Subgroups — #33073923

Order 18688.

858 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621049	C₂	1	2	yes	maximal, complemented
2	C₂×#17988992	C₂	1	1	yes	maximal, characteristic, complemented
3	#22620391	C₂	1	2	yes	maximal, complemented
4	#22609604	C₂	1	4	yes	maximal, complemented
5	#22609548	C₂	1	2	yes	maximal, complemented
6	#22609520	C₂	1	2	yes	maximal, complemented
7	C₈×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#3169	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556621	—	2	4	no	—
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	#17988990	C₂²	1	4	yes	—
15	#17988982	—	2	4	no	—
16	#17988966	C₂²	1	4	yes	—
17	#17988959	C₂²	1	4	yes	—
18	#17988933	C₂²	1	4	yes	—
19	#17988898	—	2	4	no	—
20	#17988867	—	2	4	no	—
21	#17988640	C₂²	1	1	yes	characteristic
22	C₂×#14556525	—	2	2	no	—
23	C₄×#11924012	C₂²	1	1	yes	characteristic
24	#17988309	C₂²	1	2	yes	—
25	C₈×#10596831	C₂²	1	2	yes	—
26	#532×C₇₃	C₂²	1	2	yes	—
27	#500×C₇₃	C₂²	1	1	yes	characteristic
28	#493×C₇₃	C₂²	1	1	yes	characteristic
29	#486×C₇₃	C₂²	1	2	yes	—
30	#466×C₇₃	—	2	2	no	—
31	C₄×D₈×C₇₃	—	2	2	no	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₄₆	—	2	2	no	—
34	#14556621	—	4	16	no	—
35	C₂×#11924018	—	2	8	no	—
36	C₂×#11924016	—	2	8	no	—
37	C₂×#11924012	D₄	1	2	yes	—
38	#14556591	—	2	2	no	—
39	#14556591	—	2	2	no	—
40	#14556591	—	2	2	no	—
41	#14556591	C₂³	1	4	yes	—
42	C₂×C₄⋊Dic₇₃	—	2	2	no	—
43	C₂×C₄⋊Dic₇₃	—	2	2	no	—
44	#14556525	—	2	4	no	—
45	#14556525	—	2	4	no	—
46	#14556524	—	2	8	no	—
47	C₄⋊#10596831	D₄	1	2	yes	—
48	#14556512	—	2	8	no	—
49	C₂²⋊#10596831	D₄	1	2	yes	—
50	#14556502	D₄	1	2	yes	—
51	C₄×#10596831	C₂³	1	2	yes	—
52	C₄×#10596831	C₂³	1	2	yes	—
53	C₂×#41×C₇₃	—	2	2	no	—
54	C₂×D₈×C₇₃	—	2	2	no	—
55	C₂²×C₈×C₇₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
57	#127×C₇₃	C₂³	1	2	yes	—
58	#124×C₇₃	—	2	2	no	—
59	#121×C₇₃	—	2	2	no	—
60	#121×C₇₃	—	2	2	no	—
61	C₄×D₄×C₇₃	—	2	2	no	—
62	C₄×D₄×C₇₃	—	2	2	no	—
63	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
64	#107×C₇₃	—	2	2	no	—
65	#102×C₇₃	—	2	4	no	—
66	C₄×C₈×C₇₃	D₄	1	2	yes	—
67	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₄₆	—	2	4	no	—
70	C₂×Dic₁₄₆	—	4	8	no	—
71	C₂×Dic₁₄₆	—	4	2	no	—
72	C₂²×Dic₇₃	—	4	2	no	—
73	#11924018	—	4	16	no	—
74	#11924016	—	4	16	no	—
75	#11924012	D₈	1	8	yes	complemented
76	C₂×#10596831	C₂×D₄	1	4	yes	—
77	C₂×#10596831	C₂×D₄	1	4	yes	—
78	C₄⋊Dic₇₃	—	2	4	no	—
79	C₄⋊Dic₇₃	—	2	4	no	—
80	C₄⋊Dic₇₃	—	4	2	no	—
81	C₄⋊Dic₇₃	—	4	2	no	—
82	#41×C₇₃	—	4	8	no	—
