Subgroups — #72340453

Order 4096.

At least 17538 conjugacy classes of subgroups; only the 393 normal subgroups are shown.

The subgroup lattice has 282 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72340451	C₂	1	1	yes	maximal, characteristic, complemented
2	#72340442	C₂	1	1	yes	maximal, characteristic, complemented
3	#72340439	C₂	1	1	yes	maximal, characteristic
4	#72340435	C₄	1	1	yes	characteristic, complemented
5	#72340431	C₂²	1	1	yes	characteristic
6	#72340413	C₄	1	1	yes	characteristic, complemented
7	#72340320	C₂	1	2	yes	maximal, complemented
8	C₂×#72236934	C₂	1	2	yes	maximal, complemented
9	C₂×#149742	C₂²	1	2	yes	complemented
10	#72340231	C₄	1	1	yes	characteristic, complemented
11	#72340231	C₂²	1	1	yes	characteristic
12	#72311044	C₂	1	2	yes	maximal, complemented
13	#72311038	C₄	1	1	yes	characteristic, complemented
14	#72311038	C₂²	1	1	yes	characteristic
15	C₂×#128062	C₂²	1	2	yes	complemented
16	#72310932	C₂	1	2	yes	maximal, complemented
17	#72310930	C₂²	1	2	yes	complemented
18	#72310922	C₂²	1	2	yes	complemented
19	#72286051	C₂	1	2	yes	maximal, complemented
20	#72285480	C₄	1	1	yes	characteristic, complemented
21	#72285480	C₂²	1	1	yes	characteristic
22	#72283582	C₂²	1	4	yes	complemented
23	#72265970	C₄	1	1	yes	characteristic, complemented
24	#72265970	C₂²	1	1	yes	characteristic
25	C₂×#251828	C₄	1	1	yes	characteristic, complemented
26	C₂×#251828	C₂²	1	1	yes	characteristic
27	#72236936	C₂	1	2	yes	maximal, complemented
28	#72236934	C₂²	1	4	yes	complemented
29	#72236934	C₂²	1	4	yes	complemented
30	#72236934	C₂²	1	4	yes	complemented
31	#72213758	C₂²	1	2	yes	complemented
32	#72213462	C₂²	1	2	yes	complemented
33	#72210439	C₄	1	1	yes	characteristic, complemented
34	#72210439	C₂²	1	1	yes	characteristic
35	C₂²×H(7,𝔽₂)	D₄	1	1	yes	characteristic
36	#8934366	C₂×C₄	1	1	yes	characteristic, complemented
37	#8933820	C₂×C₄	1	1	yes	characteristic, complemented
38	#8930050	C₂×C₄	1	1	yes	characteristic, complemented
39	#8926943	C₂×C₄	1	1	yes	characteristic, complemented
40	#7713426	C₂×C₄	1	1	yes	characteristic
41	C₂×#46384	C₂×C₄	1	1	yes	characteristic
42	#7630502	C₂×C₄	1	1	yes	characteristic
43	#7630502	D₄	1	1	yes	characteristic
44	#7630502	D₄	1	1	yes	characteristic
45	#6628902	C₂×C₄	1	1	yes	characteristic
46	#6628902	D₄	1	1	yes	characteristic
47	#6628902	D₄	1	1	yes	characteristic
48	#6628902	C₂³	1	1	yes	characteristic
49	#6628901	C₂×C₄	1	1	yes	characteristic
50	#6628901	D₄	1	1	yes	characteristic
51	#6628901	D₄	1	1	yes	characteristic
52	#6628901	C₂³	1	1	yes	characteristic
53	C₂×#20586	C₂×C₄	1	1	yes	characteristic
54	C₂×#20586	D₄	1	1	yes	characteristic
55	C₂×#20586	C₂³	1	1	yes	characteristic
56	#251855	D₄	1	2	yes	—
57	#251850	D₄	1	2	yes	—
58	#251833	C₂×C₄	1	1	yes	characteristic
59	#251833	C₂×C₄	1	1	yes	characteristic, complemented
60	#251833	C₂×C₄	1	1	yes	characteristic, complemented
