Subgroups — #33073994

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620858	C₂	1	2	yes	maximal, complemented
2	#22620510	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988992	C₂	1	1	yes	maximal, characteristic, complemented
4	#22620325	C₂	1	2	yes	maximal, complemented
5	#22609584	C₂	1	4	yes	maximal, complemented
6	#22609541	C₂	1	2	yes	maximal, complemented
7	#22609523	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#17989137	C₂²	1	2	yes	—
10	C₂×#14556591	C₂²	1	2	yes	—
11	#17988992	—	2	2	no	—
12	#17988992	C₂²	1	2	yes	—
13	#17988992	C₂²	1	2	yes	—
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988964	C₂²	1	1	yes	characteristic
16	#17988960	C₂²	1	4	yes	—
17	#17988947	—	2	4	no	—
18	#17988938	C₂²	1	4	yes	—
19	#17988938	C₂²	1	4	yes	—
20	#17988934	C₂²	1	2	yes	—
21	#17988934	C₂²	1	4	yes	—
22	#17988823	—	2	2	no	—
23	#17988793	—	2	4	no	—
24	#17988728	—	2	4	no	—
25	C₄×D₅₈₄	—	2	2	no	—
26	C₂×#14556525	—	2	2	no	—
27	#17988589	—	2	4	no	—
28	#431×C₇₃	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#322×C₇₃	C₂²	1	2	yes	—
32	C₈²×C₇₃	C₂²	1	2	yes	—
33	C₂×#11924028	—	2	2	no	—
34	C₂²×Dic₁₄₆	—	2	2	no	—
35	C₂×#11924017	—	2	8	no	—
36	C₂×D₅₈₄	—	2	2	no	—
37	#14556591	—	2	2	no	—
38	#14556591	—	2	2	no	—
39	#14556591	—	2	2	no	—
40	#14556591	C₂³	1	4	yes	—
41	#14556591	C₂³	1	2	yes	—
42	#14556588	C₂³	1	2	yes	—
43	#14556579	—	2	2	no	—
44	#14556565	—	2	2	no	—
45	#14556565	—	2	2	no	—
46	C₄×D₂₉₂	—	2	2	no	—
47	C₄×D₂₉₂	—	2	2	no	—
48	C₂×C₄⋊Dic₇₃	—	2	2	no	—
49	C₂×C₄⋊Dic₇₃	—	2	2	no	—
50	#14556525	—	2	4	no	—
51	#14556525	—	2	4	no	—
52	#14556525	—	2	2	no	—
53	#14556516	—	2	8	no	—
54	#14556516	—	2	8	no	—
55	#14556513	—	2	4	no	—
56	#14556513	—	2	8	no	—
57	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
58	C₂²×C₈×C₇₃	D₄	1	2	yes	—
59	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
61	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
62	#97×C₇₃	D₄	1	2	yes	—
63	C₄×C₈×C₇₃	D₄	1	2	yes	—
64	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂³	1	2	yes	—
67	C₄×C₈×C₇₃	C₂³	1	2	yes	—
68	#11924028	—	4	8	no	—
69	C₂×Dic₁₄₆	—	2	2	no	—
70	C₂×Dic₁₄₆	—	2	4	no	—
71	C₂×Dic₁₄₆	—	4	2	no	—
72	C₂×Dic₁₄₆	—	8	4	no	—
73	C₂×D₂₉₂	—	2	2	no	—
74	C₂×D₂₉₂	—	2	2	no	—
75	C₂×D₂₉₂	—	2	2	no	—
76	C₂×C₂²⋊D₇₃	—	4	2	no	—
77	C₂²×Dic₇₃	—	4	2	no	—
78	C₂×C₄×D₇₃	—	4	2	no	—
79	#11924017	—	8	8	no	—
80	D₅₈₄	—	4	4	no	—
81	C₄⋊Dic₇₃	—	2	4	no	—
82	C₄⋊Dic₇₃	—	2	2	no	—
83	C₄⋊Dic₇₃	—	2	2	no	—
84	C₄⋊Dic₇₃	—	2	4	no	—
85	C₄⋊Dic₇₃	—	4	2	no	—
86	C₄⋊Dic₇₃	—	4	2	no	—
87	#11924005	—	4	2	no	—
88	#11924005	—	4	2	no	—
89	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	M₁₆×C₇₃	—	2	4	no	—
93	C₂×C₈×C₇₃	—	2	4	no	—
94	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
97	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
98	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
102	C₂³×C₇₃	#127	1	1	yes	characteristic
