Subgroups — #22626704

Order 9344.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14556679	C₂	1	1	yes	maximal, characteristic, complemented
2	D₇₃⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989633	C₂	1	2	yes	maximal, complemented
4	#17989632	C₂	1	2	yes	maximal, complemented
5	#17989631	C₂	1	2	yes	maximal, complemented
6	#17989630	C₂	1	2	yes	maximal, complemented
7	#17989602	C₂	1	1	yes	maximal, characteristic, complemented
8	#17989586	C₂	1	1	yes	maximal, characteristic, complemented
9	#17989584	C₂	1	1	yes	maximal, characteristic, complemented
10	#17989553	C₂	1	1	yes	maximal, characteristic, complemented
11	#17989386	C₂	1	1	yes	maximal, characteristic
12	#14556679	—	2	1	no	—
13	#14556679	—	2	1	no	—
14	#14556679	C₂²	1	2	yes	complemented
15	#14556679	C₂²	1	1	yes	characteristic, complemented
16	#14556679	C₂²	1	1	yes	characteristic, complemented
17	#14556679	C₂²	1	2	yes	complemented
18	#14556678	—	2	1	no	—
19	#14556678	—	2	1	no	—
20	#14556678	C₂²	1	1	yes	characteristic, complemented
21	#14556678	C₂²	1	1	yes	characteristic, complemented
22	C₂×#11924038	—	2	1	no	—
23	C₂×#11924038	C₂²	1	1	yes	characteristic
24	D₇₃⋊M₁₆	—	2	1	no	—
25	D₇₃⋊M₁₆	C₂²	1	1	yes	characteristic
26	C₂×#11924036	—	2	1	no	—
27	C₂×#11924036	—	2	1	no	—
28	C₂×#11924036	C₂²	1	1	yes	characteristic
29	#14556670	C₂²	1	1	yes	characteristic
30	#14556662	C₂²	1	2	yes	complemented
31	#14556661	C₂²	1	2	yes	complemented
32	#14556656	C₂²	1	2	yes	complemented
33	#14556656	C₂²	1	2	yes	complemented
34	#14556656	C₂²	1	2	yes	complemented
35	#14556655	C₂²	1	2	yes	complemented
36	#14556645	C₂²	1	2	yes	complemented
37	C₈×#64408	C₂²	1	2	yes	complemented
38	C₂×#11924029	C₂²	1	1	yes	characteristic
39	#41×D₇₃	C₂²	1	1	yes	characteristic
40	#14556623	C₄	1	2	yes	complemented
41	#14556622	C₄	1	1	yes	characteristic, complemented
42	#14556619	C₄	1	1	yes	characteristic, complemented
43	#14556618	C₄	1	2	yes	complemented
44	#14556616	C₄	1	1	yes	characteristic, complemented
45	#14556614	C₂²	1	2	yes	—
46	#14556612	C₂²	1	2	yes	—
47	M₁₆×D₇₃	C₂²	1	1	yes	characteristic
48	#137×C₇₃	C₄	1	1	yes	characteristic, complemented
49	#11924038	—	2	1	no	—
50	#11924038	—	2	1	no	—
51	#11924038	—	2	1	no	—
52	#11924038	—	2	2	no	—
53	#11924038	—	2	2	no	—
54	#11924038	C₂³	1	1	yes	characteristic
55	#11924038	C₂³	1	1	yes	characteristic
56	#11924038	C₂³	1	2	yes	—
57	#11924037	—	2	1	no	—
58	#11924037	—	2	1	no	—
59	#11924037	—	2	1	no	—
60	#11924036	—	2	1	no	—
61	#11924036	—	2	1	no	—
62	#11924036	—	2	1	no	—
63	#11924036	—	2	2	no	—
64	#11924036	—	2	2	no	—
65	#11924036	D₄	1	1	yes	characteristic
66	#11924036	D₄	1	1	yes	characteristic
67	#11924036	C₂³	1	1	yes	characteristic
68	#11924036	C₂³	1	1	yes	characteristic
69	C₂×#10596833	—	2	1	no	—
70	C₂×#10596833	—	2	1	no	—
71	C₂×#10596833	—	2	1	no	—
72	C₂×#10596833	D₄	1	1	yes	characteristic
