Subgroups — #22626788

Order 9344.

464 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17989620	C₂	1	2	yes	maximal, complemented
2	#17989618	C₂	1	2	yes	maximal, complemented
3	#17989617	C₂	1	2	yes	maximal, complemented
4	#17989614	C₂	1	2	yes	maximal, complemented
5	#17989610	C₂	1	1	yes	maximal, characteristic, complemented
6	#17989605	C₂	1	1	yes	maximal, characteristic, complemented
7	#17989559	C₂	1	1	yes	maximal, characteristic, complemented
8	#17989558	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556648	C₂	1	1	yes	maximal, characteristic, complemented
10	#17989555	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₄×D₇₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#11924038	C₂²	1	2	yes	complemented
13	C₂×#11924037	C₂²	1	2	yes	complemented
14	C₂×#11924036	C₂²	1	2	yes	complemented
15	C₂²×#10596833	C₂²	1	2	yes	complemented
16	#14556658	C₂²	1	2	yes	complemented
17	#14556658	C₂²	1	2	yes	complemented
18	#14556648	C₂²	1	1	yes	characteristic
19	#14556648	C₂²	1	1	yes	characteristic
20	#14556648	C₂²	1	1	yes	characteristic
21	#14556648	C₂²	1	1	yes	characteristic
22	#14556648	C₂²	1	2	yes	complemented
23	#14556648	C₂²	1	2	yes	complemented
24	C₂²⋊#10596833	C₂²	1	2	yes	complemented
25	C₂²⋊#10596833	C₂²	1	2	yes	complemented
26	#14556646	C₂²	1	2	yes	complemented
27	C₄×#10596833	C₂²	1	2	yes	complemented
28	C₂×D₄×D₇₃	C₄	1	1	yes	characteristic
29	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
30	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
31	C₄×D₂₉₂	C₄	1	1	yes	characteristic
32	D₄×Dic₇₃	C₂²	1	1	yes	characteristic
33	#14556553	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
35	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
36	#14556551	C₂²	1	1	yes	characteristic
37	#14556551	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₇₃	C₄	1	1	yes	characteristic
39	C₂²⋊C₄×D₇₃	C₄	1	1	yes	characteristic
40	C₂²⋊C₄×D₇₃	C₄	1	1	yes	characteristic
41	C₄²×D₇₃	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₇₃	C₄	1	1	yes	characteristic
43	#11924038	—	4	2	no	—
44	#11924037	—	4	2	no	—
45	#11924036	—	4	2	no	—
46	C₂×#10596833	—	4	2	no	—
47	C₂×#10596833	C₂³	1	2	yes	—
48	C₂×#10596833	C₂³	1	2	yes	—
49	C₂×#10596833	C₂³	1	2	yes	—
50	C₂×#10596833	C₂³	1	2	yes	—
51	C₂³×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×D₂₉₂	C₂×C₄	1	1	yes	characteristic
54	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
56	D₄×D₇₃	—	2	4	no	—
57	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
58	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₇₃	—	4	2	no	—
60	C₂×C₄×D₇₃	—	4	2	no	—
61	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
71	#11924007	C₂×C₄	1	1	yes	characteristic
72	#11924007	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
75	#11924005	C₂×C₄	1	1	yes	characteristic
76	#11924005	C₂×C₄	1	1	yes	characteristic
77	#11924005	C₂×C₄	1	1	yes	characteristic
78	#11924005	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
81	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
82	C₂×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
88	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
89	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×D₇₃	—	2	2	no	—
92	C₂²×D₇₃	—	2	2	no	—
93	C₂²×D₇₃	—	2	2	no	—
94	C₂²×D₇₃	—	2	2	no	—
95	C₂²×D₇₃	—	4	1	no	—
96	C₂²×D₇₃	—	4	1	no	—
97	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
