Subgroups — #72341467

Order 2048.

At least 6430 conjugacy classes of subgroups; only the 342 normal subgroups are shown.

The subgroup lattice has 221 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72341466	C₂	1	1	yes	maximal, characteristic, complemented
2	#72341465	C₂	1	1	yes	maximal, characteristic, complemented
3	#72311266	C₂	1	1	yes	maximal, characteristic, complemented
4	#72303086	C₂	1	1	yes	maximal, characteristic, complemented
5	#72303081	C₂	1	1	yes	maximal, characteristic, complemented
6	#72302950	C₂	1	1	yes	maximal, characteristic, complemented
7	#72260657	C₂	1	1	yes	maximal, characteristic, complemented
8	#72238015	C₂	1	2	yes	maximal, complemented
9	#72238013	C₂	1	2	yes	maximal, complemented
10	#72213523	C₂	1	2	yes	maximal, complemented
11	#72213504	C₂	1	2	yes	maximal, complemented
12	#10586106	C₂²	1	1	yes	characteristic
13	#10574841	C₂²	1	1	yes	characteristic
14	#7742598	C₂²	1	1	yes	characteristic
15	#6628896	C₂²	1	2	yes	—
16	#6470234	C₂²	1	2	yes	—
17	#6470234	C₂²	1	2	yes	complemented
18	#6437630	C₂²	1	1	yes	characteristic
19	#6437453	C₂²	1	1	yes	characteristic
20	#6385981	C₂²	1	1	yes	characteristic
21	#496240	C₂²	1	2	yes	complemented
22	#494682	C₂²	1	2	yes	complemented
23	#493433	C₂²	1	1	yes	characteristic
24	#493433	C₂²	1	1	yes	characteristic
25	#493397	C₂²	1	2	yes	—
26	#493397	C₂²	1	2	yes	complemented
27	#392138	C₂²	1	1	yes	characteristic
28	#392135	C₂²	1	2	yes	—
29	#359606	C₂²	1	1	yes	characteristic
30	#359529	C₂²	1	1	yes	characteristic
31	#213854	C₂²	1	2	yes	complemented
32	#146577	C₂²	1	2	yes	complemented
33	#128079	C₂²	1	2	yes	complemented
34	#128062	C₂²	1	2	yes	complemented
35	C₂×2₋¹⁺⁶	D₄	1	1	yes	characteristic, complemented
36	C₂×H(7,𝔽₂)	C₂³	1	1	yes	characteristic
37	#62859	D₄	1	1	yes	characteristic, complemented
38	#62852	C₂³	1	1	yes	characteristic
39	#62852	C₂³	1	1	yes	characteristic
40	#62812	D₄	1	1	yes	characteristic, complemented
41	C₂×#3500	C₂³	1	1	yes	characteristic
42	C₂²×H(3,𝔽₄)	D₄	1	1	yes	characteristic, complemented
43	C₄×#585	D₄	1	1	yes	characteristic, complemented
44	C₂²×B(4,𝔽₂)	D₄	1	4	yes	complemented
45	C₂×#3022	D₄	1	1	yes	characteristic, complemented
46	C₂×#3018	D₄	1	1	yes	characteristic, complemented
47	C₂×#2888	C₂³	1	1	yes	characteristic
48	#36702	D₄	1	2	yes	complemented
49	#35346	C₂³	1	2	yes	—
50	#34366	D₄	1	1	yes	characteristic, complemented
51	C₂×#2127	D₄	1	2	yes	complemented
52	C₂²×#354	D₄	1	2	yes	complemented
53	#24058	D₄	1	4	yes	complemented
