Subgroups — #33239398

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22623067	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988705	C₂	1	2	yes	maximal, complemented
3	#22611818	C₂	1	4	yes	maximal, complemented
4	#22611813	C₂	1	4	yes	maximal, complemented
5	#22611726	C₂	1	1	yes	maximal, characteristic, complemented
6	#22611660	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556613	C₂²	1	2	yes	—
9	C₂×#14556597	—	2	2	no	—
10	#17989030	C₄	1	4	yes	complemented
11	#17989025	C₄	1	4	yes	complemented
12	#17989019	C₂²	1	4	yes	—
13	#17988705	—	2	2	no	—
14	#17988705	—	2	2	no	—
15	#17988705	C₂²	1	4	yes	complemented
16	#17988705	C₂²	1	4	yes	complemented
17	C₂×#14556527	C₂²	1	2	yes	—
18	C₂×#14556519	—	2	2	no	—
19	#17988417	C₂²	1	4	yes	complemented
20	#17988395	C₂²	1	8	yes	complemented
21	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
22	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#346×C₇₃	C₂²	1	2	yes	complemented
25	C₄×C₁₆×C₇₃	C₂²	1	2	yes	complemented
26	C₂×#11924028	—	2	2	no	—
27	#14556613	—	2	2	no	—
28	#14556613	—	2	2	no	—
29	#14556613	C₂×C₄	1	4	yes	—
30	#14556613	C₂³	1	4	yes	—
31	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
32	C₂×#11924017	—	2	2	no	—
33	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
34	#14556597	—	2	4	no	—
35	#14556597	—	2	4	no	—
36	#14556597	—	4	2	no	—
37	#14556597	—	4	2	no	—
38	C₂×#11924012	—	2	2	no	—
39	C₂×#11924011	—	4	2	no	—
40	C₂×C₈×D₇₃	—	4	2	no	—
41	Dic₅₈₄	C₂×C₄	1	4	yes	complemented
42	#14556536	C₂×C₄	1	8	yes	complemented
43	D₁₁₆₈	C₂×C₄	1	4	yes	complemented
44	#14556527	—	2	2	no	—
45	#14556527	C₂³	1	4	yes	—
46	#14556519	—	2	4	no	—
47	#14556519	—	2	4	no	—
48	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
51	M₃₂×C₇₃	—	2	2	no	—
52	M₃₂×C₇₃	C₂×C₄	1	4	yes	—
53	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
54	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
55	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
57	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	#11924028	—	2	4	no	—
60	#11924028	—	2	4	no	—
61	#11924028	—	4	2	no	—
62	#11924028	—	4	2	no	—
63	C₂×Dic₁₄₆	—	2	2	no	—
64	C₂×D₂₉₂	—	2	2	no	—
65	C₂×C₂²⋊D₇₃	—	4	2	no	—
66	C₂×C₄×D₇₃	—	4	2	no	—
67	Dic₂₉₂	—	2	2	no	—
68	Dic₂₉₂	C₂²×C₄	1	4	yes	—
69	#11924017	—	2	4	no	—
70	#11924017	—	2	4	no	—
71	D₅₈₄	—	2	2	no	—
72	D₅₈₄	C₂²×C₄	1	4	yes	—
73	#11924012	—	2	4	no	—
74	#11924012	—	4	2	no	—
75	#11924011	—	4	4	no	—
76	#11924011	—	4	4	no	—
77	C₂×#10596831	—	4	2	no	—
78	C₈×D₇₃	—	4	4	no	—
79	C₈×D₇₃	—	4	4	no	—
80	C₂²×C₄×C₇₃	—	2	1	no	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₃	D₈	1	2	yes	—
83	C₂×C₈×C₇₃	D₈	1	2	yes	—
84	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	#41	1	2	yes	—
89	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
90	C₄²×C₇₃	—	2	1	no	—
91	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₁₆×C₇₃	—	2	2	no	—
94	C₁₆×C₇₃	C₂²×C₄	1	4	yes	—
95	C₂³×C₇₃	#135	1	1	yes	characteristic
96	C₂²×D₇₃	—	4	2	no	—
97	C₂×C₄×C₇₃	—	2	1	no	—
