Subgroups — #33104515

Order 18688.

808 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618849	C₂	1	2	yes	maximal
2	C₂×#17988710	C₂	1	1	yes	maximal, characteristic
3	#22618539	C₂	1	2	yes	maximal, complemented
4	#22610721	C₂	1	2	yes	maximal, complemented
5	#22610172	C₂	1	4	yes	maximal, complemented
6	#22610106	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609982	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	1	yes	characteristic
10	#17989353	C₂²	1	4	yes	—
11	C₂×#14556597	C₂²	1	2	yes	—
12	#17988710	—	2	4	no	—
13	#17988710	C₄	1	8	yes	complemented
14	#17988705	—	2	4	no	—
15	#17988635	C₂²	1	4	yes	—
16	#17988634	—	2	4	no	—
17	C₂×#14556527	C₂²	1	2	yes	—
18	C₂×#14556527	C₂²	1	2	yes	—
19	C₂×#14556527	C₂²	1	1	yes	characteristic
20	#108×D₇₃	—	2	4	no	—
21	#17988611	C₂²	1	2	yes	—
22	#17988605	C₂²	1	4	yes	—
23	C₂×#14556519	C₂²	1	2	yes	—
24	#17988357	C₂²	1	4	yes	—
25	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
26	#427×C₇₃	C₂²	1	1	yes	characteristic
27	#405×C₇₃	C₂²	1	2	yes	—
28	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
29	#339×C₇₃	C₂²	1	2	yes	—
30	C₂×#11924028	C₂³	1	2	yes	—
31	#14556613	C₂×C₄	1	4	yes	—
32	#14556613	D₄	1	4	yes	—
33	C₂×Dic₂₉₂	Q₈	1	1	yes	characteristic
34	C₂×#11924017	Q₈	1	2	yes	—
35	C₂×D₅₈₄	Q₈	1	1	yes	characteristic
36	#14556599	—	4	4	no	—
37	#14556597	—	2	8	no	—
38	#14556597	—	2	2	no	—
39	#14556597	—	2	2	no	—
40	#14556597	C₂×C₄	1	4	yes	—
41	#14556597	C₂×C₄	1	4	yes	—
42	C₂×#11924012	C₂³	1	4	yes	—
43	C₂×#11924012	C₂³	1	2	yes	—
44	C₂×#11924011	—	2	2	no	—
45	M₁₆×D₇₃	—	2	8	no	—
46	C₂×C₈×D₇₃	—	2	2	no	—
47	#14556527	—	2	2	no	—
48	#14556527	—	2	2	no	—
49	#14556527	—	2	2	no	—
50	#14556527	—	2	2	no	—
51	#14556527	C₂×C₄	1	4	yes	—
52	#14556526	—	2	8	no	—
53	C₄⋊#10596831	—	4	2	no	—
54	#14556519	—	2	8	no	—
55	#14556519	C₂×C₄	1	8	yes	—
56	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
57	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	#117×C₇₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	#108×C₇₃	D₄	1	4	yes	—
62	#107×C₇₃	—	2	1	no	—
63	#106×C₇₃	—	2	1	no	—
64	#97×C₇₃	—	2	2	no	—
65	#97×C₇₃	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₇₃	—	2	2	no	—
67	#11924028	—	2	2	no	—
68	#11924028	—	2	2	no	—
69	#11924028	C₂²×C₄	1	4	yes	—
70	#11924028	C₂×D₄	1	4	yes	—
71	C₂×Dic₁₄₆	C₂×Q₈	1	2	yes	—
72	C₂×D₂₉₂	C₂×Q₈	1	2	yes	—
73	C₂×C₂²⋊D₇₃	—	2	2	no	—
74	C₂×C₄×D₇₃	—	2	2	no	—
75	Dic₂₉₂	C₄⋊C₄	1	4	yes	—
76	#11924017	C₄⋊C₄	1	8	yes	—
77	D₅₈₄	C₄⋊C₄	1	4	yes	—
78	#11924012	—	2	2	no	—
79	#11924012	—	2	4	no	—
80	#11924012	—	2	4	no	—
81	#11924012	C₂²×C₄	1	4	yes	—
82	#11924011	—	2	4	no	—
83	#11924011	—	2	4	no	—
84	#11924011	—	4	8	no	—
85	C₂×#10596831	—	2	2	no	—
86	C₂×#10596831	—	2	4	no	—
87	C₈×D₇₃	—	2	4	no	—
88	C₈×D₇₃	—	2	4	no	—
89	C₈×D₇₃	—	4	8	no	—
90	C₂²×C₄×C₇₃	—	2	1	no	—
91	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₇₃	—	2	2	no	—
93	M₁₆×C₇₃	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
102	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₇₃	—	2	2	no	—
104	C₂²⋊C₄×C₇₃	—	2	2	no	—
105	C₄²×C₇₃	—	2	1	no	—
106	C₄²×C₇₃	—	2	2	no	—
