Subgroups — #33241338

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988964	C₂	1	1	yes	maximal, characteristic
2	#22612174	C₂	1	4	yes	maximal, complemented
3	#22612156	C₂	1	4	yes	maximal, complemented
4	#22611646	C₂	1	4	yes	maximal, complemented
5	C₄×#14556531	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₈₄	—	2	1	no	—
8	C₂²×D₅₈₄	C₂²	1	2	yes	—
9	C₂×#14556588	—	2	1	no	—
10	C₂×#14556531	C₂²	1	2	yes	—
11	#17988964	—	2	1	no	—
12	#17988964	—	2	1	no	—
13	#17988964	C₂²	1	2	yes	—
14	#17988964	C₂²	1	2	yes	—
15	#17988387	C₂²	1	16	yes	complemented
16	#17988363	C₂²	1	2	yes	complemented
17	C₄×#11924003	C₂²	1	2	yes	complemented
18	#533×C₇₃	—	2	1	no	—
19	#500×C₇₃	C₂²	1	2	yes	complemented
20	#499×C₇₃	—	2	1	no	—
21	#499×C₇₃	C₂²	1	2	yes	complemented
22	#498×C₇₃	—	2	1	no	—
23	#427×C₇₃	—	2	1	no	—
24	#427×C₇₃	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂²×D₂₉₂	—	2	2	no	—
27	C₂²×D₂₉₂	—	4	2	no	—
28	C₂×D₅₈₄	—	2	2	no	—
29	C₂×D₅₈₄	—	2	2	no	—
30	C₂×D₅₈₄	—	2	2	no	—
31	C₂×D₅₈₄	—	2	8	no	—
32	C₂×D₅₈₄	—	4	4	no	—
33	C₂×D₅₈₄	C₂³	1	4	yes	—
34	#14556588	—	2	2	no	—
35	#14556588	—	2	2	no	—
36	#14556588	—	4	2	no	—
37	#14556531	—	2	4	no	—
38	C₂×#11924003	C₂³	1	4	yes	—
39	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₇₃	—	2	2	no	—
42	#125×C₇₃	—	2	2	no	—
43	C₂²⋊Q₈×C₇₃	—	4	2	no	—
44	C₄×Q₈×C₇₃	—	4	2	no	—
45	#117×C₇₃	—	2	2	no	—
46	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
47	#107×C₇₃	—	2	2	no	—
48	#107×C₇₃	C₂³	1	4	yes	—
49	#106×C₇₃	—	2	2	no	—
50	#106×C₇₃	—	2	2	no	—
51	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
53	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
55	C₂³×D₇₃	—	4	2	no	—
56	C₂×D₂₉₂	—	2	4	no	—
57	C₂×D₂₉₂	—	4	4	no	—
58	C₂×D₂₉₂	—	4	4	no	—
59	C₂×D₂₉₂	—	4	4	no	—
60	C₂×D₂₉₂	—	4	4	no	—
61	C₂×D₂₉₂	—	4	2	no	—
62	C₂×D₂₉₂	—	8	8	no	—
63	D₅₈₄	—	2	8	no	—
64	D₅₈₄	—	4	4	no	—
65	D₅₈₄	—	4	4	no	—
66	D₅₈₄	—	4	4	no	—
67	#11924003	—	2	4	no	—
68	C₂×Q₈×C₇₃	—	4	2	no	—
69	C₂²×C₄×C₇₃	D₈	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₃	SD₁₆	1	4	yes	—
72	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₇₃	#41	1	4	yes	—
76	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₇₃	—	2	4	no	—
78	C₄⋊C₄×C₇₃	—	4	4	no	—
79	C₄⋊C₄×C₇₃	—	4	2	no	—
80	C₂²⋊C₄×C₇₃	—	4	2	no	—
81	C₄²×C₇₃	—	4	2	no	—
82	C₄²×C₇₃	D₈	1	2	yes	—
83	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
86	C₂²×D₇₃	—	4	4	no	—
87	C₂²×D₇₃	—	8	8	no	—
88	C₂²×D₇₃	—	8	4	no	—
89	Q₈×C₇₃	—	8	4	no	—
90	C₂×C₄×C₇₃	—	2	4	no	—
91	C₂×C₄×C₇₃	—	4	4	no	—
92	C₂×C₄×C₇₃	#124	1	2	yes	—
93	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
97	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
98	D₂₉₂	—	4	4	no	—
99	D₂₉₂	—	4	4	no	—
100	D₂₉₂	—	8	8	no	—
101	D₂₉₂	—	8	4	no	—
102	D₂₉₂	—	8	8	no	—
103	C₈×C₇₃	#124	1	2	yes	—
104	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
105	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
106	C₈×C₇₃	C₂×#41	1	2	yes	—
107	C₂²×C₇₃	—	2	2	no	—
108	C₂²×C₇₃	#486	1	2	yes	—
109	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
110	C₂×D₇₃	—	8	4	no	—
111	C₂×D₇₃	—	8	8	no	—
112	C₂×D₇₃	—	16	8	no	—
113	C₄×C₇₃	—	2	2	no	—
114	C₄×C₇₃	—	8	4	no	—
115	C₄×C₇₃	#486	1	2	yes	—
116	C₄×C₇₃	#509	1	4	yes	—
117	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
118	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
119	#33704	—	73	1	no	maximal, Sylow
120	C₂×C₇₃	—	2	2	no	—
121	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
122	C₂×C₇₃	C₂×#509	1	2	yes	—
123	D₇₃	—	16	4	no	—
124	#3160	D₇₃	1	1	yes	characteristic, complemented
125	C₂×#493	—	73	1	no	—
126	#2244	—	73	4	no	—
127	#2240	—	73	4	no	—
128	#2140	—	73	4	no	—
129	C₄×M₃₂	—	73	1	no	—
130	C₇₃	#33704	1	1	yes	characteristic, complemented, Sylow
131	C₂²×D₈	—	73	2	no	—
132	C₂²×D₈	—	146	1	no	—
133	#533	—	2	1	no	—
134	C₂×#127	—	146	1	no	—
135	C₂×M₃₂	—	73	2	no	—
136	#500	C₂×D₇₃	1	2	yes	complemented