98	C₂×C₄×C₇₃	—	2	1	no	—
99	C₂×C₄×C₇₃	—	4	2	no	—
100	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₇₃	—	4	4	no	—
105	C₂²⋊D₇₃	—	4	4	no	—
106	C₂×Dic₇₃	—	4	2	no	—
107	D₂₉₂	—	2	4	no	—
108	D₂₉₂	—	4	2	no	—
109	C₄×D₇₃	—	4	4	no	—
110	C₄×D₇₃	—	4	4	no	—
111	Dic₁₄₆	—	2	4	no	—
112	Dic₁₄₆	—	4	2	no	—
113	C₈×C₇₃	C₄×D₄	1	2	yes	—
114	C₈×C₇₃	C₂×D₈	1	2	yes	—
115	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
116	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
117	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
119	#10596831	—	4	4	no	—
120	C₂²×C₇₃	—	4	1	no	—
121	C₂²×C₇₃	C₄×D₈	1	2	yes	—
122	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
123	C₂×D₇₃	—	4	4	no	—
124	C₂×D₇₃	—	8	4	no	—
125	C₄×C₇₃	—	2	2	no	—
126	C₄×C₇₃	—	4	1	no	—
127	C₄×C₇₃	C₄×D₈	1	2	yes	—
128	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
130	Dic₇₃	—	4	4	no	—
131	#33463	—	73	1	no	maximal, Sylow
132	C₂×C₇₃	—	2	1	no	—
133	C₂×C₇₃	—	2	2	no	—
134	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
135	D₇₃	—	8	4	no	—
136	#3414	—	73	1	no	—
137	C₂×#443	—	73	2	no	—
138	#2179	—	73	4	no	—
139	#2178	—	73	4	no	—
140	#2161	—	73	1	no	—
141	#2143	—	73	2	no	—
142	C₄×M₃₂	D₇₃	1	1	yes	characteristic, complemented
143	C₇₃	#33463	1	1	yes	characteristic, complemented, Sylow
144	C₂×#135	—	73	2	no	—
145	C₂×#131	—	146	2	no	—
146	#510	—	73	4	no	—
147	#509	—	73	4	no	—
148	#508	—	73	4	no	—
149	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
150	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
151	#443	—	73	4	no	—
152	#443	—	73	4	no	—
153	#443	—	146	2	no	—
154	#443	—	146	2	no	—
155	C₂×#108	—	73	2	no	—
156	C₂×#104	—	146	2	no	—
157	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
158	#368	—	73	4	no	—
159	#359	—	73	8	no	—
160	#346	C₂×D₇₃	1	2	yes	complemented
161	C₄×C₁₆	C₂×D₇₃	1	2	yes	complemented
162	C₂×#41	—	146	2	no	—
163	#135	—	73	4	no	—
164	#135	—	73	4	no	—
165	#135	—	146	2	no	—
166	#135	—	146	2	no	—
167	C₂×Q₁₆	—	73	2	no	—
168	C₂×SD₁₆	—	146	2	no	—
169	C₂×D₈	—	73	2	no	—
170	#131	—	292	2	no	—
171	#131	—	292	2	no	—
172	#131	—	146	4	no	—
173	#131	—	146	4	no	—
174	C₂×M₁₆	—	292	2	no	—
175	C₂×M₁₆	—	146	2	no	—
176	C₂²×C₈	—	292	2	no	—
177	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
179	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
180	Q₃₂	—	73	4	no	—
181	SD₃₂	—	73	8	no	—
182	D₁₆	—	73	4	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	C₄×D₇₃	1	4	yes	—
185	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
186	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
187	#108	—	73	4	no	—
188	#108	—	146	2	no	—
189	#104	—	146	4	no	—
190	#104	—	146	4	no	—
191	C₄×C₈	D₂₉₂	1	1	yes	characteristic
192	C₄×C₈	D₂₉₂	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
195	#41	—	292	2	no	—
196	#41	—	292	2	no	—
197	#41	—	146	4	no	—
198	#41	—	146	4	no	—
199	C₂×Q₈	—	146	2	no	—
200	C₂×D₄	—	292	2	no	—
201	C₂×D₄	—	146	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	292	2	no	—
204	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
