Subgroups — #72323927

Order 4096.

At least 10669 conjugacy classes of subgroups; only the 295 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	#72323901	C₂	1	4	yes	maximal, complemented
2	#72323892	C₂	1	4	yes	maximal, complemented
3	#72323219	C₂	1	1	yes	maximal, characteristic, complemented
4	#72323218	C₂²	1	4	yes	complemented
5	#72314618	C₂	1	1	yes	maximal, characteristic, complemented
6	#72314614	C₂²	1	4	yes	complemented
7	#72302137	C₂	1	1	yes	maximal, characteristic, complemented
8	#72302130	C₂²	1	2	yes	complemented
9	#72266442	C₂	1	1	yes	maximal, characteristic
10	#72266439	C₂²	1	2	yes	—
11	#72265766	C₂²	1	1	yes	characteristic
12	#72259157	C₄	1	2	yes	complemented
13	#72259147	C₄	1	2	yes	complemented
14	#72240132	C₄	1	1	yes	characteristic, complemented
15	#72237864	C₂	1	1	yes	maximal, characteristic, complemented
16	#72237816	C₂²	1	4	yes	complemented
17	#72237581	C₂	1	1	yes	maximal, characteristic, complemented
18	#72237576	C₂²	1	4	yes	complemented
19	#72234641	C₂²	1	2	yes	complemented
20	#72234625	C₂²	1	1	yes	characteristic
21	#72234108	C₂²	1	2	yes	—
22	#72231950	C₄	1	1	yes	characteristic, complemented
23	#72226043	C₂	1	1	yes	maximal, characteristic, complemented
24	#72226032	C₂²	1	2	yes	complemented
25	#72226029	C₂²	1	2	yes	complemented
26	#72225870	C₂²	1	1	yes	characteristic
27	#72219311	C₂²	1	1	yes	characteristic
28	#72219311	C₂²	1	1	yes	characteristic
29	#72219310	C₂²	1	1	yes	characteristic
30	C₂×#139716	C₄	1	1	yes	characteristic, complemented
31	#72219264	C₂²	1	1	yes	characteristic
32	C₂×#347042	C₄	1	1	yes	characteristic, complemented
33	C₂×#62817	C₂×C₄	1	1	yes	characteristic, complemented
34	C₂²×#3532	D₄	1	1	yes	characteristic
35	C₂×#34980	C₂³	1	1	yes	characteristic
36	#6499870	C₂×C₄	1	2	yes	complemented
37	#6499870	C₂³	1	2	yes	—
38	C₂×#21746	C₂×C₄	1	1	yes	characteristic, complemented
39	#5025789	C₂×C₄	1	1	yes	characteristic
40	#5025782	C₂³	1	1	yes	characteristic
41	#5021140	D₄	1	1	yes	characteristic
42	#5021138	D₄	1	1	yes	characteristic
43	#4422949	C₂×C₄	1	1	yes	characteristic
44	#4420586	C₂³	1	1	yes	characteristic
45	#763333	D₄	1	1	yes	characteristic
46	#686187	D₄	1	1	yes	characteristic
47	#647656	D₄	1	1	yes	characteristic
48	#616961	C₂×C₄	1	1	yes	characteristic
49	#616960	D₄	1	1	yes	characteristic
50	C₂×#15947	C₂×C₄	1	1	yes	characteristic
51	#552670	D₄	1	1	yes	characteristic
52	#392109	D₄	1	2	yes	complemented
53	#347042	C₂×C₄	1	2	yes	—
54	#347042	C₂×C₄	1	2	yes	complemented
55	#346456	C₂×C₄	1	2	yes	complemented
56	#346446	C₂×C₄	1	2	yes	—
57	#190116	D₄	1	2	yes	complemented
58	C₂×#8547	C₂×C₄	1	1	yes	characteristic, complemented
59	C₂×#8541	C₂×C₄	1	1	yes	characteristic, complemented
60	#139720	C₂×C₄	1	2	yes	complemented
61	#139718	C₂×C₄	1	2	yes	—
62	#139716	C₂×C₄	1	2	yes	—
