Subgroups — #33239030

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989003	C₂	1	1	yes	maximal, characteristic
2	#22618528	C₂	1	2	yes	maximal
3	#22611720	C₂	1	2	yes	maximal, complemented
4	#22611716	C₂	1	4	yes	maximal, complemented
5	#22611714	C₂	1	2	yes	maximal, complemented
6	C₂×#17988415	C₂	1	1	yes	maximal, characteristic, complemented
7	#367×D₇₃	C₂	1	2	yes	maximal, complemented
8	C₇₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556597	C₂²	1	1	yes	characteristic
10	#17989067	—	2	2	no	—
11	#17989003	C₄	1	8	yes	complemented
12	C₂×#14556531	C₂²	1	1	yes	characteristic
13	C₂×#14556530	C₂²	1	2	yes	—
14	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
15	#17988630	—	2	2	no	—
16	#17988626	—	2	4	no	—
17	#17988624	C₂²	1	4	yes	—
18	C₂×#14556525	C₂²	1	2	yes	—
19	#107×D₇₃	C₂²	1	4	yes	—
20	#17988415	C₂²	1	2	yes	—
21	#17988415	C₂²	1	2	yes	—
22	#17988414	C₂²	1	4	yes	—
23	#17988412	C₂²	1	4	yes	—
24	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	#427×C₇₃	C₂²	1	2	yes	—
26	#367×C₇₃	C₂²	1	1	yes	characteristic
27	#367×C₇₃	C₂²	1	1	yes	characteristic
28	#366×C₇₃	C₂²	1	2	yes	—
29	C₂×#11924028	D₄	1	1	yes	characteristic
30	#14556597	C₂×C₄	1	4	yes	—
31	#14556597	Q₈	1	4	yes	—
32	C₂×#11924012	D₄	1	1	yes	characteristic
33	C₂×#11924011	D₄	1	2	yes	—
34	C₂×C₈×D₇₃	C₂³	1	2	yes	—
35	C₄×Dic₁₄₆	—	4	2	no	—
36	C₄×D₂₉₂	—	4	2	no	—
37	#14556551	—	4	4	no	—
38	#14556549	—	2	4	no	—
39	C₂×C₄⋊Dic₇₃	—	2	2	no	—
40	C₄⋊C₄×D₇₃	—	2	4	no	—
41	#14556531	C₂×C₄	1	4	yes	complemented
42	#14556530	C₂×C₄	1	8	yes	complemented
43	C₂×#11924003	C₂³	1	2	yes	—
44	C₁₆×D₇₃	C₂×C₄	1	8	yes	complemented
45	#14556525	—	2	2	no	—
46	#14556525	C₂³	1	4	yes	—
47	#14556524	—	2	4	no	—
48	#14556522	—	2	4	no	—
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	#117×C₇₃	—	2	2	no	—
51	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
52	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
54	#107×C₇₃	C₂³	1	2	yes	—
55	#107×C₇₃	C₂³	1	2	yes	—
56	#106×C₇₃	—	2	2	no	—
57	#11924028	C₄⋊C₄	1	4	yes	—
58	#11924028	Q₁₆	1	4	yes	—
59	C₂×Dic₁₄₆	D₈	1	1	yes	characteristic
60	C₂×D₂₉₂	D₈	1	1	yes	characteristic
61	C₂×C₂²⋊D₇₃	D₈	1	2	yes	—
62	C₂²×Dic₇₃	—	4	2	no	—
63	C₂×C₄×D₇₃	—	4	4	no	—
64	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
65	#11924012	C₄⋊C₄	1	4	yes	—
66	#11924011	C₄⋊C₄	1	8	yes	—
67	C₂×#10596831	C₂×D₄	1	2	yes	—
68	C₈×D₇₃	C₂²×C₄	1	4	yes	—
69	C₈×D₇₃	C₂×Q₈	1	4	yes	—
70	C₄⋊Dic₇₃	—	2	4	no	—
71	C₄⋊Dic₇₃	—	4	2	no	—
72	#11924007	—	4	4	no	—
73	#11924005	—	4	4	no	—
74	C₄×Dic₇₃	—	4	4	no	—
75	#11924003	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
80	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₇₃	—	2	2	no	—
82	C₄⋊C₄×C₇₃	—	2	2	no	—
83	C₂²⋊C₄×C₇₃	—	4	2	no	—
84	C₄²×C₇₃	—	4	2	no	—
85	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
86	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
