Subgroups — #33073973

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621075	C₂	1	2	yes	maximal
2	#22620517	C₂	1	1	yes	maximal, characteristic
3	#22620311	C₂	1	2	yes	maximal
4	#22609576	C₂	1	4	yes	maximal, complemented
5	#22609540	C₂	1	2	yes	maximal, complemented
6	#22609529	C₂	1	2	yes	maximal, complemented
7	C₈×#11924012	C₂	1	1	yes	maximal, characteristic
8	C₂×#500×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#17989144	C₂²	1	2	yes	—
10	#17988992	C₂²	1	1	yes	characteristic
11	#17988983	C₂²	1	4	yes	—
12	#17988967	C₂²	1	1	yes	characteristic
13	#17988955	—	2	4	no	—
14	#17988939	C₂²	1	4	yes	—
15	#17988938	C₂²	1	2	yes	—
16	#17988937	C₂²	1	4	yes	—
17	#17988937	C₂²	1	4	yes	—
18	#17988917	—	2	4	no	—
19	#17988869	—	2	4	no	—
20	#17988828	—	2	2	no	—
21	C₄×Dic₂₉₂	—	2	2	no	—
22	#17988640	C₂²	1	1	yes	characteristic
23	#17988596	—	2	4	no	—
24	C₄×#11924012	C₂²	1	1	yes	characteristic
25	#17988309	C₂²	1	2	yes	—
26	C₈×#10596831	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
28	#500×C₇₃	—	2	2	no	—
29	#500×C₇₃	C₂²	1	2	yes	—
30	#500×C₇₃	C₂²	1	2	yes	—
31	C₂×#107×C₇₃	—	2	2	no	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₉₂	—	2	2	no	—
34	C₂×#11924012	D₄	1	2	yes	—
35	#14556591	C₂³	1	2	yes	—
36	#14556591	C₂³	1	2	yes	—
37	#14556583	—	2	2	no	—
38	#14556570	—	2	2	no	—
39	#14556570	—	2	2	no	—
40	C₄×Dic₁₄₆	—	2	2	no	—
41	C₄×Dic₁₄₆	—	2	2	no	—
42	#14556550	—	2	2	no	—
43	#14556525	—	2	2	no	—
44	#14556522	—	2	8	no	—
45	C₄⋊#10596831	D₄	1	2	yes	—
46	#14556516	—	2	4	no	—
47	#14556515	—	2	8	no	—
48	#14556515	—	2	8	no	—
49	#14556514	—	2	8	no	—
50	C₂²⋊#10596831	D₄	1	2	yes	—
51	#14556502	D₄	1	2	yes	—
52	C₄×#10596831	C₂³	1	2	yes	—
53	C₄×#10596831	C₂³	1	2	yes	—
54	C₂²×Q₈×C₇₃	—	2	2	no	—
55	C₂²×C₈×C₇₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₇₃	—	2	2	no	—
57	C₄⋊Q₈×C₇₃	—	2	2	no	—
58	C₄⋊Q₈×C₇₃	—	2	2	no	—
59	C₄⋊Q₈×C₇₃	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
62	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
63	#107×C₇₃	—	2	4	no	—
64	#107×C₇₃	—	2	4	no	—
65	C₄×C₈×C₇₃	D₄	1	2	yes	—
66	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₄₆	—	2	2	no	—
69	C₂×Dic₁₄₆	—	2	2	no	—
70	Dic₂₉₂	—	4	4	no	—
71	#11924012	—	2	4	no	—
72	C₂×#10596831	C₂×D₄	1	4	yes	—
73	C₂×#10596831	C₂×D₄	1	4	yes	—
74	C₄⋊Dic₇₃	—	2	2	no	—
75	C₄⋊Dic₇₃	—	2	2	no	—
76	C₄⋊Dic₇₃	—	2	2	no	—
77	C₄⋊Dic₇₃	—	2	2	no	—
78	#11924007	—	4	2	no	—
79	#11924007	—	4	2	no	—
80	C₂²⋊Dic₇₃	—	4	2	no	—
81	C₄×Dic₇₃	—	4	2	no	—
82	C₂×Q₈×C₇₃	—	2	4	no	—
83	C₂×Q₈×C₇₃	—	4	2	no	—
84	C₂×Q₈×C₇₃	—	4	8	no	—
85	C₂²×C₄×C₇₃	—	4	2	no	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	—	2	4	no	—
90	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₇₃	—	2	4	no	—
93	C₄⋊C₄×C₇₃	—	2	4	no	—
94	C₄⋊C₄×C₇₃	—	4	2	no	—
95	C₄⋊C₄×C₇₃	—	4	2	no	—
96	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
100	C₂³×C₇₃	#127	1	1	yes	characteristic
101	Q₈×C₇₃	—	4	8	no	—
102	Q₈×C₇₃	—	8	4	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	4	4	no	—
106	C₂×C₄×C₇₃	—	8	4	no	—
107	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#135	1	4	yes	—
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	4	2	no	—
116	Dic₁₄₆	—	4	4	no	—
117	Dic₁₄₆	—	4	4	no	—
118	C₈×C₇₃	—	2	2	no	—
119	C₈×C₇₃	#137	1	4	yes	—
120	#10596831	—	2	4	no	—
121	#10596831	—	2	4	no	—
122	C₂²×C₇₃	—	2	1	no	—
123	C₂²×C₇₃	—	2	1	no	—
124	C₂²×C₇₃	#495	1	2	yes	—
125	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
126	C₄×C₇₃	—	2	1	no	—
127	C₄×C₇₃	—	2	1	no	—
128	C₄×C₇₃	—	8	4	no	—
129	C₄×C₇₃	#495	1	2	yes	—
130	C₄×C₇₃	C₂×#135	1	2	yes	—
131	C₄×C₇₃	C₂×#137	1	2	yes	—
132	C₄×C₇₃	#578	1	2	yes	—
133	Dic₇₃	—	8	2	no	—
134	Dic₇₃	—	8	2	no	—
135	#19636	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	—	2	2	no	—
137	C₂×C₇₃	C₂×#495	1	1	yes	characteristic
138	C₂×C₇₃	#3152	1	1	yes	characteristic
