Subgroups — #33090375

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618575	C₂	1	2	yes	maximal, complemented
2	#22610820	C₂	1	2	yes	maximal
3	#22610680	C₂	1	2	yes	maximal, complemented
4	#22610637	C₂	1	4	yes	maximal, complemented
5	C₄×#14556527	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14556518	C₂	1	2	yes	maximal, complemented
7	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#17988648	—	2	4	no	—
10	#17988640	—	2	4	no	—
11	C₂×#14556527	C₂²	1	2	yes	—
12	#17988614	C₂²	1	4	yes	—
13	C₂×#14556518	C₂²	1	4	yes	—
14	#17988554	C₂²	1	4	yes	—
15	C₄×#11924012	C₂²	1	2	yes	—
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
18	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
19	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
20	#17988352	C₂²	1	2	yes	—
21	#17988346	C₂²	1	4	yes	—
22	C₂×#135×C₇₃	C₂²	1	1	yes	characteristic
23	C₄×#41×C₇₃	C₂²	1	2	yes	—
24	C₄×Q₁₆×C₇₃	C₄	1	2	yes	complemented
25	C₄×SD₁₆×C₇₃	C₄	1	4	yes	complemented
26	C₄×D₈×C₇₃	C₄	1	2	yes	complemented
27	#427×C₇₃	C₂²	1	1	yes	characteristic
28	#416×C₇₃	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×#11924012	C₂³	1	4	yes	—
31	C₂²×#10596831	C₂³	1	2	yes	—
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
34	#14556527	—	2	4	no	—
35	C₄⋊#10596831	—	2	8	no	—
36	#14556518	—	2	8	no	—
37	C₂²⋊#10596831	—	2	8	no	—
38	#14556502	—	2	2	no	—
39	C₄×#10596831	—	2	2	no	—
40	C₄×#10596831	D₄	1	2	yes	—
41	C₄×#10596831	D₄	1	2	yes	—
42	C₈×Dic₇₃	—	2	4	no	—
43	C₈×Dic₇₃	—	2	4	no	—
44	C₂×#41×C₇₃	C₂³	1	2	yes	—
45	#135×C₇₃	—	2	4	no	—
46	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₇₃	C₂×C₄	1	2	yes	—
48	C₂×D₈×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₇₃	C₂×C₄	1	4	yes	—
52	C₄×D₄×C₇₃	—	2	4	no	—
53	C₄×D₄×C₇₃	C₂×C₄	1	4	yes	—
54	#117×C₇₃	—	2	2	no	—
55	#117×C₇₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₇₃	—	2	2	no	—
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	#107×C₇₃	C₂×C₄	1	2	yes	—
59	#106×C₇₃	C₂×C₄	1	2	yes	—
60	#103×C₇₃	C₂×C₄	1	4	yes	—
61	#102×C₇₃	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
64	C₂²×Dic₇₃	—	2	2	no	—
65	#11924012	—	2	4	no	—
66	#11924012	—	2	4	no	—
67	C₂×#10596831	—	2	2	no	—
68	C₂×#10596831	—	2	2	no	—
69	C₂×#10596831	—	2	4	no	—
70	C₂×#10596831	C₂×D₄	1	4	yes	—
71	C₄×Dic₇₃	—	2	8	no	—
72	C₄×Dic₇₃	—	2	2	no	—
73	C₄×Dic₇₃	#41	1	4	yes	—
74	#41×C₇₃	—	2	4	no	—
75	#41×C₇₃	—	2	4	no	—
76	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
77	C₂×D₄×C₇₃	—	2	2	no	—
78	C₂×D₄×C₇₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	—	2	4	no	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	#41	1	2	yes	—
82	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
83	Q₁₆×C₇₃	—	2	2	no	—
84	SD₁₆×C₇₃	—	2	4	no	—
85	D₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	—	2	2	no	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₃	#41	1	4	yes	—
91	C₄⋊C₄×C₇₃	—	2	4	no	—
92	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
