Subgroups — #33240695

Order 18688.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620470	C₂	1	1	yes	maximal, characteristic, complemented
2	#22618808	C₂	1	1	yes	maximal, characteristic
3	#22612096	C₂	1	2	yes	maximal, complemented
4	#22612089	C₂	1	2	yes	maximal, complemented
5	#22612084	C₂	1	2	yes	maximal, complemented
6	#22612069	C₂	1	2	yes	maximal, complemented
7	#22611719	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#17988413	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611644	C₂	1	1	yes	maximal, characteristic
10	#22611639	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556613	C₂²	1	1	yes	characteristic
13	#17989215	—	2	1	no	—
14	#17989067	—	2	1	no	—
15	#17988996	—	2	2	no	—
16	#17988992	C₂²	1	2	yes	—
17	#17988989	C₂²	1	2	yes	—
18	#17988689	—	2	2	no	—
19	C₄×Dic₂₉₂	C₂²	1	2	yes	—
20	C₄×D₅₈₄	C₂²	1	2	yes	—
21	#17988629	—	2	1	no	—
22	#17988625	C₂²	1	1	yes	characteristic
23	C₂×#14556525	C₂²	1	1	yes	characteristic
24	C₂×#14556525	C₂²	1	1	yes	characteristic
25	#17988589	—	2	1	no	—
26	#17988415	C₂²	1	2	yes	—
27	#17988413	C₂²	1	2	yes	—
28	#17988413	C₂²	1	2	yes	—
29	#17988413	C₂²	1	2	yes	—
30	#17988392	C₂²	1	1	yes	characteristic
31	#17988392	C₂²	1	2	yes	—
32	#17988392	C₂²	1	1	yes	characteristic
33	#17988389	C₂²	1	1	yes	characteristic
34	#17988389	C₂²	1	2	yes	—
35	#17988389	C₂²	1	1	yes	characteristic
36	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
38	#405×C₇₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
40	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
41	C₂×#11924028	—	2	1	no	—
42	#14556613	—	2	2	no	—
43	#14556613	Q₈	1	4	yes	complemented
44	C₂×Dic₂₉₂	C₂³	1	1	yes	characteristic
45	C₂×#11924017	—	2	1	no	—
46	C₂×D₅₈₄	C₂³	1	1	yes	characteristic
47	#14556591	—	2	2	no	—
48	#14556590	—	4	1	no	—
49	#14556583	—	2	2	no	—
50	#14556582	—	4	1	no	—
51	#14556568	—	4	2	no	—
52	C₄×Dic₁₄₆	—	2	2	no	—
53	C₄×D₂₉₂	—	2	2	no	—
54	#14556551	—	4	2	no	—
55	#14556550	—	2	1	no	—
56	#14556549	—	4	1	no	—
57	C₂×C₄⋊Dic₇₃	—	2	1	no	—
58	C₂×C₄⋊Dic₇₃	—	2	1	no	—
59	C₄⋊C₄×D₇₃	—	4	1	no	—
60	#14556525	—	2	1	no	—
61	#14556525	—	2	1	no	—
62	#14556525	—	2	2	no	—
63	#14556525	C₂³	1	2	yes	—
64	#14556525	C₂³	1	2	yes	—
65	#14556525	C₂³	1	1	yes	characteristic
66	#14556525	C₂³	1	1	yes	characteristic
67	#14556523	—	2	1	no	—
68	#14556523	—	2	2	no	—
69	#14556516	—	2	2	no	—
70	#14556513	—	2	2	no	—
71	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
73	#117×C₇₃	D₄	1	1	yes	characteristic
74	M₃₂×C₇₃	—	2	2	no	—
75	C₂×C₁₆×C₇₃	—	2	2	no	—
76	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
77	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
78	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
