Subgroups — #22618837

Order 9344.

458 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989385	C₂	1	1	yes	maximal, characteristic
2	#17989354	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989353	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988706	C₂	1	4	yes	maximal, complemented
5	#17988701	C₂	1	4	yes	maximal, complemented
6	#17988635	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14556519	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#104×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11924028	C₂²	1	1	yes	characteristic
11	#14556622	C₄	1	2	yes	complemented
12	#14556621	C₄	1	2	yes	complemented
13	#14556617	C₄	1	2	yes	complemented
14	#14556616	C₄	1	2	yes	complemented
15	#14556611	C₂²	1	4	yes	—
16	#14556599	C₂²	1	4	yes	complemented
17	#14556598	—	2	2	no	—
18	#14556598	C₂²	1	4	yes	complemented
19	#14556597	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	C₂²	1	1	yes	characteristic
22	C₂×#11924012	C₂²	1	1	yes	characteristic
23	C₂×#11924012	C₂²	1	1	yes	characteristic
24	M₁₆×D₇₃	—	2	2	no	—
25	#14556526	C₂²	1	4	yes	complemented
26	#14556519	C₂²	1	2	yes	complemented
27	#14556519	C₂²	1	2	yes	complemented
28	#14556502	C₂²	1	2	yes	complemented
29	C₄×#10596831	C₂²	1	2	yes	complemented
30	C₂×#41×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
33	#104×C₇₃	C₂²	1	2	yes	complemented
34	#104×C₇₃	C₂²	1	2	yes	complemented
35	#11924028	—	2	1	no	—
36	#11924028	—	2	1	no	—
37	#11924028	C₂×C₄	1	2	yes	—
38	#11924028	C₂³	1	2	yes	—
39	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
40	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₇₃	—	2	1	no	—
42	C₂×C₄×D₇₃	—	2	1	no	—
43	#11924018	C₂×C₄	1	4	yes	complemented
44	#11924016	C₂×C₄	1	4	yes	complemented
45	#11924015	C₂×C₄	1	4	yes	complemented
46	#11924013	C₂×C₄	1	4	yes	complemented
47	#11924012	—	2	1	no	—
48	#11924012	—	2	4	no	—
49	#11924012	C₂×C₄	1	2	yes	—
50	#11924012	C₂×C₄	1	2	yes	—
51	#11924012	D₄	1	2	yes	—
52	#11924012	D₄	1	2	yes	—
53	#11924012	C₂³	1	2	yes	—
54	#11924011	—	2	4	no	—
55	C₂×#10596831	—	2	1	no	—
56	C₂×#10596831	—	2	4	no	—
57	C₂×#10596831	C₂³	1	2	yes	—
58	C₂×#10596831	C₂³	1	2	yes	—
59	C₈×D₇₃	—	2	4	no	—
60	#41×C₇₃	—	2	1	no	—
61	#41×C₇₃	—	2	1	no	—
62	#41×C₇₃	C₂×C₄	1	2	yes	—
63	#41×C₇₃	C₂³	1	2	yes	—
64	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	C₂×D₄×C₇₃	—	2	1	no	—
66	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	—	2	1	no	—
68	C₂²×C₄×C₇₃	—	2	1	no	—
69	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
70	M₁₆×C₇₃	—	2	1	no	—
71	M₁₆×C₇₃	C₂³	1	2	yes	—
72	C₂×C₈×C₇₃	—	2	1	no	—
73	C₄²×C₇₃	—	2	1	no	—
74	C₄²×C₇₃	C₂³	1	1	yes	characteristic
75	C₄²×C₇₃	C₂³	1	1	yes	characteristic
76	C₂³×C₇₃	—	2	1	no	—
77	C₂³×C₇₃	#41	1	1	yes	characteristic
78	C₂²×D₇₃	—	2	1	no	—
79	Q₈×C₇₃	—	2	1	no	—
80	Q₈×C₇₃	C₂²×C₄	1	2	yes	—
