Subgroups — #72342829

Order 2048.

At least 7283 conjugacy classes of subgroups; only the 470 normal subgroups are shown.

The subgroup lattice has 280 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72342824	C₂	1	1	yes	maximal, characteristic, complemented
2	#72342821	C₂	1	1	yes	maximal, characteristic, complemented
3	#72342819	C₂	1	1	yes	maximal, characteristic, complemented
4	#72342809	C₂	1	1	yes	maximal, characteristic, complemented
5	#72342796	C₂	1	1	yes	maximal, characteristic, complemented
6	#72342777	C₂	1	1	yes	maximal, characteristic, complemented
7	#72317394	C₂	1	1	yes	maximal, characteristic, complemented
8	#72317374	C₂	1	1	yes	maximal, characteristic, complemented
9	#72317368	C₂	1	1	yes	maximal, characteristic, complemented
10	#72317361	C₂	1	1	yes	maximal, characteristic, complemented
11	#72308105	C₂	1	1	yes	maximal, characteristic, complemented
12	#72240519	C₂	1	1	yes	maximal, characteristic, complemented
13	#72240482	C₂	1	1	yes	maximal, characteristic, complemented
14	#72233438	C₂	1	1	yes	maximal, characteristic, complemented
15	#72218347	C₂	1	1	yes	maximal, characteristic
16	#10586169	C₂²	1	1	yes	characteristic
17	#10574874	C₂²	1	1	yes	characteristic
18	#10574299	C₂²	1	1	yes	characteristic, complemented
19	#6499875	C₄	1	4	yes	complemented
20	#6499874	C₄	1	4	yes	complemented
21	#6437628	C₂²	1	1	yes	characteristic
22	#6437628	C₂²	1	1	yes	characteristic
23	#6437616	C₂²	1	1	yes	characteristic
24	#6437616	C₂²	1	1	yes	characteristic
25	#6374194	C₂²	1	1	yes	characteristic, complemented
26	#6374186	C₂²	1	1	yes	characteristic
27	#493402	C₂²	1	1	yes	characteristic, complemented
28	#493402	C₂²	1	1	yes	characteristic, complemented
29	#493390	C₂²	1	1	yes	characteristic
30	#493390	C₂²	1	1	yes	characteristic, complemented
31	#493352	C₂²	1	1	yes	characteristic, complemented
32	#359562	C₂²	1	1	yes	characteristic, complemented
33	#359556	C₂²	1	1	yes	characteristic
34	#359556	C₂²	1	1	yes	characteristic, complemented
35	#359556	C₂²	1	1	yes	characteristic, complemented
36	#359551	C₂²	1	1	yes	characteristic
37	#359551	C₂²	1	1	yes	characteristic, complemented
38	#359507	C₂²	1	1	yes	characteristic, complemented
39	#359426	C₂²	1	1	yes	characteristic, complemented
40	#188071	C₂²	1	1	yes	characteristic, complemented
41	#188004	C₂²	1	1	yes	characteristic
42	#188004	C₂²	1	1	yes	characteristic, complemented
43	#188004	C₂²	1	1	yes	characteristic, complemented
44	#186930	C₂²	1	1	yes	characteristic, complemented
45	#146483	C₂²	1	1	yes	characteristic, complemented
46	#146461	C₂²	1	1	yes	characteristic, complemented
47	#128040	C₂²	1	1	yes	characteristic, complemented
48	#128036	C₂²	1	1	yes	characteristic, complemented
49	#128024	C₂²	1	1	yes	characteristic, complemented
50	#128018	C₂²	1	1	yes	characteristic, complemented
51	#108236	C₂²	1	1	yes	characteristic, complemented
52	#106146	C₂²	1	1	yes	characteristic, complemented
53	#63103	D₄	1	1	yes	characteristic
54	#62820	C₂×C₄	1	2	yes	—
55	#62817	C₂×C₄	1	2	yes	—
56	#62811	C₂³	1	1	yes	characteristic
57	#62811	C₂³	1	1	yes	characteristic
58	#62811	C₂³	1	1	yes	characteristic
59	#62811	C₂³	1	1	yes	characteristic
60	C₂²×#525	C₂³	1	1	yes	characteristic
61	#60401	D₄	1	4	yes	—