79	#97×C₇₃	D₄	1	2	yes	—
80	C₄×C₈×C₇₃	C₂³	1	2	yes	—
81	#11924028	—	2	4	no	—
82	#11924028	—	4	2	no	—
83	C₂×Dic₁₄₆	—	2	1	no	—
84	C₂×Dic₁₄₆	—	4	2	no	—
85	C₂×D₂₉₂	—	2	1	no	—
86	C₂×C₂²⋊D₇₃	—	4	1	no	—
87	C₂²×Dic₇₃	—	4	1	no	—
88	C₂²×Dic₇₃	—	4	1	no	—
89	C₂×C₄×D₇₃	—	4	1	no	—
90	C₂×C₄×D₇₃	—	4	2	no	—
91	Dic₂₉₂	—	2	1	no	—
92	Dic₂₉₂	C₂×Q₈	1	2	yes	—
93	#11924017	—	2	4	no	—
94	D₅₈₄	—	2	1	no	—
95	D₅₈₄	C₂×Q₈	1	2	yes	—
96	C₄⋊Dic₇₃	—	2	2	no	—
97	C₄⋊Dic₇₃	—	2	1	no	—
98	C₄⋊Dic₇₃	—	2	2	no	—
99	C₄⋊Dic₇₃	—	2	1	no	—
100	C₄⋊Dic₇₃	—	4	1	no	—
101	C₄⋊Dic₇₃	—	4	1	no	—
102	C₄⋊Dic₇₃	—	4	2	no	—
103	#11924007	—	4	2	no	—
104	#11924007	—	4	2	no	—
105	#11924007	—	4	2	no	—
106	C₂²⋊Dic₇₃	—	4	1	no	—
107	#11924005	—	4	2	no	—
108	C₄×Dic₇₃	—	4	2	no	—
109	C₄×Dic₇₃	—	4	1	no	—
110	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
111	M₁₆×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₇₃	#41	1	2	yes	—
116	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
118	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
119	C₂²⋊C₄×C₇₃	D₈	1	2	yes	—
120	C₄²×C₇₃	C₂×D₄	1	2	yes	—
121	C₁₆×C₇₃	—	2	1	no	—
122	C₁₆×C₇₃	—	2	1	no	—
123	C₁₆×C₇₃	—	2	2	no	—
124	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
125	C₂²×D₇₃	—	4	1	no	—
126	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₇₃	#137	1	2	yes	—
131	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₇₃	—	4	4	no	—
133	C₂×Dic₇₃	—	4	1	no	—
134	C₂×Dic₇₃	—	4	2	no	—
135	C₂×Dic₇₃	—	4	1	no	—
136	C₂×Dic₇₃	—	4	1	no	—
137	C₂×Dic₇₃	—	4	2	no	—
138	C₂×Dic₇₃	—	16	1	no	—
139	C₂×Dic₇₃	—	16	1	no	—
140	D₂₉₂	—	2	2	no	—
141	D₂₉₂	—	4	1	no	—
142	C₄×D₇₃	—	4	4	no	—
143	C₄×D₇₃	—	16	2	no	—
144	Dic₁₄₆	—	2	2	no	—
145	Dic₁₄₆	—	4	1	no	—
146	Dic₁₄₆	—	16	2	no	—
147	C₈×C₇₃	—	2	2	no	—
148	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
149	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
151	C₂²×C₇₃	—	2	1	no	—
152	C₂²×C₇₃	—	2	1	no	—
153	C₂²×C₇₃	#474	1	2	yes	—
154	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₇₃	—	4	2	no	—
156	C₂×D₇₃	—	8	1	no	—
157	C₂×D₇₃	—	8	1	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	#474	1	2	yes	—
161	C₄×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
163	Dic₇₃	—	4	2	no	—
164	Dic₇₃	—	8	1	no	—
165	Dic₇₃	—	8	2	no	—
166	Dic₇₃	—	8	2	no	—
167	Dic₇₃	—	8	1	no	—
168	#33633	—	73	1	no	maximal, Sylow
169	C₂×C₇₃	—	2	2	no	—
170	C₂×C₇₃	C₂×#474	1	1	yes	characteristic
171	C₂×C₇₃	#3415	1	1	yes	characteristic
172	C₂×C₇₃	#3417	1	1	yes	characteristic
173	D₇₃	—	8	2	no	—
174	#3164	—	73	1	no	—
175	#2951	—	73	1	no	—
176	#2229	—	73	2	no	—
177	#2228	—	73	2	no	—
178	#2227	—	73	2	no	—
179	#2224	—	73	2	no	—
180	#2159	—	73	1	no	—
181	C₂×#366	—	73	1	no	—
182	#2151	D₇₃	1	1	yes	characteristic, complemented