93	C₂²⋊C₄×C₇₃	—	2	2	no	—
94	C₂²⋊C₄×C₇₃	—	2	4	no	—
95	C₄²×C₇₃	—	2	4	no	—
96	C₄²×C₇₃	—	2	2	no	—
97	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂³×C₇₃	—	2	2	no	—
101	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
102	Q₈×C₇₃	—	2	2	no	—
103	Q₈×C₇₃	—	2	2	no	—
104	D₄×C₇₃	—	2	2	no	—
105	D₄×C₇₃	—	2	2	no	—
106	D₄×C₇₃	—	4	4	no	—
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	4	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	4	4	no	—
111	C₂×C₄×C₇₃	—	4	8	no	—
112	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
117	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
118	C₂×Dic₇₃	—	2	8	no	—
119	C₂×Dic₇₃	—	2	4	no	—
120	C₈×C₇₃	—	2	2	no	—
121	C₈×C₇₃	#117	1	2	yes	—
122	C₈×C₇₃	#117	1	2	yes	—
123	#10596831	—	2	4	no	—
124	#10596831	—	4	4	no	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	—	4	4	no	—
127	C₂²×C₇₃	—	4	2	no	—
128	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₇₃	#529	1	2	yes	—
130	C₄×C₇₃	—	2	2	no	—
131	C₄×C₇₃	—	4	4	no	—
132	C₄×C₇₃	—	4	2	no	—
133	C₄×C₇₃	#443	1	4	yes	—
134	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
135	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
136	C₄×C₇₃	#532	1	2	yes	—
137	Dic₇₃	—	2	8	no	—
138	#21292	—	73	1	no	maximal, Sylow
139	C₂×C₇₃	—	2	2	no	—
140	C₂×C₇₃	—	8	2	no	—
141	C₂×C₇₃	#2293	1	1	yes	characteristic
142	C₂×C₇₃	C₂×#443	1	2	yes	—
143	C₄×#135	D₇₃	1	1	yes	characteristic
144	#2922	—	73	2	no	—
145	#1974	—	73	2	no	—
146	#1945	—	73	2	no	—
147	#1937	—	73	4	no	—
148	C₄×#108	—	73	1	no	—
149	C₄×#104	—	73	2	no	—
150	C₄²×C₈	—	73	1	no	—
151	C₇₃	#21292	1	1	yes	characteristic, complemented, Sylow
152	C₂×#135	C₂×D₇₃	1	1	yes	characteristic
153	C₄×#41	C₂×D₇₃	1	2	yes	—
154	C₄×Q₁₆	Dic₇₃	1	2	yes	complemented
155	C₄×SD₁₆	Dic₇₃	1	4	yes	complemented
156	C₄×D₈	Dic₇₃	1	2	yes	complemented
157	#433	—	146	4	no	—
158	#431	—	146	4	no	—
159	C₂×#108	—	73	2	no	—
160	#427	C₂×D₇₃	1	1	yes	characteristic
161	#424	—	73	4	no	—
162	C₂×#104	—	73	4	no	—
163	#416	C₂×D₇₃	1	2	yes	—
164	#408	—	73	4	no	—
165	C₄×M₁₆	—	73	2	no	—
166	C₂×C₄×C₈	—	73	1	no	—
167	C₂×C₄×C₈	—	73	2	no	—
168	C₂×C₄×C₈	—	73	2	no	—
169	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
170	C₄³	—	73	1	no	—
171	#341	—	73	2	no	—
172	#339	—	73	4	no	—
173	C₂×#41	C₂²×D₇₃	1	2	yes	—
174	#135	—	2	4	no	—
175	C₂×Q₁₆	C₂×Dic₇₃	1	1	yes	characteristic
176	C₂×SD₁₆	C₂×Dic₇₃	1	2	yes	—
177	C₂×D₈	C₂×Dic₇₃	1	1	yes	characteristic
178	C₂×M₁₆	—	73	4	no	—
179	C₂²×C₈	—	73	2	no	—
180	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
182	C₄×Q₈	C₂×Dic₇₃	1	4	yes	—
183	C₄×D₄	—	2	4	no	—
184	C₄×D₄	C₂×Dic₇₃	1	4	yes	—
185	#117	—	2	2	no	—
186	#117	C₂²×D₇₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	73	2	no	—
189	C₂×C₄²	—	146	2	no	—
190	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
191	#108	—	146	4	no	—
192	#107	C₂×Dic₇₃	1	2	yes	—
193	#106	C₂×Dic₇₃	1	2	yes	—
194	C₄⋊C₈	—	146	8	no	—
195	#104	—	146	8	no	—
