Subgroups — #22623719

Order 9344.

1540 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14556638	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989244	C₂	1	4	yes	maximal, complemented
3	#17989231	C₂	1	12	yes	maximal, complemented
4	#17989158	C₂	1	3	yes	maximal, complemented
5	#17989154	C₂	1	3	yes	maximal, complemented
6	#17989133	C₂	1	3	yes	maximal, complemented
7	C₂×#14556578	C₂	1	3	yes	maximal, complemented
8	#41×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×H(5,𝔽₂)×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#14556638	—	2	8	no	—
11	C₂×#11924028	C₂²	1	3	yes	complemented
12	C₂×#11924027	C₂²	1	6	yes	complemented
13	C₂²×Dic₁₄₆	C₂²	1	3	yes	complemented
14	C₂×Q₈×D₇₃	C₂²	1	2	yes	complemented
15	#14556589	C₂²	1	3	yes	complemented
16	#14556587	C₂²	1	3	yes	complemented
17	#14556580	C₂²	1	1	yes	characteristic, complemented
18	#14556578	C₂²	1	3	yes	complemented
19	#14556578	C₂²	1	6	yes	complemented
20	#14556578	C₂²	1	12	yes	complemented
21	#14556572	C₂²	1	12	yes	complemented
22	#14556570	C₂²	1	12	yes	complemented
23	#14556563	C₂²	1	12	yes	complemented
24	#14556562	C₂²	1	12	yes	complemented
25	#14556560	C₂²	1	12	yes	complemented
26	Q₈×Dic₇₃	C₂²	1	2	yes	complemented
27	D₄×Dic₇₃	C₂²	1	6	yes	complemented
28	C₄×C₂²⋊D₇₃	C₂²	1	12	yes	complemented
29	#14556550	C₂²	1	3	yes	complemented
30	C₂×C₂²⋊Dic₇₃	C₂²	1	12	yes	complemented
31	C₂×C₄×Dic₇₃	C₂²	1	3	yes	complemented
32	H(5,𝔽₂)×C₇₃	—	2	8	no	—
33	C₂×#41×C₇₃	C₂²	1	3	yes	—
34	C₂×#41×C₇₃	C₂²	1	2	yes	—
35	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₇₃	C₂²	1	3	yes	—
37	C₂²×D₄×C₇₃	C₂²	1	6	yes	—
38	#11924028	—	2	12	no	—
39	#11924027	—	2	24	no	—
40	C₂×Dic₁₄₆	—	2	12	no	—
41	C₂×Dic₁₄₆	C₂³	1	6	yes	—
42	C₂×Dic₁₄₆	C₂³	1	3	yes	—
43	Q₈×D₇₃	—	2	8	no	—
44	C₂×D₂₉₂	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₇₃	C₂³	1	6	yes	—
46	C₂×C₂²⋊D₇₃	C₂³	1	12	yes	—
47	C₂²×Dic₇₃	C₂³	1	6	yes	—
48	C₂²×Dic₇₃	C₂³	1	6	yes	—
49	C₂×C₄×D₇₃	C₂³	1	6	yes	—
50	C₄⋊Dic₇₃	C₂³	1	6	yes	—
51	#11924007	C₂³	1	12	yes	—
52	C₂²⋊Dic₇₃	C₂³	1	12	yes	—