205	Q₁₆	—	73	4	no	—
206	Q₁₆	—	146	2	no	—
207	SD₁₆	—	146	4	no	—
208	SD₁₆	—	146	4	no	—
209	D₈	—	73	4	no	—
210	D₈	—	146	2	no	—
211	M₁₆	—	292	2	no	—
212	M₁₆	—	292	4	no	—
213	M₁₆	—	292	4	no	—
214	M₁₆	—	146	4	no	—
215	C₂×C₈	—	292	2	no	—
216	C₂×C₈	—	292	4	no	—
217	C₂×C₈	—	292	4	no	—
218	C₂×C₈	D₅₈₄	1	2	yes	—
219	C₂×C₈	D₅₈₄	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
221	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
222	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
223	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
224	C₂×C₈	#11924028	1	2	yes	—
225	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
226	C₄²	—	2	1	no	—
227	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
228	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
229	C₁₆	—	2	2	no	—
230	C₁₆	C₂×C₄×D₇₃	1	4	yes	—
231	C₂³	—	292	2	no	—
232	C₂³	#14556613	1	1	yes	characteristic
233	Q₈	—	292	2	no	—
234	Q₈	—	146	4	no	—
235	D₄	—	292	4	no	—
236	D₄	—	292	4	no	—
237	D₄	—	292	2	no	—
238	D₄	—	146	4	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	292	2	no	—
243	C₂×C₄	—	292	4	no	—
244	C₂×C₄	—	292	4	no	—
245	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
246	C₂×C₄	C₂×D₅₈₄	1	2	yes	—
247	C₂×C₄	#14556613	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
249	C₈	—	292	4	no	—
250	C₈	C₄×D₂₉₂	1	2	yes	—
251	C₈	C₂×D₅₈₄	1	2	yes	—
252	C₈	C₂×D₅₈₄	1	1	yes	characteristic
253	C₈	C₂×D₅₈₄	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
255	C₈	C₂×#11924028	1	1	yes	characteristic
256	C₂²	—	4	1	no	—
257	C₂²	—	292	4	no	—
258	C₂²	—	584	4	no	—
259	C₂²	C₄×D₅₈₄	1	2	yes	—
260	C₂²	C₂×#14556613	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	1	no	—
263	C₄	—	292	4	no	—
264	C₄	C₄×D₅₈₄	1	2	yes	—
265	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
266	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
267	C₂	—	2	1	no	—
268	C₂	—	2	2	no	—
269	C₂	—	584	4	no	—
270	C₂	C₂×C₄×D₅₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₅₈₄
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×M₃₂×C₇₃	C₂
socle	C₂×C₇₃	C₂×C₄×D₈
prime radical	#22623067	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₄×M₃₂	D₇₃
hypocenter	C₇₃	#33463
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#17988705	C₂	semidirect
3. #22611818	C₂	semidirect
4. #22611813	C₂	semidirect
5. #22611726	C₂	semidirect
6. #22611660	C₂	semidirect
7. C₄×M₃₂×C₇₃	C₂	semidirect
10. #17989030	C₄	semidirect
11. #17989025	C₄	semidirect
15. #17988705	C₂²	semidirect
16. #17988705	C₂²	semidirect
19. #17988417	C₂²	semidirect
20. #17988395	C₂²	semidirect
24. #346×C₇₃	C₂²	semidirect
25. C₄×C₁₆×C₇₃	C₂²	semidirect
41. Dic₅₈₄	C₂×C₄	semidirect
42. #14556536	C₂×C₄	semidirect
43. D₁₁₆₈	C₂×C₄	semidirect
142. C₄×M₃₂	D₇₃	semidirect
143. C₇₃	#33463	semidirect
160. #346	C₂×D₇₃	semidirect
161. C₄×C₁₆	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22623067	2
C₂×#17988705	2
#22611818	2
#22611813	2
#22611726	2
#22611660	2
C₄×M₃₂×C₇₃	2
#33463	73