196	#103	C₂×Dic₇₃	1	4	yes	—
197	#102	C₂×Dic₇₃	1	4	yes	—
198	C₂²⋊C₈	—	146	8	no	—
199	#97	—	146	2	no	—
200	C₄×C₈	—	73	2	no	—
201	C₄×C₈	—	73	2	no	—
202	C₄×C₈	—	146	4	no	—
203	C₄×C₈	—	146	4	no	—
204	C₄×C₈	—	146	2	no	—
205	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
206	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	C₂×Q₈	C₂²×Dic₇₃	1	2	yes	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	C₂²×Dic₇₃	1	2	yes	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	146	2	no	—
215	C₂²×C₄	#11924027	1	2	yes	—
216	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
217	Q₁₆	—	2	2	no	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	2	no	—
220	M₁₆	—	146	4	no	—
221	M₁₆	—	146	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	73	4	no	—
224	C₂×C₈	—	146	4	no	—
225	C₂×C₈	—	146	2	no	—
226	C₂×C₈	—	146	2	no	—
227	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
228	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
230	C₂×C₈	#11924028	1	4	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	73	4	no	—
238	C₄²	—	146	8	no	—
239	C₄²	—	146	2	no	—
240	C₄²	C₂²×Dic₇₃	1	2	yes	—
241	C₄²	D₄×D₇₃	1	1	yes	characteristic
242	C₄²	D₄×D₇₃	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	C₂×#11924027	1	1	yes	characteristic
245	Q₈	—	2	2	no	—
246	Q₈	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	8	no	—
255	C₂×C₄	—	146	8	no	—
256	C₂×C₄	—	146	4	no	—
257	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
258	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
259	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
261	C₂×C₄	#41×D₇₃	1	2	yes	—
262	C₂×C₄	C₂×#11924028	1	2	yes	—
263	C₈	—	2	2	no	—
264	C₈	—	292	4	no	—
265	C₈	—	146	4	no	—
266	C₈	#14556550	1	2	yes	—
267	C₈	#14556550	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
272	C₂²	#17989119	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	146	8	no	—
277	C₄	#17988704	1	4	yes	—
278	C₄	C₂×#14556550	1	1	yes	characteristic
279	C₄	#41×Dic₇₃	1	1	yes	characteristic
280	C₄	#17989134	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	8	2	no	—
283	C₂	#22612396	1	1	yes	characteristic
284	C₂	C₂×#17988704	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₄×#135×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#135×C₇₃	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄×#135	D₇₃
hypocenter	C₇₃	#21292
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #22618575	C₂	semidirect
3. #22610680	C₂	semidirect
4. #22610637	C₂	semidirect
5. C₄×#14556527	C₂	semidirect
6. C₄×#14556518	C₂	semidirect
7. C₄×C₈×Dic₇₃	C₂	semidirect
24. C₄×Q₁₆×C₇₃	C₄	semidirect
25. C₄×SD₁₆×C₇₃	C₄	semidirect
26. C₄×D₈×C₇₃	C₄	semidirect
151. C₇₃	#21292	semidirect
154. C₄×Q₁₆	Dic₇₃	semidirect
155. C₄×SD₁₆	Dic₇₃	semidirect
156. C₄×D₈	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22618575	2
#22610820	2
#22610680	2
#22610637	2
C₄×#14556527	2
C₄×#14556518	2
C₄×C₈×Dic₇₃	2
C₄×#135×C₇₃	2
#21292	73