81	D₄×C₇₃	—	2	2	no	—
82	D₄×C₇₃	—	2	2	no	—
83	D₄×C₇₃	—	2	1	no	—
84	D₄×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₇₃	—	2	1	no	—
86	C₂×C₄×C₇₃	—	2	1	no	—
87	C₂×C₄×C₇₃	—	2	2	no	—
88	C₂×C₄×C₇₃	—	2	2	no	—
89	C₂×C₄×C₇₃	—	2	1	no	—
90	C₂×C₄×C₇₃	—	4	2	no	—
91	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
96	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₇₃	—	2	2	no	—
98	C₂²⋊D₇₃	—	2	2	no	—
99	C₂×Dic₇₃	—	2	1	no	—
100	D₂₉₂	—	2	1	no	—
101	D₂₉₂	C₂²×C₄	1	2	yes	—
102	C₄×D₇₃	—	2	2	no	—
103	C₄×D₇₃	—	2	2	no	—
104	Dic₁₄₆	—	2	1	no	—
105	Dic₁₄₆	C₂²×C₄	1	2	yes	—
106	C₈×C₇₃	—	2	2	no	—
107	#10596831	—	2	2	no	—
108	#10596831	C₂²×C₄	1	4	yes	—
109	#10596831	C₂×D₄	1	4	yes	—
110	C₂²×C₇₃	—	2	2	no	—
111	C₂²×C₇₃	—	4	1	no	—
112	C₂²×C₇₃	—	8	1	no	—
113	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
114	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
116	C₂×D₇₃	—	2	2	no	—
117	C₂×D₇₃	—	8	1	no	—
118	C₄×C₇₃	—	2	2	no	—
119	C₄×C₇₃	—	2	2	no	—
120	C₄×C₇₃	—	4	1	no	—
121	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
122	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	Dic₇₃	—	2	2	no	—
126	C₂×C₇₃	—	2	1	no	—
127	C₂×C₇₃	—	2	1	no	—
128	C₂×C₇₃	—	2	1	no	—
129	C₂×C₇₃	—	4	2	no	—
130	C₂×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
131	D₇₃	—	4	2	no	—
132	#2956	—	73	1	no	maximal, Sylow
133	C₇₃	#2956	1	1	yes	characteristic, complemented, Sylow
134	#575	—	73	1	no	—
135	#568	—	73	1	no	—
136	#568	—	73	1	no	—
137	#444	—	73	4	no	—
138	#443	—	73	4	no	—
139	#430	—	73	1	no	—
140	C₂×#104	—	73	1	no	—
141	C₂×#104	D₇₃	1	1	yes	characteristic, complemented
142	C₄×M₁₆	—	73	1	no	—
143	C₂×#41	—	73	1	no	—
144	C₂×#41	C₂×D₇₃	1	1	yes	characteristic
145	#137	—	73	2	no	—
146	#137	—	73	2	no	—
147	AGL(1,ℤ/8)	—	73	2	no	—
148	AGL(1,ℤ/8)	—	73	2	no	—
149	#135	—	73	4	no	—
150	#131	—	73	4	no	—
151	#131	—	73	4	no	—
152	#131	—	146	2	no	—
153	#131	—	146	2	no	—
154	C₂×M₁₆	—	73	1	no	—
155	C₂×M₁₆	—	73	1	no	—
156	C₂×M₁₆	—	73	1	no	—
157	C₂×M₁₆	—	146	2	no	—
158	C₂×M₁₆	—	146	2	no	—
159	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
160	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
161	#108	—	73	4	no	—
162	#104	—	73	2	no	—
163	#104	—	73	2	no	—
164	#104	C₂×D₇₃	1	2	yes	complemented
165	#104	C₂×D₇₃	1	2	yes	complemented
166	#97	—	73	2	no	—
167	C₄×C₈	—	73	2	no	—
168	#41	—	2	1	no	—
169	#41	—	2	1	no	—
170	#41	—	73	2	no	—
171	#41	—	73	2	no	—
172	#41	—	146	1	no	—
173	#41	—	146	1	no	—
174	#41	C₄×D₇₃	1	2	yes	—
175	#41	C₂²×D₇₃	1	2	yes	—
176	C₂×Q₈	—	73	1	no	—
177	C₂×Q₈	C₄×D₇₃	1	1	yes	characteristic
178	C₂×D₄	—	2	1	no	—
179	C₂×D₄	—	73	1	no	—
180	C₂×D₄	—	146	1	no	—
181	C₂×D₄	C₄×D₇₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	146	1	no	—
185	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
186	Q₁₆	—	73	4	no	—
187	SD₁₆	—	73	4	no	—
