Subgroups — #33073922

Order 18688.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621054	C₂	1	2	yes	maximal, complemented
2	#22620510	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988992	C₂	1	1	yes	maximal, characteristic, complemented
4	#22620390	C₂	1	2	yes	maximal, complemented
5	#22609602	C₂	1	4	yes	maximal, complemented
6	#22609549	C₂	1	2	yes	maximal, complemented
7	#22609523	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556624	—	2	4	no	—
10	#17989137	C₂²	1	2	yes	—
11	C₂×#14556591	C₂²	1	2	yes	—
12	#17988992	—	2	2	no	—
13	#17988992	C₂²	1	2	yes	—
14	#17988992	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988992	C₂²	1	4	yes	—
17	#17988981	—	2	4	no	—
18	#17988967	C₂²	1	4	yes	—
19	#17988964	C₂²	1	1	yes	characteristic
20	#17988960	C₂²	1	4	yes	—
21	#17988934	C₂²	1	2	yes	—
22	#17988934	C₂²	1	4	yes	—
23	#17988901	—	2	4	no	—
24	#17988868	—	2	4	no	—
25	#17988823	—	2	2	no	—
26	C₄×D₅₈₄	—	2	2	no	—
27	C₂×#14556525	—	2	2	no	—
28	#431×C₇₃	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#322×C₇₃	C₂²	1	2	yes	—
32	C₈²×C₇₃	C₂²	1	2	yes	—
33	C₂×#11924028	—	2	2	no	—
34	C₂²×Dic₁₄₆	—	2	2	no	—
35	#14556624	—	4	16	no	—
36	C₂×Dic₂₉₂	—	2	8	no	—
37	C₂×#11924017	—	2	8	no	—
38	C₂×D₅₈₄	—	2	2	no	—
39	#14556591	—	2	2	no	—
40	#14556591	—	2	2	no	—
41	#14556591	—	2	2	no	—
42	#14556591	C₂³	1	4	yes	—
43	#14556591	C₂³	1	2	yes	—
44	#14556588	C₂³	1	2	yes	—
45	#14556579	—	2	2	no	—
46	#14556565	—	2	2	no	—
47	#14556565	—	2	2	no	—
48	C₄×D₂₉₂	—	2	2	no	—
49	C₄×D₂₉₂	—	2	2	no	—
50	C₂×C₄⋊Dic₇₃	—	2	2	no	—
51	C₂×C₄⋊Dic₇₃	—	2	2	no	—
52	#14556525	—	2	8	no	—
53	#14556525	—	2	4	no	—
54	#14556525	—	2	4	no	—
55	#14556525	—	2	2	no	—
56	#14556513	—	2	8	no	—
57	#14556513	—	2	4	no	—
58	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
59	C₂²×C₈×C₇₃	D₄	1	2	yes	—
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
62	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
63	#97×C₇₃	D₄	1	2	yes	—
64	C₄×C₈×C₇₃	D₄	1	2	yes	—
65	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	2	yes	—
68	C₄×C₈×C₇₃	C₂³	1	2	yes	—
69	#11924028	—	4	8	no	—
70	C₂×Dic₁₄₆	—	2	2	no	—
71	C₂×Dic₁₄₆	—	2	4	no	—
72	C₂×Dic₁₄₆	—	4	8	no	—
73	C₂×Dic₁₄₆	—	4	2	no	—
74	C₂×D₂₉₂	—	2	2	no	—
75	C₂×D₂₉₂	—	2	2	no	—
76	C₂×D₂₉₂	—	2	2	no	—
77	C₂×C₂²⋊D₇₃	—	4	2	no	—
78	C₂²×Dic₇₃	—	4	2	no	—
79	C₂×C₄×D₇₃	—	4	2	no	—
80	Dic₂₉₂	—	4	16	no	—
81	#11924017	—	4	16	no	—
82	D₅₈₄	—	4	4	no	—
83	C₄⋊Dic₇₃	—	2	4	no	—
84	C₄⋊Dic₇₃	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	4	no	—
86	C₄⋊Dic₇₃	—	2	2	no	—
87	C₄⋊Dic₇₃	—	4	2	no	—
88	C₄⋊Dic₇₃	—	4	2	no	—
89	#11924005	—	4	2	no	—
90	#11924005	—	4	2	no	—
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
94	M₁₆×C₇₃	D₈	1	8	yes	complemented
95	C₂×C₈×C₇₃	—	2	4	no	—
96	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
98	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
99	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
100	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
104	C₂³×C₇₃	#127	1	1	yes	characteristic
