Subgroups — #72238656

Order 4096.

At least 9007 conjugacy classes of subgroups; only the 445 normal subgroups are shown.

The subgroup lattice has 228 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72238644	C₂	1	1	yes	maximal, characteristic, complemented
2	#72238614	C₂	1	1	yes	maximal, characteristic, complemented
3	#72238601	C₂	1	1	yes	maximal, characteristic, complemented
4	#72238364	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237997	C₂	1	1	yes	maximal, characteristic, complemented
6	#72237937	C₂	1	4	yes	maximal, complemented
7	#72237935	C₂²	1	2	yes	complemented
8	#72237883	C₂	1	4	yes	maximal, complemented
9	#72237881	C₂²	1	2	yes	complemented
10	#72237878	C₂²	1	4	yes	—
11	#72237874	C₂²	1	2	yes	complemented
12	#72237872	C₂²	1	4	yes	complemented
13	#72237747	C₂²	1	4	yes	complemented
14	#72237616	C₂²	1	2	yes	complemented
15	#72237596	C₂	1	1	yes	maximal, characteristic, complemented
16	#72237594	C₂²	1	4	yes	—
17	#72235718	C₂²	1	1	yes	characteristic
18	#72234717	C₂	1	1	yes	maximal, characteristic
19	#72234716	C₂²	1	1	yes	characteristic
20	#72234715	C₄	1	2	yes	complemented
21	#72234714	C₄	1	2	yes	complemented
22	#72234712	C₂²	1	2	yes	—
23	#72234711	C₄	1	2	yes	complemented
24	#72234710	C₄	1	2	yes	complemented
25	#72234668	C₂²	1	1	yes	characteristic
26	#72232556	C₂²	1	2	yes	—
27	#72225870	C₂²	1	1	yes	characteristic
28	#72225713	C₂²	1	1	yes	characteristic
29	#72224136	C₂²	1	1	yes	characteristic
30	#72219311	C₂²	1	1	yes	characteristic
31	#10586404	D₄	1	1	yes	characteristic
32	C₂×#47031	C₂³	1	1	yes	characteristic
33	C₂×#34980	C₂³	1	1	yes	characteristic
34	#7626202	D₄	1	1	yes	characteristic
35	#6499871	D₄	1	2	yes	—
36	#6499870	C₂×C₄	1	2	yes	—
37	#6499869	C₂×C₄	1	2	yes	—
38	#6499867	D₄	1	2	yes	—
39	#6451874	C₂³	1	2	yes	—
40	C₂×#23984	D₄	1	2	yes	—
41	#6374251	D₄	1	1	yes	characteristic
42	#6374128	D₄	1	1	yes	characteristic
43	#6374128	D₄	1	1	yes	characteristic
44	C₂×#21746	D₄	1	1	yes	characteristic
45	#5025772	C₂×C₄	1	1	yes	characteristic, complemented
46	#5020747	C₂×C₄	1	1	yes	characteristic, complemented
47	#4435227	C₂×C₄	1	1	yes	characteristic, complemented
48	#686193	C₂×C₄	1	1	yes	characteristic, complemented
49	#576189	C₂³	1	1	yes	characteristic
50	#485079	C₂×C₄	1	4	yes	complemented
51	#485044	C₂×C₄	1	4	yes	complemented
52	#480250	C₂³	1	2	yes	—
53	#480250	C₂³	1	2	yes	—
54	#392109	D₄	1	2	yes	—
55	#391639	D₄	1	4	yes	complemented
56	#391095	D₄	1	4	yes	complemented
57	#388990	D₄	1	4	yes	complemented
58	#382707	C₂³	1	2	yes	—
59	#378118	D₄	1	4	yes	complemented
60	#359705	D₄	1	2	yes	—
61	#359705	C₂³	1	2	yes	—
62	#190116	C₂³	1	2	yes	—
63	#139625	C₂×C₄	1	4	yes	complemented
64	#139623	C₂×C₄	1	4	yes	complemented
65	#134036	C₂×C₄	1	2	yes	complemented
66	#134035	C₂×C₄	1	2	yes	complemented
67	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
68	#62820	C₂²⋊C₄	1	2	yes	—
69	#62817	C₂²⋊C₄	1	2	yes	—
70	C₂×#3490	C₂×D₄	1	2	yes	—
71	C₂²×#534	C₂×D₄	1	1	yes	characteristic
72	C₂²×#534	C₂×D₄	1	1	yes	characteristic
73	C₂³×#120	C₂×D₄	1	1	yes	characteristic
74	C₂×#2815	C₂²×C₄	1	1	yes	characteristic, complemented
75	C₂×#2849	C₂²×C₄	1	1	yes	characteristic
76	C₂×#2849	#41	1	1	yes	characteristic
77	C₂×#2846	C₂×D₄	1	1	yes	characteristic
78	C₂×#2846	#41	1	1	yes	characteristic
79	C₂×#2607	C₂²×C₄	1	1	yes	characteristic, complemented
80	C₂×#2613	C₂²×C₄	1	1	yes	characteristic
81	C₂×#2613	C₂×D₄	1	1	yes	characteristic
82	C₂×#2613	#41	1	1	yes	characteristic