139	C₂×C₇₃	#3156	1	1	yes	characteristic
140	#3229	—	73	2	no	—
141	#3170	—	73	1	no	—
142	C₂×#500	D₇₃	1	1	yes	characteristic
143	#3139	—	73	2	no	—
144	#1718	—	73	4	no	—
145	#1710	—	73	2	no	—
146	#1707	—	73	2	no	—
147	C₈×M₁₆	—	73	1	no	—
148	C₇₃	#19636	1	1	yes	characteristic, complemented, Sylow
149	#533	—	73	2	no	—
150	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
151	#500	—	2	2	no	—
152	#500	—	73	1	no	—
153	#500	C₂×D₇₃	1	2	yes	—
154	#500	C₂×D₇₃	1	2	yes	—
155	#498	—	73	4	no	—
156	#494	—	73	1	no	—
157	#491	—	146	4	no	—
158	#487	—	73	4	no	—
159	#487	—	73	4	no	—
160	#487	—	73	4	no	—
161	#487	—	73	2	no	—
162	#483	—	146	4	no	—
163	#475	—	146	4	no	—
164	#467	—	146	2	no	—
165	C₄×Q₁₆	—	146	2	no	—
166	#431	—	73	1	no	—
167	C₂×#107	—	2	2	no	—
168	#419	—	146	4	no	—
169	C₄×M₁₆	—	73	1	no	—
170	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
171	#322	—	73	2	no	—
172	C₈²	—	73	2	no	—
173	C₂²×Q₈	—	2	2	no	—
174	C₂×Q₁₆	—	146	2	no	—
175	C₂×M₁₆	—	73	2	no	—
176	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	73	2	no	—
181	C₄⋊Q₈	—	73	2	no	—
182	C₄⋊Q₈	C₂²×D₇₃	1	4	yes	—
183	#125	—	146	2	no	—
184	C₂²⋊Q₈	—	146	2	no	—
185	C₂²⋊Q₈	—	146	2	no	—
186	C₄×Q₈	—	146	2	no	—
187	C₄×Q₈	—	146	2	no	—
188	#117	—	146	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	—	2	4	no	—
194	#107	—	146	2	no	—
195	#106	—	146	8	no	—
196	C₄⋊C₈	—	73	2	no	—
197	#103	—	146	8	no	—
198	#103	—	146	8	no	—
199	#103	—	146	8	no	—
200	#103	—	146	4	no	—
201	C₂²⋊C₈	—	73	2	no	—
202	#97	—	73	2	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
206	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	8	no	—
211	C₂×Q₈	—	146	2	no	—
212	C₂×Q₈	—	146	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
217	Q₁₆	—	292	4	no	—
218	M₁₆	—	146	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	73	4	no	—
221	C₂×C₈	—	73	4	no	—
222	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	292	2	no	—
229	C₄⋊C₄	—	292	2	no	—
230	C₄⋊C₄	—	146	2	no	—
231	C₄⋊C₄	—	146	2	no	—
232	C₄⋊C₄	—	146	2	no	—
233	C₄⋊C₄	—	146	2	no	—
234	C₂²⋊C₄	—	292	2	no	—
235	C₄²	—	292	2	no	—
236	C₄²	D₄×D₇₃	1	1	yes	characteristic
237	C₄²	D₄×D₇₃	1	1	yes	characteristic
238	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
239	C₄²	C₂³×D₇₃	1	1	yes	characteristic
240	C₂³	#14556587	1	1	yes	characteristic
241	Q₈	—	4	8	no	—
242	Q₈	—	8	4	no	—
243	Q₈	—	292	4	no	—
244	Q₈	—	292	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	292	2	no	—
250	C₂×C₄	—	292	2	no	—
251	C₂×C₄	#14556587	1	1	yes	characteristic
252	C₂×C₄	#14556587	1	1	yes	characteristic
253	C₂×C₄	#14556587	1	1	yes	characteristic
254	C₂×C₄	#14556610	1	4	yes	—
255	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	146	4	no	—
260	C₈	—	146	4	no	—
261	C₈	#14556622	1	4	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	#17988970	1	2	yes	—
265	C₂²	C₂×#14556587	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	4	no	—
269	C₄	—	584	2	no	—
270	C₄	—	584	2	no	—
271	C₄	#17988970	1	2	yes	—
272	C₄	C₂×#14556610	1	2	yes	—
273	C₄	C₂×#14556622	1	2	yes	—
274	C₄	#17989400	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	C₂×#17988970	1	1	yes	characteristic
277	C₂	#22620388	1	1	yes	characteristic
278	C₂	#22620420	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556587
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#500×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	C₂³×C₇₃	#127
hypercenter	C₂×#500	D₇₃
hypocenter	C₇₃	#19636
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #22609576	C₂	semidirect
5. #22609540	C₂	semidirect
6. #22609529	C₂	semidirect
148. C₇₃	#19636	semidirect

Maximal subgroups

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

subgroup	index
#22621075	2
#22620517	2
#22620311	2
#22609576	2
#22609540	2
#22609529	2
C₈×#11924012	2
C₂×#500×C₇₃	2
#19636	73