107	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
108	C₂³×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂²×D₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	4	2	no	—
112	C₂×C₄×C₇₃	—	8	1	no	—
113	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²⋊D₇₃	—	2	4	no	—
120	C₂²⋊D₇₃	—	2	4	no	—
121	C₂×Dic₇₃	—	2	2	no	—
122	D₂₉₂	—	2	2	no	—
123	D₂₉₂	C₂×C₄⋊C₄	1	4	yes	—
124	C₄×D₇₃	—	2	4	no	—
125	C₄×D₇₃	—	2	4	no	—
126	Dic₁₄₆	—	2	2	no	—
127	Dic₁₄₆	C₂×C₄⋊C₄	1	4	yes	—
128	C₈×C₇₃	—	2	4	no	—
129	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₈×C₇₃	C₄×D₄	1	2	yes	—
132	C₈×C₇₃	C₄×D₄	1	2	yes	—
133	#10596831	—	2	4	no	—
134	#10596831	—	4	4	no	—
135	C₂²×C₇₃	—	4	1	no	—
136	C₂²×C₇₃	#520	1	1	yes	characteristic
137	C₂²×C₇₃	#547	1	1	yes	characteristic
138	C₂²×C₇₃	D₄×Q₈	1	1	yes	characteristic
139	C₂×D₇₃	—	2	4	no	—
140	C₂×D₇₃	—	8	2	no	—
141	C₄×C₇₃	—	4	2	no	—
142	C₄×C₇₃	—	8	2	no	—
143	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₇₃	D₄²	1	1	yes	characteristic
146	C₄×C₇₃	#554	1	1	yes	characteristic
147	Dic₇₃	—	2	4	no	—
148	#22695	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	1	no	—
150	C₂×C₇₃	—	2	1	no	—
151	C₂×C₇₃	—	2	1	no	—
152	C₂×C₇₃	D₄×C₄⋊C₄	1	1	yes	characteristic
153	D₇₃	—	4	4	no	—
154	#2957	—	73	2	no	—
155	C₂×#444	—	73	1	no	—
156	#2916	—	73	2	no	—
157	#1953	—	73	2	no	—
158	#1861	—	73	4	no	—
159	#1849	—	73	1	no	—
160	#1814	—	73	2	no	—
161	#1773	D₇₃	1	1	yes	characteristic, complemented
162	C₇₃	#22695	1	1	yes	characteristic, complemented, Sylow
163	C₂×#135	—	73	1	no	—
164	#568	—	73	4	no	—
165	C₂×#131	—	73	2	no	—
166	#444	—	73	8	no	—
167	#444	—	146	4	no	—
168	#443	—	146	4	no	—
169	#430	—	73	4	no	—
170	#430	—	146	4	no	—
171	C₂×#108	—	73	1	no	—
172	C₂×#108	—	73	2	no	—
173	C₂×#108	—	73	2	no	—
174	C₂×#108	—	146	4	no	—
175	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
176	#427	C₂×D₇₃	1	1	yes	characteristic
177	#423	—	73	2	no	—
178	#421	—	73	4	no	—
179	C₂×#104	—	73	2	no	—
180	#405	C₂×D₇₃	1	2	yes	—
181	C₂×#97	C₂×D₇₃	1	1	yes	characteristic
182	#344	—	73	4	no	—
183	#339	C₂×D₇₃	1	2	yes	—
184	C₂×#41	—	73	2	no	—
185	#135	—	73	4	no	—
186	#135	—	73	4	no	—
187	C₂×Q₁₆	—	73	1	no	—
188	C₂×SD₁₆	—	73	2	no	—
189	C₂×D₈	—	73	1	no	—
190	#131	—	292	4	no	—
191	#131	—	73	4	no	—
192	#131	—	73	4	no	—
193	#131	—	146	8	no	—
194	#131	—	146	2	no	—
195	#131	—	146	2	no	—
196	C₂×M₁₆	—	73	2	no	—
197	C₂×M₁₆	—	73	4	no	—
198	C₂×M₁₆	—	146	8	no	—
199	C₂×M₁₆	—	146	2	no	—
200	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
201	C₂²×C₈	—	146	2	no	—
202	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
204	#117	C₂²×D₇₃	1	2	yes	—
205	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
206	#108	—	73	4	no	—
207	#108	—	146	8	no	—
208	#108	—	146	2	no	—
209	#108	—	146	2	no	—
210	#108	—	146	2	no	—
211	#108	—	146	2	no	—
212	#108	D₂₉₂	1	4	yes	—
213	#107	—	2	1	no	—
214	#106	—	2	1	no	—
215	C₄⋊C₈	—	292	2	no	—
216	#104	—	73	8	no	—
217	#104	—	146	8	no	—
218	#97	—	2	2	no	—
219	#97	C₄×D₇₃	1	4	yes	—
220	C₄×C₈	—	2	2	no	—
221	#41	—	73	4	no	—
222	#41	—	73	4	no	—
223	#41	—	146	2	no	—
224	#41	—	146	2	no	—
