Subgroups — #72317758

Order 2048.

At least 6110 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

The subgroup lattice has 212 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72317728	C₂	1	1	yes	maximal, characteristic, complemented
2	#72317693	C₂	1	1	yes	maximal, characteristic, complemented
3	#72317663	C₂	1	1	yes	maximal, characteristic, complemented
4	#72317626	C₂	1	1	yes	maximal, characteristic, complemented
5	#72317524	C₂	1	1	yes	maximal, characteristic, complemented
6	#72317501	C₂	1	1	yes	maximal, characteristic, complemented
7	#72317472	C₂	1	1	yes	maximal, characteristic, complemented
8	#72317466	C₂	1	1	yes	maximal, characteristic, complemented
9	#72317394	C₂	1	1	yes	maximal, characteristic, complemented
10	#72317374	C₂	1	1	yes	maximal, characteristic, complemented
11	#72317357	C₂	1	1	yes	maximal, characteristic, complemented
12	#72286087	C₂	1	1	yes	maximal, characteristic, complemented
13	#72258308	C₂	1	1	yes	maximal, characteristic, complemented
14	#72233438	C₂	1	1	yes	maximal, characteristic, complemented
15	#72216294	C₂	1	1	yes	maximal, characteristic
16	#10586169	C₂²	1	1	yes	characteristic
17	#10574299	C₂²	1	1	yes	characteristic, complemented
18	#6437616	C₂²	1	1	yes	characteristic
19	#6437463	C₂²	1	1	yes	characteristic
20	#6431743	C₂²	1	1	yes	characteristic
21	#493402	C₂²	1	1	yes	characteristic
22	#493402	C₂²	1	1	yes	characteristic, complemented
23	#493390	C₂²	1	1	yes	characteristic
24	#493352	C₂²	1	1	yes	characteristic
25	#382440	C₂²	1	1	yes	characteristic
26	#382440	C₂²	1	1	yes	characteristic, complemented
27	#373690	C₂²	1	1	yes	characteristic, complemented
28	#373643	C₂²	1	1	yes	characteristic, complemented
29	#359562	C₂²	1	1	yes	characteristic
30	#359562	C₂²	1	1	yes	characteristic
31	#359556	C₂²	1	1	yes	characteristic
32	#359556	C₂²	1	1	yes	characteristic
33	#359551	C₂²	1	1	yes	characteristic
34	#359507	C₂²	1	1	yes	characteristic
35	#359507	C₂²	1	1	yes	characteristic
36	#359426	C₂²	1	1	yes	characteristic
37	#188071	C₂²	1	1	yes	characteristic
38	#188004	C₂²	1	1	yes	characteristic
39	#188004	C₂²	1	1	yes	characteristic
40	#188004	C₂²	1	1	yes	characteristic, complemented
41	#146478	C₂²	1	1	yes	characteristic, complemented
42	#146478	C₂²	1	1	yes	characteristic, complemented
43	#146463	C₂²	1	1	yes	characteristic, complemented
44	#128042	C₂²	1	1	yes	characteristic, complemented
45	#128036	C₂²	1	1	yes	characteristic, complemented
46	#128030	C₂²	1	1	yes	characteristic, complemented
47	#128022	C₂²	1	1	yes	characteristic, complemented
48	#108236	C₂²	1	1	yes	characteristic, complemented
49	#107152	C₂²	1	1	yes	characteristic, complemented
50	#106146	C₂²	1	1	yes	characteristic, complemented
51	#63090	D₄	1	1	yes	characteristic
52	#62811	C₂³	1	1	yes	characteristic
53	#62811	C₂³	1	1	yes	characteristic
54	#62811	C₂³	1	1	yes	characteristic
55	C₂²×#525	C₂³	1	1	yes	characteristic
56	#60387	D₄	1	2	yes	—
57	#60093	D₄	1	1	yes	characteristic
58	#60093	D₄	1	1	yes	characteristic
59	#60064	C₂³	1	1	yes	characteristic
60	#60064	C₂³	1	1	yes	characteristic
61	C₂²×#408	C₂³	1	1	yes	characteristic
62	#36614	D₄	1	2	yes	—
63	#21012	D₄	1	2	yes	—
64	#21009	D₄	1	2	yes	complemented
65	#20989	D₄	1	2	yes	complemented
66	#20983	D₄	1	2	yes	—
67	#20947	D₄	1	2	yes	complemented
68	#20895	D₄	1	2	yes	complemented
69	#20591	C₂³	1	1	yes	characteristic
