Subgroups — #22626705

Order 9344.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989666	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989665	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989634	C₂	1	2	yes	maximal, complemented
4	#17989632	C₂	1	2	yes	maximal, complemented
5	#17989630	C₂	1	2	yes	maximal, complemented
6	#17989629	C₂	1	2	yes	maximal, complemented
7	#17989602	C₂	1	1	yes	maximal, characteristic, complemented
8	D₇₃⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556655	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#64408	C₂	1	1	yes	maximal, characteristic, complemented
11	#17989383	C₂	1	1	yes	maximal, characteristic
12	#14556679	C₂²	1	2	yes	complemented
13	#14556678	—	2	1	no	—
14	#14556678	—	2	1	no	—
15	#14556678	—	2	2	no	—
16	#14556678	C₂²	1	2	yes	complemented
17	#14556678	C₂²	1	2	yes	complemented
18	#14556678	C₂²	1	2	yes	complemented
19	C₂×#11924038	—	2	1	no	—
20	C₂×#11924038	C₂²	1	1	yes	characteristic
21	D₇₃⋊M₁₆	—	2	1	no	—
22	D₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
23	D₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
24	C₂×#11924037	—	2	2	no	—
25	C₂×C₄×#64408	C₂²	1	1	yes	characteristic
26	#14556662	C₂²	1	2	yes	complemented
27	#14556661	C₂²	1	2	yes	complemented
28	#14556656	C₂²	1	1	yes	characteristic, complemented
29	#14556656	C₂²	1	1	yes	characteristic, complemented
30	#14556655	C₂²	1	2	yes	complemented
31	#14556655	C₂²	1	1	yes	characteristic, complemented
32	#14556655	C₂²	1	2	yes	complemented
33	#14556655	C₂²	1	1	yes	characteristic, complemented
34	#14556645	C₂²	1	2	yes	complemented
35	C₈×#64408	C₂²	1	2	yes	complemented
36	C₂×#11924027	C₂²	1	1	yes	characteristic
37	#41×D₇₃	C₂²	1	1	yes	characteristic
38	#14556624	C₄	1	1	yes	characteristic, complemented
39	#14556621	C₄	1	1	yes	characteristic, complemented
40	#14556620	C₄	1	2	yes	complemented
41	#14556617	C₄	1	1	yes	characteristic, complemented
42	#14556615	C₄	1	2	yes	complemented
43	#14556612	C₂²	1	2	yes	—
44	#14556610	C₂²	1	2	yes	—
45	M₁₆×D₇₃	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₇₃	C₄	1	1	yes	characteristic, complemented
47	C₂²×#64408	—	2	1	no	—
48	#11924038	—	2	1	no	—
49	#11924038	—	2	2	no	—
50	#11924038	D₄	1	1	yes	characteristic
51	#11924038	D₄	1	1	yes	characteristic
52	#11924038	C₂³	1	1	yes	characteristic
53	#11924038	C₂³	1	1	yes	characteristic
54	#11924038	C₂³	1	2	yes	—
55	#11924037	—	2	1	no	—
56	#11924037	—	2	2	no	—
57	#11924037	—	2	2	no	—
58	#11924037	—	2	2	no	—
59	#11924037	D₄	1	2	yes	—
60	#11924036	—	2	1	no	—
61	#11924036	—	2	2	no	—
62	#11924036	C₂³	1	2	yes	—
63	C₂×#10596833	—	2	1	no	—
64	C₂×#10596833	—	2	2	no	—
65	C₂×#10596833	—	2	2	no	—
66	C₂×#10596833	—	2	2	no	—
67	C₄×#64408	—	2	1	no	—
68	C₄×#64408	C₂³	1	1	yes	characteristic
69	C₄×#64408	C₂³	1	1	yes	characteristic
70	#11924029	C₂³	1	1	yes	characteristic
71	#11924028	—	2	1	no	—