100	D₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	4	1	no	—
102	C₂×C₄×C₇₃	—	4	1	no	—
103	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²⋊D₇₃	—	2	2	no	—
109	C₂²⋊D₇₃	M₁₆	1	4	yes	complemented
110	C₂×Dic₇₃	—	2	2	no	—
111	C₂×Dic₇₃	—	2	2	no	—
112	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
117	D₂₉₂	—	2	2	no	—
118	C₄×D₇₃	—	2	2	no	—
119	C₄×D₇₃	—	2	2	no	—
120	C₄×D₇₃	—	2	2	no	—
121	C₄×D₇₃	—	2	2	no	—
122	C₄×D₇₃	—	2	2	no	—
123	C₄×D₇₃	—	4	1	no	—
124	C₄×D₇₃	—	4	1	no	—
125	C₄×D₇₃	—	4	1	no	—
126	C₄×D₇₃	—	4	1	no	—
127	#10596833	—	2	2	no	—
128	#10596833	—	2	2	no	—
129	#10596833	—	2	2	no	—
130	#10596833	—	2	2	no	—
131	C₂²×C₇₃	—	2	1	no	—
132	C₂²×C₇₃	—	2	1	no	—
133	C₂²×C₇₃	—	2	1	no	—
134	C₂²×C₇₃	—	2	1	no	—
135	C₂²×C₇₃	—	2	1	no	—
136	C₂²×C₇₃	C₂×M₁₆	1	2	yes	—
137	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×D₇₃	—	2	1	no	—
139	C₂×D₇₃	—	2	1	no	—
140	C₂×D₇₃	—	2	1	no	—
141	C₂×D₇₃	—	2	1	no	—
142	C₂×D₇₃	—	2	1	no	—
143	C₂×D₇₃	—	2	1	no	—
144	C₂×D₇₃	—	2	1	no	—
145	C₂×D₇₃	—	4	4	no	—
146	C₂×D₇₃	—	4	4	no	—
147	C₂×D₇₃	C₂×M₁₆	1	2	yes	—
148	C₂×D₇₃	#131	1	2	yes	—
149	C₄×C₇₃	—	2	1	no	—
150	C₄×C₇₃	—	2	1	no	—
151	C₄×C₇₃	—	2	1	no	—
152	C₄×C₇₃	—	2	1	no	—
153	C₄×C₇₃	—	2	1	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	2	1	no	—
156	Dic₇₃	—	2	1	no	—
157	Dic₇₃	C₂×M₁₆	1	2	yes	—
158	Dic₇₃	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₇₃	—	2	2	no	—
160	C₂×C₇₃	—	4	1	no	—
161	C₂×C₇₃	#518	1	1	yes	characteristic
162	C₂×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
163	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
164	D₇₃	—	2	2	no	—
165	D₇₃	—	2	2	no	—
166	D₇₃	—	4	1	no	—
167	#2974	—	73	1	no	maximal, Sylow
168	C₇₃	#2974	1	1	yes	characteristic, complemented, Sylow
169	C₂×C₄×D₄	—	73	1	no	—
170	#436	—	73	2	no	—
171	#435	—	73	2	no	—
172	#435	—	73	2	no	—
173	C₈×D₄	—	73	2	no	—
174	#432	—	73	1	no	—
175	#431	—	73	1	no	—
176	#407	—	73	1	no	—
177	#407	—	73	1	no	—
178	C₂×C₂²⋊C₈	—	73	1	no	—
179	C₂×C₂²⋊C₈	—	73	1	no	—
180	C₂²×D₄	—	73	1	no	—
181	C₂³×C₄	—	73	1	no	—
182	C₂³×C₄	—	73	1	no	—
183	C₂×M₁₆	—	73	2	no	—
184	C₂×M₁₆	—	73	2	no	—
185	C₂²×C₈	—	73	2	no	—
186	C₂²×C₈	—	73	2	no	—
187	C₄×D₄	—	73	1	no	—
188	C₄×D₄	—	73	1	no	—
189	C₄×D₄	—	73	1	no	—
190	C₄×D₄	—	73	1	no	—
191	C₄×D₄	—	73	1	no	—
192	C₄×D₄	—	73	1	no	—
193	C₄×D₄	—	73	1	no	—
194	C₄×D₄	#64408	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	73	1	no	—
196	C₂×C₂²⋊C₄	—	73	1	no	—
197	C₂×C₂²⋊C₄	—	73	1	no	—
198	C₂×C₄²	—	73	1	no	—
199	C₄⋊C₈	—	73	2	no	—
200	C₄⋊C₈	—	73	2	no	—
201	C₂²⋊C₈	—	73	1	no	—
202	C₂²⋊C₈	—	73	1	no	—
203	C₂²⋊C₈	—	73	1	no	—
204	C₂²⋊C₈	—	73	2	no	—
205	C₂²⋊C₈	—	73	2	no	—
206	C₂²⋊C₈	—	73	1	no	—
207	C₂²⋊C₈	—	73	2	no	—
208	C₂²⋊C₈	—	73	2	no	—
209	#97	—	73	2	no	—
210	C₄×C₈	—	73	2	no	—
211	C₂⁴	—	73	1	no	—
212	C₂⁴	—	73	1	no	—
213	C₂×D₄	—	73	1	no	—
214	C₂×D₄	—	73	1	no	—
215	C₂×D₄	—	73	1	no	—
216	C₂×D₄	—	146	4	no	—
217	C₂×D₄	C₂×#64408	1	1	yes	characteristic
218	C₂²×C₄	—	292	2	no	—
219	C₂²×C₄	—	292	2	no	—
220	C₂²×C₄	—	73	1	no	—
221	C₂²×C₄	—	73	1	no	—
222	C₂²×C₄	—	73	1	no	—
223	C₂²×C₄	—	73	1	no	—
224	C₂²×C₄	—	73	1	no	—
225	C₂²×C₄	—	73	1	no	—
226	C₂²×C₄	—	73	1	no	—
227	C₂²×C₄	—	73	1	no	—
228	C₂²×C₄	—	73	1	no	—
229	C₂²×C₄	—	73	1	no	—
230	C₂²×C₄	—	73	1	no	—
231	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
232	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