61	#251833	C₂³	1	1	yes	characteristic
62	#251828	C₂×C₄	1	1	yes	characteristic
63	#251828	C₂×C₄	1	1	yes	characteristic
64	#251828	C₂×C₄	1	1	yes	characteristic
65	#251828	C₂×C₄	1	1	yes	characteristic, complemented
66	#251828	C₂×C₄	1	1	yes	characteristic, complemented
67	#251828	C₂×C₄	1	1	yes	characteristic, complemented
68	#251828	C₂×C₄	1	1	yes	characteristic, complemented
69	#251828	C₂×C₄	1	1	yes	characteristic, complemented
70	#251828	C₂×C₄	1	1	yes	characteristic, complemented
71	#251828	D₄	1	2	yes	—
72	#251828	D₄	1	2	yes	—
73	#251828	C₂³	1	1	yes	characteristic
74	#251828	C₂³	1	1	yes	characteristic
75	#251828	C₂³	1	1	yes	characteristic
76	#238082	D₄	1	2	yes	—
77	C₂×#10392	D₄	1	2	yes	—
78	#233549	D₄	1	2	yes	—
79	C₂×#10336	D₄	1	2	yes	—
80	#213863	C₂×C₄	1	1	yes	characteristic, complemented
81	#213863	C₂×C₄	1	1	yes	characteristic, complemented
82	#213860	C₂×C₄	1	1	yes	characteristic, complemented
83	#213860	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂×#8748	C₂×C₄	1	1	yes	characteristic
85	C₂×#8748	C₂×C₄	1	1	yes	characteristic
86	#149742	C₂³	1	4	yes	—
87	#128062	C₂³	1	4	yes	—
88	C₂×H(7,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×H(7,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
91	C₂×H(7,𝔽₂)	C₂×D₄	1	1	yes	characteristic
92	C₂³×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×H(5,𝔽₂)	C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
95	C₂²×#535	C₂²⋊C₄	1	1	yes	characteristic
96	#47563	C₂²⋊C₄	1	1	yes	characteristic, complemented
97	#47195	C₂²⋊C₄	1	1	yes	characteristic, complemented
98	#46998	C₂²×C₄	1	1	yes	characteristic
99	#46384	C₂²⋊C₄	1	1	yes	characteristic, complemented
100	#46384	C₂²⋊C₄	1	1	yes	characteristic, complemented
101	#46384	C₂²×C₄	1	1	yes	characteristic
102	#46384	C₂²×C₄	1	1	yes	characteristic
103	#46384	C₂²×C₄	1	1	yes	characteristic
104	#37475	C₂²⋊C₄	1	1	yes	characteristic, complemented
105	#37436	C₂²⋊C₄	1	1	yes	characteristic, complemented
106	C₂×#2842	C₂²⋊C₄	1	1	yes	characteristic, complemented
107	#35616	C₂²⋊C₄	1	1	yes	characteristic
108	#35616	C₂²⋊C₄	1	1	yes	characteristic
109	#35616	C₂²⋊C₄	1	1	yes	characteristic
110	#35616	C₂²×C₄	1	1	yes	characteristic
111	#35616	C₂×D₄	1	1	yes	characteristic
112	#35616	C₂×D₄	1	1	yes	characteristic
113	#35527	C₂²⋊C₄	1	1	yes	characteristic
114	#35527	C₂²⋊C₄	1	1	yes	characteristic
115	#35527	C₂²⋊C₄	1	1	yes	characteristic
116	#35527	C₂²×C₄	1	1	yes	characteristic
117	#35527	C₂×D₄	1	1	yes	characteristic
118	#35527	C₂×D₄	1	1	yes	characteristic
119	C₂×#2403	C₂²⋊C₄	1	1	yes	characteristic, complemented
120	C₂×#2309	C₂²⋊C₄	1	1	yes	characteristic
121	C₂×#2309	C₂²×C₄	1	1	yes	characteristic
122	C₂×#2309	C₂×D₄	1	1	yes	characteristic
123	#20586	C₂²⋊C₄	1	1	yes	characteristic
124	#20586	C₂²⋊C₄	1	1	yes	characteristic
125	#20586	C₂²⋊C₄	1	1	yes	characteristic
126	#20586	C₂²⋊C₄	1	1	yes	characteristic
127	#20586	C₂²×C₄	1	1	yes	characteristic