72	#11924028	C₂×C₄	1	1	yes	characteristic
73	#11924027	—	2	1	no	—
74	#11924027	—	2	1	no	—
75	#11924027	—	2	1	no	—
76	#11924027	C₂×C₄	1	2	yes	—
77	#11924027	C₂³	1	1	yes	characteristic
78	#11924027	C₂³	1	2	yes	—
79	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
80	Q₈×D₇₃	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₇₃	—	2	1	no	—
82	D₄×D₇₃	—	2	1	no	—
83	D₄×D₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×Dic₇₃	—	2	1	no	—
85	C₂×C₄×D₇₃	—	2	1	no	—
86	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
87	Dic₂₉₂	C₂×C₄	1	2	yes	complemented
88	#11924018	C₂×C₄	1	2	yes	complemented
89	#11924017	C₂×C₄	1	2	yes	complemented
90	#11924016	C₂×C₄	1	2	yes	complemented
91	#11924016	C₂×C₄	1	2	yes	complemented
92	#11924013	C₂×C₄	1	2	yes	complemented
93	#11924012	C₂×C₄	1	1	yes	characteristic
94	#11924011	C₂×C₄	1	2	yes	—
95	C₈×D₇₃	C₂³	1	2	yes	—
96	#41×C₇₃	C₂×C₄	1	1	yes	characteristic
97	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
98	SD₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
99	D₈×C₇₃	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
101	C₂³×C₇₃	—	2	1	no	—
102	C₂²×D₇₃	—	2	1	no	—
103	C₂²×D₇₃	#41	1	1	yes	characteristic
104	C₂×#64408	—	2	1	no	—
105	C₂×#64408	—	2	1	no	—
106	C₂×#64408	—	4	2	no	—
107	Q₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	D₄×C₇₃	—	2	1	no	—
109	D₄×C₇₃	—	2	1	no	—
110	D₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	D₄×C₇₃	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²⋊D₇₃	—	2	1	no	—
115	C₂²⋊D₇₃	—	2	2	no	—
116	C₂²⋊D₇₃	—	2	1	no	—
117	C₂²⋊D₇₃	—	2	2	no	—
118	C₂²⋊D₇₃	—	2	1	no	—
119	C₂×Dic₇₃	—	2	1	no	—
120	C₂×Dic₇₃	—	2	2	no	—
121	C₂×Dic₇₃	—	2	1	no	—
122	C₂×Dic₇₃	—	2	2	no	—
123	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
124	D₂₉₂	—	2	1	no	—
125	D₂₉₂	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₇₃	—	2	1	no	—
127	C₄×D₇₃	—	2	1	no	—
128	C₄×D₇₃	—	2	1	no	—
129	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
131	C₄×D₇₃	#41	1	1	yes	characteristic
132	C₄×D₇₃	C₂⁴	1	1	yes	characteristic
133	Dic₁₄₆	—	2	1	no	—
134	Dic₁₄₆	—	2	1	no	—
135	Dic₁₄₆	C₂²×C₄	1	1	yes	characteristic
136	Dic₁₄₆	C₂²×C₄	1	2	yes	—
137	#10596833	—	2	1	no	—
138	#10596833	—	2	1	no	—
139	#10596833	—	2	2	no	—
140	#10596833	C₂×D₄	1	2	yes	—
141	#10596833	C₂×D₄	1	2	yes	—
142	C₈×C₇₃	C₂²×C₄	1	2	yes	—
143	#10596831	C₂²×C₄	1	2	yes	—
144	C₂²×C₇₃	—	2	2	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	—	8	1	no	—
147	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
148	C₂×D₇₃	—	2	1	no	—
149	C₂×D₇₃	—	4	1	no	—
150	C₂×D₇₃	—	8	1	no	—
151	C₂×D₇₃	C₄×D₄	1	1	yes	characteristic
152	C₂×D₇₃	C₂×#41	1	1	yes	characteristic
153	#64408	—	2	2	no	—
154	#64408	—	4	1	no	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
157	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
158	Dic₇₃	—	2	1	no	—
159	Dic₇₃	—	2	2	no	—
160	Dic₇₃	C₄×D₄	1	1	yes	characteristic