73	C₂×#10596833	D₄	1	1	yes	characteristic
74	C₄⋊#64408	—	2	1	no	—
75	C₂²⋊#64408	—	2	1	no	—
76	C₄×#64408	C₂³	1	2	yes	—
77	#11924029	—	2	1	no	—
78	#11924029	—	2	1	no	—
79	#11924029	C₂×C₄	1	2	yes	—
80	#11924029	C₂³	1	2	yes	—
81	#11924029	C₂³	1	1	yes	characteristic
82	#11924028	—	2	1	no	—
83	#11924028	C₂×C₄	1	1	yes	characteristic
84	#11924027	—	2	1	no	—
85	#11924027	C₂³	1	1	yes	characteristic
86	Q₈×D₇₃	C₂×C₄	1	1	yes	characteristic
87	C₂×D₂₉₂	—	2	1	no	—
88	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
89	D₄×D₇₃	—	2	1	no	—
90	D₄×D₇₃	C₂×C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	—	2	1	no	—
92	C₂×C₄×D₇₃	—	2	1	no	—
93	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
94	#11924018	C₂×C₄	1	2	yes	—
95	#11924017	C₂×C₄	1	2	yes	—
96	#11924015	C₂×C₄	1	2	yes	—
97	#11924015	C₂×C₄	1	2	yes	—
98	D₅₈₄	C₂×C₄	1	2	yes	—
99	#11924013	C₂×C₄	1	2	yes	—
100	#11924012	C₂×C₄	1	1	yes	characteristic
101	#11924011	C₂×C₄	1	2	yes	—
102	C₈×D₇₃	C₂³	1	2	yes	—
103	#41×C₇₃	C₂×C₄	1	1	yes	characteristic
104	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
105	Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
106	SD₁₆×C₇₃	C₂×C₄	1	2	yes	—
107	M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
108	C₂²×D₇₃	—	2	1	no	—
109	C₂²×D₇₃	—	2	1	no	—
110	C₂²×D₇₃	#41	1	1	yes	characteristic
111	C₂×#64408	—	2	2	no	—
112	Q₈×C₇₃	—	2	1	no	—
113	Q₈×C₇₃	C₂²×C₄	1	2	yes	—
114	Q₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	D₄×C₇₃	—	2	1	no	—
116	D₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₂²⋊D₇₃	—	2	1	no	—
121	C₂²⋊D₇₃	—	2	1	no	—
122	C₂²⋊D₇₃	—	2	1	no	—
123	C₂×Dic₇₃	—	2	1	no	—
124	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
125	D₂₉₂	—	2	2	no	—
126	D₂₉₂	—	2	1	no	—
127	D₂₉₂	—	2	2	no	—
128	D₂₉₂	—	2	1	no	—
129	D₂₉₂	C₂²×C₄	1	2	yes	—
130	D₂₉₂	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₇₃	—	2	1	no	—
132	C₄×D₇₃	—	2	1	no	—
133	C₄×D₇₃	—	2	2	no	—
134	C₄×D₇₃	—	2	1	no	—
135	C₄×D₇₃	—	2	2	no	—
136	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
137	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
138	C₄×D₇₃	#41	1	1	yes	characteristic
139	C₄×D₇₃	C₂⁴	1	1	yes	characteristic
140	Dic₁₄₆	—	2	1	no	—
141	Dic₁₄₆	C₂²×C₄	1	1	yes	characteristic
142	#10596833	—	2	1	no	—
143	#10596833	—	2	1	no	—
144	#10596833	—	2	2	no	—
145	#10596833	C₂×D₄	1	1	yes	characteristic
146	#10596833	C₂×D₄	1	1	yes	characteristic
147	#10596833	C₂×D₄	1	1	yes	characteristic
148	#10596833	C₂×D₄	1	1	yes	characteristic
149	C₈×C₇₃	C₂²×C₄	1	2	yes	—
150	#10596831	C₂²×C₄	1	2	yes	—
151	C₂²×C₇₃	—	2	1	no	—
152	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
153	C₂×D₇₃	—	2	2	no	—
154	C₂×D₇₃	—	2	1	no	—
155	C₂×D₇₃	—	4	1	no	—
156	C₂×D₇₃	—	4	2	no	—
157	C₂×D₇₃	—	4	2	no	—
158	C₂×D₇₃	C₄×D₄	1	1	yes	characteristic
159	C₂×D₇₃	C₂×#41	1	1	yes	characteristic
160	#64408	—	4	2	no	—
161	C₄×C₇₃	—	2	1	no	—
