Subgroups — #33068625

Order 18688.

586 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620445	C₂	1	1	yes	maximal, characteristic
2	#22619970	C₂	1	2	yes	maximal
3	#22609459	C₂	1	2	yes	maximal, complemented
4	#22609449	C₂	1	2	yes	maximal, complemented
5	#22609437	C₂	1	4	yes	maximal, complemented
6	#22608679	C₂	1	1	yes	maximal, characteristic
7	#22608589	C₂	1	2	yes	maximal
8	C₂×#446×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989144	C₂²	1	2	yes	—
10	#17988992	C₂²	1	1	yes	characteristic
11	#17988992	C₂²	1	1	yes	characteristic
12	#17988989	—	2	1	no	—
13	#17988989	—	2	1	no	—
14	#17988985	C₂²	1	2	yes	—
15	#17988922	—	2	2	no	—
16	#17988882	C₂²	1	4	yes	—
17	#17988797	C₂²	1	4	yes	—
18	#17988743	—	2	2	no	—
19	C₄×#11924018	—	2	4	no	—
20	#17988640	C₂²	1	2	yes	—
21	#17988625	—	2	1	no	—
22	#17988625	—	2	1	no	—
23	#17988340	C₂²	1	2	yes	—
24	#17988338	C₂²	1	2	yes	—
25	#17988335	C₂²	1	4	yes	—
26	#17988332	C₂²	1	4	yes	—
27	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
28	#446×C₇₃	—	2	1	no	—
29	#446×C₇₃	—	2	1	no	—
30	#446×C₇₃	C₂²	1	2	yes	—
31	#446×C₇₃	C₂²	1	2	yes	—
32	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
33	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×#97×C₇₃	—	2	1	no	—
35	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₂×#11924018	—	2	4	no	—
37	#14556591	C₂³	1	2	yes	—
38	#14556591	C₂³	1	2	yes	—
39	#14556583	—	2	2	no	—
40	#14556570	—	2	2	no	—
41	#14556570	—	2	2	no	—
42	C₄×Dic₁₄₆	—	2	2	no	—
43	C₄×Dic₁₄₆	—	2	2	no	—
44	#14556550	—	2	2	no	—
45	#14556525	—	2	1	no	—
46	#14556525	—	2	1	no	—
47	#14556525	—	2	1	no	—
48	#14556525	—	2	1	no	—
49	#14556524	—	2	4	no	—
50	#14556523	—	2	2	no	—
51	#14556523	—	2	2	no	—
52	C₄⋊#10596831	C₂³	1	4	yes	—
53	#14556515	—	2	4	no	—
54	#14556514	—	2	4	no	—
55	C₂²⋊#10596831	—	2	2	no	—
56	C₄×#10596831	—	2	2	no	—
57	C₂²×Q₈×C₇₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₃	—	2	1	no	—
59	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
61	C₄×Q₈×C₇₃	—	2	1	no	—
62	C₄×Q₈×C₇₃	—	2	1	no	—
63	C₄×Q₈×C₇₃	D₄	1	2	yes	—
64	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
66	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₇₃	—	2	1	no	—
68	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₇₃	—	2	2	no	—
70	C₄⋊C₈×C₇₃	—	2	2	no	—
71	C₄⋊C₈×C₇₃	—	2	1	no	—
72	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
73	#97×C₇₃	—	2	2	no	—
74	#97×C₇₃	—	2	2	no	—
75	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₄₆	—	2	2	no	—
80	C₂×Dic₁₄₆	—	2	2	no	—
81	#11924018	—	4	8	no	—
82	C₂×#10596831	—	2	4	no	—
83	C₄⋊Dic₇₃	—	2	2	no	—
84	C₄⋊Dic₇₃	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	2	2	no	—
87	#11924007	—	4	2	no	—
88	#11924007	—	4	2	no	—
89	C₂²⋊Dic₇₃	—	4	2	no	—
90	C₄×Dic₇₃	—	4	2	no	—
91	C₂×Q₈×C₇₃	—	2	1	no	—
92	C₂×Q₈×C₇₃	—	4	2	no	—
93	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₇₃	—	2	1	no	—
95	C₂²×C₄×C₇₃	—	2	1	no	—
96	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
99	C₂×C₈×C₇₃	—	2	2	no	—
100	C₂×C₈×C₇₃	—	2	2	no	—
101	C₂×C₈×C₇₃	—	4	2	no	—
102	C₂×C₈×C₇₃	Q₁₆	1	4	yes	complemented
