Subgroups — #72238616

Order 4096.

At least 9151 conjugacy classes of subgroups; only the 457 normal subgroups are shown.

The subgroup lattice has 231 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72238614	C₂	1	1	yes	maximal, characteristic, complemented
2	#72238601	C₂	1	1	yes	maximal, characteristic, complemented
3	#72237908	C₂	1	1	yes	maximal, characteristic, complemented
4	#72237886	C₂	1	1	yes	maximal, characteristic, complemented
5	#72237775	C₂	1	4	yes	maximal, complemented
6	#72237773	C₂²	1	2	yes	complemented
7	#72237750	C₂	1	4	yes	maximal, complemented
8	#72237749	C₂²	1	4	yes	—
9	#72237747	C₂²	1	4	yes	complemented
10	#72237747	C₂²	1	4	yes	complemented
11	#72237432	C₂	1	1	yes	maximal, characteristic, complemented
12	#72237425	C₂²	1	4	yes	—
13	#72235740	C₂	1	1	yes	maximal, characteristic, complemented
14	#72235732	C₂²	1	2	yes	complemented
15	#72235718	C₂²	1	1	yes	characteristic
16	#72234668	C₂²	1	1	yes	characteristic
17	#72234639	C₂	1	1	yes	maximal, characteristic
18	#72234638	C₄	1	2	yes	complemented
19	#72234637	C₄	1	2	yes	complemented
20	#72234636	C₂²	1	1	yes	characteristic
21	#72234598	C₂²	1	2	yes	—
22	#72234597	C₂²	1	2	yes	—
23	#72231809	C₂²	1	2	yes	complemented
24	#72231809	C₂²	1	2	yes	complemented
25	#72225870	C₂²	1	1	yes	characteristic
26	#72225713	C₂²	1	1	yes	characteristic
27	#72224136	C₂²	1	1	yes	characteristic
28	#72219311	C₂²	1	1	yes	characteristic
29	#72211186	C₄	1	2	yes	complemented
30	#72210578	C₄	1	2	yes	complemented
31	#10586404	D₄	1	1	yes	characteristic
32	C₂×#46996	C₂³	1	1	yes	characteristic
33	#7630939	D₄	1	2	yes	—
34	C₂×#34980	C₂³	1	1	yes	characteristic
35	#7626202	D₄	1	1	yes	characteristic, complemented
36	#6499870	C₂×C₄	1	2	yes	—
37	#6499870	D₄	1	2	yes	—
38	#6499867	C₂×C₄	1	2	yes	—
39	#6499867	D₄	1	2	yes	—
40	#6374251	D₄	1	1	yes	characteristic, complemented
41	#6374210	C₂³	1	2	yes	—
42	#6374128	D₄	1	1	yes	characteristic
43	#6374128	D₄	1	1	yes	characteristic, complemented
44	C₂×#21746	D₄	1	1	yes	characteristic, complemented
45	#5024691	C₂×C₄	1	1	yes	characteristic, complemented
46	#5020751	C₂×C₄	1	1	yes	characteristic, complemented
47	#4430405	C₂×C₄	1	1	yes	characteristic, complemented
48	#648360	C₂×C₄	1	1	yes	characteristic, complemented
49	#576160	C₂³	1	1	yes	characteristic
50	#480250	C₂³	1	2	yes	—
51	#480250	C₂³	1	2	yes	—
52	#388001	D₄	1	4	yes	complemented
53	#388001	D₄	1	4	yes	complemented
54	#359756	D₄	1	2	yes	complemented
55	#359705	D₄	1	2	yes	complemented
56	#359705	C₂³	1	2	yes	—
57	#359705	C₂³	1	2	yes	—
58	#357285	D₄	1	4	yes	complemented
59	#356055	D₄	1	4	yes	complemented
60	#238077	C₂×C₄	1	4	yes	complemented
61	#233541	C₂×C₄	1	4	yes	complemented
62	#190116	C₂³	1	2	yes	—
63	#115690	C₂×C₄	1	2	yes	complemented
64	#115631	C₂×C₄	1	2	yes	complemented
65	#110180	C₂×C₄	1	4	yes	complemented
66	#110086	C₂×C₄	1	4	yes	complemented
67	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
68	#62817	C₂²⋊C₄	1	2	yes	—
69	#62817	C₂²⋊C₄	1	2	yes	—
70	C₂×#3484	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×#2840	C₂²×C₄	1	1	yes	characteristic, complemented
75	C₂×#2849	#41	1	1	yes	characteristic
76	C₂×#2849	#41	1	1	yes	characteristic
77	C₂×#2846	C₂²⋊C₄	1	4	yes	—
78	C₂×#2846	C₂²×C₄	1	1	yes	characteristic
79	C₂×#2846	C₂×D₄	1	1	yes	characteristic
80	#34980	C₂²⋊C₄	1	4	yes	—