63	#139716	C₂×C₄	1	2	yes	complemented
64	#134037	C₂³	1	2	yes	—
65	#133825	C₂×C₄	1	2	yes	complemented
66	#133825	C₂³	1	2	yes	—
67	#115690	D₄	1	2	yes	complemented
68	#115633	C₂³	1	2	yes	—
69	#115631	D₄	1	2	yes	complemented
70	#115631	C₂³	1	2	yes	—
71	#115631	C₂³	1	2	yes	—
72	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
73	#62817	C₂²×C₄	1	2	yes	—
74	C₂²×#551	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×D₄²	C₂×D₄	1	1	yes	characteristic
76	C₂²×H(3,𝔽₄)	C₂×D₄	1	1	yes	characteristic
77	C₂²×#560	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×#534	C₂×D₄	1	1	yes	characteristic
80	C₂²×#534	C₂×D₄	1	1	yes	characteristic
81	C₂³×#120	C₂²⋊C₄	1	1	yes	characteristic
82	C₂³×#120	C₂²×C₄	1	1	yes	characteristic
83	C₂×#2849	C₂×D₄	1	1	yes	characteristic
84	C₂×#2849	#41	1	1	yes	characteristic
85	C₂×#2846	C₂×D₄	1	1	yes	characteristic
86	C₂×#2846	C₂×D₄	1	1	yes	characteristic
87	C₂×#2613	C₂²×C₄	1	1	yes	characteristic
88	C₂×#2613	C₂×D₄	1	1	yes	characteristic
89	C₂×#2613	#41	1	1	yes	characteristic
90	C₂×#2613	C₂⁴	1	1	yes	characteristic
91	C₂×#2308	C₂×D₄	1	1	yes	characteristic
92	C₂×#2308	#41	1	1	yes	characteristic
93	C₂²×#379	#41	1	1	yes	characteristic
94	#21746	C₂²×C₄	1	2	yes	—
95	C₂×#1781	C₂×D₄	1	4	yes	complemented
96	#15948	C₂×D₄	1	1	yes	characteristic
97	#15948	C₂×D₄	1	1	yes	characteristic
98	#15947	C₂²⋊C₄	1	1	yes	characteristic
99	#15947	C₂²⋊C₄	1	1	yes	characteristic
100	#15947	C₂²⋊C₄	1	1	yes	characteristic
101	#15947	C₂²⋊C₄	1	1	yes	characteristic
102	#15947	C₂²×C₄	1	1	yes	characteristic
103	#15947	C₂²×C₄	1	1	yes	characteristic
104	#15262	C₂×D₄	1	1	yes	characteristic
105	#15261	C₂×D₄	1	1	yes	characteristic
106	#15259	C₂²⋊C₄	1	1	yes	characteristic
107	#15259	C₂²⋊C₄	1	1	yes	characteristic
108	#15259	C₂²×C₄	1	1	yes	characteristic
109	#15258	C₂²⋊C₄	1	1	yes	characteristic
110	#15258	C₂²⋊C₄	1	1	yes	characteristic
111	#15258	C₂²×C₄	1	1	yes	characteristic
112	#14658	C₂²⋊C₄	1	1	yes	characteristic
113	#14657	C₂×D₄	1	1	yes	characteristic
114	#14335	C₂²⋊C₄	1	1	yes	characteristic
115	#14167	C₂²⋊C₄	1	1	yes	characteristic
116	#14134	C₂×D₄	1	1	yes	characteristic
117	C₂×#1438	C₂²⋊C₄	1	1	yes	characteristic
118	#8547	C₂²×C₄	1	2	yes	—
119	#8541	C₂²×C₄	1	2	yes	—
120	#8101	C₂×D₄	1	4	yes	complemented
121	C₂⁷	#99	1	1	yes	characteristic
122	C₂⁷	C₄×D₄	1	1	yes	characteristic
123	C₂⁷	#120	1	1	yes	characteristic
124	C₂³×#41	#99	1	1	yes	characteristic
125	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂⁴×D₄	C₄×D₄	1	1	yes	characteristic
127	C₂⁴×D₄	#120	1	1	yes	characteristic
128	C₂⁴×D₄	#121	1	1	yes	characteristic
129	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
130	C₂⁵×C₄	#121	1	1	yes	characteristic
131	C₂²×#127	#99	1	1	yes	characteristic
132	C₂²×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×#124	#99	1	1	yes	characteristic