233	M₁₆	—	292	2	no	—
234	M₁₆	—	292	2	no	—
235	C₂×C₈	—	292	2	no	—
236	C₂×C₈	—	292	2	no	—
237	C₂×C₈	—	73	2	no	—
238	C₂×C₈	—	73	2	no	—
239	C₂×C₈	—	73	2	no	—
240	C₂×C₈	—	73	2	no	—
241	C₄⋊C₄	—	73	1	no	—
242	C₄⋊C₄	—	73	1	no	—
243	C₄⋊C₄	—	73	1	no	—
244	C₄⋊C₄	C₂×#64408	1	1	yes	characteristic
245	C₂²⋊C₄	—	73	1	no	—
246	C₂²⋊C₄	—	73	1	no	—
247	C₂²⋊C₄	—	73	1	no	—
248	C₂²⋊C₄	—	73	1	no	—
249	C₂²⋊C₄	—	73	1	no	—
250	C₂²⋊C₄	—	73	1	no	—
251	C₂²⋊C₄	C₂×#64408	1	1	yes	characteristic
252	C₂²⋊C₄	C₂×#64408	1	1	yes	characteristic
253	C₄²	—	73	1	no	—
254	C₄²	—	73	1	no	—
255	C₄²	—	73	1	no	—
256	C₄²	C₂×#64408	1	1	yes	characteristic
257	C₂³	—	292	1	no	—
258	C₂³	—	292	1	no	—
259	C₂³	—	73	1	no	—
260	C₂³	—	73	1	no	—
261	C₂³	—	73	1	no	—
262	C₂³	—	146	2	no	—
263	C₂³	—	146	2	no	—
264	C₂³	—	146	2	no	—
265	C₂³	—	146	2	no	—
266	C₂³	C₂²×#64408	1	1	yes	characteristic
267	C₂³	C₂²×#64408	1	1	yes	characteristic
268	D₄	—	2	2	no	—
269	D₄	—	73	4	no	—
270	D₄	—	146	2	no	—
271	D₄	—	146	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	292	1	no	—
275	C₂×C₄	—	292	1	no	—
276	C₂×C₄	—	292	1	no	—
277	C₂×C₄	—	292	1	no	—
278	C₂×C₄	—	73	1	no	—
279	C₂×C₄	—	73	1	no	—
280	C₂×C₄	—	73	1	no	—
281	C₂×C₄	—	73	1	no	—
282	C₂×C₄	—	73	1	no	—
283	C₂×C₄	—	146	2	no	—
284	C₂×C₄	—	146	2	no	—
285	C₂×C₄	—	146	2	no	—
286	C₂×C₄	—	146	2	no	—
287	C₂×C₄	—	146	2	no	—
288	C₂×C₄	—	146	2	no	—
289	C₂×C₄	—	146	2	no	—
290	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
291	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
292	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
293	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
294	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
295	C₈	—	146	2	no	—
296	C₈	—	146	2	no	—
297	C₈	—	146	2	no	—
298	C₈	—	146	2	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	292	4	no	—
305	C₂²	—	292	4	no	—
306	C₂²	—	73	2	no	—
307	C₂²	—	73	2	no	—
308	C₂²	—	146	1	no	—
309	C₂²	—	146	1	no	—
310	C₂²	—	146	1	no	—
311	C₂²	—	146	1	no	—
312	C₂²	—	146	1	no	—
313	C₂²	—	146	1	no	—
314	C₂²	—	146	1	no	—
315	C₂²	D₇₃⋊M₁₆	1	2	yes	—
316	C₂²	C₂³×#64408	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	73	2	no	—
323	C₄	—	73	2	no	—
324	C₄	—	146	1	no	—
325	C₄	—	146	1	no	—
326	C₄	—	146	1	no	—
327	C₂	—	2	2	no	—
328	C₂	—	4	1	no	—
329	C₂	—	292	1	no	—
330	C₂	—	146	2	no	—
331	C₂	—	146	2	no	—
332	C₂	#17989654	1	1	yes	characteristic
333	C₂	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
334	C₂	D₇₃⋊#131	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₄×D₄×C₇₃	C₄
hypercenter	C₄×D₄	#64408
hypocenter	C₇₃	#2974
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #17989620	C₂	semidirect
2. #17989618	C₂	semidirect
3. #17989617	C₂	semidirect
4. #17989614	C₂	semidirect
5. #17989610	C₂	semidirect
6. #17989605	C₂	semidirect
7. #17989559	C₂	semidirect
8. #17989558	C₂	semidirect
9. C₂×#14556648	C₂	semidirect
10. #17989555	C₂	semidirect
12. C₂×#11924038	C₂²	semidirect
13. C₂×#11924037	C₂²	semidirect
14. C₂×#11924036	C₂²	semidirect
15. C₂²×#10596833	C₂²	semidirect
16. #14556658	C₂²	semidirect
17. #14556658	C₂²	semidirect
22. #14556648	C₂²	semidirect
23. #14556648	C₂²	semidirect
24. C₂²⋊#10596833	C₂²	semidirect
25. C₂²⋊#10596833	C₂²	semidirect
26. #14556646	C₂²	semidirect
27. C₄×#10596833	C₂²	semidirect
109. C₂²⋊D₇₃	M₁₆	semidirect
168. C₇₃	#2974	semidirect