Subgroups — #33240425

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989019	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#22619513	C₂	1	2	yes	maximal, complemented
4	#22612028	C₂	1	4	yes	maximal, complemented
5	#22612023	C₂	1	2	yes	maximal, complemented
6	#22612018	C₂	1	2	yes	maximal, complemented
7	#22611636	C₂	1	2	yes	maximal, complemented
8	C₇₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	2	yes	—
10	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	#17989099	—	2	2	no	—
12	#17989019	—	2	4	no	—
13	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic, complemented
14	C₂×#14556536	C₂²	1	2	yes	complemented
15	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic, complemented
16	C₂²×#11924003	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₇₃	—	2	4	no	—
18	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
19	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
20	#17988829	C₂²	1	4	yes	—
21	#17988824	C₂²	1	4	yes	—
22	#17988817	—	2	4	no	—
23	#17988581	—	2	2	no	—
24	#17988390	C₂²	1	4	yes	complemented
25	#17988387	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	#427×C₇₃	C₂²	1	2	yes	—
28	#367×C₇₃	C₂²	1	2	yes	complemented
29	#366×C₇₃	C₂²	1	2	yes	complemented
30	C₂×#11924028	—	2	2	no	—
31	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
32	#14556613	—	2	4	no	—
33	#14556613	—	2	4	no	—
34	C₂×Dic₂₉₂	C₂³	1	2	yes	—
35	C₂×#11924017	—	2	2	no	—
36	C₂×D₅₈₄	C₂³	1	2	yes	—
37	C₂²×#10596831	D₄	1	1	yes	characteristic
38	C₂×C₈×D₇₃	—	2	2	no	—
39	C₂×C₈×D₇₃	—	2	2	no	—
40	C₂×C₈×D₇₃	D₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂³	1	2	yes	—
42	#14556575	—	4	2	no	—
43	#14556571	—	2	4	no	—
44	#14556566	—	2	4	no	—
45	#14556561	—	4	2	no	—
46	C₄×D₂₉₂	—	4	4	no	—
47	Dic₅₈₄	—	2	2	no	—
48	#14556536	—	2	4	no	—
49	D₁₁₆₈	—	2	2	no	—
50	C₂×#11924003	—	2	2	no	—
51	C₂×#11924003	C₂³	1	2	yes	—
52	C₁₆×D₇₃	—	2	4	no	—
53	C₁₆×D₇₃	—	2	4	no	—
54	#14556514	—	2	4	no	—
55	#14556511	—	2	4	no	—
56	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
57	#117×C₇₃	—	2	2	no	—
58	C₂×C₁₆×C₇₃	D₄	1	4	yes	complemented
59	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
61	#107×C₇₃	C₂³	1	4	yes	—
62	#106×C₇₃	—	2	2	no	—
63	C₂³×D₇₃	D₈	1	1	yes	characteristic
64	#11924028	—	4	4	no	—
65	#11924028	—	4	4	no	—
66	C₂×Dic₁₄₆	—	2	2	no	—
67	C₂×D₂₉₂	—	2	2	no	—
68	C₂×D₂₉₂	—	4	4	no	—
69	C₂×C₂²⋊D₇₃	—	4	2	no	—
70	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
71	C₂×C₄×D₇₃	—	2	2	no	—
72	C₂×C₄×D₇₃	—	2	2	no	—
73	C₂×C₄×D₇₃	—	4	2	no	—
74	C₂×C₄×D₇₃	D₈	1	2	yes	—
75	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
76	Dic₂₉₂	—	2	2	no	—
77	Dic₂₉₂	—	2	2	no	—
78	#11924017	—	4	4	no	—
79	D₅₈₄	—	2	2	no	—
80	D₅₈₄	—	2	2	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	C₂×D₄	1	2	yes	—
83	C₈×D₇₃	—	2	4	no	—
84	C₈×D₇₃	—	2	2	no	—
85	C₈×D₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	4	4	no	—
87	#11924005	—	4	4	no	—
88	#11924005	—	4	4	no	—
89	#11924003	—	2	2	no	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₇₃	#41	1	2	yes	—
94	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
95	C₄⋊C₄×C₇₃	—	2	4	no	—
96	C₂²⋊C₄×C₇₃	—	4	2	no	—
97	C₄²×C₇₃	—	4	2	no	—
98	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