162	C₄×C₇₃	—	2	2	no	—
163	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
164	C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
165	Dic₇₃	—	2	1	no	—
166	Dic₇₃	C₄×D₄	1	1	yes	characteristic
167	Dic₇₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₇₃	—	2	1	no	—
169	C₂×C₇₃	—	4	1	no	—
170	C₂×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
171	D₇₃	—	2	1	no	—
172	D₇₃	—	2	1	no	—
173	D₇₃	—	4	2	no	—
174	D₇₃	—	4	1	no	—
175	#2955	—	73	1	no	maximal, Sylow
176	C₇₃	#2955	1	1	yes	characteristic, complemented, Sylow
177	#575	—	73	1	no	—
178	C₂×#131	—	73	1	no	—
179	C₂×#131	—	73	1	no	—
180	#444	—	73	2	no	—
181	#444	—	73	2	no	—
182	#443	—	73	2	no	—
183	#443	—	73	2	no	—
184	#430	—	73	1	no	—
185	#421	—	73	1	no	—
186	#421	—	73	1	no	—
187	#405	—	73	1	no	—
188	C₂×#41	—	73	1	no	—
189	C₂×#41	—	73	1	no	—
190	#137	—	73	1	no	—
191	#137	—	73	2	no	—
192	#137	#64408	1	1	yes	characteristic, complemented
193	AGL(1,ℤ/8)	—	73	1	no	—
194	AGL(1,ℤ/8)	—	73	2	no	—
195	AGL(1,ℤ/8)	—	73	1	no	—
196	#135	—	73	2	no	—
197	#135	—	73	2	no	—
198	#131	—	73	1	no	—
199	#131	—	73	1	no	—
200	#131	—	73	1	no	—
201	#131	—	73	2	no	—
202	#131	—	73	2	no	—
203	#131	—	73	1	no	—
204	#131	—	146	1	no	—
205	#131	—	146	1	no	—
206	#131	—	146	1	no	—
207	#131	—	146	1	no	—
208	C₂×M₁₆	—	73	1	no	—
209	C₂×M₁₆	—	73	1	no	—
210	C₂×M₁₆	—	73	1	no	—
211	C₂×M₁₆	—	146	1	no	—
212	C₂×M₁₆	—	146	1	no	—
213	C₂²×C₈	—	73	1	no	—
214	C₂²×C₈	—	146	1	no	—
215	C₂²×C₈	—	146	1	no	—
216	#117	—	73	1	no	—
217	#108	—	73	2	no	—
218	#108	—	73	2	no	—
219	#104	—	73	2	no	—
220	#104	—	73	2	no	—
221	#104	—	73	2	no	—
222	#104	—	73	2	no	—
223	#97	—	73	2	no	—
224	C₄×C₈	—	73	2	no	—
225	#41	—	73	2	no	—
226	#41	—	73	1	no	—
227	#41	—	73	2	no	—
228	#41	—	73	1	no	—
229	#41	—	73	1	no	—
230	#41	—	146	1	no	—
231	#41	—	146	1	no	—
232	#41	—	146	1	no	—
233	#41	—	146	1	no	—
234	#41	C₂×#64408	1	1	yes	characteristic
235	C₂×Q₈	—	73	1	no	—
236	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
237	C₂×D₄	—	73	1	no	—
238	C₂×D₄	—	73	1	no	—
239	C₂×D₄	—	146	1	no	—
240	C₂×D₄	—	146	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	146	1	no	—
243	C₂²×C₄	—	146	1	no	—
244	Q₁₆	—	73	2	no	—
245	Q₁₆	C₂×#64408	1	2	yes	—
246	SD₁₆	—	73	2	no	—
247	SD₁₆	—	73	2	no	—
248	SD₁₆	—	73	2	no	—
249	SD₁₆	C₂×#64408	1	2	yes	—
250	D₈	—	73	2	no	—
251	D₈	—	73	2	no	—
252	M₁₆	—	73	1	no	—
253	M₁₆	—	73	2	no	—
254	M₁₆	—	73	1	no	—
255	M₁₆	—	73	1	no	—
256	M₁₆	—	73	2	no	—
257	M₁₆	—	146	1	no	—
258	M₁₆	—	146	1	no	—
259	M₁₆	—	146	1	no	—
260	M₁₆	—	146	2	no	—
261	M₁₆	—	146	1	no	—
262	M₁₆	—	146	2	no	—
263	M₁₆	—	146	1	no	—
264	M₁₆	—	146	1	no	—
265	M₁₆	C₂×#64408	1	1	yes	characteristic
266	C₂×C₈	—	73	1	no	—
267	C₂×C₈	—	73	1	no	—
268	C₂×C₈	—	73	1	no	—
269	C₂×C₈	—	73	1	no	—
270	C₂×C₈	—	73	1	no	—
271	C₂×C₈	—	73	1	no	—
272	C₂×C₈	—	73	2	no	—
273	C₂×C₈	—	146	1	no	—
274	C₂×C₈	—	146	1	no	—