128	#20586	C₂²×C₄	1	1	yes	characteristic
129	#20586	C₂²×C₄	1	1	yes	characteristic
130	#20586	C₂×D₄	1	1	yes	characteristic
131	#20586	C₂×D₄	1	2	yes	—
132	#20586	C₂×D₄	1	2	yes	—
133	#20586	C₂×D₄	1	1	yes	characteristic
134	#20586	C₂×D₄	1	1	yes	characteristic
135	#20586	C₂×D₄	1	1	yes	characteristic
136	#20586	C₂⁴	1	1	yes	characteristic
137	C₂×#1781	C₂×D₄	1	2	yes	—
138	C₂×#1781	C₂×D₄	1	2	yes	—
139	#10392	C₂²⋊C₄	1	2	yes	complemented
140	#10392	C₂²⋊C₄	1	2	yes	complemented
141	#10392	C₂²⋊C₄	1	2	yes	complemented
142	#10392	C₂×D₄	1	2	yes	—
143	#10392	C₂×D₄	1	2	yes	—
144	#10392	C₂×D₄	1	2	yes	—
145	#10390	C₂²⋊C₄	1	2	yes	complemented
146	#10390	C₂×D₄	1	2	yes	—
147	#10338	C₂²⋊C₄	1	2	yes	complemented
148	#10338	C₂×D₄	1	2	yes	—
149	#10336	C₂²⋊C₄	1	2	yes	complemented
150	#10336	C₂²⋊C₄	1	2	yes	complemented
151	#10336	C₂²⋊C₄	1	2	yes	complemented
152	#10336	C₂×D₄	1	2	yes	—
153	#10336	C₂×D₄	1	2	yes	—
154	#10336	C₂×D₄	1	2	yes	—
155	#8748	C₂²⋊C₄	1	2	yes	complemented
156	#8748	C₂²⋊C₄	1	2	yes	complemented
157	#8748	C₂²×C₄	1	1	yes	characteristic
158	#8748	C₂²×C₄	1	1	yes	characteristic
159	#8748	C₂²×C₄	1	1	yes	characteristic
160	#8748	C₂²×C₄	1	1	yes	characteristic
161	C₂×#1304	C₂²⋊C₄	1	2	yes	complemented
162	C₂×#1304	C₂²⋊C₄	1	2	yes	complemented
163	C₂²×H(5,𝔽₂)	#99	1	2	yes	—
164	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
165	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
168	C₂²×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
170	C₂²×H(5,𝔽₂)	C₂²×D₄	1	1	yes	characteristic
171	C₂³×#41	#99	1	1	yes	characteristic
172	C₂⁴×D₄	#99	1	2	yes	—
173	C₂⁴×D₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂⁴×D₄	#120	1	2	yes	—
175	C₂²×#127	#99	1	1	yes	characteristic
176	C₂²×#127	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×#124	#99	1	1	yes	characteristic
178	C₂²×#124	C₂×C₂²⋊C₄	1	1	yes	characteristic
179	C₂²×#117	#99	1	1	yes	characteristic
180	#2842	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
181	#2842	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
182	#2842	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
183	#2840	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
184	#2483	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
185	#2403	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
186	#2403	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
187	#2403	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented
188	#2309	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	#2309	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	#2309	C₂×C₂²⋊C₄	1	1	yes	characteristic
191	#2309	C₂×C₂²⋊C₄	1	1	yes	characteristic
192	#2309	C₂×C₂²⋊C₄	1	1	yes	characteristic
193	#2309	C₂×C₂²⋊C₄	1	1	yes	characteristic
194	#2309	#120	1	2	yes	—
195	#2309	C₂³×C₄	1	1	yes	characteristic
196	#2309	C₂²×D₄	1	1	yes	characteristic