53	C₂²⋊Dic₇₃	C₂³	1	24	yes	—
54	C₂²⋊Dic₇₃	C₂³	1	6	yes	—
55	#11924005	C₂³	1	4	yes	—
56	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
57	C₄×Dic₇₃	C₂³	1	3	yes	—
58	C₄×Dic₇₃	C₂³	1	6	yes	—
59	C₂⁴×C₇₃	C₂³	1	6	yes	—
60	#41×C₇₃	—	2	8	no	—
61	#41×C₇₃	—	2	12	no	—
62	#41×C₇₃	D₄	1	8	yes	complemented
63	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
64	C₂×Q₈×C₇₃	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₇₃	—	2	24	no	—
66	C₂×D₄×C₇₃	—	2	12	no	—
67	C₂×D₄×C₇₃	C₂³	1	6	yes	—
68	C₂×D₄×C₇₃	C₂³	1	6	yes	—
69	C₂×D₄×C₇₃	C₂³	1	3	yes	—
70	C₂×D₄×C₇₃	C₂³	1	3	yes	—
71	C₂²×C₄×C₇₃	C₂³	1	6	yes	—
72	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
73	C₂³×C₇₃	—	2	12	no	—
74	C₂³×C₇₃	—	4	12	no	—
75	C₂³×C₇₃	C₂⁴	1	6	yes	—
76	C₂³×C₇₃	C₂⁴	1	3	yes	—
77	C₂²×D₇₃	C₂⁴	1	2	yes	—
78	Q₈×C₇₃	—	2	2	no	—
79	Q₈×C₇₃	C₂×D₄	1	4	yes	—
80	D₄×C₇₃	—	2	12	no	—
81	D₄×C₇₃	—	2	6	no	—
82	D₄×C₇₃	—	2	12	no	—
83	D₄×C₇₃	C₂×D₄	1	12	yes	—
84	C₂×C₄×C₇₃	—	2	12	no	—
85	C₂×C₄×C₇₃	C₂×D₄	1	12	yes	—
86	C₂×C₄×C₇₃	C₂⁴	1	2	yes	—
87	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
88	C₂×C₄×C₇₃	C₂⁴	1	3	yes	—
89	C₂²⋊D₇₃	—	2	12	no	—
90	C₂²⋊D₇₃	—	4	12	no	—
91	C₂×Dic₇₃	—	2	12	no	—
92	C₂×Dic₇₃	—	4	6	no	—
93	C₂×Dic₇₃	C₂⁴	1	2	yes	—
94	C₂×Dic₇₃	C₂⁴	1	6	yes	—
95	C₂×Dic₇₃	C₂⁴	1	6	yes	—
96	D₂₉₂	—	2	2	no	—
97	C₄×D₇₃	—	2	12	no	—
98	Dic₁₄₆	—	2	12	no	—
99	Dic₁₄₆	—	2	6	no	—
100	C₂²×C₇₃	—	2	6	no	—
101	C₂²×C₇₃	—	2	6	no	—
102	C₂²×C₇₃	—	2	12	no	—
103	C₂²×C₇₃	—	8	12	no	—
104	C₂²×C₇₃	C₂²×D₄	1	6	yes	—
105	C₂²×C₇₃	C₂⁵	1	1	yes	characteristic
106	C₂×D₇₃	—	2	2	no	—
107	C₂×D₇₃	—	2	4	no	—
108	C₄×C₇₃	—	2	2	no	—
109	C₄×C₇₃	C₂²×D₄	1	2	yes	—
110	C₄×C₇₃	C₂²×D₄	1	6	yes	—
111	Dic₇₃	—	2	6	no	—
112	Dic₇₃	—	2	2	no	—
113	Dic₇₃	—	2	6	no	—
114	C₂×C₇₃	—	2	6	no	—
115	C₂×C₇₃	—	4	6	no	—
116	C₂×C₇₃	C₂³×D₄	1	1	yes	characteristic
117	C₂×C₇₃	#585	1	2	yes	—
118	D₇₃	—	4	2	no	—
119	#3500	—	73	1	no	maximal, Sylow
120	C₇₃	#3500	1	1	yes	characteristic, complemented, Sylow
121	C₂×2₋¹⁺⁴	—	73	1	no	—
122	C₂×H(5,𝔽₂)	D₇₃	1	1	yes	characteristic, complemented
123	#548	—	73	4	no	—