83	C₂×#2613	#41	1	1	yes	characteristic
84	#34176	C₂×D₄	1	2	yes	—
85	C₂×#2308	C₂×D₄	1	1	yes	characteristic
86	C₂×#2308	C₂⁴	1	1	yes	characteristic
87	C₂²×#379	C₂×D₄	1	1	yes	characteristic
88	#23987	#41	1	8	yes	—
89	#23985	C₂²×C₄	1	4	yes	—
90	#23984	C₂²⋊C₄	1	4	yes	—
91	#23984	C₂²⋊C₄	1	4	yes	—
92	#23984	C₂×D₄	1	4	yes	—
93	#23468	C₂×D₄	1	4	yes	—
94	#23444	C₂×D₄	1	4	yes	—
95	#21747	C₂²⋊C₄	1	2	yes	—
96	#21746	C₂²⋊C₄	1	2	yes	—
97	C₂×#1819	C₂×D₄	1	2	yes	—
98	C₂×#1819	C₂×D₄	1	2	yes	—
99	C₂×#1781	C₂×D₄	1	2	yes	—
100	C₂×#1781	C₂×D₄	1	2	yes	—
101	#13828	C₂²×C₄	1	1	yes	characteristic, complemented
102	#13807	C₂²×C₄	1	1	yes	characteristic, complemented
103	#11500	C₂²×C₄	1	4	yes	complemented
104	#11166	C₂×D₄	1	4	yes	—
105	#11166	C₂×D₄	1	4	yes	—
106	#10938	C₂×D₄	1	4	yes	—
107	#10938	C₂×D₄	1	4	yes	—
108	C₂⁷	#120	1	1	yes	characteristic
109	C₂⁷	#121	1	1	yes	characteristic
110	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
111	C₂⁴×D₄	#120	1	1	yes	characteristic
112	C₂⁴×D₄	#120	1	1	yes	characteristic
113	C₂⁴×D₄	#121	1	1	yes	characteristic
114	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
115	C₂⁵×C₄	C₄×D₄	1	1	yes	characteristic
116	#3490	C₂×C₂²⋊C₄	1	4	yes	—
117	#3490	#123	1	4	yes	—
118	C₂×#543	#120	1	2	yes	—
119	C₂×#534	#120	1	2	yes	—
120	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂²×#123	C₄×D₄	1	1	yes	characteristic
122	C₂²×#123	C₄×D₄	1	1	yes	characteristic
123	C₂²×#123	#123	1	1	yes	characteristic
124	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂²×#120	C₄×D₄	1	1	yes	characteristic
126	C₂²×#120	C₄×D₄	1	1	yes	characteristic
127	C₂²×#120	#121	1	1	yes	characteristic
128	C₂²×#120	#121	1	1	yes	characteristic
129	C₂²×#120	#121	1	1	yes	characteristic
130	C₂²×#120	#121	1	1	yes	characteristic
131	C₂²×#120	#123	1	1	yes	characteristic
132	C₂³×C₂²⋊C₄	#120	1	1	yes	characteristic
133	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
134	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
135	C₂²×#99	#120	1	2	yes	—
136	C₂²×#99	#120	1	4	yes	—
137	C₂²×#99	#120	1	2	yes	—
138	#2308	C₂²×D₄	1	4	yes	—
139	#2308	C₂×#41	1	4	yes	—
140	C₂×#381	C₄×D₄	1	1	yes	characteristic
141	C₂×#381	#121	1	1	yes	characteristic
142	C₂×#381	C₂³×C₄	1	1	yes	characteristic
143	C₂×#381	C₂×#41	1	1	yes	characteristic
144	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×#379	C₄×D₄	1	1	yes	characteristic
147	C₂×#379	#121	1	1	yes	characteristic
148	C₂×#379	#123	1	1	yes	characteristic
149	C₂×#379	#123	1	1	yes	characteristic
150	C₂×#379	C₂²×D₄	1	1	yes	characteristic
151	C₂×#379	C₂×#41	1	1	yes	characteristic
152	#2273	#120	1	2	yes	—
153	#2273	#120	1	2	yes	—
154	#2030	C₄×D₄	1	8	yes	—
155	#2029	#121	1	8	yes	—
156	#2022	C₄×D₄	1	8	yes	—
157	#2021	#121	1	8	yes	—
158	#1819	C₂×C₂²⋊C₄	1	4	yes	—
159	#1819	#123	1	4	yes	—
160	C₂×#342	#120	1	4	yes	—
161	C₂×#342	#120	1	4	yes	—
162	C₂⁶	#386	1	1	yes	characteristic
163	C₂⁶	#386	1	1	yes	characteristic
164	C₂⁶	#386	1	1	yes	characteristic
165	C₂⁶	B(4,𝔽₂)	1	1	yes	characteristic
166	C₂⁶	C₂×#120	1	1	yes	characteristic
167	C₂⁶	C₂×#121	1	1	yes	characteristic
168	C₂³×D₄	#386	1	1	yes	characteristic
169	C₂³×D₄	#386	1	1	yes	characteristic
170	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
171	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
172	C₂³×D₄	C₂×#120	1	2	yes	—
173	C₂³×D₄	#525	1	2	yes	—
174	C₂⁴×C₄	#386	1	1	yes	characteristic
175	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
176	C₂⁴×C₄	C₂×C₄×D₄	1	1	yes	characteristic
177	#543	#386	1	4	yes	—