183	#2139	—	73	1	no	—
184	#2138	—	73	1	no	—
185	C₇₃	#33633	1	1	yes	characteristic, complemented, Sylow
186	C₂×#135	—	73	1	no	—
187	#544	—	146	1	no	—
188	#520	—	146	1	no	—
189	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
190	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
191	#501	—	146	2	no	—
192	#500	—	73	2	no	—
193	#499	—	73	2	no	—
194	#440	—	146	2	no	—
195	C₄×Q₁₆	—	73	2	no	—
196	C₄×D₈	—	73	2	no	—
197	#428	—	146	1	no	—
198	#427	—	73	1	no	—
199	C₂×#107	—	73	1	no	—
200	C₂×#107	—	73	1	no	—
201	#417	—	146	1	no	—
202	#405	C₂×D₇₃	1	1	yes	characteristic
203	#367	—	73	2	no	—
204	#366	—	73	2	no	—
205	#366	—	73	2	no	—
206	#366	—	73	2	no	—
207	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
208	C₄⋊C₁₆	C₂×D₇₃	1	2	yes	—
209	#358	—	73	1	no	—
210	#358	—	73	1	no	—
211	#358	—	73	2	no	—
212	#357	—	73	1	no	—
213	#357	—	73	1	no	—
214	#357	—	73	2	no	—
215	C₂×#41	—	146	1	no	—
216	#135	—	73	4	no	—
217	#135	—	146	2	no	—
218	C₂×Q₁₆	—	73	1	no	—
219	C₂×SD₁₆	—	146	1	no	—
220	C₂×D₈	—	73	1	no	—
221	C₂×M₁₆	D₂₉₂	1	1	yes	characteristic
222	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
223	C₄⋊Q₈	—	292	1	no	—
224	C₄⋊Q₈	—	146	2	no	—
225	#125	—	292	1	no	—
226	#125	—	146	2	no	—
227	C₂²⋊Q₈	—	292	2	no	—
228	C₄×Q₈	—	146	2	no	—
229	C₄×D₄	—	292	2	no	—
230	C₄×D₄	—	146	2	no	—
231	#117	—	292	1	no	—
232	#117	—	146	1	no	—
233	#117	D₂₉₂	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	292	1	no	—
235	C₂×C₄⋊C₄	—	146	1	no	—
236	C₂×C₄⋊C₄	—	146	1	no	—
237	M₃₂	—	2	2	no	—
238	C₂×C₁₆	—	2	2	no	—
239	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
240	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
241	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
242	#107	—	73	2	no	—
243	#107	—	73	1	no	—
244	#107	—	73	1	no	—
245	#107	—	73	2	no	—
246	#107	—	146	1	no	—
247	#107	—	146	1	no	—
248	#107	—	146	2	no	—
249	#106	—	146	1	no	—
250	#106	—	146	2	no	—
251	#103	—	146	2	no	—
252	#102	—	146	2	no	—
253	#97	D₂₉₂	1	2	yes	—
254	C₄×C₈	C₂²×D₇₃	1	2	yes	—
255	#41	—	292	2	no	—
256	#41	—	146	4	no	—
257	C₂×Q₈	—	292	2	no	—
258	C₂×Q₈	—	146	1	no	—
259	C₂×D₄	—	292	1	no	—
260	C₂×D₄	—	146	1	no	—
261	C₂²×C₄	—	292	1	no	—
262	C₂²×C₄	—	292	1	no	—
263	C₂²×C₄	—	292	1	no	—
264	C₂²×C₄	—	292	2	no	—
265	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
266	Q₁₆	—	73	2	no	—
267	Q₁₆	—	146	1	no	—
268	SD₁₆	—	146	4	no	—
269	D₈	—	73	2	no	—
270	D₈	—	146	1	no	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
273	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
274	C₂×C₈	Q₈×D₇₃	1	2	yes	—
275	C₂×C₈	#11924027	1	2	yes	—
276	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
277	C₄⋊C₄	—	292	2	no	—
278	C₄⋊C₄	—	292	2	no	—