161	Dic₇₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₇₃	—	2	1	no	—
163	C₂×C₇₃	—	4	1	no	—
164	C₂×C₇₃	—	4	2	no	—
165	C₂×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
166	D₇₃	—	2	1	no	—
167	D₇₃	—	2	1	no	—
168	D₇₃	—	4	1	no	—
169	#2956	—	73	1	no	maximal, Sylow
170	C₇₃	#2956	1	1	yes	characteristic, complemented, Sylow
171	#575	—	73	1	no	—
172	#568	—	73	1	no	—
173	#568	—	73	1	no	—
174	#444	—	73	2	no	—
175	#444	—	73	2	no	—
176	#443	—	73	2	no	—
177	#443	—	73	2	no	—
178	#430	—	73	1	no	—
179	C₂×#104	—	73	1	no	—
180	C₂×#104	—	73	1	no	—
181	C₄×M₁₆	—	73	1	no	—
182	C₂×#41	—	73	1	no	—
183	C₂×#41	—	73	1	no	—
184	#137	—	73	1	no	—
185	#137	—	73	2	no	—
186	#137	—	73	1	no	—
187	AGL(1,ℤ/8)	—	73	1	no	—
188	AGL(1,ℤ/8)	—	73	2	no	—
189	AGL(1,ℤ/8)	#64408	1	1	yes	characteristic, complemented
190	#135	—	73	2	no	—
191	#135	—	73	2	no	—
192	#131	—	73	2	no	—
193	#131	—	73	2	no	—
194	#131	—	73	2	no	—
195	#131	—	73	2	no	—
196	#131	—	146	1	no	—
197	#131	—	146	1	no	—
198	#131	—	146	2	no	—
199	C₂×M₁₆	—	73	1	no	—
200	C₂×M₁₆	—	73	1	no	—
201	C₂×M₁₆	—	73	1	no	—
202	C₂×M₁₆	—	73	1	no	—
203	C₂×M₁₆	—	146	2	no	—
204	C₂×M₁₆	—	146	1	no	—
205	C₂×M₁₆	—	146	1	no	—
206	C₂×C₄²	—	73	1	no	—
207	#108	—	73	2	no	—
208	#108	—	73	2	no	—
209	#104	—	73	2	no	—
210	#104	—	73	1	no	—
211	#104	—	73	1	no	—
212	#104	—	73	2	no	—
213	#104	—	73	1	no	—
214	#104	—	73	1	no	—
215	#97	—	73	2	no	—
216	C₄×C₈	—	73	2	no	—
217	#41	—	73	1	no	—
218	#41	—	73	2	no	—
219	#41	—	73	2	no	—
220	#41	—	73	1	no	—
221	#41	—	73	1	no	—
222	#41	—	146	1	no	—
223	#41	—	146	1	no	—
224	#41	—	146	1	no	—
225	#41	—	146	1	no	—
226	#41	C₂×#64408	1	1	yes	characteristic
227	C₂×Q₈	—	73	1	no	—
228	C₂×Q₈	—	73	1	no	—
229	C₂×D₄	—	73	1	no	—
230	C₂×D₄	—	146	1	no	—
231	C₂×D₄	—	146	1	no	—
232	C₂×D₄	C₂×#64408	1	1	yes	characteristic
233	C₂²×C₄	—	73	1	no	—
234	C₂²×C₄	—	146	1	no	—
235	C₂²×C₄	—	146	1	no	—
236	C₂²×C₄	—	146	1	no	—
237	Q₁₆	—	73	2	no	—
238	Q₁₆	—	73	2	no	—
239	SD₁₆	—	73	2	no	—
240	SD₁₆	—	73	2	no	—
241	SD₁₆	—	73	2	no	—
242	SD₁₆	C₂×#64408	1	2	yes	complemented
243	D₈	—	73	2	no	—
244	D₈	C₂×#64408	1	2	yes	complemented
245	M₁₆	—	73	2	no	—
246	M₁₆	—	73	1	no	—
247	M₁₆	—	73	1	no	—
248	M₁₆	—	73	1	no	—
249	M₁₆	—	73	2	no	—
250	M₁₆	—	73	1	no	—
251	M₁₆	—	73	1	no	—
252	M₁₆	—	73	2	no	—
253	M₁₆	—	146	2	no	—
254	M₁₆	—	146	2	no	—
255	M₁₆	—	146	2	no	—
256	M₁₆	—	146	2	no	—
257	M₁₆	—	146	1	no	—
258	M₁₆	—	146	1	no	—
259	M₁₆	C₂×#64408	1	1	yes	characteristic
260	C₂×C₈	—	73	2	no	—
261	C₂×C₈	—	73	2	no	—
262	C₂×C₈	—	146	2	no	—
263	C₂×C₈	—	146	2	no	—
264	C₂×C₈	—	146	2	no	—
265	C₂×C₈	—	146	2	no	—
266	C₂×C₈	—	146	1	no	—
267	C₂×C₈	—	146	1	no	—
268	C₄²	—	73	1	no	—
269	C₄²	—	73	1	no	—
270	C₄²	—	146	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	73	1	no	—
273	C₂³	—	146	1	no	—
274	Q₈	—	73	1	no	—
275	Q₈	—	73	2	no	—
276	Q₈	—	146	1	no	—
277	Q₈	—	146	1	no	—
278	Q₈	C₂²×#64408	1	1	yes	characteristic