124	#546	—	73	12	no	—
125	#536	—	73	3	no	—
126	#535	—	73	3	no	—
127	#532	—	73	3	no	—
128	C₂×#124	—	73	3	no	—
129	C₄×#41	—	73	1	no	—
130	2₋¹⁺⁴	—	146	8	no	—
131	H(5,𝔽₂)	—	2	8	no	—
132	C₂×#41	—	73	6	no	—
133	C₂×#41	—	73	3	no	—
134	C₂×#41	C₂×D₇₃	1	3	yes	—
135	C₂×#41	C₂×D₇₃	1	2	yes	—
136	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
137	C₂²×Q₈	—	73	2	no	—
138	C₂²×Q₈	—	73	3	no	—
139	C₂²×D₄	C₂×D₇₃	1	3	yes	—
140	C₂²×D₄	C₂×D₇₃	1	6	yes	—
141	C₄⋊Q₈	—	73	3	no	—
142	#127	—	73	3	no	—
143	#124	—	73	1	no	—
144	#124	—	73	3	no	—
145	#124	—	73	6	no	—
146	#124	—	73	12	no	—
147	#123	—	73	12	no	—
148	C₂²⋊Q₈	—	73	12	no	—
149	#121	—	73	12	no	—
150	#121	—	73	12	no	—
151	#120	—	73	12	no	—
152	C₄×Q₈	—	73	2	no	—
153	C₄×D₄	—	73	6	no	—
154	C₄×D₄	—	73	12	no	—
155	#117	—	73	3	no	—
156	C₂×C₂²⋊C₄	—	73	12	no	—
157	C₂×C₄²	—	73	3	no	—
158	C₂⁴	C₂²×D₇₃	1	6	yes	—
159	#41	—	2	8	no	—
160	#41	—	2	12	no	—
161	#41	—	146	24	no	—
162	#41	—	146	12	no	—
163	#41	C₂²⋊D₇₃	1	8	yes	complemented
164	C₂×Q₈	—	73	6	no	—
165	C₂×Q₈	—	73	3	no	—
166	C₂×Q₈	—	146	8	no	—
167	C₂×Q₈	—	146	12	no	—
168	C₂×Q₈	C₂²×D₇₃	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₇₃	1	1	yes	characteristic
170	C₂×D₄	—	2	24	no	—
171	C₂×D₄	—	2	12	no	—
172	C₂×D₄	—	73	12	no	—
173	C₂×D₄	—	73	6	no	—
174	C₂×D₄	—	73	1	no	—
175	C₂×D₄	C₂²×D₇₃	1	6	yes	—
176	C₂×D₄	C₂²×D₇₃	1	6	yes	—
177	C₂×D₄	C₂²×D₇₃	1	3	yes	—
178	C₂×D₄	C₂²×D₇₃	1	3	yes	—
179	C₂²×C₄	—	73	6	no	—
180	C₂²×C₄	—	73	6	no	—
181	C₂²×C₄	—	73	6	no	—
182	C₂²×C₄	C₂²×D₇₃	1	6	yes	—
183	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
184	C₄⋊C₄	—	73	12	no	—
185	C₄⋊C₄	—	73	6	no	—
186	C₂²⋊C₄	—	73	12	no	—
187	C₂²⋊C₄	—	73	24	no	—
188	C₂²⋊C₄	—	73	6	no	—
189	C₂²⋊C₄	—	73	4	no	—
190	C₄²	—	73	1	no	—
191	C₄²	—	73	3	no	—
192	C₄²	—	73	6	no	—
193	C₂³	—	2	12	no	—
194	C₂³	—	4	12	no	—
195	C₂³	—	73	2	no	—
196	C₂³	C₂³×D₇₃	1	6	yes	—
197	C₂³	C₂³×D₇₃	1	3	yes	—
198	Q₈	—	2	2	no	—
199	Q₈	—	146	12	no	—
200	Q₈	—	146	6	no	—
201	Q₈	C₂×C₂²⋊D₇₃	1	4	yes	—
202	D₄	—	2	12	no	—
203	D₄	—	2	12	no	—
204	D₄	—	2	6	no	—
205	D₄	—	292	12	no	—