103	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₇₃	—	2	2	no	—
108	C₄⋊C₄×C₇₃	—	2	1	no	—
109	C₄⋊C₄×C₇₃	—	2	2	no	—
110	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
111	C₄²×C₇₃	—	2	2	no	—
112	C₄²×C₇₃	—	2	2	no	—
113	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₃	#41	1	1	yes	characteristic
116	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
117	C₂³×C₇₃	#121	1	1	yes	characteristic
118	Q₈×C₇₃	—	2	4	no	—
119	Q₈×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	4	1	no	—
124	C₂×C₄×C₇₃	—	4	2	no	—
125	C₂×C₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
128	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
130	C₂×C₄×C₇₃	#137	1	2	yes	—
131	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
134	C₂×Dic₇₃	—	4	2	no	—
135	C₂×Dic₇₃	—	4	2	no	—
136	Dic₁₄₆	—	4	4	no	—
137	Dic₁₄₆	—	8	2	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	—	4	2	no	—
141	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
142	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
143	#10596831	—	4	4	no	—
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	#462	1	2	yes	—
147	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	2	1	no	—
150	C₄×C₇₃	—	2	1	no	—
151	C₄×C₇₃	—	4	2	no	—
152	C₄×C₇₃	—	4	2	no	—
153	C₄×C₇₃	#462	1	2	yes	—
154	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
155	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
156	C₄×C₇₃	#575	1	2	yes	—
157	Dic₇₃	—	8	2	no	—
158	Dic₇₃	—	8	2	no	—
159	#19150	—	73	1	no	maximal, Sylow
160	C₂×C₇₃	—	2	2	no	—
161	C₂×C₇₃	C₂×#462	1	1	yes	characteristic
162	C₂×C₇₃	#3054	1	1	yes	characteristic
163	C₂×C₇₃	#3067	1	1	yes	characteristic
164	#3160	—	73	1	no	—
165	#3104	—	73	2	no	—
166	C₂×#446	D₇₃	1	1	yes	characteristic
167	#1696	—	73	2	no	—
168	#1694	—	73	2	no	—
169	#1692	—	73	4	no	—
170	#1564	—	73	1	no	—
171	#1542	—	73	2	no	—
172	C₇₃	#19150	1	1	yes	characteristic, complemented, Sylow
173	#533	—	73	2	no	—
174	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
175	#500	—	73	1	no	—
176	#500	—	73	1	no	—
177	#499	—	146	1	no	—
178	#499	—	146	1	no	—
179	#498	—	73	2	no	—
180	#484	—	146	2	no	—
181	#477	—	73	4	no	—
182	#462	—	73	4	no	—
183	#451	—	146	2	no	—
184	#446	—	2	1	no	—
185	#446	—	2	1	no	—
186	#446	C₂×D₇₃	1	2	yes	—
187	#446	C₂×D₇₃	1	2	yes	—
188	C₄×Q₁₆	—	146	4	no	—
189	#431	—	73	2	no	—
190	#427	—	146	1	no	—
191	#427	—	146	1	no	—
192	C₂×C₄⋊C₈	—	2	1	no	—
193	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
194	C₂×#97	—	2	1	no	—
195	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
196	#335	—	73	2	no	—
197	#334	—	73	2	no	—
198	#333	—	73	4	no	—
199	#332	—	73	4	no	—
200	C₂²×Q₈	D₂₉₂	1	1	yes	characteristic
201	C₂×Q₁₆	—	146	4	no	—
202	C₂²×C₈	—	2	1	no	—
203	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
205	C₄⋊Q₈	—	73	2	no	—
206	C₄⋊Q₈	—	73	2	no	—
207	#125	—	146	2	no	—
208	C₂²⋊Q₈	—	146	2	no	—
209	C₂²⋊Q₈	—	146	2	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	146	2	no	—
213	C₄×Q₈	—	146	2	no	—
214	C₄×Q₈	D₂₉₂	1	2	yes	—
215	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
216	#117	—	146	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