62	#60399	C₂×C₄	1	4	yes	—
63	#60398	D₄	1	2	yes	—
64	#60070	D₄	1	1	yes	characteristic
65	#60070	D₄	1	1	yes	characteristic
66	#60070	D₄	1	1	yes	characteristic
67	#60064	C₂³	1	1	yes	characteristic
68	C₂²×#408	C₂³	1	1	yes	characteristic
69	#36614	D₄	1	2	yes	—
70	#23901	C₂×C₄	1	2	yes	—
71	#23901	C₂×C₄	1	2	yes	—
72	#23901	D₄	1	2	yes	—
73	#23900	C₂×C₄	1	2	yes	—
74	#23900	C₂×C₄	1	2	yes	—
75	#23900	D₄	1	2	yes	—
76	#23896	D₄	1	4	yes	—
77	#23895	C₂×C₄	1	4	yes	—
78	#23895	C₂×C₄	1	4	yes	—
79	#21776	C₂×C₄	1	4	yes	—
80	#21017	D₄	1	2	yes	complemented
81	#21014	D₄	1	2	yes	—
82	#20989	D₄	1	2	yes	complemented
83	#20983	D₄	1	2	yes	—
84	#20964	D₄	1	2	yes	complemented
85	#20895	D₄	1	2	yes	complemented
86	#20591	C₂³	1	1	yes	characteristic
87	#20591	C₂³	1	1	yes	characteristic
88	#20591	C₂³	1	1	yes	characteristic
89	#20591	C₂³	1	1	yes	characteristic
90	#20584	C₂³	1	1	yes	characteristic
91	#20584	C₂³	1	1	yes	characteristic
92	#20584	C₂³	1	1	yes	characteristic
93	C₂×#1780	C₂³	1	1	yes	characteristic
94	H(7,𝔽₂)	C₂²⋊C₄	1	4	yes	—
95	C₂×#585	C₂×D₄	1	1	yes	characteristic
96	C₂×#585	C₂×D₄	1	1	yes	characteristic
97	C₂×#585	C₂×D₄	1	1	yes	characteristic
98	C₂×#585	C₂×D₄	1	1	yes	characteristic
99	C₂³×#41	C₂⁴	1	1	yes	characteristic
100	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂×#568	C₂×D₄	1	1	yes	characteristic, complemented
102	C₂×#568	C₂×D₄	1	1	yes	characteristic, complemented
103	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic, complemented
104	C₂⁴×C₈	C₂×D₄	1	1	yes	characteristic, complemented
105	#3485	C₂×D₄	1	2	yes	—
106	#3485	C₂×D₄	1	2	yes	—
107	#3485	C₂×D₄	1	2	yes	—
108	#3485	C₂×D₄	1	2	yes	—
109	#3485	C₂×D₄	1	2	yes	—
110	#3485	#41	1	2	yes	—
111	#3484	C₂²×C₄	1	2	yes	—
112	#3484	C₂²×C₄	1	2	yes	—
113	#3484	C₂²×C₄	1	2	yes	—
114	#3484	C₂²×C₄	1	2	yes	—
115	#3484	C₂×D₄	1	2	yes	—
116	#3484	#41	1	2	yes	—
117	#3449	C₂×D₄	1	2	yes	—
118	#3449	C₂×D₄	1	2	yes	—
119	C₂×#525	C₂×D₄	1	2	yes	—
120	C₂×#525	C₂×D₄	1	2	yes	—
121	C₂²×#123	C₂²×C₄	1	1	yes	characteristic
122	C₂²×#123	C₂×D₄	1	1	yes	characteristic
123	C₂²×#120	C₂²×C₄	1	1	yes	characteristic
124	C₂²×#120	C₂×D₄	1	1	yes	characteristic
125	#3428	C₂×D₄	1	1	yes	characteristic
126	#3428	C₂×D₄	1	1	yes	characteristic
127	#3428	C₂×D₄	1	1	yes	characteristic
128	#3428	C₂×D₄	1	1	yes	characteristic
129	C₂²×#117	C₂×D₄	1	1	yes	characteristic
130	#3026	C₂²⋊C₄	1	8	yes	—
131	C₂×#413	C₂×D₄	1	2	yes	complemented
132	#2883	C₂²⋊C₄	1	4	yes	—
133	#2883	C₂²⋊C₄	1	4	yes	—
134	#2883	C₂²⋊C₄	1	4	yes	—
135	C₂×#410	C₂×D₄	1	2	yes	—
136	C₂×#407	C₂×D₄	1	2	yes	complemented
137	#2849	C₂²⋊C₄	1	4	yes	—
138	#2846	C₂²⋊C₄	1	4	yes	—
139	#2280	C₂×D₄	1	2	yes	—
140	#1913	C₂²⋊C₄	1	8	yes	—
141	#1898	C₂²⋊C₄	1	4	yes	—
142	#1898	C₂²⋊C₄	1	4	yes	—
143	#1782	C₂×D₄	1	2	yes	—
144	#1781	C₂²×C₄	1	2	yes	—
145	#1781	C₂²×C₄	1	2	yes	—
146	#1781	C₂×D₄	1	2	yes	—
147	C₂⁶	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×H(5,𝔽₂)	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×H(5,𝔽₂)	#120	1	2	yes	—
153	C₂×H(5,𝔽₂)	#123	1	2	yes	—
154	C₂²×#41	#120	1	2	yes	—
155	C₂²×#41	#120	1	2	yes	—
156	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
157	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
158	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
