Subgroups — #33116866

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618863	C₂	1	1	yes	maximal, characteristic, complemented
2	#22618835	C₂	1	1	yes	maximal, characteristic
3	#22611512	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611507	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611383	C₂	1	2	yes	maximal, complemented
6	#22611238	C₂	1	1	yes	maximal, characteristic
7	#22611219	C₂	1	1	yes	maximal, characteristic
8	#22611102	C₂	1	2	yes	maximal, complemented
9	#22609770	C₂	1	2	yes	maximal, complemented
10	#22609732	C₂	1	2	yes	maximal, complemented
11	C₇₃×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989387	C₂²	1	1	yes	characteristic
13	C₂×#14556597	C₂²	1	2	yes	—
14	#17989140	C₂²	1	1	yes	characteristic
15	#17988715	—	2	2	no	—
16	C₈×Dic₁₄₆	—	2	2	no	—
17	#17988710	—	2	2	no	—
18	#17988705	—	2	2	no	—
19	#17988636	C₂²	1	1	yes	characteristic
20	#17988615	C₂²	1	1	yes	characteristic
21	#17988614	C₂²	1	1	yes	characteristic
22	#17988610	C₂²	1	1	yes	characteristic
23	#17988605	C₂²	1	2	yes	—
24	C₂×#14556517	C₂²	1	2	yes	—
25	#17988581	C₂²	1	2	yes	—
26	#17988569	C₂²	1	2	yes	—
27	#17988560	C₂²	1	2	yes	—
28	#17988542	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
30	#17988359	C₂²	1	2	yes	—
31	#17988357	C₂²	1	2	yes	—
32	#17988351	C₂²	1	2	yes	—
33	#17988346	C₂²	1	2	yes	—
34	#405×C₇₃	C₂²	1	2	yes	—
35	#405×C₇₃	C₂²	1	1	yes	characteristic
36	#338×C₇₃	C₂²	1	1	yes	characteristic
37	H(3,ℤ/4)×C₇₃	C₂²	1	1	yes	characteristic
38	H(3,ℤ/4)×C₇₃	C₂²	1	2	yes	—
39	C₂×#11924028	C₂³	1	2	yes	—
40	#14556624	—	2	2	no	—
41	#14556619	—	2	2	no	—
42	#14556613	—	2	2	no	—
43	#14556597	—	2	2	no	—
44	#14556597	—	2	2	no	—
45	#14556597	—	2	2	no	—
46	#14556597	—	2	2	no	—
47	C₂×#11924012	C₂³	1	1	yes	characteristic
48	C₂×#11924012	C₂³	1	2	yes	—
49	C₂×#11924011	—	2	2	no	—
50	C₂²×#10596831	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₇₃	—	2	2	no	—
52	#14556590	—	2	1	no	—
53	#14556582	—	2	1	no	—
54	#14556568	—	2	2	no	—
55	C₄×Dic₁₄₆	—	2	2	no	—
56	#14556550	C₂³	1	1	yes	characteristic
57	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
58	#14556527	—	2	2	no	—
59	C₄⋊#10596831	—	2	2	no	—
60	#14556520	—	2	2	no	—
61	#14556520	—	2	2	no	—
62	#14556519	—	2	4	no	—
63	#14556517	—	2	2	no	—
64	#14556517	—	2	2	no	—
65	#14556514	—	2	2	no	—
66	#14556511	—	2	2	no	—
67	#14556509	—	2	2	no	—
68	#14556507	—	2	2	no	—
69	#14556505	—	2	4	no	—
70	#14556501	—	2	1	no	—
71	C₈×Dic₇₃	—	2	1	no	—
72	C₈×Dic₇₃	—	2	1	no	—
73	C₈×Dic₇₃	—	2	1	no	—
74	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
75	C₂×M₁₆×C₇₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
77	#117×C₇₃	C₂³	1	2	yes	—
78	#117×C₇₃	C₂³	1	1	yes	characteristic
79	#97×C₇₃	—	2	2	no	—