88	C₂²×D₇₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₇₃	—	4	2	no	—
90	C₂×C₄×C₇₃	—	4	2	no	—
91	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
94	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
95	C₂²⋊D₇₃	#107	1	8	yes	—
96	C₂×Dic₇₃	—	4	4	no	—
97	C₂×Dic₇₃	—	16	2	no	—
98	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
99	D₂₉₂	#107	1	4	yes	—
100	C₄×D₇₃	—	16	4	no	—
101	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
102	C₄×D₇₃	C₂×Q₁₆	1	4	yes	—
103	Dic₁₄₆	#107	1	4	yes	—
104	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
105	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
106	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
107	#10596831	C₂×C₄⋊C₄	1	4	yes	—
108	C₂²×C₇₃	—	2	2	no	—
109	C₂²×C₇₃	C₂×#107	1	2	yes	—
110	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
111	C₂×D₇₃	—	2	4	no	—
112	C₂×D₇₃	C₂×#107	1	4	yes	—
113	C₄×C₇₃	—	8	2	no	—
114	C₄×C₇₃	—	8	2	no	—
115	C₄×C₇₃	C₂×#107	1	2	yes	—
116	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
117	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
118	Dic₇₃	—	8	4	no	—
119	Dic₇₃	C₂×#107	1	4	yes	—
120	#33411	—	73	1	no	maximal, Sylow
121	C₂×C₇₃	—	2	2	no	—
122	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
123	C₂×C₇₃	#3416	1	2	yes	—
124	D₇₃	—	2	4	no	—
125	C₂×#504	—	73	1	no	—
126	#2913	—	73	2	no	—
127	#2159	—	73	2	no	—
128	#2159	—	73	4	no	—
129	#2158	—	73	2	no	—
130	#2158	D₇₃	1	1	yes	characteristic, complemented
131	C₂×#367	—	73	1	no	—
132	C₂×#367	—	73	2	no	—
133	C₇₃	#33411	1	1	yes	characteristic, complemented, Sylow
134	C₂×#131	—	73	1	no	—
135	#520	—	146	2	no	—
136	#504	—	73	8	no	—
137	C₂×M₃₂	—	73	1	no	—
138	C₂×M₃₂	—	73	2	no	—
139	C₂²×C₁₆	—	73	2	no	—
140	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
141	#428	—	146	4	no	—
142	#428	—	146	2	no	—
143	#427	—	73	4	no	—
144	#427	C₂×D₇₃	1	2	yes	—
145	C₂×#107	—	73	2	no	—
146	C₂×#107	—	73	4	no	—
147	#367	—	73	2	no	—
148	#367	—	73	2	no	—
149	#367	—	73	4	no	—
150	#367	C₂×D₇₃	1	1	yes	characteristic
151	#367	C₂×D₇₃	1	1	yes	characteristic
152	#366	—	73	4	no	—
153	#366	C₂×D₇₃	1	2	yes	—
154	C₂×#41	—	73	1	no	—
155	#131	—	73	4	no	—
156	#131	—	73	4	no	—
157	C₂×M₁₆	—	73	1	no	—
158	C₂×M₁₆	—	73	2	no	—
159	C₂²×C₈	—	73	2	no	—
160	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
161	C₄×Q₈	—	292	2	no	—
162	C₄×D₄	—	292	2	no	—
163	C₄×D₄	—	292	4	no	—
164	#117	—	2	2	no	—
165	#117	—	146	4	no	—
166	C₂×C₄⋊C₄	—	146	4	no	—
167	C₂×C₄⋊C₄	—	146	2	no	—
168	M₃₂	—	73	4	no	—
169	M₃₂	—	73	8	no	—
170	C₂×C₁₆	—	73	2	no	—
171	C₂×C₁₆	—	73	8	no	—
172	C₂×C₁₆	C₄×D₇₃	1	4	yes	complemented
173	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
174	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
175	#107	—	73	4	no	—
176	#107	—	146	4	no	—
177	#107	—	146	2	no	—
178	#107	C₂²×D₇₃	1	2	yes	—
179	#107	C₂²×D₇₃	1	2	yes	—
180	#106	—	2	2	no	—
181	#106	—	146	4	no	—