105	C₂²×D₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
112	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂²⋊D₇₃	—	8	4	no	—
116	C₂×Dic₇₃	—	4	2	no	—
117	C₂×Dic₇₃	—	4	4	no	—
118	C₂×Dic₇₃	—	8	4	no	—
119	D₂₉₂	—	4	4	no	—
120	D₂₉₂	—	4	4	no	—
121	D₂₉₂	—	4	4	no	—
122	C₄×D₇₃	—	8	4	no	—
123	Dic₁₄₆	—	4	8	no	—
124	Dic₁₄₆	—	4	4	no	—
125	Dic₁₄₆	—	8	4	no	—
126	C₈×C₇₃	—	2	2	no	—
127	C₈×C₇₃	—	2	4	no	—
128	C₈×C₇₃	C₂×D₈	1	8	yes	—
129	C₈×C₇₃	#137	1	4	yes	complemented
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	—	2	1	no	—
132	C₂²×C₇₃	#493	1	2	yes	—
133	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
134	C₂×D₇₃	—	8	2	no	—
135	C₂×D₇₃	—	8	2	no	—
136	C₂×D₇₃	—	8	2	no	—
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	1	no	—
139	C₄×C₇₃	#493	1	2	yes	—
140	C₄×C₇₃	C₂²×D₈	1	2	yes	—
141	C₄×C₇₃	C₂×#137	1	2	yes	—
142	C₄×C₇₃	#578	1	2	yes	—
143	Dic₇₃	—	8	4	no	—
144	Dic₇₃	—	8	2	no	—
145	#19630	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	—	2	2	no	—
147	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
148	C₂×C₇₃	#3152	1	1	yes	characteristic
149	C₂×C₇₃	#3156	1	1	yes	characteristic
150	D₇₃	—	16	2	no	—
151	#3226	—	73	2	no	—
152	#3169	—	73	1	no	—
153	C₂×#500	—	73	1	no	—
154	#3152	—	73	2	no	—
155	#1722	—	73	4	no	—
156	#1713	—	73	2	no	—
157	#1706	—	73	2	no	—
158	C₈×M₁₆	D₇₃	1	1	yes	characteristic, complemented
159	C₇₃	#19630	1	1	yes	characteristic, complemented, Sylow
160	C₂×#137	—	146	4	no	—
161	#532	—	73	2	no	—
162	C₂×C₄⋊Q₈	—	73	2	no	—
163	#500	—	73	2	no	—
164	#500	—	73	2	no	—
165	#500	—	73	1	no	—
166	#500	—	73	4	no	—
167	#500	—	146	2	no	—
168	#497	—	146	4	no	—
169	#494	—	73	4	no	—
170	#493	—	73	1	no	—
171	#492	—	73	4	no	—
172	#486	—	73	2	no	—
173	#486	—	73	4	no	—
174	#480	—	146	4	no	—
175	#474	—	146	4	no	—
176	#466	—	146	2	no	—
177	C₄×D₈	—	146	2	no	—
178	#431	C₂×D₇₃	1	1	yes	characteristic
179	C₂×#107	—	146	2	no	—
180	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
181	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
182	#322	C₂×D₇₃	1	2	yes	—
183	C₈²	C₂×D₇₃	1	2	yes	—
184	C₂×#41	—	146	2	no	—
185	C₂²×Q₈	—	146	2	no	—
186	#137	—	292	16	no	—
187	C₂×Q₁₆	—	146	8	no	—
188	C₂×SD₁₆	—	146	8	no	—
189	C₂×D₈	—	146	2	no	—
190	C₂×M₁₆	D₂₉₂	1	2	yes	—
191	C₂²×C₈	D₂₉₂	1	2	yes	—
192	C₄⋊Q₈	—	73	4	no	—
193	C₄⋊Q₈	—	73	2	no	—
194	C₄⋊Q₈	—	146	2	no	—
195	C₄⋊Q₈	—	146	2	no	—
196	C₄⋊Q₈	—	146	2	no	—
197	#127	—	73	2	no	—
198	#124	—	146	2	no	—
199	#121	—	146	2	no	—
200	#121	—	146	2	no	—
201	C₄×D₄	—	146	2	no	—
202	C₄×D₄	—	146	2	no	—
203	C₂×C₄⋊C₄	—	146	2	no	—
204	C₂×C₄⋊C₄	—	146	2	no	—
205	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
206	#107	—	146	4	no	—
207	#107	—	146	4	no	—
208	#107	—	146	2	no	—
209	#107	—	146	8	no	—
210	C₄⋊C₈	D₂₉₂	1	2	yes	—
211	#102	—	146	4	no	—
212	#102	—	146	8	no	—
213	C₂²⋊C₈	D₂₉₂	1	2	yes	—
214	#97	D₂₉₂	1	2	yes	—
215	C₄×C₈	D₂₉₂	1	2	yes	—
216	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₇₃	1	2	yes	—
218	C₄×C₈	C₂²×D₇₃	1	2	yes	—
219	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
220	#41	—	292	8	no	—