81	C₂×#2403	C₂²×C₄	1	1	yes	characteristic, complemented
82	C₂×#2613	C₂²×C₄	1	1	yes	characteristic
83	C₂×#2613	C₂×D₄	1	1	yes	characteristic
84	C₂×#2613	#41	1	1	yes	characteristic
85	C₂×#2613	#41	1	1	yes	characteristic
86	#34176	C₂×D₄	1	2	yes	—
87	#34176	C₂×D₄	1	2	yes	—
88	#34171	C₂×D₄	1	2	yes	—
89	C₂×#2308	C₂×D₄	1	1	yes	characteristic
90	C₂×#2308	C₂⁴	1	1	yes	characteristic
91	C₂²×#379	C₂×D₄	1	1	yes	characteristic
92	#21749	#41	1	8	yes	—
93	#21746	C₂²⋊C₄	1	2	yes	—
94	#21746	C₂²⋊C₄	1	2	yes	—
95	#21746	C₂²×C₄	1	4	yes	—
96	#21746	C₂×D₄	1	4	yes	—
97	#20689	C₂×D₄	1	4	yes	—
98	C₂×#1819	C₂×D₄	1	2	yes	—
99	C₂×#1819	C₂×D₄	1	2	yes	—
100	#20585	C₂×D₄	1	4	yes	—
101	#13827	C₂²×C₄	1	1	yes	characteristic, complemented
102	C₂²⋊C₄²	C₂²×C₄	1	1	yes	characteristic, complemented
103	#10938	C₂×D₄	1	4	yes	—
104	#10938	C₂×D₄	1	4	yes	—
105	#10938	C₂×D₄	1	4	yes	—
106	#10938	C₂×D₄	1	4	yes	—
107	#8101	C₂²×C₄	1	4	yes	complemented
108	C₂⁷	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂⁷	#120	1	1	yes	characteristic
110	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
111	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂⁴×D₄	#120	1	1	yes	characteristic
113	C₂⁴×D₄	#120	1	1	yes	characteristic
114	C₂⁴×D₄	#123	1	1	yes	characteristic
115	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
116	C₂⁵×C₄	#123	1	1	yes	characteristic
117	C₂×#534	#123	1	4	yes	—
118	C₂²×#123	C₄×D₄	1	1	yes	characteristic
119	C₂²×#123	#121	1	1	yes	characteristic
120	C₂²×#123	#123	1	1	yes	characteristic
121	C₂²×#123	#123	1	1	yes	characteristic
122	C₂²×#120	C₂×C₂²⋊C₄	1	4	yes	—
123	C₂²×#120	C₂×C₂²⋊C₄	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	C₄×D₄	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	C₂×#518	#120	1	2	yes	—
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	#2309	#123	1	4	yes	—
136	#2308	#120	1	4	yes	—
137	#2308	C₂²×D₄	1	4	yes	—
138	#2308	C₂×#41	1	4	yes	—
139	C₂×#381	C₄×D₄	1	1	yes	characteristic
140	C₂×#381	#121	1	1	yes	characteristic
141	C₂×#381	C₂×#41	1	1	yes	characteristic
142	C₂×#381	C₂×#41	1	1	yes	characteristic
143	C₂×#379	C₂×C₂²⋊C₄	1	4	yes	—
144	C₂×#379	C₄×D₄	1	1	yes	characteristic
145	C₂×#379	C₄×D₄	1	1	yes	characteristic
146	C₂×#379	C₄×D₄	1	1	yes	characteristic
147	C₂×#379	#121	1	1	yes	characteristic
148	C₂×#379	#121	1	1	yes	characteristic
149	C₂×#379	#121	1	1	yes	characteristic
150	C₂×#379	C₂³×C₄	1	1	yes	characteristic
151	C₂×#379	C₂²×D₄	1	1	yes	characteristic
152	#2273	#120	1	2	yes	—
153	#2273	#120	1	4	yes	—
154	#2273	#120	1	2	yes	—
155	#1819	C₄×D₄	1	8	yes	—
156	#1819	#121	1	8	yes	—
157	#1781	C₄×D₄	1	8	yes	—
158	#1781	#121	1	8	yes	—
159	C₂×#342	#120	1	2	yes	—
160	C₂×#342	#120	1	4	yes	—
161	C₂×#342	#120	1	2	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	C₂²×C₂²⋊C₄	1	1	yes	characteristic
167	C₂⁶	C₂×#120	1	1	yes	characteristic
168	C₂³×D₄	#386	1	1	yes	characteristic
169	C₂³×D₄	#386	1	1	yes	characteristic
170	C₂³×D₄	#386	1	4	yes	—
171	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
172	C₂³×D₄	B(4,𝔽₂)	1	1	yes	characteristic
173	C₂³×D₄	C₂×#120	1	2	yes	—
174	C₂³×D₄	C₂×#123	1	2	yes	—
175	C₂⁴×C₄	#386	1	1	yes	characteristic
176	C₂⁴×C₄	B(4,𝔽₂)	1	1	yes	characteristic
177	C₂⁴×C₄	C₂×#123	1	1	yes	characteristic