206	D₄	—	146	12	no	—
207	D₄	—	146	2	no	—
208	D₄	C₂×C₂²⋊D₇₃	1	12	yes	—
209	C₂×C₄	—	2	12	no	—
210	C₂×C₄	—	292	6	no	—
211	C₂×C₄	—	73	2	no	—
212	C₂×C₄	—	73	6	no	—
213	C₂×C₄	—	73	6	no	—
214	C₂×C₄	—	146	12	no	—
215	C₂×C₄	—	146	12	no	—
216	C₂×C₄	C₂×C₂²⋊D₇₃	1	12	yes	—
217	C₂×C₄	C₂³×D₇₃	1	2	yes	—
218	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
219	C₂×C₄	C₂³×D₇₃	1	3	yes	—
220	C₂²	—	2	6	no	—
221	C₂²	—	2	6	no	—
222	C₂²	—	2	12	no	—
223	C₂²	—	8	12	no	—
224	C₂²	—	146	2	no	—
225	C₂²	—	146	4	no	—
226	C₂²	C₂²×C₂²⋊D₇₃	1	6	yes	—
227	C₂²	C₂⁴×D₇₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	146	6	no	—
230	C₄	—	146	2	no	—
231	C₄	—	146	6	no	—
232	C₄	C₂²×C₂²⋊D₇₃	1	6	yes	—
233	C₄	C₂²×C₂²⋊D₇₃	1	2	yes	—
234	C₂	—	2	6	no	—
235	C₂	—	4	6	no	—
236	C₂	—	292	2	no	—
237	C₂	C₂³×C₂²⋊D₇₃	1	1	yes	characteristic
238	C₂	#17989422	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₇₃
commutator, socle	C₂²×C₇₃	C₂⁵
Fitting	C₂×H(5,𝔽₂)×C₇₃	C₂
hypercenter	C₂×H(5,𝔽₂)	D₇₃
hypocenter	C₇₃	#3500
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×#14556638	C₂	semidirect
2. #17989244	C₂	semidirect
3. #17989231	C₂	semidirect
4. #17989158	C₂	semidirect
5. #17989154	C₂	semidirect
6. #17989133	C₂	semidirect
7. C₂×#14556578	C₂	semidirect
8. #41×Dic₇₃	C₂	semidirect
9. C₂×H(5,𝔽₂)×C₇₃	C₂	semidirect
11. C₂×#11924028	C₂²	semidirect
12. C₂×#11924027	C₂²	semidirect
13. C₂²×Dic₁₄₆	C₂²	semidirect
14. C₂×Q₈×D₇₃	C₂²	semidirect
15. #14556589	C₂²	semidirect
16. #14556587	C₂²	semidirect
17. #14556580	C₂²	semidirect
18. #14556578	C₂²	semidirect
19. #14556578	C₂²	semidirect
20. #14556578	C₂²	semidirect
21. #14556572	C₂²	semidirect
22. #14556570	C₂²	semidirect
23. #14556563	C₂²	semidirect
24. #14556562	C₂²	semidirect
25. #14556560	C₂²	semidirect
26. Q₈×Dic₇₃	C₂²	semidirect
27. D₄×Dic₇₃	C₂²	semidirect
28. C₄×C₂²⋊D₇₃	C₂²	semidirect
29. #14556550	C₂²	semidirect
30. C₂×C₂²⋊Dic₇₃	C₂²	semidirect
31. C₂×C₄×Dic₇₃	C₂²	semidirect
62. #41×C₇₃	D₄	semidirect
120. C₇₃	#3500	semidirect
122. C₂×H(5,𝔽₂)	D₇₃	semidirect
163. #41	C₂²⋊D₇₃	semidirect