99	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
100	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
101	C₂²×D₇₃	—	2	2	no	—
102	C₂²×D₇₃	—	2	4	no	—
103	C₂²×D₇₃	—	4	2	no	—
104	C₂²×D₇₃	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₇₃	—	4	4	no	—
106	C₂×C₄×C₇₃	#121	1	2	yes	—
107	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#135	1	2	yes	—
109	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
110	C₂²⋊D₇₃	—	8	4	no	—
111	C₂×Dic₇₃	—	2	2	no	—
112	C₂×Dic₇₃	—	4	2	no	—
113	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
114	D₂₉₂	—	4	2	no	—
115	D₂₉₂	—	4	2	no	—
116	D₂₉₂	—	8	8	no	—
117	C₄×D₇₃	—	2	2	no	—
118	C₄×D₇₃	—	2	2	no	—
119	C₄×D₇₃	—	2	4	no	—
120	C₄×D₇₃	—	8	4	no	—
121	Dic₁₄₆	—	4	2	no	—
122	Dic₁₄₆	—	4	2	no	—
123	C₈×C₇₃	#121	1	2	yes	—
124	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
126	#10596831	—	2	2	no	—
127	C₂²×C₇₃	—	2	2	no	—
128	C₂²×C₇₃	#466	1	2	yes	—
129	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
130	C₂×D₇₃	—	2	8	no	—
131	C₂×D₇₃	—	2	2	no	—
132	C₂×D₇₃	—	2	2	no	—
133	C₂×D₇₃	—	8	4	no	—
134	C₂×D₇₃	—	8	2	no	—
135	C₂×D₇₃	#508	1	4	yes	complemented
136	C₄×C₇₃	—	8	4	no	—
137	C₄×C₇₃	#466	1	2	yes	—
138	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
139	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
140	Dic₇₃	—	2	2	no	—
141	Dic₇₃	—	8	2	no	—
142	#33600	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	2	no	—
144	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
145	C₂×C₇₃	C₂×#508	1	2	yes	—
146	D₇₃	—	2	4	no	—
147	D₇₃	—	16	2	no	—
148	C₂×#508	—	73	1	no	—
149	C₂³×C₁₆	—	73	1	no	—
150	#3050	—	73	2	no	—
151	#2217	—	73	4	no	—
152	#2216	—	73	2	no	—
153	#2215	—	73	2	no	—
154	#2158	D₇₃	1	1	yes	characteristic, complemented
155	#2138	—	73	2	no	—
156	C₇₃	#33600	1	1	yes	characteristic, complemented, Sylow
157	C₂×#135	—	73	2	no	—
158	C₂³×C₈	—	73	1	no	—
159	#525	—	146	2	no	—
160	#508	—	146	4	no	—
161	C₂×Q₃₂	—	73	1	no	—
162	C₂×SD₃₂	—	73	2	no	—
163	C₂×D₁₆	—	73	1	no	—
164	C₂²×C₁₆	—	73	1	no	—
165	C₂²×C₁₆	—	73	2	no	—
166	C₂²×C₁₆	—	73	2	no	—
167	C₂²×C₁₆	—	146	4	no	—
168	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
169	#467	—	73	4	no	—
170	#466	—	73	4	no	—
171	#465	—	146	4	no	—
172	#427	C₂×D₇₃	1	2	yes	—
173	#416	—	146	2	no	—
174	#367	C₂×D₇₃	1	2	yes	complemented
175	#366	C₂×D₇₃	1	2	yes	complemented
176	#358	—	73	4	no	—
177	#357	—	73	4	no	—
178	C₂×#41	—	146	2	no	—
179	C₂³×C₄	—	73	1	no	—
180	#135	—	146	4	no	—
181	#135	—	146	4	no	—
182	C₂×Q₁₆	—	73	2	no	—
183	C₂×SD₁₆	—	146	2	no	—
184	C₂×D₈	—	73	2	no	—
185	C₂²×C₈	—	73	1	no	—
186	C₂²×C₈	—	73	2	no	—
187	C₂²×C₈	—	73	2	no	—
188	C₂²×C₈	—	146	2	no	—
189	C₂²×C₈	—	146	2	no	—
190	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
191	#123	—	292	2	no	—
192	C₂²⋊Q₈	—	146	4	no	—
193	#121	—	146	4	no	—
194	#120	—	292	2	no	—
195	C₄×D₄	—	292	4	no	—
196	#117	—	2	2	no	—
197	Q₃₂	—	146	2	no	—
198	SD₃₂	—	146	4	no	—
199	D₁₆	—	146	2	no	—
200	C₂×C₁₆	—	73	2	no	—
201	C₂×C₁₆	—	146	2	no	—
202	C₂×C₁₆	—	146	4	no	—
203	C₂×C₁₆	—	146	4	no	—
204	C₂×C₁₆	D₂₉₂	1	4	yes	complemented
205	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
207	#107	C₂²×D₇₃	1	4	yes	—
208	#106	—	2	2	no	—
209	#103	—	146	4	no	—
210	#102	—	146	4	no	—