103	C₂²×D₇₃	—	4	2	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#135	1	4	yes	—
110	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
113	C₂²⋊D₇₃	—	8	4	no	—
114	C₂×Dic₇₃	—	4	2	no	—
115	C₂×Dic₇₃	—	4	4	no	—
116	C₂×Dic₇₃	—	8	4	no	—
117	D₂₉₂	—	4	4	no	—
118	D₂₉₂	—	4	4	no	—
119	D₂₉₂	—	4	4	no	—
120	C₄×D₇₃	—	8	4	no	—
121	Dic₁₄₆	—	4	4	no	—
122	Dic₁₄₆	—	4	8	no	—
123	Dic₁₄₆	—	8	4	no	—
124	C₈×C₇₃	—	2	4	no	—
125	C₈×C₇₃	—	2	2	no	—
126	C₈×C₇₃	—	2	4	no	—
127	C₈×C₇₃	#137	1	4	yes	—
128	C₂²×C₇₃	—	2	1	no	—
129	C₂²×C₇₃	—	2	1	no	—
130	C₂²×C₇₃	#495	1	2	yes	—
131	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
132	C₂×D₇₃	—	8	2	no	—
133	C₂×D₇₃	—	8	2	no	—
134	C₂×D₇₃	—	8	2	no	—
135	C₄×C₇₃	—	2	1	no	—
136	C₄×C₇₃	—	2	1	no	—
137	C₄×C₇₃	#495	1	2	yes	—
138	C₄×C₇₃	C₂×#135	1	2	yes	—
139	C₄×C₇₃	C₂×#137	1	2	yes	—
140	C₄×C₇₃	#576	1	2	yes	—
141	Dic₇₃	—	8	2	no	—
142	Dic₇₃	—	8	4	no	—
143	#19638	—	73	1	no	maximal, Sylow
144	C₂×C₇₃	—	2	2	no	—
145	C₂×C₇₃	C₂×#495	1	1	yes	characteristic
146	C₂×C₇₃	#3152	1	1	yes	characteristic
147	C₂×C₇₃	#3155	1	1	yes	characteristic
148	D₇₃	—	16	2	no	—
149	#3206	—	73	2	no	—
150	#3169	—	73	1	no	—
151	C₂×#500	—	73	1	no	—
152	#3141	—	73	2	no	—
153	#1719	—	73	4	no	—
154	#1710	—	73	2	no	—
155	#1706	—	73	2	no	—
156	C₈×M₁₆	D₇₃	1	1	yes	characteristic, complemented
157	C₇₃	#19638	1	1	yes	characteristic, complemented, Sylow
158	#532	—	73	2	no	—
159	C₂×C₄⋊Q₈	—	73	2	no	—
160	#500	—	73	2	no	—
161	#500	—	73	2	no	—
162	#500	—	73	1	no	—
163	#500	—	146	2	no	—
164	#493	—	73	1	no	—
165	#492	—	73	4	no	—
166	#489	—	146	4	no	—
167	#487	—	73	4	no	—
168	#487	—	73	4	no	—
169	#486	—	73	4	no	—
170	#486	—	73	2	no	—
171	#466	—	146	2	no	—
172	#461	—	146	4	no	—
173	#448	—	146	4	no	—
174	C₄×D₈	—	146	2	no	—
175	#431	C₂×D₇₃	1	1	yes	characteristic
176	C₂×#107	—	146	2	no	—
177	#417	—	146	4	no	—
178	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
180	#322	C₂×D₇₃	1	2	yes	—
181	C₈²	C₂×D₇₃	1	2	yes	—
182	C₂×#41	—	146	2	no	—
183	C₂²×Q₈	—	146	2	no	—
184	C₂×SD₁₆	—	146	8	no	—
185	C₂×D₈	—	146	2	no	—
186	C₂×M₁₆	D₂₉₂	1	2	yes	—
187	C₂²×C₈	D₂₉₂	1	2	yes	—
188	C₄⋊Q₈	—	73	4	no	—
189	C₄⋊Q₈	—	73	2	no	—
190	C₄⋊Q₈	—	146	2	no	—
191	C₄⋊Q₈	—	146	2	no	—
192	C₄⋊Q₈	—	146	2	no	—
193	#127	—	73	2	no	—
194	#124	—	146	2	no	—
195	#121	—	146	2	no	—
196	#121	—	146	2	no	—
197	C₄×D₄	—	146	2	no	—
198	C₄×D₄	—	146	2	no	—
199	C₂×C₄⋊C₄	—	146	2	no	—
200	C₂×C₄⋊C₄	—	146	2	no	—
201	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
202	#107	—	146	4	no	—
203	#107	—	146	4	no	—
204	#107	—	146	2	no	—
205	C₄⋊C₈	D₂₉₂	1	2	yes	—
206	#103	—	146	8	no	—
207	#103	—	146	8	no	—
208	#102	—	146	8	no	—
209	#102	—	146	4	no	—
210	C₂²⋊C₈	D₂₉₂	1	2	yes	—
211	#97	D₂₉₂	1	2	yes	—
212	C₄×C₈	D₂₉₂	1	2	yes	—
213	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₇₃	1	2	yes	—
215	C₄×C₈	C₂²×D₇₃	1	2	yes	—
216	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