275	C₂×C₈	—	146	1	no	—
276	C₂×C₈	—	146	2	no	—
277	C₂×C₈	—	146	1	no	—
278	C₂×C₈	—	146	2	no	—
279	C₂×C₈	—	146	1	no	—
280	C₂×C₈	—	146	1	no	—
281	C₄⋊C₄	—	146	1	no	—
282	C₂²⋊C₄	—	146	1	no	—
283	C₄²	—	73	2	no	—
284	C₂³	—	73	1	no	—
285	C₂³	—	146	1	no	—
286	C₂³	—	146	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	73	1	no	—
289	Q₈	—	146	1	no	—
290	Q₈	C₂²×#64408	1	1	yes	characteristic
291	Q₈	C₂²×#64408	1	2	yes	—
292	D₄	—	2	1	no	—
293	D₄	—	73	2	no	—
294	D₄	—	73	1	no	—
295	D₄	—	146	1	no	—
296	D₄	—	146	2	no	—
297	D₄	—	146	2	no	—
298	D₄	—	146	1	no	—
299	D₄	—	146	1	no	—
300	D₄	—	146	1	no	—
301	D₄	—	146	1	no	—
302	D₄	C₂²×#64408	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	73	1	no	—
306	C₂×C₄	—	73	1	no	—
307	C₂×C₄	—	73	1	no	—
308	C₂×C₄	—	73	1	no	—
309	C₂×C₄	—	73	1	no	—
310	C₂×C₄	—	146	1	no	—
311	C₂×C₄	—	146	2	no	—
312	C₂×C₄	—	146	2	no	—
313	C₂×C₄	—	146	1	no	—
314	C₂×C₄	—	146	2	no	—
315	C₂×C₄	—	146	1	no	—
316	C₂×C₄	—	146	1	no	—
317	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
318	C₈	—	73	1	no	—
319	C₈	—	73	1	no	—
320	C₈	—	73	1	no	—
321	C₈	—	73	1	no	—
322	C₈	—	73	2	no	—
323	C₈	—	146	1	no	—
324	C₈	—	146	1	no	—
325	C₈	—	146	2	no	—
326	C₈	C₂²×#64408	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	292	2	no	—
329	C₂²	—	292	1	no	—
330	C₂²	—	292	2	no	—
331	C₂²	—	73	1	no	—
332	C₂²	—	73	1	no	—
333	C₂²	—	146	2	no	—
334	C₂²	—	146	1	no	—
335	C₂²	D₄×#64408	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	292	2	no	—
339	C₄	—	73	1	no	—
340	C₄	—	73	1	no	—
341	C₄	—	146	1	no	—
342	C₄	D₄×#64408	1	1	yes	characteristic
343	C₄	C₂³×#64408	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	1	no	—
346	C₂	—	292	2	no	—
347	C₂	—	292	1	no	—
348	C₂	—	146	1	no	—
349	C₂	—	146	1	no	—
350	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	#137×C₇₃	C₄
socle	C₂×C₇₃	C₂×C₄×D₄
hypercenter	#137	#64408
hypocenter	C₇₃	#2955
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. C₂×#14556679	C₂	semidirect
2. D₇₃⋊#131	C₂	semidirect
3. #17989633	C₂	semidirect
4. #17989632	C₂	semidirect
5. #17989631	C₂	semidirect
6. #17989630	C₂	semidirect
7. #17989602	C₂	semidirect
8. #17989586	C₂	semidirect
9. #17989584	C₂	semidirect
10. #17989553	C₂	semidirect
14. #14556679	C₂²	semidirect
15. #14556679	C₂²	semidirect
16. #14556679	C₂²	semidirect
17. #14556679	C₂²	semidirect
20. #14556678	C₂²	semidirect
21. #14556678	C₂²	semidirect
30. #14556662	C₂²	semidirect
31. #14556661	C₂²	semidirect
32. #14556656	C₂²	semidirect
33. #14556656	C₂²	semidirect
34. #14556656	C₂²	semidirect
35. #14556655	C₂²	semidirect
36. #14556645	C₂²	semidirect
37. C₈×#64408	C₂²	semidirect
40. #14556623	C₄	semidirect
41. #14556622	C₄	semidirect
42. #14556619	C₄	semidirect
43. #14556618	C₄	semidirect
44. #14556616	C₄	semidirect
48. #137×C₇₃	C₄	semidirect
176. C₇₃	#2955	semidirect
192. #137	#64408	semidirect