211	C₂⁴	—	73	1	no	—
212	#41	—	292	4	no	—
213	#41	—	292	4	no	—
214	C₂×Q₈	—	146	2	no	—
215	C₂×D₄	—	292	4	no	—
216	C₂×D₄	—	292	2	no	—
217	C₂×D₄	—	146	2	no	—
218	C₂²×C₄	—	292	2	no	—
219	C₂²×C₄	—	73	1	no	—
220	C₂²×C₄	—	73	2	no	—
221	C₂²×C₄	—	73	2	no	—
222	C₂²×C₄	—	146	2	no	—
223	C₂²×C₄	—	146	2	no	—
224	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
225	Q₁₆	—	146	2	no	—
226	Q₁₆	—	146	2	no	—
227	SD₁₆	—	292	4	no	—
228	D₈	—	146	2	no	—
229	D₈	—	146	2	no	—
230	C₂×C₈	—	73	2	no	—
231	C₂×C₈	—	146	2	no	—
232	C₂×C₈	—	146	4	no	—
233	C₂×C₈	—	146	2	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
236	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
237	C₂×C₈	#11924029	1	2	yes	—
238	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	292	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	292	4	no	—
243	C₂²⋊C₄	—	292	4	no	—
244	C₄²	—	4	2	no	—
245	C₁₆	—	146	2	no	—
246	C₁₆	C₂×D₂₉₂	1	2	yes	—
247	C₁₆	C₂×D₂₉₂	1	2	yes	—
248	C₂³	—	292	2	no	—
249	C₂³	—	73	2	no	—
250	C₂³	—	146	4	no	—
251	C₂³	—	146	2	no	—
252	C₂³	D₈×D₇₃	1	1	yes	characteristic
253	Q₈	—	292	2	no	—
254	Q₈	—	292	2	no	—
255	D₄	—	292	2	no	—
256	D₄	—	292	2	no	—
257	D₄	—	584	8	no	—
258	D₄	—	584	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	292	2	no	—
261	C₂×C₄	—	584	4	no	—
262	C₂×C₄	—	73	2	no	—
263	C₂×C₄	—	146	2	no	—
264	C₂×C₄	—	146	4	no	—
265	C₂×C₄	—	146	2	no	—
266	C₂×C₄	—	146	2	no	—
267	C₂×C₄	#14556566	1	2	yes	—
268	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
269	C₂×C₄	#14556614	1	2	yes	—
270	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
271	C₈	—	146	2	no	—
272	C₈	#14556566	1	2	yes	—
273	C₈	C₂²×D₂₉₂	1	1	yes	characteristic
274	C₈	C₂×#11924029	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	584	4	no	—
277	C₂²	—	584	2	no	—
278	C₂²	—	73	4	no	—
279	C₂²	—	146	8	no	—
280	C₂²	—	146	2	no	—
281	C₂²	—	146	2	no	—
282	C₂²	#17988824	1	2	yes	—
283	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
284	C₄	—	8	4	no	—
285	C₄	—	584	2	no	—
286	C₄	—	146	2	no	—
287	C₄	#17988824	1	2	yes	—
288	C₄	C₂×#14556566	1	1	yes	characteristic
289	C₄	C₂×#14556614	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	1168	2	no	—
292	C₂	—	146	4	no	—
293	C₂	C₂×#17988824	1	1	yes	characteristic
294	C₂	#508×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2158	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#2158	D₇₃
hypocenter	C₇₃	#33600
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
1. C₂×#17989019	C₂	semidirect
2. C₂²×C₁₆×D₇₃	C₂	semidirect
3. #22619513	C₂	semidirect
4. #22612028	C₂	semidirect
5. #22612023	C₂	semidirect
6. #22612018	C₂	semidirect
7. #22611636	C₂	semidirect
8. C₇₃×#2158	C₂	semidirect
13. C₂×Dic₅₈₄	C₂²	semidirect
14. C₂×#14556536	C₂²	semidirect
15. C₂×D₁₁₆₈	C₂²	semidirect
24. #17988390	C₂²	semidirect
25. #17988387	C₂²	semidirect
28. #367×C₇₃	C₂²	semidirect
29. #366×C₇₃	C₂²	semidirect
58. C₂×C₁₆×C₇₃	D₄	semidirect
135. C₂×D₇₃	#508	semidirect
154. #2158	D₇₃	semidirect
156. C₇₃	#33600	semidirect
174. #367	C₂×D₇₃	semidirect
175. #366	C₂×D₇₃	semidirect
204. C₂×C₁₆	D₂₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989019	2
C₂²×C₁₆×D₇₃	2
#22619513	2
#22612028	2
#22612023	2
#22612018	2
#22611636	2
C₇₃×#2158	2
#33600	73