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	73	1	no	—
282	D₄	—	146	1	no	—
283	D₄	—	146	1	no	—
284	D₄	—	146	1	no	—
285	D₄	—	146	2	no	—
286	D₄	—	146	2	no	—
287	D₄	—	146	1	no	—
288	D₄	C₂²×#64408	1	2	yes	—
289	D₄	C₂²×#64408	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	292	2	no	—
292	C₂×C₄	—	73	1	no	—
293	C₂×C₄	—	73	1	no	—
294	C₂×C₄	—	73	1	no	—
295	C₂×C₄	—	73	1	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	146	1	no	—
298	C₂×C₄	—	146	1	no	—
299	C₂×C₄	—	146	1	no	—
300	C₂×C₄	—	146	1	no	—
301	C₂×C₄	—	146	2	no	—
302	C₂×C₄	—	146	2	no	—
303	C₂×C₄	—	146	1	no	—
304	C₂×C₄	—	146	1	no	—
305	C₂×C₄	—	146	1	no	—
306	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
307	C₈	—	73	2	no	—
308	C₈	—	73	2	no	—
309	C₈	—	73	2	no	—
310	C₈	—	146	1	no	—
311	C₈	—	146	1	no	—
312	C₈	—	146	2	no	—
313	C₈	C₂²×#64408	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	292	1	no	—
318	C₂²	—	584	1	no	—
319	C₂²	—	73	1	no	—
320	C₂²	—	73	1	no	—
321	C₂²	—	146	1	no	—
322	C₂²	D₄×#64408	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	292	1	no	—
325	C₄	—	73	1	no	—
326	C₄	—	73	1	no	—
327	C₄	—	146	2	no	—
328	C₄	—	146	1	no	—
329	C₄	—	146	2	no	—
330	C₄	D₄×#64408	1	1	yes	characteristic
331	C₄	C₂³×#64408	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	1	no	—
335	C₂	—	292	1	no	—
336	C₂	—	146	1	no	—
337	C₂	—	146	1	no	—
338	C₂	C₂×D₄×#64408	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#64408
commutator	C₄×C₇₃	C₂³×C₄
Frattini	C₄	C₂³×#64408
Fitting	AGL(1,ℤ/8)×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₄
hypercenter	AGL(1,ℤ/8)	#64408
hypocenter	C₇₃	#2956
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #17989666	C₂	semidirect
2. #17989665	C₂	semidirect
3. #17989634	C₂	semidirect
4. #17989632	C₂	semidirect
5. #17989630	C₂	semidirect
6. #17989629	C₂	semidirect
7. #17989602	C₂	semidirect
8. D₇₃⋊#104	C₂	semidirect
9. C₂×#14556655	C₂	semidirect
10. M₁₆×#64408	C₂	semidirect
12. #14556679	C₂²	semidirect
16. #14556678	C₂²	semidirect
17. #14556678	C₂²	semidirect
18. #14556678	C₂²	semidirect
26. #14556662	C₂²	semidirect
27. #14556661	C₂²	semidirect
28. #14556656	C₂²	semidirect
29. #14556656	C₂²	semidirect
30. #14556655	C₂²	semidirect
31. #14556655	C₂²	semidirect
32. #14556655	C₂²	semidirect
33. #14556655	C₂²	semidirect
34. #14556645	C₂²	semidirect
35. C₈×#64408	C₂²	semidirect
38. #14556624	C₄	semidirect
39. #14556621	C₄	semidirect
40. #14556620	C₄	semidirect
41. #14556617	C₄	semidirect
42. #14556615	C₄	semidirect
46. AGL(1,ℤ/8)×C₇₃	C₄	semidirect
87. Dic₂₉₂	C₂×C₄	semidirect
88. #11924018	C₂×C₄	semidirect
89. #11924017	C₂×C₄	semidirect
90. #11924016	C₂×C₄	semidirect
91. #11924016	C₂×C₄	semidirect
92. #11924013	C₂×C₄	semidirect
98. SD₁₆×C₇₃	C₂×C₄	semidirect
99. D₈×C₇₃	C₂×C₄	semidirect
170. C₇₃	#2956	semidirect
189. AGL(1,ℤ/8)	#64408	semidirect
242. SD₁₆	C₂×#64408	semidirect
244. D₈	C₂×#64408	semidirect