54	#23900	D₄	1	4	yes	complemented
55	#21794	D₄	1	2	yes	complemented
56	#21751	D₄	1	2	yes	complemented
57	#20955	D₄	1	1	yes	characteristic, complemented
58	#20952	D₄	1	1	yes	characteristic, complemented
59	#20609	D₄	1	4	yes	complemented
60	#20592	C₂³	1	1	yes	characteristic
61	#20586	C₂³	1	1	yes	characteristic
62	#20585	C₂³	1	2	yes	—
63	#20585	C₂³	1	2	yes	—
64	#20585	C₂³	1	2	yes	—
65	#10989	D₄	1	2	yes	complemented
66	#10977	D₄	1	2	yes	complemented
67	C₂×#585	C₂×D₄	1	1	yes	characteristic
68	C₂×#585	C₂×D₄	1	1	yes	characteristic
69	C₂×#585	C₂×D₄	1	1	yes	characteristic
70	C₂×#585	C₂×D₄	1	1	yes	characteristic
71	C₂²×2₋¹⁺⁴	C₂×D₄	1	1	yes	characteristic
72	C₂²×H(5,𝔽₂)	C₂×D₄	1	4	yes	—
73	C₂²×H(5,𝔽₂)	C₂⁴	1	1	yes	characteristic
74	#3484	C₂×D₄	1	4	yes	—
75	C₂×#536	C₂×D₄	1	1	yes	characteristic
76	C₂×#536	C₂×D₄	1	1	yes	characteristic
77	C₂×#536	#41	1	1	yes	characteristic
78	C₂×#535	C₂×D₄	1	1	yes	characteristic
79	C₂×#535	C₂×D₄	1	1	yes	characteristic
80	C₂×#535	C₂×D₄	1	1	yes	characteristic
81	C₂×#532	C₂×D₄	1	1	yes	characteristic
82	C₂×#532	C₂×D₄	1	1	yes	characteristic
83	C₂×#532	#41	1	1	yes	characteristic
84	C₂²×#124	C₂×D₄	1	1	yes	characteristic
85	C₂²×#124	C₂×D₄	1	1	yes	characteristic
86	C₂²×#124	C₂×D₄	1	1	yes	characteristic
87	C₂²×#120	C₂×D₄	1	2	yes	—
88	C₂²×#120	C₂×D₄	1	2	yes	—
89	C₂²×#120	C₂×D₄	1	2	yes	—
90	#3428	C₂×D₄	1	1	yes	characteristic
91	#3428	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×#41	C₂×D₄	1	1	yes	characteristic
93	C₂×#413	C₂×D₄	1	1	yes	characteristic
94	C₂×#413	C₂×D₄	1	1	yes	characteristic
95	C₂²×#100	C₂×D₄	1	2	yes	—
96	C₂²×#99	C₂×D₄	1	4	yes	—
97	C₂²×#99	C₂×D₄	1	4	yes	—
98	#2309	C₂×D₄	1	2	yes	—
99	#2308	C₂×D₄	1	2	yes	—
100	#2308	C₂×D₄	1	2	yes	—
101	#2308	C₂×D₄	1	2	yes	—
102	#1782	C₂×D₄	1	4	yes	—
103	#1781	C₂×D₄	1	4	yes	—
104	C₂⁶	#120	1	2	yes	—
105	C₂×H(5,𝔽₂)	#120	1	2	yes	—
106	C₂×H(5,𝔽₂)	#120	1	2	yes	—
107	C₂×H(5,𝔽₂)	#120	1	2	yes	—
108	C₂²×#41	#120	1	1	yes	characteristic
109	C₂²×#41	#120	1	1	yes	characteristic
110	C₂²×#41	#121	1	1	yes	characteristic
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	2	yes	—
116	C₂³×Q₈	#120	1	1	yes	characteristic
117	C₂³×Q₈	#120	1	1	yes	characteristic
118	C₂³×Q₈	#121	1	1	yes	characteristic
119	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
120	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
121	C₂³×D₄	C₂²×D₄	1	1	yes	characteristic
122	C₂³×D₄	C₂²×D₄	1	2	yes	—
123	C₂³×D₄	C₂²×D₄	1	2	yes	—
124	C₂³×D₄	H(5,𝔽₂)	1	2	yes	—