178	C₂²×C₂²⋊C₄	#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	C₂×C₄×D₄	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×#121	1	1	yes	characteristic
187	C₂²×C₂²⋊C₄	#525	1	2	yes	—
188	C₂²×C₂²⋊C₄	#525	1	2	yes	—
189	#381	#525	1	4	yes	—
190	#381	#525	1	4	yes	—
191	#379	#386	1	8	yes	—
192	#379	C₂×C₄×D₄	1	4	yes	—
193	#379	C₂×#121	1	4	yes	—
194	C₂⁵	#1889	1	1	yes	characteristic
195	C₂⁵	#1889	1	1	yes	characteristic
196	C₂⁵	#1911	1	4	yes	complemented
197	C₂⁵	C₂×#386	1	1	yes	characteristic
198	C₂⁵	C₂×#386	1	2	yes	—
199	C₂⁵	#2846	1	1	yes	characteristic
200	C₂⁵	#2849	1	1	yes	characteristic
201	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
202	C₂⁵	C₂×B(4,𝔽₂)	1	1	yes	characteristic
203	C₂⁵	#3026	1	1	yes	characteristic
204	C₂⁵	#3026	1	1	yes	characteristic
205	C₂³×C₄	#1889	1	1	yes	characteristic
206	C₂³×C₄	#1889	1	1	yes	characteristic
207	C₂³×C₄	#1911	1	4	yes	complemented
208	C₂³×C₄	#2849	1	2	yes	—
209	C₂×C₂²⋊C₄	C₄×#120	1	2	yes	—
210	C₂×C₂²⋊C₄	C₄×#120	1	2	yes	—
211	C₂×C₂²⋊C₄	#2847	1	2	yes	—
212	C₂×C₂²⋊C₄	#2847	1	2	yes	—
213	C₂⁴	#14694	1	1	yes	characteristic
214	C₂⁴	C₂×#1889	1	1	yes	characteristic
215	C₂⁴	#20960	1	1	yes	characteristic
216	C₂⁴	C₂×#1911	1	4	yes	—
217	C₂⁴	#32898	1	1	yes	characteristic
218	C₂⁴	#32898	1	1	yes	characteristic
219	C₂⁴	#33020	1	1	yes	characteristic
220	C₂⁴	#33020	1	1	yes	characteristic
221	C₂³	#392658	1	1	yes	characteristic
222	C₂³	#392696	1	2	yes	—
223	C₂³	#511790	1	1	yes	characteristic
224	C₂³	#511791	1	1	yes	characteristic
225	C₂³	#511793	1	2	yes	—
226	C₂²	#72235143	1	1	yes	characteristic
227	C₂²	#72235186	1	1	yes	characteristic
228	C₂²	#72235250	1	1	yes	characteristic
229	C₂	#72235909	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72235909
commutator	C₂×#379	C₂³×C₄
Frattini, cosocle	C₂×#2308	C₂⁴
prime radical	#72234639	C₂
LCS	G ▷ C₂×#379 ▷ 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. #72238614	C₂	semidirect
2. #72238601	C₂	semidirect
3. #72237908	C₂	semidirect
4. #72237886	C₂	semidirect
5. #72237775	C₂	semidirect
6. #72237773	C₂²	semidirect
7. #72237750	C₂	semidirect
9. #72237747	C₂²	semidirect
10. #72237747	C₂²	semidirect
11. #72237432	C₂	semidirect
13. #72235740	C₂	semidirect
14. #72235732	C₂²	semidirect
18. #72234638	C₄	semidirect
19. #72234637	C₄	semidirect
23. #72231809	C₂²	semidirect
24. #72231809	C₂²	semidirect
29. #72211186	C₄	semidirect
30. #72210578	C₄	semidirect
35. #7626202	D₄	semidirect
40. #6374251	D₄	semidirect
43. #6374128	D₄	semidirect
44. C₂×#21746	D₄	semidirect
45. #5024691	C₂×C₄	semidirect
46. #5020751	C₂×C₄	semidirect
47. #4430405	C₂×C₄	semidirect
48. #648360	C₂×C₄	semidirect
52. #388001	D₄	semidirect
53. #388001	D₄	semidirect
54. #359756	D₄	semidirect
55. #359705	D₄	semidirect
58. #357285	D₄	semidirect
59. #356055	D₄	semidirect
60. #238077	C₂×C₄	semidirect
61. #233541	C₂×C₄	semidirect
63. #115690	C₂×C₄	semidirect
64. #115631	C₂×C₄	semidirect
65. #110180	C₂×C₄	semidirect
66. #110086	C₂×C₄	semidirect
74. C₂×#2840	C₂²×C₄	semidirect
81. C₂×#2403	C₂²×C₄	semidirect
101. #13827	C₂²×C₄	semidirect
102. C₂²⋊C₄²	C₂²×C₄	semidirect
107. #8101	C₂²×C₄	semidirect
196. C₂⁵	#1911	semidirect
207. C₂³×C₄	#1911	semidirect

Maximal subgroups

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

subgroup	index
#72238614	2
#72238601	2
#72237908	2
#72237886	2
#72237775	2
#72237750	2
#72237432	2
#72235740	2
#72234639	2