279	C₄⋊C₄	—	292	2	no	—
280	C₄⋊C₄	—	292	1	no	—
281	C₄⋊C₄	—	292	1	no	—
282	C₄⋊C₄	—	292	2	no	—
283	C₄⋊C₄	—	146	2	no	—
284	C₄⋊C₄	—	146	1	no	—
285	C₄⋊C₄	—	146	2	no	—
286	C₄⋊C₄	—	146	1	no	—
287	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
288	C₄⋊C₄	D₅₈₄	1	1	yes	characteristic
289	C₂²⋊C₄	—	292	1	no	—
290	C₂²⋊C₄	—	292	2	no	—
291	C₂²⋊C₄	D₅₈₄	1	2	yes	—
292	C₄²	—	292	2	no	—
293	C₄²	—	292	1	no	—
294	C₄²	C₂×D₂₉₂	1	2	yes	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₂³	—	292	1	no	—
299	C₂³	C₂×D₅₈₄	1	1	yes	characteristic
300	Q₈	—	292	1	no	—
301	Q₈	—	1168	2	no	—
302	Q₈	—	146	2	no	—
303	D₄	—	292	4	no	—
304	D₄	—	292	1	no	—
305	D₄	—	146	2	no	—
306	C₂×C₄	—	292	1	no	—
307	C₂×C₄	—	292	4	no	—
308	C₂×C₄	—	292	1	no	—
309	C₂×C₄	—	292	2	no	—
310	C₂×C₄	—	292	1	no	—
311	C₂×C₄	—	292	2	no	—
312	C₂×C₄	—	1168	2	no	—
313	C₂×C₄	—	1168	1	no	—
314	C₂×C₄	—	1168	1	no	—
315	C₂×C₄	#14556571	1	2	yes	—
316	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
317	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
318	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
319	C₂×C₄	#14556624	1	2	yes	—
320	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#14556571	1	2	yes	—
323	C₈	C₂×Q₈×D₇₃	1	1	yes	characteristic
324	C₈	C₂×#11924027	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	292	2	no	—
328	C₂²	—	584	1	no	—
329	C₂²	—	584	1	no	—
330	C₂²	#17988868	1	2	yes	—
331	C₂²	C₂²×D₅₈₄	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	292	2	no	—
335	C₄	—	584	2	no	—
336	C₄	—	584	2	no	—
337	C₄	—	584	1	no	—
338	C₄	—	584	1	no	—
339	C₄	#17988868	1	2	yes	—
340	C₄	C₂×#14556571	1	1	yes	characteristic
341	C₄	C₂×#14556624	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	584	2	no	—
344	C₂	C₂×#17988868	1	1	yes	characteristic
345	C₂	#22623072	1	1	yes	characteristic
346	C₂	#22623084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2151	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#14556613	C₂²
hypercenter	#2151	D₇₃
hypocenter	C₇₃	#33633
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2151

Complements

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

N	H ≅ G/N	type
1. #22620470	C₂	semidirect
3. #22612096	C₂	semidirect
4. #22612089	C₂	semidirect
5. #22612084	C₂	semidirect
6. #22612069	C₂	semidirect
7. #22611719	C₂	semidirect
8. C₂×#17988413	C₂	semidirect
11. C₇₃×#2151	C₂	semidirect
43. #14556613	Q₈	semidirect
182. #2151	D₇₃	semidirect
185. C₇₃	#33633	semidirect

Maximal subgroups

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

subgroup	index
#22620470	2
#22618808	2
#22612096	2
#22612089	2
#22612084	2
#22612069	2
#22611719	2
C₂×#17988413	2
#22611644	2
#22611639	2
C₇₃×#2151	2
#33633	73