134	C₂²×#124	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×#123	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂²×#123	#121	1	1	yes	characteristic
137	C₂²×#123	#123	1	1	yes	characteristic
138	C₂²×#123	C₂²×D₄	1	1	yes	characteristic
139	C₂²×#120	#99	1	1	yes	characteristic
140	C₂²×#120	#99	1	1	yes	characteristic
141	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×#120	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×#120	C₄×D₄	1	1	yes	characteristic
146	C₂²×#120	C₄×D₄	1	1	yes	characteristic
147	C₂²×#120	#120	1	1	yes	characteristic
148	C₂²×#120	#121	1	1	yes	characteristic
149	C₂²×#120	C₂³×C₄	1	1	yes	characteristic
150	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
151	C₂²×#117	#99	1	1	yes	characteristic
152	C₂³×C₂²⋊C₄	#99	1	1	yes	characteristic
153	C₂³×C₂²⋊C₄	C₄×D₄	1	1	yes	characteristic
154	C₂³×C₂²⋊C₄	#121	1	1	yes	characteristic
155	C₂³×C₂²⋊C₄	#123	1	1	yes	characteristic
156	C₂×#381	C₄×D₄	1	1	yes	characteristic
157	C₂×#381	#121	1	1	yes	characteristic
158	C₂×#381	#121	1	1	yes	characteristic
159	C₂×#381	C₂×#41	1	1	yes	characteristic
160	C₂×#379	C₄×D₄	1	1	yes	characteristic
161	C₂×#379	C₄×D₄	1	1	yes	characteristic
162	C₂×#379	#120	1	1	yes	characteristic
163	C₂×#379	#121	1	1	yes	characteristic
164	C₂×#379	#123	1	1	yes	characteristic
165	C₂×#379	#123	1	1	yes	characteristic
166	C₂×#379	C₂²×D₄	1	1	yes	characteristic
167	C₂×#379	C₂×#41	1	1	yes	characteristic
168	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	#1438	C₂×C₂²⋊C₄	1	1	yes	characteristic
172	C₂⁶	#386	1	1	yes	characteristic
173	C₂⁶	#386	1	1	yes	characteristic
174	C₂⁶	#386	1	1	yes	characteristic
175	C₂⁶	C₂×#99	1	1	yes	characteristic
176	C₂⁶	C₂×#99	1	1	yes	characteristic
177	C₂⁶	C₂×#99	1	1	yes	characteristic
178	C₂⁶	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂⁶	C₂×#120	1	1	yes	characteristic
180	C₂³×Q₈	C₂×#99	1	1	yes	characteristic
181	C₂³×D₄	#386	1	1	yes	characteristic
182	C₂³×D₄	C₂×#99	1	1	yes	characteristic
183	C₂³×D₄	C₂×#99	1	1	yes	characteristic
184	C₂³×D₄	C₂×#99	1	1	yes	characteristic
185	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
186	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
187	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
188	C₂⁴×C₄	#386	1	1	yes	characteristic
189	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
190	C₂⁴×C₄	C₂×#121	1	1	yes	characteristic
191	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
192	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
193	C₂²×C₂²⋊C₄	#386	1	1	yes	characteristic
194	C₂²×C₂²⋊C₄	C₂×#99	1	1	yes	characteristic
195	C₂²×C₂²⋊C₄	C₂×#99	1	1	yes	characteristic
196	C₂²×C₂²⋊C₄	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×C₂²⋊C₄	C₂×#121	1	1	yes	characteristic
198	C₂²×C₂²⋊C₄	C₂×#123	1	1	yes	characteristic