80	#97×C₇₃	D₄	1	2	yes	—
81	C₄×C₈×C₇₃	—	2	2	no	—
82	C₄×C₈×C₇₃	D₄	1	2	yes	—
83	#11924028	—	2	2	no	—
84	#11924028	—	2	2	no	—
85	#11924028	—	2	2	no	—
86	#11924028	—	2	2	no	—
87	C₂×Dic₁₄₆	—	2	2	no	—
88	C₂×Dic₁₄₆	#41	1	2	yes	—
89	C₂×D₂₉₂	#41	1	2	yes	—
90	C₂×C₂²⋊D₇₃	—	2	2	no	—
91	C₂²×Dic₇₃	—	2	1	no	—
92	C₂×C₄×D₇₃	—	2	2	no	—
93	Dic₂₉₂	—	2	2	no	—
94	#11924017	—	2	4	no	—
95	D₅₈₄	—	2	2	no	—
96	#11924012	—	2	2	no	—
97	#11924012	—	2	2	no	—
98	#11924012	—	2	2	no	—
99	#11924011	—	4	2	no	—
100	#11924011	—	4	2	no	—
101	C₂×#10596831	—	2	1	no	—
102	C₂×#10596831	—	2	1	no	—
103	C₂×#10596831	—	2	2	no	—
104	C₂×#10596831	—	2	2	no	—
105	C₈×D₇₃	—	4	2	no	—
106	C₈×D₇₃	—	4	2	no	—
107	C₄⋊Dic₇₃	—	2	2	no	—
108	C₄⋊Dic₇₃	—	2	1	no	—
109	#11924007	—	2	2	no	—
110	#11924007	—	2	2	no	—
111	C₂²⋊Dic₇₃	#41	1	2	yes	—
112	C₄×Dic₇₃	—	2	1	no	—
113	C₄×Dic₇₃	—	2	1	no	—
114	C₄×Dic₇₃	#41	1	1	yes	characteristic
115	C₄×Dic₇₃	#41	1	1	yes	characteristic
116	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
117	M₁₆×C₇₃	—	2	2	no	—
118	M₁₆×C₇₃	—	2	2	no	—
119	M₁₆×C₇₃	—	2	2	no	—
120	C₂×C₈×C₇₃	—	2	2	no	—
121	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₇₃	#41	1	2	yes	—
125	C₂×C₈×C₇₃	#41	1	2	yes	—
126	C₄⋊C₄×C₇₃	—	2	1	no	—
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₂²⋊C₄×C₇₃	—	2	1	no	—
129	C₂²⋊C₄×C₇₃	—	2	2	no	—
130	C₄²×C₇₃	—	2	2	no	—
131	C₄²×C₇₃	C₂×D₄	1	2	yes	—
132	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
133	C₂²×D₇₃	—	2	2	no	—
134	C₂×C₄×C₇₃	—	2	2	no	—
135	C₂×C₄×C₇₃	—	4	2	no	—
136	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
138	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
140	C₂²⋊D₇₃	—	4	2	no	—
141	C₂²⋊D₇₃	—	4	2	no	—
142	C₂×Dic₇₃	—	2	1	no	—
143	C₂×Dic₇₃	—	2	1	no	—
144	C₂×Dic₇₃	—	2	2	no	—
145	C₂×Dic₇₃	—	2	2	no	—
146	C₂×Dic₇₃	—	4	2	no	—
147	D₂₉₂	—	2	2	no	—
148	D₂₉₂	—	2	2	no	—
149	C₄×D₇₃	—	4	2	no	—
150	C₄×D₇₃	—	4	2	no	—
151	Dic₁₄₆	—	2	2	no	—
152	Dic₁₄₆	—	2	2	no	—
153	Dic₁₄₆	—	4	4	no	—
154	C₈×C₇₃	—	2	2	no	—
155	C₈×C₇₃	—	4	2	no	—
156	C₈×C₇₃	#124	1	2	yes	—
157	C₈×C₇₃	#124	1	1	yes	characteristic
158	C₈×C₇₃	#124	1	1	yes	characteristic
159	#10596831	—	4	1	no	—
160	#10596831	—	4	1	no	—
161	#10596831	—	4	2	no	—
162	C₂²×C₇₃	—	4	1	no	—
163	C₂²×C₇₃	#525	1	1	yes	characteristic
164	C₂²×C₇₃	#529	1	2	yes	—
165	C₂×D₇₃	—	4	2	no	—
166	C₂×D₇₃	—	8	2	no	—
167	C₄×C₇₃	—	4	2	no	—
168	C₄×C₇₃	—	8	2	no	—
169	C₄×C₇₃	#525	1	1	yes	characteristic
170	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
171	C₄×C₇₃	#529	1	2	yes	—
172	Dic₇₃	—	2	2	no	—
173	Dic₇₃	—	4	2	no	—
174	Dic₇₃	—	4	1	no	—
175	Dic₇₃	—	4	2	no	—
176	#23833	—	73	1	no	maximal, Sylow
177	C₂×C₇₃	—	2	1	no	—
178	C₂×C₇₃	—	2	2	no	—
179	C₂×C₇₃	#2398	1	1	yes	characteristic
180	D₇₃	—	8	2	no	—