197	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×#379	C₂×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×#379	#120	1	2	yes	—
200	#1781	C₂×C₂²⋊C₄	1	2	yes	—
201	#1781	C₂×C₂²⋊C₄	1	2	yes	—
202	#1781	#120	1	2	yes	—
203	#1781	#120	1	2	yes	—
204	#1781	#120	1	2	yes	—
205	#1781	#120	1	2	yes	—
206	#1781	C₂²×D₄	1	2	yes	—
207	#1781	C₂²×D₄	1	2	yes	—
208	#1304	C₂×C₂²⋊C₄	1	2	yes	complemented
209	#1304	C₂×C₂²⋊C₄	1	2	yes	complemented
210	#1304	C₂×C₂²⋊C₄	1	2	yes	complemented
211	#1304	C₂×C₂²⋊C₄	1	2	yes	complemented
212	C₂⁶	AΣL(1,𝔽₁₆)	1	2	yes	—
213	C₂⁶	#379	1	1	yes	characteristic
214	C₂⁶	C₂×#99	1	1	yes	characteristic
215	C₂⁶	#453	1	2	yes	—
216	C₂³×Q₈	C₂×#99	1	1	yes	characteristic
217	C₂³×D₄	AΣL(1,𝔽₁₆)	1	2	yes	—
218	C₂³×D₄	AΣL(1,𝔽₁₆)	1	2	yes	—
219	C₂³×D₄	#379	1	2	yes	—
220	C₂³×D₄	#379	1	1	yes	characteristic
221	C₂³×D₄	C₂×#99	1	1	yes	characteristic
222	C₂³×D₄	C₂×#99	1	1	yes	characteristic
223	C₂³×D₄	C₂×#99	1	2	yes	—
224	C₂³×D₄	C₂×#99	1	2	yes	—
225	C₂³×D₄	C₂×#99	1	1	yes	characteristic
226	C₂³×D₄	#453	1	2	yes	—
227	C₂³×D₄	#455	1	2	yes	—
228	C₂³×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
229	C₂³×D₄	C₂×#120	1	2	yes	—
230	C₂⁴×C₄	AΣL(1,𝔽₁₆)	1	2	yes	—
231	C₂⁴×C₄	C₂×#99	1	1	yes	characteristic
232	C₂⁴×C₄	#455	1	2	yes	—
233	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
234	C₂²×C₄²	C₂×#99	1	1	yes	characteristic
235	#379	#379	1	1	yes	characteristic
236	#379	#379	1	1	yes	characteristic
237	#379	#379	1	2	yes	—
238	#379	C₂²×C₂²⋊C₄	1	1	yes	characteristic
239	#379	C₂²×C₂²⋊C₄	1	1	yes	characteristic
240	#379	C₂×#120	1	2	yes	—
241	C₂⁵	#1781	1	1	yes	characteristic
242	C₂⁵	#1781	1	1	yes	characteristic
243	C₂⁵	#1781	1	1	yes	characteristic
244	C₂⁵	#1792	1	2	yes	—
245	C₂⁵	#1792	1	2	yes	—
246	C₂⁵	#1792	1	2	yes	—
247	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
248	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
249	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
250	C₂⁵	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
251	C₂⁵	C₂×#379	1	1	yes	characteristic
252	C₂⁵	C₂²×#99	1	1	yes	characteristic
253	C₂⁵	C₂×#453	1	2	yes	—
254	C₂³×C₄	#1781	1	1	yes	characteristic
255	C₂³×C₄	#1792	1	2	yes	—
256	C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
257	C₂³×C₄	C₂×AΣL(1,𝔽₁₆)	1	2	yes	—
258	C₂³×C₄	C₂²×#99	1	1	yes	characteristic
259	C₂³×C₄	C₂×#455	1	2	yes	—
260	C₂⁴	#8453	1	1	yes	characteristic
261	C₂⁴	#8453	1	2	yes	—
262	C₂⁴	#8453	1	1	yes	characteristic
263	C₂⁴	#12687	1	1	yes	characteristic
264	C₂⁴	#12687	1	2	yes	—
265	C₂⁴	#12687	1	1	yes	characteristic
266	C₂⁴	#12707	1	1	yes	characteristic
267	C₂⁴	#12707	1	1	yes	characteristic
268	C₂⁴	C₂×#1781	1	1	yes	characteristic
269	C₂⁴	C₂×#1792	1	2	yes	—
270	C₂⁴	C₂²×AΣL(1,𝔽₁₆)	1	2	yes	—
271	C₂³	#131967	1	1	yes	characteristic
272	C₂³	#131967	1	1	yes	characteristic
273	C₂³	#131967	1	1	yes	characteristic
274	C₂³	#131967	1	1	yes	characteristic
275	C₂³	C₂×#8453	1	1	yes	characteristic