159	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
160	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
161	C₂²×#41	C₂×#41	1	1	yes	characteristic
162	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
164	C₂³×Q₈	H(5,𝔽₂)	1	1	yes	characteristic
165	C₂³×D₄	C₂³×C₄	1	1	yes	characteristic
166	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
167	C₂³×D₄	C₂×#41	1	1	yes	characteristic
168	C₂⁴×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
170	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
171	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
172	#534	C₂×C₂²⋊C₄	1	4	yes	—
173	#534	C₄×D₄	1	4	yes	—
174	#534	#120	1	4	yes	—
175	#534	#121	1	4	yes	—
176	C₂×#123	#120	1	4	yes	—
177	C₂×#123	#121	1	4	yes	—
178	C₂×#120	C₂×C₂²⋊C₄	1	4	yes	—
179	C₂×#120	C₄×D₄	1	4	yes	—
180	#518	C₂×C₂²⋊C₄	1	2	yes	—
181	#518	C₂×C₂²⋊C₄	1	2	yes	—
182	#518	C₂×C₂²⋊C₄	1	2	yes	—
183	#518	C₂×C₂²⋊C₄	1	2	yes	—
184	#518	#120	1	2	yes	—
185	#518	#123	1	2	yes	—
186	C₂×#117	#120	1	2	yes	—
187	C₂×#117	#120	1	2	yes	—
188	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×#99	C₂×C₂²⋊C₄	1	4	yes	—
191	#379	C₂×C₂²⋊C₄	1	4	yes	—
192	C₂⁵	C₂²×C₂²⋊C₄	1	1	yes	characteristic
193	C₂⁵	C₂×C₄×D₄	1	1	yes	characteristic
194	C₂⁵	#546	1	1	yes	characteristic
195	C₂²×Q₈	B(4,𝔽₂)	1	2	yes	—
196	C₂²×Q₈	#458	1	2	yes	—
197	C₂²×Q₈	#534	1	1	yes	characteristic
198	C₂²×Q₈	#534	1	1	yes	characteristic
199	C₂²×Q₈	D₄²	1	1	yes	characteristic
200	C₂²×Q₈	#546	1	1	yes	characteristic
201	C₂²×D₄	#379	1	4	yes	—
202	C₂²×D₄	#386	1	4	yes	—
203	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
204	C₂²×D₄	#458	1	2	yes	—
205	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×D₄	C₂²×C₂²⋊C₄	1	1	yes	characteristic
207	C₂²×D₄	C₂×C₄×D₄	1	1	yes	characteristic
208	C₂²×D₄	#518	1	1	yes	characteristic
209	C₂²×D₄	C₂×#120	1	1	yes	characteristic
210	C₂²×D₄	C₂×#120	1	1	yes	characteristic
211	C₂²×D₄	C₂×#121	1	1	yes	characteristic
212	C₂²×D₄	C₂×#123	1	1	yes	characteristic
213	C₂²×D₄	D₄²	1	1	yes	characteristic
214	C₂²×D₄	#546	1	1	yes	characteristic
215	C₂²×D₄	#546	1	1	yes	characteristic
216	C₂²×D₄	#546	1	1	yes	characteristic
217	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
218	C₂³×C₄	B(4,𝔽₂)	1	4	yes	—
219	C₂³×C₄	#518	1	1	yes	characteristic
220	C₂³×C₄	C₂×#120	1	1	yes	characteristic
221	C₂³×C₄	C₂×#120	1	1	yes	characteristic
222	C₂³×C₄	C₂×#120	1	1	yes	characteristic
223	C₂³×C₄	C₂×#121	1	1	yes	characteristic
224	C₂³×C₄	#534	1	1	yes	characteristic
225	C₂³×C₄	#534	1	1	yes	characteristic
226	C₂³×C₄	D₄²	1	1	yes	characteristic
227	C₂³×C₄	D₄²	1	1	yes	characteristic
228	C₂³×C₄	D₄²	1	1	yes	characteristic
229	C₂³×C₄	D₄²	1	1	yes	characteristic
230	C₂³×C₄	D₄²	1	1	yes	characteristic
231	C₂³×C₄	D₄²	1	1	yes	characteristic
232	C₂³×C₄	#546	1	1	yes	characteristic
233	C₂³×C₄	#546	1	1	yes	characteristic
234	C₂³×C₄	#546	1	1	yes	characteristic
235	C₂×C₂²⋊C₄	#379	1	4	yes	—
236	C₂×C₂²⋊C₄	#386	1	4	yes	—
237	C₂⁴	#1790	1	4	yes	—
238	C₂⁴	#1889	1	4	yes	—
239	C₂⁴	C₂×#379	1	1	yes	characteristic
240	C₂⁴	C₂×#386	1	1	yes	characteristic
241	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
242	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
243	C₂⁴	D₄×C₂²⋊C₄	1	1	yes	characteristic