225	C₂×Q₈	—	73	2	no	—
226	C₂×D₄	—	73	2	no	—
227	C₂×D₄	—	146	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	146	2	no	—
230	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
231	Q₁₆	—	73	4	no	—
232	SD₁₆	—	73	8	no	—
233	D₈	—	73	4	no	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	292	8	no	—
236	M₁₆	—	73	4	no	—
237	M₁₆	—	146	4	no	—
238	M₁₆	—	146	4	no	—
239	M₁₆	—	146	2	no	—
240	M₁₆	—	146	4	no	—
241	M₁₆	—	146	4	no	—
242	M₁₆	C₂×D₂₉₂	1	4	yes	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	292	8	no	—
245	C₂×C₈	—	146	4	no	—
246	C₂×C₈	—	146	2	no	—
247	C₂×C₈	—	146	4	no	—
248	C₂×C₈	—	146	4	no	—
249	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
250	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
251	C₂×C₈	D₄×D₇₃	1	2	yes	—
252	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
253	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
254	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
255	C₂×C₈	#11924028	1	1	yes	characteristic
256	C₂×C₈	#11924028	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂×C₄×D₇₃	1	2	yes	—
262	C₂³	—	146	2	no	—
263	C₂³	#14556638	1	1	yes	characteristic
264	Q₈	—	73	4	no	—
265	Q₈	—	146	2	no	—
266	D₄	—	73	4	no	—
267	D₄	—	146	4	no	—
268	D₄	—	146	4	no	—
269	D₄	—	146	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	146	2	no	—
274	C₂×C₄	—	146	4	no	—
275	C₂×C₄	—	146	4	no	—
276	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
280	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
281	C₂×C₄	#14556637	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	292	4	no	—
284	C₈	—	146	4	no	—
285	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
286	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
287	C₈	C₄×D₂₉₂	1	2	yes	—
288	C₈	C₄×D₂₉₂	1	2	yes	—
289	C₂²	—	4	1	no	—
290	C₂²	—	584	2	no	—
291	C₂²	—	146	4	no	—
292	C₂²	#17989067	1	1	yes	characteristic
293	C₂²	#17989242	1	1	yes	characteristic
294	C₂²	Q₈×D₂₉₂	1	1	yes	characteristic
295	C₄	—	4	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	146	4	no	—
298	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
299	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
300	C₄	D₄×D₂₉₂	1	1	yes	characteristic
301	C₄	#17989297	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	292	4	no	—
306	C₂	C₄⋊C₄×D₂₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1773	C₂
socle	C₂×C₇₃	D₄×C₄⋊C₄
prime radical	C₂×#14556613	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1773	D₇₃
hypocenter	C₇₃	#22695
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1773

Complements

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

N	H ≅ G/N	type
3. #22618539	C₂	semidirect
4. #22610721	C₂	semidirect
5. #22610172	C₂	semidirect
6. #22610106	C₂	semidirect
7. #22609982	C₂	semidirect
8. C₇₃×#1773	C₂	semidirect
13. #17988710	C₄	semidirect
161. #1773	D₇₃	semidirect
162. C₇₃	#22695	semidirect

Maximal subgroups

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

subgroup	index
#22618849	2
C₂×#17988710	2
#22618539	2
#22610721	2
#22610172	2
#22610106	2
#22609982	2
C₇₃×#1773	2
#22695	73