182	#41	—	73	4	no	—
183	#41	—	73	4	no	—
184	C₂×Q₈	—	73	1	no	—
185	C₂×D₄	—	73	2	no	—
186	C₂×D₄	—	73	1	no	—
187	C₂²×C₄	—	292	4	no	—
188	C₂²×C₄	—	292	2	no	—
189	C₂²×C₄	—	73	2	no	—
190	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
191	M₁₆	—	73	4	no	—
192	M₁₆	—	73	8	no	—
193	C₂×C₈	—	73	2	no	—
194	C₂×C₈	—	73	4	no	—
195	C₂×C₈	—	73	4	no	—
196	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
197	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
198	C₂×C₈	Q₈×D₇₃	1	2	yes	—
199	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	292	4	no	—
203	C₄⋊C₄	—	292	2	no	—
204	C₄⋊C₄	—	146	4	no	—
205	C₂²⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	292	4	no	—
207	C₄²	—	4	2	no	—
208	C₄²	—	292	4	no	—
209	C₁₆	—	73	4	no	—
210	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
211	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
212	C₂³	—	73	2	no	—
213	C₂³	D₈×D₇₃	1	1	yes	characteristic
214	Q₈	—	73	4	no	—
215	D₄	—	73	8	no	—
216	D₄	—	73	4	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	292	4	no	—
220	C₂×C₄	—	73	2	no	—
221	C₂×C₄	—	73	4	no	—
222	C₂×C₄	—	73	4	no	—
223	C₂×C₄	—	1168	4	no	—
224	C₂×C₄	—	1168	2	no	—
225	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
226	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
227	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
228	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
229	C₈	—	73	4	no	—
230	C₈	C₄⋊C₄×D₇₃	1	2	yes	—
231	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
232	C₈	C₂×Q₈×D₇₃	1	1	yes	characteristic
233	C₂²	—	2	2	no	—
234	C₂²	—	73	4	no	—
235	C₂²	—	146	4	no	—
236	C₂²	#107×D₇₃	1	2	yes	—
237	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
238	C₄	—	8	2	no	—
239	C₄	—	8	2	no	—
240	C₄	—	584	4	no	—
241	C₄	—	73	4	no	—
242	C₄	#107×D₇₃	1	2	yes	—
243	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
244	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
245	C₂	—	2	2	no	—
246	C₂	—	146	4	no	—
247	C₂	C₂×#107×D₇₃	1	1	yes	characteristic
248	C₂	#22623074	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2158	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂×#11924028	D₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2158	D₇₃
hypocenter	C₇₃	#33411
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #22611720	C₂	semidirect
4. #22611716	C₂	semidirect
5. #22611714	C₂	semidirect
6. C₂×#17988415	C₂	semidirect
7. #367×D₇₃	C₂	semidirect
8. C₇₃×#2158	C₂	semidirect
11. #17989003	C₄	semidirect
41. #14556531	C₂×C₄	semidirect
42. #14556530	C₂×C₄	semidirect
44. C₁₆×D₇₃	C₂×C₄	semidirect
51. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
130. #2158	D₇₃	semidirect
133. C₇₃	#33411	semidirect
172. C₂×C₁₆	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989003	2
#22618528	2
#22611720	2
#22611716	2
#22611714	2
C₂×#17988415	2
#367×D₇₃	2
C₇₃×#2158	2
#33411	73