
#72266134	2
#72266023	2
#72236870	2
#72225904	2