221	C₂×Q₈	—	292	2	no	—
222	C₂×Q₈	—	292	8	no	—
223	C₂×Q₈	—	146	2	no	—
224	C₂×Q₈	—	146	4	no	—
225	C₂×D₄	—	292	2	no	—
226	C₂×D₄	—	146	2	no	—
227	C₂×D₄	—	146	2	no	—
228	C₂×D₄	—	146	2	no	—
229	C₂²×C₄	—	292	2	no	—
230	C₂²×C₄	—	292	2	no	—
231	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
234	Q₁₆	—	292	16	no	—
235	SD₁₆	—	292	16	no	—
236	D₈	—	292	4	no	—
237	M₁₆	D₅₈₄	1	8	yes	complemented
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
240	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
241	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
242	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
243	C₄⋊C₄	—	292	2	no	—
244	C₄⋊C₄	—	292	2	no	—
245	C₄⋊C₄	—	146	4	no	—
246	C₄⋊C₄	—	146	2	no	—
247	C₄⋊C₄	—	146	4	no	—
248	C₄⋊C₄	—	146	2	no	—
249	C₂²⋊C₄	—	292	2	no	—
250	C₂²⋊C₄	—	292	2	no	—
251	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
252	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
253	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
254	C₄²	C₂³×D₇₃	1	1	yes	characteristic
255	C₂³	—	292	2	no	—
256	C₂³	#14556588	1	1	yes	characteristic
257	Q₈	—	292	8	no	—
258	Q₈	—	292	4	no	—
259	Q₈	—	584	4	no	—
260	D₄	—	292	4	no	—
261	D₄	—	292	4	no	—
262	D₄	—	292	4	no	—
263	D₄	—	584	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	292	2	no	—
267	C₂×C₄	—	292	4	no	—
268	C₂×C₄	—	584	4	no	—
269	C₂×C₄	—	584	4	no	—
270	C₂×C₄	#14556588	1	1	yes	characteristic
271	C₂×C₄	#14556588	1	1	yes	characteristic
272	C₂×C₄	#14556588	1	1	yes	characteristic
273	C₂×C₄	C₂×D₅₈₄	1	4	yes	—
274	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
275	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	4	no	—
279	C₈	C₂×D₅₈₄	1	8	yes	—
280	C₈	#14556624	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	584	2	no	—
284	C₂²	—	584	2	no	—
285	C₂²	—	584	2	no	—
286	C₂²	#17988964	1	2	yes	—
287	C₂²	C₂×#14556588	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	584	2	no	—
291	C₄	—	584	4	no	—
292	C₄	#17988964	1	2	yes	—
293	C₄	C₂²×D₅₈₄	1	2	yes	—
294	C₄	C₂×#14556624	1	2	yes	—
295	C₄	#17989404	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	1168	2	no	—
298	C₂	C₂×#17988964	1	1	yes	characteristic
299	C₂	#22620390	1	1	yes	characteristic
300	C₂	#22620425	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556588
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#22620510	C₂
hypercenter	C₈×M₁₆	D₇₃
hypocenter	C₇₃	#19630
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #22621054	C₂	semidirect
3. C₂×#17988992	C₂	semidirect
4. #22620390	C₂	semidirect
5. #22609602	C₂	semidirect
6. #22609549	C₂	semidirect
7. #22609523	C₂	semidirect
8. C₈×M₁₆×C₇₃	C₂	semidirect
94. M₁₆×C₇₃	D₈	semidirect
129. C₈×C₇₃	#137	semidirect
158. C₈×M₁₆	D₇₃	semidirect
159. C₇₃	#19630	semidirect
237. M₁₆	D₅₈₄	semidirect
280. C₈	#14556624	semidirect

Maximal subgroups

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

subgroup	index
#22621054	2
#22620510	2
C₂×#17988992	2
#22620390	2
#22609602	2
#22609549	2
#22609523	2
C₈×M₁₆×C₇₃	2
#19630	73