217	#41	—	292	8	no	—
218	C₂×Q₈	—	292	2	no	—
219	C₂×Q₈	—	584	4	no	—
220	C₂×Q₈	—	146	2	no	—
221	C₂×Q₈	—	146	4	no	—
222	C₂×D₄	—	292	2	no	—
223	C₂×D₄	—	146	2	no	—
224	C₂×D₄	—	146	2	no	—
225	C₂×D₄	—	146	2	no	—
226	C₂²×C₄	—	292	2	no	—
227	C₂²×C₄	—	292	2	no	—
228	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
231	SD₁₆	—	584	8	no	—
232	D₈	—	292	4	no	—
233	M₁₆	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
236	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
237	C₂×C₈	C₂×D₂₉₂	1	4	yes	—
238	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
239	C₄⋊C₄	—	292	2	no	—
240	C₄⋊C₄	—	292	2	no	—
241	C₄⋊C₄	—	146	4	no	—
242	C₄⋊C₄	—	146	2	no	—
243	C₄⋊C₄	—	146	2	no	—
244	C₄⋊C₄	—	146	4	no	—
245	C₂²⋊C₄	—	292	2	no	—
246	C₂²⋊C₄	—	292	2	no	—
247	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
248	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
249	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
250	C₄²	C₂³×D₇₃	1	1	yes	characteristic
251	C₂³	—	292	2	no	—
252	C₂³	#14556588	1	1	yes	characteristic
253	Q₈	—	292	8	no	—
254	Q₈	—	292	4	no	—
255	Q₈	—	584	4	no	—
256	D₄	—	292	4	no	—
257	D₄	—	292	4	no	—
258	D₄	—	292	4	no	—
259	D₄	—	584	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	292	2	no	—
263	C₂×C₄	—	292	4	no	—
264	C₂×C₄	—	584	4	no	—
265	C₂×C₄	—	584	4	no	—
266	C₂×C₄	#14556588	1	1	yes	characteristic
267	C₂×C₄	#14556588	1	1	yes	characteristic
268	C₂×C₄	#14556588	1	1	yes	characteristic
269	C₂×C₄	#14556613	1	4	yes	—
270	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
272	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	#14556624	1	4	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	584	2	no	—
280	C₂²	—	584	2	no	—
281	C₂²	—	584	2	no	—
282	C₂²	#17988971	1	2	yes	—
283	C₂²	C₂×#14556588	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	584	4	no	—
287	C₄	—	584	2	no	—
288	C₄	#17988971	1	2	yes	—
289	C₄	C₂×#14556613	1	2	yes	—
290	C₄	C₂×#14556624	1	2	yes	—
291	C₄	#17989392	1	2	yes	—
292	C₂	—	2	2	no	—
293	C₂	—	1168	2	no	—
294	C₂	C₂×#17988971	1	1	yes	characteristic
295	C₂	#22620390	1	1	yes	characteristic
296	C₂	#22620417	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556588
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×M₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#22620510	C₂
hypercenter	C₈×M₁₆	D₇₃
hypocenter	C₇₃	#19638
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. #22620858	C₂	semidirect
3. C₂×#17988992	C₂	semidirect
4. #22620325	C₂	semidirect
5. #22609584	C₂	semidirect
6. #22609541	C₂	semidirect
7. #22609523	C₂	semidirect
8. C₈×M₁₆×C₇₃	C₂	semidirect
156. C₈×M₁₆	D₇₃	semidirect
157. C₇₃	#19638	semidirect

Maximal subgroups

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

subgroup	index
#22620858	2
#22620510	2
C₂×#17988992	2
#22620325	2
#22609584	2
#22609541	2
#22609523	2
C₈×M₁₆×C₇₃	2
#19638	73