244	C₂⁴	#2405	1	1	yes	characteristic
245	C₂⁴	#2410	1	1	yes	characteristic
246	C₂²×C₄	#1790	1	4	yes	—
247	C₂²×C₄	#1889	1	4	yes	—
248	C₂²×C₄	#2308	1	1	yes	characteristic
249	C₂²×C₄	D₄×C₂²⋊C₄	1	1	yes	characteristic
250	C₂²×C₄	#2403	1	1	yes	characteristic
251	C₂²×C₄	#2403	1	1	yes	characteristic
252	C₂²×C₄	#2405	1	1	yes	characteristic
253	C₂²×C₄	#2410	1	1	yes	characteristic
254	C₂²×C₄	#2410	1	1	yes	characteristic
255	C₂²×C₄	#2410	1	1	yes	characteristic
256	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
257	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
258	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
259	C₂²×C₄	C₂×#458	1	2	yes	—
260	C₂³	#13867	1	1	yes	characteristic
261	C₂³	C₂×#1790	1	2	yes	—
262	C₂³	C₂×#1889	1	2	yes	—
263	C₂×C₄	#23895	1	1	yes	characteristic
264	C₂×C₄	#23895	1	1	yes	characteristic
265	C₂×C₄	#23900	1	2	yes	—
266	C₂×C₄	#23900	1	1	yes	characteristic
267	C₂×C₄	#23900	1	1	yes	characteristic
268	C₂×C₄	#23900	1	1	yes	characteristic
269	C₂×C₄	#23901	1	1	yes	characteristic
270	C₂×C₄	#24056	1	2	yes	—
271	C₂²	#359501	1	1	yes	characteristic
272	C₂²	#359549	1	1	yes	characteristic
273	C₂²	#359552	1	1	yes	characteristic
274	C₄	#245225	1	1	yes	characteristic
275	C₄	#245225	1	1	yes	characteristic
276	C₄	#245226	1	1	yes	characteristic
277	C₄	#245226	1	1	yes	characteristic
278	C₂	#72236754	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#245225
commutator	C₂³×D₄	C₂³×C₄
Frattini, cosocle	C₂³×#41	C₂⁴
socle	C₂	#72236754
prime radical	#72218347	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72342824	C₂	semidirect
2. #72342821	C₂	semidirect
3. #72342819	C₂	semidirect
4. #72342809	C₂	semidirect
5. #72342796	C₂	semidirect
6. #72342777	C₂	semidirect
7. #72317394	C₂	semidirect
8. #72317374	C₂	semidirect
9. #72317368	C₂	semidirect
10. #72317361	C₂	semidirect
11. #72308105	C₂	semidirect
12. #72240519	C₂	semidirect
13. #72240482	C₂	semidirect
14. #72233438	C₂	semidirect
18. #10574299	C₂²	semidirect
19. #6499875	C₄	semidirect
20. #6499874	C₄	semidirect
25. #6374194	C₂²	semidirect
27. #493402	C₂²	semidirect
28. #493402	C₂²	semidirect
30. #493390	C₂²	semidirect
31. #493352	C₂²	semidirect
32. #359562	C₂²	semidirect
34. #359556	C₂²	semidirect
35. #359556	C₂²	semidirect
37. #359551	C₂²	semidirect
38. #359507	C₂²	semidirect
39. #359426	C₂²	semidirect
40. #188071	C₂²	semidirect
42. #188004	C₂²	semidirect
43. #188004	C₂²	semidirect
44. #186930	C₂²	semidirect
45. #146483	C₂²	semidirect
46. #146461	C₂²	semidirect
47. #128040	C₂²	semidirect
48. #128036	C₂²	semidirect
49. #128024	C₂²	semidirect
50. #128018	C₂²	semidirect
51. #108236	C₂²	semidirect
52. #106146	C₂²	semidirect
80. #21017	D₄	semidirect
82. #20989	D₄	semidirect
84. #20964	D₄	semidirect
85. #20895	D₄	semidirect
101. C₂×#568	C₂×D₄	semidirect
102. C₂×#568	C₂×D₄	semidirect
103. C₂³×M₁₆	C₂×D₄	semidirect
104. C₂⁴×C₈	C₂×D₄	semidirect
131. C₂×#413	C₂×D₄	semidirect
136. C₂×#407	C₂×D₄	semidirect

Maximal subgroups

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

subgroup	index
#72342824	2
#72342821	2
#72342819	2
#72342809	2
#72342796	2
#72342777	2
#72317394	2
#72317374	2
#72317368	2
#72317361	2
#72308105	2
#72240519	2
#72240482	2
#72233438	2
#72218347	2