125	C₂×C₄⋊Q₈	#121	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	#121	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	#121	1	1	yes	characteristic
128	C₂×#127	#120	1	1	yes	characteristic
129	C₂×#127	#121	1	1	yes	characteristic
130	C₂×#127	#127	1	1	yes	characteristic
131	C₂×#124	#120	1	1	yes	characteristic
132	C₂×#124	#120	1	1	yes	characteristic
133	C₂×#124	#120	1	1	yes	characteristic
134	C₂×#124	#121	1	1	yes	characteristic
135	C₂×#124	#121	1	1	yes	characteristic
136	C₂×#124	#121	1	1	yes	characteristic
137	C₂×#124	#121	1	1	yes	characteristic
138	C₂×#124	#127	1	1	yes	characteristic
139	C₂×#124	#127	1	1	yes	characteristic
140	C₂×#124	#127	1	1	yes	characteristic
141	#518	#120	1	2	yes	—
142	C₂×#117	#120	1	1	yes	characteristic
143	C₂×#117	#120	1	1	yes	characteristic
144	C₂×#117	#121	1	1	yes	characteristic
145	C₂²×C₂²⋊C₄	#120	1	2	yes	—
146	C₂²×C₂²⋊C₄	#120	1	2	yes	—
147	C₂²×C₂²⋊C₄	#120	1	2	yes	—
148	C₂²×C₄²	#120	1	1	yes	characteristic
149	C₂²×C₄²	#120	1	1	yes	characteristic
150	C₂²×C₄²	#121	1	1	yes	characteristic
151	C₂×#99	#120	1	8	yes	—
152	C₂⁵	C₂×#120	1	2	yes	—
153	C₂⁵	D₄²	1	2	yes	—
154	C₂⁵	D₄²	1	2	yes	—
155	H(5,𝔽₂)	B(4,𝔽₂)	1	4	yes	—
156	C₂²×Q₈	#392	1	1	yes	characteristic
157	C₂²×Q₈	#392	1	1	yes	characteristic
158	C₂²×Q₈	#392	1	1	yes	characteristic
159	C₂²×Q₈	C₂×#121	1	1	yes	characteristic
160	C₂²×Q₈	#534	1	1	yes	characteristic
161	C₂²×D₄	#392	1	1	yes	characteristic
162	C₂²×D₄	C₂×#120	1	1	yes	characteristic
163	C₂²×D₄	C₂×#120	1	1	yes	characteristic
164	C₂²×D₄	C₂×#120	1	2	yes	—
165	C₂²×D₄	C₂×#120	1	1	yes	characteristic
166	C₂²×D₄	C₂×#121	1	1	yes	characteristic
167	C₂²×D₄	C₂×#127	1	1	yes	characteristic
168	C₂²×D₄	#534	1	1	yes	characteristic
169	C₂²×D₄	#534	1	2	yes	—
170	C₂²×D₄	D₄²	1	2	yes	—
171	C₂²×D₄	D₄²	1	2	yes	—
172	C₂²×D₄	D₄²	1	2	yes	—
173	C₂²×D₄	#546	1	2	yes	—
174	C₂³×C₄	C₂×#120	1	1	yes	characteristic
175	C₂³×C₄	C₂×#120	1	1	yes	characteristic
176	C₂³×C₄	C₂×#121	1	1	yes	characteristic
177	C₂³×C₄	#535	1	2	yes	—
178	C₂³×C₄	D₄²	1	2	yes	—
179	C₂³×C₄	#546	1	2	yes	—
180	#124	B(4,𝔽₂)	1	2	yes	—
181	#124	B(4,𝔽₂)	1	2	yes	—
182	#124	B(4,𝔽₂)	1	4	yes	—
183	C₂×C₄²	#392	1	1	yes	characteristic
184	C₂×C₄²	#392	1	1	yes	characteristic
185	C₂×C₄²	#392	1	1	yes	characteristic
186	C₂×C₄²	#392	1	1	yes	characteristic
187	C₂⁴	#2403	1	1	yes	characteristic
188	C₂⁴	#2403	1	2	yes	—
189	C₂⁴	#2410	1	2	yes	—
190	C₂⁴	#2410	1	1	yes	characteristic
191	C₂⁴	#2433	1	2	yes	—
192	C₂⁴	#2846	1	1	yes	characteristic
193	C₂×Q₈	#2021	1	2	yes	—