181	#2960	—	73	1	no	—
182	#2955	—	73	1	no	—
183	#2103	—	73	1	no	—
184	#2102	—	73	1	no	—
185	#2080	—	73	2	no	—
186	#2053	—	73	1	no	—
187	#2050	—	73	1	no	—
188	#2031	—	73	2	no	—
189	#1764	—	73	2	no	—
190	#1757	—	73	2	no	—
191	#1747	D₇₃	1	1	yes	characteristic, complemented
192	C₇₃	#23833	1	1	yes	characteristic, complemented, Sylow
193	#575	—	73	1	no	—
194	C₂×#131	—	73	2	no	—
195	#533	—	73	1	no	—
196	#446	—	146	2	no	—
197	C₈×Q₈	—	146	2	no	—
198	#444	—	146	2	no	—
199	#443	—	146	2	no	—
200	#430	—	73	1	no	—
201	#424	—	73	1	no	—
202	#424	—	73	1	no	—
203	#423	—	73	1	no	—
204	#421	—	73	2	no	—
205	C₂×#104	—	73	2	no	—
206	#416	—	73	2	no	—
207	#413	—	73	2	no	—
208	#410	—	73	2	no	—
209	#405	—	73	1	no	—
210	#405	C₂×D₇₃	1	2	yes	—
211	#405	C₂×D₇₃	1	1	yes	characteristic
212	C₂×C₄×C₈	—	73	1	no	—
213	#344	—	73	2	no	—
214	#344	—	73	2	no	—
215	#341	—	73	2	no	—
216	#339	—	73	2	no	—
217	#338	C₂×D₇₃	1	1	yes	characteristic
218	H(3,ℤ/4)	C₂×D₇₃	1	1	yes	characteristic
219	H(3,ℤ/4)	C₂×D₇₃	1	2	yes	—
220	C₂×#41	—	73	2	no	—
221	#137	—	146	2	no	—
222	AGL(1,ℤ/8)	—	146	2	no	—
223	#135	—	146	2	no	—
224	#131	—	146	2	no	—
225	#131	—	146	2	no	—
226	#131	—	146	2	no	—
227	#131	—	146	2	no	—
228	C₂×M₁₆	—	73	2	no	—
229	C₂×M₁₆	—	73	1	no	—
230	C₂×M₁₆	—	146	2	no	—
231	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
232	C₂×M₁₆	C₂²×D₇₃	1	1	yes	characteristic
233	C₂²×C₈	—	73	1	no	—
234	C₂²×C₈	—	146	2	no	—
235	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
236	C₄⋊Q₈	—	146	1	no	—
237	#125	—	146	1	no	—
238	C₂²⋊Q₈	—	146	2	no	—
239	C₄×Q₈	—	146	2	no	—
240	#117	—	73	1	no	—
241	#117	C₂²×D₇₃	1	2	yes	—
242	#117	C₂²×D₇₃	1	1	yes	characteristic
243	C₂×C₄²	—	73	1	no	—
244	#108	—	146	2	no	—
245	C₄⋊C₈	—	146	2	no	—
246	C₄⋊C₈	—	146	2	no	—
247	C₄⋊C₈	—	146	2	no	—
248	#104	—	146	2	no	—
249	#104	—	146	2	no	—
250	#104	—	146	4	no	—
251	#103	—	146	2	no	—
252	#102	—	146	2	no	—
253	#101	—	146	2	no	—
254	#100	—	146	2	no	—
255	#99	—	146	4	no	—
256	#97	—	2	2	no	—
257	#97	—	146	1	no	—
258	#97	D₂₉₂	1	2	yes	—
259	C₄×C₈	—	2	2	no	—
260	C₄×C₈	—	146	1	no	—
261	C₄×C₈	—	146	1	no	—
262	C₄×C₈	—	146	1	no	—
263	C₄×C₈	D₂₉₂	1	2	yes	—
264	#41	—	146	2	no	—
265	#41	—	146	2	no	—
266	#41	—	146	2	no	—
267	#41	—	146	2	no	—
268	C₂×Q₈	—	73	2	no	—
269	C₂×Q₈	—	146	2	no	—
270	C₂×D₄	—	73	2	no	—
271	C₂×D₄	—	146	2	no	—
272	C₂²×C₄	—	146	1	no	—
273	C₂²×C₄	—	146	2	no	—
274	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
275	Q₁₆	—	146	2	no	—
276	SD₁₆	—	146	4	no	—
277	D₈	—	146	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	292	2	no	—
282	M₁₆	—	292	2	no	—
283	M₁₆	—	146	2	no	—
284	M₁₆	—	146	2	no	—
285	M₁₆	—	146	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	292	2	no	—
288	C₂×C₈	—	292	2	no	—
289	C₂×C₈	—	146	1	no	—