276	C₂³	C₂×#12687	1	1	yes	characteristic
277	C₂³	C₂×#12707	1	1	yes	characteristic
278	C₂²	#72210482	1	2	yes	—
279	C₂²	C₂×#131967	1	1	yes	characteristic
280	C₂	C₂×#72210482	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂×#72210482
commutator	#2309	C₂³×C₄
Frattini, cosocle	#20586	C₂⁴
prime radical	#72340439	C₂
LCS	G ▷ #2309 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁵ ◁ C₂⁴×D₄ ◁ #72340431 ◁ G
LECS	G ▷ #20586 ▷ C₂⁵ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72340451	C₂	semidirect
2. #72340442	C₂	semidirect
4. #72340435	C₄	semidirect
6. #72340413	C₄	semidirect
7. #72340320	C₂	semidirect
8. C₂×#72236934	C₂	semidirect
9. C₂×#149742	C₂²	semidirect
10. #72340231	C₄	semidirect
12. #72311044	C₂	semidirect
13. #72311038	C₄	semidirect
15. C₂×#128062	C₂²	semidirect
16. #72310932	C₂	semidirect
17. #72310930	C₂²	semidirect
18. #72310922	C₂²	semidirect
19. #72286051	C₂	semidirect
20. #72285480	C₄	semidirect
22. #72283582	C₂²	semidirect
23. #72265970	C₄	semidirect
25. C₂×#251828	C₄	semidirect
27. #72236936	C₂	semidirect
28. #72236934	C₂²	semidirect
29. #72236934	C₂²	semidirect
30. #72236934	C₂²	semidirect
31. #72213758	C₂²	semidirect
32. #72213462	C₂²	semidirect
33. #72210439	C₄	semidirect
36. #8934366	C₂×C₄	semidirect
37. #8933820	C₂×C₄	semidirect
38. #8930050	C₂×C₄	semidirect
39. #8926943	C₂×C₄	semidirect
59. #251833	C₂×C₄	semidirect
60. #251833	C₂×C₄	semidirect
65. #251828	C₂×C₄	semidirect
66. #251828	C₂×C₄	semidirect
67. #251828	C₂×C₄	semidirect
68. #251828	C₂×C₄	semidirect
69. #251828	C₂×C₄	semidirect
70. #251828	C₂×C₄	semidirect
80. #213863	C₂×C₄	semidirect
81. #213863	C₂×C₄	semidirect
82. #213860	C₂×C₄	semidirect
83. #213860	C₂×C₄	semidirect
96. #47563	C₂²⋊C₄	semidirect
97. #47195	C₂²⋊C₄	semidirect
99. #46384	C₂²⋊C₄	semidirect
100. #46384	C₂²⋊C₄	semidirect
104. #37475	C₂²⋊C₄	semidirect
105. #37436	C₂²⋊C₄	semidirect
106. C₂×#2842	C₂²⋊C₄	semidirect
119. C₂×#2403	C₂²⋊C₄	semidirect
139. #10392	C₂²⋊C₄	semidirect
140. #10392	C₂²⋊C₄	semidirect
141. #10392	C₂²⋊C₄	semidirect
145. #10390	C₂²⋊C₄	semidirect
147. #10338	C₂²⋊C₄	semidirect
149. #10336	C₂²⋊C₄	semidirect
150. #10336	C₂²⋊C₄	semidirect
151. #10336	C₂²⋊C₄	semidirect
155. #8748	C₂²⋊C₄	semidirect
156. #8748	C₂²⋊C₄	semidirect
161. C₂×#1304	C₂²⋊C₄	semidirect
162. C₂×#1304	C₂²⋊C₄	semidirect
180. #2842	C₂×C₂²⋊C₄	semidirect
181. #2842	C₂×C₂²⋊C₄	semidirect
182. #2842	C₂×C₂²⋊C₄	semidirect
183. #2840	C₂×C₂²⋊C₄	semidirect
184. #2483	C₂×C₂²⋊C₄	semidirect
185. #2403	C₂×C₂²⋊C₄	semidirect
186. #2403	C₂×C₂²⋊C₄	semidirect
187. #2403	C₂×C₂²⋊C₄	semidirect
208. #1304	C₂×C₂²⋊C₄	semidirect
209. #1304	C₂×C₂²⋊C₄	semidirect
210. #1304	C₂×C₂²⋊C₄	semidirect
211. #1304	C₂×C₂²⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#72340451	2
#72340442	2
#72340439	2
#72340320	2
C₂×#72236934	2
#72311044	2
#72310932	2
#72286051	2
#72236936	2