194	C₂×Q₈	#2021	1	2	yes	—
195	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
196	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
197	C₂×D₄	C₂×B(4,𝔽₂)	1	2	yes	—
198	C₂×D₄	#3026	1	2	yes	—
199	C₂²×C₄	C₂×#392	1	1	yes	characteristic
200	C₂²×C₄	#2403	1	1	yes	characteristic
201	C₂²×C₄	#2408	1	2	yes	—
202	C₂²×C₄	#2410	1	1	yes	characteristic
203	C₂²×C₄	#2613	1	1	yes	characteristic
204	C₄²	#2021	1	1	yes	characteristic
205	C₄²	#2021	1	1	yes	characteristic
206	C₄²	#2021	1	2	yes	—
207	C₂³	#14715	1	1	yes	characteristic
208	C₂³	#23895	1	2	yes	—
209	C₂³	#23900	1	4	yes	—
210	C₂³	#24404	1	2	yes	—
211	C₂³	#32898	1	2	yes	—
212	C₂×C₄	C₂×#2021	1	1	yes	characteristic
213	C₂×C₄	C₂×#2021	1	1	yes	characteristic
214	C₂×C₄	#23900	1	1	yes	characteristic
215	C₂×C₄	#23903	1	1	yes	characteristic
216	C₂²	#394058	1	1	yes	characteristic
217	C₂²	#394064	1	1	yes	characteristic
218	C₂²	#394066	1	1	yes	characteristic
219	C₂	#72210600	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72210600
commutator, Frattini, cosocle	C₂²×H(5,𝔽₂)	C₂⁴
LCS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×H(5,𝔽₂) ▷ C₂³ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #72341466	C₂	semidirect
2. #72341465	C₂	semidirect
3. #72311266	C₂	semidirect
4. #72303086	C₂	semidirect
5. #72303081	C₂	semidirect
6. #72302950	C₂	semidirect
7. #72260657	C₂	semidirect
8. #72238015	C₂	semidirect
9. #72238013	C₂	semidirect
10. #72213523	C₂	semidirect
11. #72213504	C₂	semidirect
17. #6470234	C₂²	semidirect
21. #496240	C₂²	semidirect
22. #494682	C₂²	semidirect
26. #493397	C₂²	semidirect
31. #213854	C₂²	semidirect
32. #146577	C₂²	semidirect
33. #128079	C₂²	semidirect
34. #128062	C₂²	semidirect
35. C₂×2₋¹⁺⁶	D₄	semidirect
37. #62859	D₄	semidirect
40. #62812	D₄	semidirect
42. C₂²×H(3,𝔽₄)	D₄	semidirect
43. C₄×#585	D₄	semidirect
44. C₂²×B(4,𝔽₂)	D₄	semidirect
45. C₂×#3022	D₄	semidirect
46. C₂×#3018	D₄	semidirect
48. #36702	D₄	semidirect
50. #34366	D₄	semidirect
51. C₂×#2127	D₄	semidirect
52. C₂²×#354	D₄	semidirect
53. #24058	D₄	semidirect
54. #23900	D₄	semidirect
55. #21794	D₄	semidirect
56. #21751	D₄	semidirect
57. #20955	D₄	semidirect
58. #20952	D₄	semidirect
59. #20609	D₄	semidirect
65. #10989	D₄	semidirect
66. #10977	D₄	semidirect

Maximal subgroups

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

subgroup	index
#72341466	2
#72341465	2
#72311266	2
#72303086	2
#72303081	2
#72302950	2
#72260657	2
#72238015	2
#72238013	2
#72213523	2
#72213504	2