188	SD₁₆	—	73	4	no	—
189	D₈	—	73	4	no	—
190	M₁₆	—	2	1	no	—
191	M₁₆	—	73	2	no	—
192	M₁₆	—	73	2	no	—
193	M₁₆	—	73	2	no	—
194	M₁₆	—	73	2	no	—
195	M₁₆	—	73	2	no	—
196	M₁₆	—	146	4	no	—
197	M₁₆	—	146	4	no	—
198	M₁₆	—	146	1	no	—
199	M₁₆	C₂²×D₇₃	1	2	yes	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	73	2	no	—
202	C₂×C₈	—	73	2	no	—
203	C₂×C₈	—	146	4	no	—
204	C₂×C₈	—	146	4	no	—
205	C₂×C₈	—	146	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	C₂²×D₇₃	1	1	yes	characteristic
208	C₄²	C₂²×D₇₃	1	1	yes	characteristic
209	C₂³	—	2	1	no	—
210	C₂³	—	146	1	no	—
211	C₂³	#11924027	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	73	2	no	—
214	Q₈	—	146	1	no	—
215	Q₈	C₂×C₄×D₇₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	73	2	no	—
220	D₄	—	146	2	no	—
221	D₄	—	146	2	no	—
222	D₄	—	146	1	no	—
223	D₄	C₂×C₄×D₇₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	146	1	no	—
231	C₂×C₄	—	146	2	no	—
232	C₂×C₄	—	146	2	no	—
233	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
235	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
236	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
237	C₂×C₄	#11924027	1	1	yes	characteristic
238	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
239	C₈	—	2	2	no	—
240	C₈	—	73	4	no	—
241	C₈	—	73	4	no	—
242	C₈	—	146	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	584	1	no	—
247	C₂²	—	146	2	no	—
248	C₂²	#14556551	1	1	yes	characteristic
249	C₂²	#14556551	1	1	yes	characteristic
250	C₂²	C₂×#11924027	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	146	2	no	—
255	C₄	#14556551	1	1	yes	characteristic
256	C₄	#14556551	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
258	C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	4	2	no	—
263	C₂	—	292	2	no	—
264	C₂	C₂×#14556551	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	C₂×#104×C₇₃	C₂
socle	C₂×C₇₃	C₂×C₄×D₄
hypercenter	C₂×#104	D₇₃
hypocenter	C₇₃	#2956
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. #17989354	C₂	semidirect
3. #17989353	C₂	semidirect
4. #17988706	C₂	semidirect
5. #17988701	C₂	semidirect
6. #17988635	C₂	semidirect
7. C₂×#14556519	C₂	semidirect
8. C₄×#11924012	C₂	semidirect
9. C₂×#104×C₇₃	C₂	semidirect
11. #14556622	C₄	semidirect
12. #14556621	C₄	semidirect
13. #14556617	C₄	semidirect
14. #14556616	C₄	semidirect
16. #14556599	C₂²	semidirect
18. #14556598	C₂²	semidirect
25. #14556526	C₂²	semidirect
26. #14556519	C₂²	semidirect
27. #14556519	C₂²	semidirect
28. #14556502	C₂²	semidirect
29. C₄×#10596831	C₂²	semidirect
33. #104×C₇₃	C₂²	semidirect
34. #104×C₇₃	C₂²	semidirect
43. #11924018	C₂×C₄	semidirect
44. #11924016	C₂×C₄	semidirect
45. #11924015	C₂×C₄	semidirect
46. #11924013	C₂×C₄	semidirect
133. C₇₃	#2956	semidirect
141. C₂×#104	D₇₃	semidirect
164. #104	C₂×D₇₃	semidirect
165. #104	C₂×D₇₃	semidirect