199	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
200	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
201	C₂⁵	#1889	1	1	yes	characteristic
202	C₂⁵	#1889	1	1	yes	characteristic
203	C₂⁵	#1898	1	1	yes	characteristic
204	C₂⁵	#1898	1	1	yes	characteristic
205	C₂⁵	#1898	1	1	yes	characteristic
206	C₂⁵	#1913	1	1	yes	characteristic
207	C₂⁵	#1913	1	1	yes	characteristic
208	C₂⁵	C₂×#386	1	1	yes	characteristic
209	C₂⁵	C₂²×#99	1	1	yes	characteristic
210	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
211	C₂³×C₄	#1889	1	2	yes	—
212	C₂³×C₄	#1898	1	1	yes	characteristic
213	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
214	C₂⁴	#9495	1	1	yes	characteristic
215	C₂⁴	#9495	1	1	yes	characteristic
216	C₂⁴	#9495	1	1	yes	characteristic
217	C₂⁴	#9495	1	1	yes	characteristic
218	C₂⁴	C₂×#1889	1	1	yes	characteristic
219	C₂⁴	C₂×#1898	1	1	yes	characteristic
220	C₂⁴	C₂×#1913	1	1	yes	characteristic
221	C₂⁴	#33551	1	4	yes	—
222	C₂³	#139750	1	1	yes	characteristic
223	C₂³	#139750	1	1	yes	characteristic
224	C₂³	C₂×#9495	1	1	yes	characteristic
225	C₂³	#494575	1	2	yes	—
226	C₂²	C₂×#139750	1	1	yes	characteristic
227	C₂²	#72237037	1	1	yes	characteristic
228	C₂²	#72255841	1	1	yes	characteristic
229	C₂	#72266319	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72266319
commutator	C₂²×#120	C₂³×C₄
Frattini, cosocle	C₂×#2613	C₂⁴
prime radical	#72266442	C₂
LCS	G ▷ C₂²×#120 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁶ ◁ C₂×#34980 ◁ G
LECS	G ▷ C₂×#2613 ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72323901	C₂	semidirect
2. #72323892	C₂	semidirect
3. #72323219	C₂	semidirect
4. #72323218	C₂²	semidirect
5. #72314618	C₂	semidirect
6. #72314614	C₂²	semidirect
7. #72302137	C₂	semidirect
8. #72302130	C₂²	semidirect
12. #72259157	C₄	semidirect
13. #72259147	C₄	semidirect
14. #72240132	C₄	semidirect
15. #72237864	C₂	semidirect
16. #72237816	C₂²	semidirect
17. #72237581	C₂	semidirect
18. #72237576	C₂²	semidirect
19. #72234641	C₂²	semidirect
22. #72231950	C₄	semidirect
23. #72226043	C₂	semidirect
24. #72226032	C₂²	semidirect
25. #72226029	C₂²	semidirect
30. C₂×#139716	C₄	semidirect
32. C₂×#347042	C₄	semidirect
33. C₂×#62817	C₂×C₄	semidirect
36. #6499870	C₂×C₄	semidirect
38. C₂×#21746	C₂×C₄	semidirect
52. #392109	D₄	semidirect
54. #347042	C₂×C₄	semidirect
55. #346456	C₂×C₄	semidirect
57. #190116	D₄	semidirect
58. C₂×#8547	C₂×C₄	semidirect
59. C₂×#8541	C₂×C₄	semidirect
60. #139720	C₂×C₄	semidirect
63. #139716	C₂×C₄	semidirect
65. #133825	C₂×C₄	semidirect
67. #115690	D₄	semidirect
69. #115631	D₄	semidirect
95. C₂×#1781	C₂×D₄	semidirect
120. #8101	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72323901	2
#72323892	2
#72323219	2
#72314618	2
#72302137	2
#72266442	2
#72237864	2
#72237581	2
#72226043	2