70	#20591	C₂³	1	1	yes	characteristic
71	#20591	C₂³	1	1	yes	characteristic
72	#20591	C₂³	1	1	yes	characteristic
73	#20591	C₂³	1	1	yes	characteristic
74	#20584	C₂³	1	1	yes	characteristic
75	#20584	C₂³	1	1	yes	characteristic
76	C₂×#1780	C₂³	1	1	yes	characteristic
77	#20338	D₄	1	1	yes	characteristic
78	C₂×#585	C₂×D₄	1	1	yes	characteristic
79	C₂×#585	C₂×D₄	1	1	yes	characteristic
80	C₂×#585	C₂×D₄	1	1	yes	characteristic
81	C₂³×#41	C₂⁴	1	1	yes	characteristic
82	C₂⁵×C₄	C₂×D₄	1	1	yes	characteristic
83	C₂×#568	C₂×D₄	1	1	yes	characteristic
84	C₂×#568	C₂×D₄	1	1	yes	characteristic
85	C₂×#568	C₂×D₄	1	1	yes	characteristic
86	C₂×#568	C₂×D₄	1	1	yes	characteristic
87	C₂³×M₁₆	C₂×D₄	1	1	yes	characteristic
88	C₂⁴×C₈	C₂×D₄	1	1	yes	characteristic
89	#3485	#41	1	2	yes	—
90	#3484	#41	1	2	yes	—
91	#3449	C₂×D₄	1	2	yes	—
92	#3449	C₂×D₄	1	2	yes	—
93	C₂×#525	C₂×D₄	1	2	yes	—
94	C₂×#525	C₂×D₄	1	2	yes	—
95	C₂²×#123	#41	1	1	yes	characteristic
96	C₂²×#123	#41	1	1	yes	characteristic
97	C₂²×#120	#41	1	1	yes	characteristic
98	C₂²×#120	#41	1	1	yes	characteristic
99	#3428	C₂×D₄	1	1	yes	characteristic
100	#3428	C₂×D₄	1	1	yes	characteristic
101	#3428	C₂×D₄	1	1	yes	characteristic
102	C₂²×#117	C₂×D₄	1	1	yes	characteristic
103	C₂×#413	C₂×D₄	1	2	yes	—
104	C₂×#410	C₂×D₄	1	2	yes	—
105	C₂×#407	C₂×D₄	1	2	yes	—
106	#2280	C₂×D₄	1	2	yes	—
107	C₂⁶	#123	1	1	yes	characteristic
108	C₂×H(5,𝔽₂)	#123	1	2	yes	—
109	C₂²×#41	#120	1	2	yes	—
110	C₂²×#41	#120	1	2	yes	—
111	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
112	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
113	C₂²×#41	C₂²×D₄	1	1	yes	characteristic
114	C₂²×#41	C₂×#41	1	1	yes	characteristic
115	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
116	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×#41	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂³×Q₈	H(5,𝔽₂)	1	1	yes	characteristic
120	C₂³×D₄	C₂×#41	1	1	yes	characteristic
121	C₂³×D₄	C₂×#41	1	1	yes	characteristic
122	C₂³×D₄	C₂×#41	1	1	yes	characteristic
123	C₂⁴×C₄	#123	1	1	yes	characteristic
124	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
125	C₂⁴×C₄	C₂²×D₄	1	1	yes	characteristic
126	C₂⁴×C₄	H(5,𝔽₂)	1	1	yes	characteristic
127	#518	#123	1	2	yes	—
128	C₂×#117	#120	1	2	yes	—
129	C₂×#117	#120	1	2	yes	—
130	C₂²×C₂²⋊C₄	#123	1	1	yes	characteristic
131	C₂²×C₂²⋊C₄	#123	1	1	yes	characteristic
132	C₂⁵	C₂×#123	1	1	yes	characteristic
133	C₂⁵	#525	1	1	yes	characteristic
134	C₂⁵	#551	1	1	yes	characteristic
135	C₂²×Q₈	B(4,𝔽₂)	1	2	yes	—
136	C₂²×Q₈	#458	1	2	yes	—
137	C₂²×Q₈	#534	1	1	yes	characteristic
138	C₂²×Q₈	#535	1	1	yes	characteristic
139	C₂²×Q₈	#546	1	1	yes	characteristic
140	C₂²×Q₈	#560	1	1	yes	characteristic
141	C₂²×D₄	B(4,𝔽₂)	1	2	yes	—
142	C₂²×D₄	#458	1	2	yes	—
143	C₂²×D₄	C₂×#123	1	1	yes	characteristic
144	C₂²×D₄	#525	1	1	yes	characteristic
145	C₂²×D₄	#538	1	1	yes	characteristic
146	C₂²×D₄	#538	1	1	yes	characteristic
147	C₂²×D₄	#546	1	1	yes	characteristic
148	C₂²×D₄	#546	1	1	yes	characteristic
149	C₂²×D₄	#546	1	1	yes	characteristic
150	C₂²×D₄	#546	1	1	yes	characteristic
151	C₂²×D₄	#546	1	1	yes	characteristic
152	C₂²×D₄	#546	1	1	yes	characteristic
153	C₂²×D₄	#546	1	1	yes	characteristic
154	C₂²×D₄	#551	1	1	yes	characteristic
155	C₂³×C₄	C₂×#120	1	1	yes	characteristic
156	C₂³×C₄	C₂×#120	1	1	yes	characteristic
157	C₂³×C₄	#534	1	1	yes	characteristic
158	C₂³×C₄	#534	1	1	yes	characteristic