290	C₂×C₈	—	146	1	no	—
291	C₂×C₈	—	146	2	no	—
292	C₂×C₈	—	146	2	no	—
293	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
294	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
295	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
296	C₂×C₈	#11924028	1	2	yes	—
297	C₂×C₈	#11924028	1	2	yes	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	146	2	no	—
301	C₄⋊C₄	—	146	2	no	—
302	C₄⋊C₄	—	146	2	no	—
303	C₄⋊C₄	—	146	1	no	—
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	73	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	73	1	no	—
309	C₄²	—	73	1	no	—
310	C₄²	—	146	1	no	—
311	C₄²	—	146	1	no	—
312	C₄²	C₂×D₂₉₂	1	2	yes	—
313	C₂³	—	146	2	no	—
314	C₂³	#41×D₇₃	1	1	yes	characteristic
315	Q₈	—	292	4	no	—
316	Q₈	—	146	2	no	—
317	Q₈	—	146	2	no	—
318	D₄	—	292	2	no	—
319	D₄	—	292	2	no	—
320	D₄	—	146	2	no	—
321	D₄	—	146	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	292	2	no	—
325	C₂×C₄	—	292	2	no	—
326	C₂×C₄	—	292	2	no	—
327	C₂×C₄	—	146	1	no	—
328	C₂×C₄	—	146	2	no	—
329	C₂×C₄	—	146	2	no	—
330	C₂×C₄	—	146	1	no	—
331	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
332	C₂×C₄	#41×D₇₃	1	2	yes	—
333	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#11924028	1	2	yes	—
335	C₈	—	2	2	no	—
336	C₈	—	4	2	no	—
337	C₈	—	292	1	no	—
338	C₈	—	292	1	no	—
339	C₈	—	292	2	no	—
340	C₈	#14556579	1	2	yes	—
341	C₈	#14556579	1	1	yes	characteristic
342	C₈	#14556579	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	292	2	no	—
345	C₂²	—	584	2	no	—
346	C₂²	#17989099	1	1	yes	characteristic
347	C₂²	#17989115	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	8	2	no	—
350	C₄	—	292	1	no	—
351	C₄	—	292	2	no	—
352	C₄	—	292	2	no	—
353	C₄	—	146	2	no	—
354	C₄	#17989099	1	1	yes	characteristic
355	C₄	C₂×#14556579	1	1	yes	characteristic
356	C₄	#17989115	1	2	yes	—
357	C₂	—	2	1	no	—
358	C₂	—	2	2	no	—
359	C₂	—	584	2	no	—
360	C₂	#22613217	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556579
commutator, cosocle	C₂²×C₄×C₇₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1747	C₂
socle	C₂×C₇₃	#2398
prime radical	#17989387	C₂²
hypercenter	#1747	D₇₃
hypocenter	C₇₃	#23833
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
1. #22618863	C₂	semidirect
3. #22611512	C₂	semidirect
4. #22611507	C₂	semidirect
5. #22611383	C₂	semidirect
8. #22611102	C₂	semidirect
9. #22609770	C₂	semidirect
10. #22609732	C₂	semidirect
11. C₇₃×#1747	C₂	semidirect
191. #1747	D₇₃	semidirect
192. C₇₃	#23833	semidirect

Maximal subgroups

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

subgroup	index
#22618863	2
#22618835	2
#22611512	2
#22611507	2
#22611383	2
#22611238	2
#22611219	2
#22611102	2
#22609770	2
#22609732	2
C₇₃×#1747	2
#23833	73