221	#107	—	146	1	no	—
222	#107	—	146	1	no	—
223	#107	—	146	1	no	—
224	#107	—	146	1	no	—
225	#107	—	146	4	no	—
226	#106	—	146	2	no	—
227	#106	—	146	2	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	2	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	73	4	no	—
232	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
233	#103	—	146	4	no	—
234	#103	—	146	4	no	—
235	C₂²⋊C₈	—	146	2	no	—
236	#97	—	2	2	no	—
237	#97	—	2	2	no	—
238	C₄×C₈	—	146	2	no	—
239	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
240	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
241	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	146	2	no	—
246	C₂×Q₈	—	146	2	no	—
247	C₂×Q₈	C₂×D₂₉₂	1	2	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
252	C₂²×C₄	#11924028	1	1	yes	characteristic
253	Q₁₆	—	292	8	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	146	4	no	—
258	C₂×C₈	#11924018	1	4	yes	complemented
259	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	292	2	no	—
267	C₄⋊C₄	—	292	2	no	—
268	C₄⋊C₄	—	146	2	no	—
269	C₄⋊C₄	—	146	2	no	—
270	C₄⋊C₄	—	146	2	no	—
271	C₄⋊C₄	—	146	2	no	—
272	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
273	C₂²⋊C₄	—	292	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	292	2	no	—
277	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
278	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
279	C₄²	#11924028	1	1	yes	characteristic
280	C₄²	C₂³×D₇₃	1	1	yes	characteristic
281	C₂³	#14556565	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	292	4	no	—
285	Q₈	—	584	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	292	2	no	—
293	C₂×C₄	—	292	2	no	—
294	C₂×C₄	#14556565	1	1	yes	characteristic
295	C₂×C₄	#14556565	1	1	yes	characteristic
296	C₂×C₄	#14556565	1	1	yes	characteristic
297	C₂×C₄	C₂×#11924018	1	2	yes	—
298	C₂×C₄	#14556621	1	2	yes	—
299	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
301	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	4	2	no	—
305	C₈	—	292	4	no	—
306	C₈	C₂×#11924018	1	2	yes	—
307	C₈	C₂×#11924018	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#17988797	1	2	yes	—
311	C₂²	C₂×#14556565	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	584	2	no	—
318	C₄	—	584	2	no	—
319	C₄	#17988797	1	2	yes	—
320	C₄	C₂²×#11924018	1	1	yes	characteristic
321	C₄	C₂×#14556621	1	1	yes	characteristic
322	C₄	#17989387	1	2	yes	—
323	C₂	—	2	2	no	—
324	C₂	C₂×#17988797	1	1	yes	characteristic
325	C₂	#22619540	1	1	yes	characteristic
326	C₂	#22619651	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556565
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#446×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂³×C₇₃	#121
hypercenter	C₂×#446	D₇₃
hypocenter	C₇₃	#19150
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
3. #22609459	C₂	semidirect
4. #22609449	C₂	semidirect
5. #22609437	C₂	semidirect
102. C₂×C₈×C₇₃	Q₁₆	semidirect
172. C₇₃	#19150	semidirect
258. C₂×C₈	#11924018	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22620445	2
#22619970	2
#22609459	2
#22609449	2
#22609437	2
#22608679	2
#22608589	2
C₂×#446×C₇₃	2
#19150	73