137	#499	—	2	1	no	—
138	#499	C₂×D₇₃	1	2	yes	complemented
139	#498	—	2	1	no	—
140	#493	—	73	2	no	—
141	#493	—	73	2	no	—
142	#493	—	146	1	no	—
143	#493	—	146	1	no	—
144	#427	—	2	1	no	—
145	#427	C₂×D₇₃	1	2	yes	—
146	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
147	#357	—	73	16	no	—
148	#346	—	73	2	no	—
149	C₄×C₁₆	—	73	2	no	—
150	C₂²×D₄	—	292	2	no	—
151	C₂²×D₄	—	146	2	no	—
152	C₂×D₈	—	292	4	no	—
153	C₂×D₈	—	73	4	no	—
154	C₂×D₈	—	146	2	no	—
155	C₂×D₈	—	146	2	no	—
156	C₂×D₈	—	146	2	no	—
157	C₂×D₈	—	146	8	no	—
158	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
159	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
160	C₄⋊Q₈	—	2	2	no	—
161	#127	—	292	2	no	—
162	#127	—	146	2	no	—
163	#127	—	146	2	no	—
164	#125	—	2	2	no	—
165	C₂²⋊Q₈	—	4	2	no	—
166	C₄×Q₈	—	4	2	no	—
167	#117	—	2	2	no	—
168	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
169	M₃₂	—	146	4	no	—
170	C₂×C₁₆	—	73	4	no	—
171	#107	—	2	2	no	—
172	#107	C₂²×D₇₃	1	4	yes	—
173	#106	—	2	2	no	—
174	#106	—	2	2	no	—
175	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
176	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
177	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
178	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
179	C₂⁴	—	292	2	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×D₄	—	292	4	no	—
182	C₂×D₄	—	292	2	no	—
183	C₂×D₄	—	292	4	no	—
184	C₂×D₄	—	292	4	no	—
185	C₂×D₄	—	292	4	no	—
186	C₂×D₄	—	584	8	no	—
187	C₂×D₄	—	146	4	no	—
188	C₂²×C₄	#11924013	1	2	yes	—
189	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
190	D₈	—	292	4	no	—
191	D₈	—	292	4	no	—
192	D₈	—	292	4	no	—
193	D₈	—	146	8	no	—
194	C₂×C₈	#11924015	1	4	yes	—
195	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
197	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
198	C₂×C₈	#11924029	1	4	yes	—
199	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	—	4	2	no	—
205	C₄²	#11924013	1	2	yes	—
206	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
208	C₁₆	—	146	4	no	—
209	C₂³	—	292	4	no	—
210	C₂³	—	584	8	no	—
211	C₂³	—	584	4	no	—
212	C₂³	C₂×#11924013	1	1	yes	characteristic
213	Q₈	—	8	4	no	—
214	D₄	—	292	4	no	—
215	D₄	—	292	4	no	—
216	D₄	—	584	8	no	—
217	D₄	—	584	4	no	—
218	D₄	—	584	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	#14556580	1	2	yes	—
222	C₂×C₄	C₂×#11924013	1	2	yes	—
223	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
224	C₂×C₄	C₂×#11924013	1	2	yes	—
225	C₂×C₄	C₂×#11924015	1	2	yes	—
226	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
227	C₈	#14556580	1	2	yes	—
228	C₈	C₂×#11924015	1	2	yes	—
229	C₈	C₂×#11924015	1	2	yes	—
230	C₈	C₂×#11924029	1	2	yes	—
231	C₂²	—	2	2	no	—
232	C₂²	—	584	8	no	—
233	C₂²	—	584	4	no	—
234	C₂²	—	1168	8	no	—
235	C₂²	#17988935	1	2	yes	—
236	C₂²	C₂²×#11924013	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	8	4	no	—
239	C₄	#17988935	1	2	yes	—
240	C₄	#17989022	1	4	yes	—
241	C₄	C₂×#14556580	1	1	yes	characteristic
242	C₄	C₂²×#11924015	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	—	1168	4	no	—
245	C₂	C₂×#17988935	1	1	yes	characteristic
246	C₂	C₂×#17989022	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3160	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#17988964	C₂
hypercenter	#3160	D₇₃
hypocenter	C₇₃	#33704
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #22612174	C₂	semidirect
3. #22612156	C₂	semidirect
4. #22611646	C₂	semidirect
5. C₄×#14556531	C₂	semidirect
6. C₇₃×#3160	C₂	semidirect
15. #17988387	C₂²	semidirect
16. #17988363	C₂²	semidirect
17. C₄×#11924003	C₂²	semidirect
19. #500×C₇₃	C₂²	semidirect
21. #499×C₇₃	C₂²	semidirect
124. #3160	D₇₃	semidirect
130. C₇₃	#33704	semidirect
136. #500	C₂×D₇₃	semidirect
138. #499	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988964	2
#22612174	2
#22612156	2
#22611646	2
C₄×#14556531	2
C₇₃×#3160	2
#33704	73