159	C₂³×C₄	#535	1	1	yes	characteristic
160	C₂³×C₄	#538	1	1	yes	characteristic
161	C₂³×C₄	#538	1	1	yes	characteristic
162	C₂³×C₄	D₄²	1	1	yes	characteristic
163	C₂³×C₄	D₄²	1	1	yes	characteristic
164	C₂³×C₄	D₄²	1	1	yes	characteristic
165	C₂³×C₄	D₄²	1	1	yes	characteristic
166	C₂³×C₄	#546	1	1	yes	characteristic
167	C₂³×C₄	#546	1	1	yes	characteristic
168	C₂³×C₄	#546	1	1	yes	characteristic
169	C₂³×C₄	#546	1	1	yes	characteristic
170	C₂³×C₄	#560	1	1	yes	characteristic
171	C₂×C₄²	B(4,𝔽₂)	1	4	yes	—
172	C₂×C₄²	B(4,𝔽₂)	1	4	yes	—
173	C₂⁴	#2404	1	1	yes	characteristic
174	C₂⁴	#2405	1	1	yes	characteristic
175	C₂⁴	#2418	1	1	yes	characteristic
176	C₂⁴	#2418	1	1	yes	characteristic
177	C₂⁴	#2418	1	1	yes	characteristic
178	C₂⁴	#2418	1	1	yes	characteristic
179	C₂⁴	#2697	1	1	yes	characteristic
180	C₂²×C₄	#2403	1	1	yes	characteristic
181	C₂²×C₄	#2404	1	1	yes	characteristic
182	C₂²×C₄	#2405	1	1	yes	characteristic
183	C₂²×C₄	#2678	1	1	yes	characteristic
184	C₂²×C₄	#2678	1	1	yes	characteristic
185	C₂²×C₄	#2678	1	1	yes	characteristic
186	C₂²×C₄	#2678	1	1	yes	characteristic
187	C₂²×C₄	#2697	1	1	yes	characteristic
188	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
189	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
190	C₂²×C₄	C₂×B(4,𝔽₂)	1	2	yes	—
191	C₂²×C₄	C₂×#458	1	2	yes	—
192	C₂³	#15948	1	1	yes	characteristic
193	C₂³	#23984	1	2	yes	—
194	C₂³	#24056	1	2	yes	—
195	C₂×C₄	#23895	1	1	yes	characteristic
196	C₂×C₄	#23896	1	1	yes	characteristic
197	C₂×C₄	#23900	1	1	yes	characteristic
198	C₂×C₄	#23900	1	1	yes	characteristic
199	C₂×C₄	#23900	1	1	yes	characteristic
200	C₂×C₄	#23901	1	1	yes	characteristic
201	C₂×C₄	#23984	1	2	yes	—
202	C₂×C₄	#24056	1	2	yes	—
203	C₂²	#449501	1	1	yes	characteristic
204	C₂²	#449505	1	1	yes	characteristic
205	C₂²	#449521	1	1	yes	characteristic
206	C₄	#245226	1	1	yes	characteristic
207	C₄	#245229	1	1	yes	characteristic
208	C₄	#245230	1	1	yes	characteristic
209	C₄	#245233	1	1	yes	characteristic
210	C₂	#72317430	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#245233
commutator, Frattini, cosocle	C₂³×#41	C₂⁴
socle	C₂	#72317430
prime radical	#72216294	C₂
LCS	G ▷ C₂³×#41 ▷ 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. #72317728	C₂	semidirect
2. #72317693	C₂	semidirect
3. #72317663	C₂	semidirect
4. #72317626	C₂	semidirect
5. #72317524	C₂	semidirect
6. #72317501	C₂	semidirect
7. #72317472	C₂	semidirect
8. #72317466	C₂	semidirect
9. #72317394	C₂	semidirect
10. #72317374	C₂	semidirect
11. #72317357	C₂	semidirect
12. #72286087	C₂	semidirect
13. #72258308	C₂	semidirect
14. #72233438	C₂	semidirect
17. #10574299	C₂²	semidirect
22. #493402	C₂²	semidirect
26. #382440	C₂²	semidirect
27. #373690	C₂²	semidirect
28. #373643	C₂²	semidirect
40. #188004	C₂²	semidirect
41. #146478	C₂²	semidirect
42. #146478	C₂²	semidirect
43. #146463	C₂²	semidirect
44. #128042	C₂²	semidirect
45. #128036	C₂²	semidirect
46. #128030	C₂²	semidirect
47. #128022	C₂²	semidirect
48. #108236	C₂²	semidirect
49. #107152	C₂²	semidirect
50. #106146	C₂²	semidirect
64. #21009	D₄	semidirect
65. #20989	D₄	semidirect
67. #20947	D₄	semidirect
68. #20895	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72317728	2
#72317693	2
#72317663	2
#72317626	2
#72317524	2
#72317501	2
#72317472	2
#72317466	2
#72317394	2
#72317374	2
#72317357	2
#72286087	2
#72258308	2
#72233438	2
#72216294	2