178	#534	#386	1	4	yes	—
179	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
180	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
181	C₂²×C₂²⋊C₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
182	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
183	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
184	C₂²×C₂²⋊C₄	C₂×#120	1	2	yes	—
185	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
186	C₂²×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
187	C₂²×C₂²⋊C₄	C₂×#123	1	2	yes	—
188	C₂²×C₂²⋊C₄	#525	1	2	yes	—
189	#381	C₂×C₄×D₄	1	4	yes	—
190	#381	#525	1	4	yes	—
191	#379	C₂×#121	1	4	yes	—
192	#379	#525	1	4	yes	—
193	#342	#386	1	8	yes	—
194	C₂⁵	#1889	1	1	yes	characteristic
195	C₂⁵	#1889	1	1	yes	characteristic
196	C₂⁵	C₂×#386	1	1	yes	characteristic
197	C₂⁵	#2846	1	1	yes	characteristic
198	C₂⁵	#2847	1	2	yes	—
199	C₂⁵	#2849	1	1	yes	characteristic
200	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
201	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
202	C₂⁵	#3026	1	1	yes	characteristic
203	C₂⁵	#3026	1	1	yes	characteristic
204	C₂³×C₄	#1889	1	1	yes	characteristic
205	C₂³×C₄	#1889	1	1	yes	characteristic
206	C₂³×C₄	C₄×#120	1	2	yes	—
207	C₂×C₂²⋊C₄	C₂×#386	1	2	yes	—
208	C₂×C₂²⋊C₄	C₄×#120	1	2	yes	—
209	C₂×C₂²⋊C₄	#2847	1	2	yes	—
210	C₂×C₂²⋊C₄	#2849	1	2	yes	—
211	C₂⁴	#14694	1	1	yes	characteristic
212	C₂⁴	C₂×#1889	1	1	yes	characteristic
213	C₂⁴	#20960	1	1	yes	characteristic
214	C₂⁴	#32898	1	1	yes	characteristic
215	C₂⁴	#32898	1	1	yes	characteristic
216	C₂⁴	#33020	1	1	yes	characteristic
217	C₂⁴	#33020	1	1	yes	characteristic
218	C₂³	#392658	1	1	yes	characteristic
219	C₂³	#392712	1	2	yes	—
220	C₂³	#511790	1	1	yes	characteristic
221	C₂³	#511790	1	2	yes	—
222	C₂³	#511791	1	1	yes	characteristic
223	C₂²	#72218104	1	1	yes	characteristic
224	C₂²	#72235178	1	1	yes	characteristic
225	C₂²	#72235423	1	1	yes	characteristic
226	C₂	#72235948	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72235948
commutator	C₂×#381	C₂³×C₄
Frattini, cosocle	C₂×#2308	C₂⁴
prime radical	#72234717	C₂
LCS	G ▷ C₂×#381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2308 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72238644	C₂	semidirect
2. #72238614	C₂	semidirect
3. #72238601	C₂	semidirect
4. #72238364	C₂	semidirect
5. #72237997	C₂	semidirect
6. #72237937	C₂	semidirect
7. #72237935	C₂²	semidirect
8. #72237883	C₂	semidirect
9. #72237881	C₂²	semidirect
11. #72237874	C₂²	semidirect
12. #72237872	C₂²	semidirect
13. #72237747	C₂²	semidirect
14. #72237616	C₂²	semidirect
15. #72237596	C₂	semidirect
20. #72234715	C₄	semidirect
21. #72234714	C₄	semidirect
23. #72234711	C₄	semidirect
24. #72234710	C₄	semidirect
45. #5025772	C₂×C₄	semidirect
46. #5020747	C₂×C₄	semidirect
47. #4435227	C₂×C₄	semidirect
48. #686193	C₂×C₄	semidirect
50. #485079	C₂×C₄	semidirect
51. #485044	C₂×C₄	semidirect
55. #391639	D₄	semidirect
56. #391095	D₄	semidirect
57. #388990	D₄	semidirect
59. #378118	D₄	semidirect
63. #139625	C₂×C₄	semidirect
64. #139623	C₂×C₄	semidirect
65. #134036	C₂×C₄	semidirect
66. #134035	C₂×C₄	semidirect
74. C₂×#2815	C₂²×C₄	semidirect
79. C₂×#2607	C₂²×C₄	semidirect
101. #13828	C₂²×C₄	semidirect
102. #13807	C₂²×C₄	semidirect
103. #11500	C₂²×C₄	semidirect

Maximal subgroups

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

subgroup	index
#72238644	2
#72238614	2
#72238601	2
#72238364	2
#72237997	2
#72237937	2
#72237883	2
#72237596	2
#72234717	2