83	C₂×Q₈×C₇₃	—	2	2	no	—
84	C₂×D₄×C₇₃	—	2	2	no	—
85	C₂×D₄×C₇₃	—	2	2	no	—
86	C₂×D₄×C₇₃	—	2	2	no	—
87	C₂×D₄×C₇₃	—	4	2	no	—
88	C₂²×C₄×C₇₃	—	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	D₈×C₇₃	—	4	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	2	yes	—
96	C₄⋊C₄×C₇₃	—	2	2	no	—
97	C₄⋊C₄×C₇₃	—	2	2	no	—
98	C₂²⋊C₄×C₇₃	—	4	2	no	—
99	C₂²⋊C₄×C₇₃	—	4	2	no	—
100	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
104	C₂³×C₇₃	—	4	2	no	—
105	C₂³×C₇₃	#127	1	1	yes	characteristic
106	Q₈×C₇₃	—	4	4	no	—
107	D₄×C₇₃	—	4	4	no	—
108	D₄×C₇₃	—	4	4	no	—
109	D₄×C₇₃	—	4	4	no	—
110	D₄×C₇₃	—	8	4	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	4	2	no	—
114	C₂×C₄×C₇₃	—	8	4	no	—
115	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×D₈	1	4	yes	—
119	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₇₃	—	4	4	no	—
123	C₂×Dic₇₃	—	8	4	no	—
124	Dic₁₄₆	—	4	8	no	—
125	Dic₁₄₆	—	8	4	no	—
126	C₈×C₇₃	—	2	2	no	—
127	C₈×C₇₃	#137	1	4	yes	complemented
128	#10596831	—	2	4	no	—
129	#10596831	C₂×D₈	1	8	yes	—
130	C₂²×C₇₃	—	2	1	no	—
131	C₂²×C₇₃	—	2	1	no	—
132	C₂²×C₇₃	—	8	2	no	—
133	C₂²×C₇₃	—	8	2	no	—
134	C₂²×C₇₃	—	8	2	no	—
135	C₂²×C₇₃	#493	1	2	yes	—
136	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	1	no	—
139	C₄×C₇₃	—	8	2	no	—
140	C₄×C₇₃	#493	1	2	yes	—
141	C₄×C₇₃	C₂²×D₈	1	2	yes	—
142	C₄×C₇₃	C₂×#137	1	2	yes	—
143	C₄×C₇₃	#578	1	2	yes	—
144	Dic₇₃	—	8	4	no	—
145	#19630	—	73	1	no	maximal, Sylow
146	C₂×C₇₃	—	2	2	no	—
147	C₂×C₇₃	—	16	2	no	—
148	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
149	C₂×C₇₃	#3152	1	1	yes	characteristic
150	C₂×C₇₃	#3156	1	1	yes	characteristic
151	#3226	—	73	2	no	—
152	#3169	D₇₃	1	1	yes	characteristic
153	C₂×#500	—	73	1	no	—
154	#3152	—	73	2	no	—
155	#1722	—	73	4	no	—
156	#1713	—	73	2	no	—
157	#1706	—	73	2	no	—
158	C₈×M₁₆	—	73	1	no	—
159	C₇₃	#19630	1	1	yes	characteristic, complemented, Sylow
160	C₂×#137	—	146	4	no	—
161	#532	C₂×D₇₃	1	2	yes	—
162	C₂×C₄⋊Q₈	—	73	2	no	—
163	#500	—	73	4	no	—
164	#500	—	73	2	no	—
165	#500	—	73	2	no	—
166	#500	—	146	2	no	—
167	#500	C₂×D₇₃	1	1	yes	characteristic
168	#497	—	146	4	no	—
169	#494	—	73	4	no	—
170	#493	C₂×D₇₃	1	1	yes	characteristic
171	#492	—	73	4	no	—
172	#486	—	73	4	no	—
173	#486	C₂×D₇₃	1	2	yes	—
174	#480	—	146	4	no	—
175	#474	—	146	4	no	—
176	#466	—	2	2	no	—
177	C₄×D₈	—	2	2	no	—
178	#431	—	73	1	no	—
179	C₂×#107	—	146	2	no	—
180	C₄×M₁₆	—	73	1	no	—
181	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
182	#322	—	73	2	no	—
183	C₈²	—	73	2	no	—
184	C₂×#41	—	2	2	no	—
185	C₂²×Q₈	—	146	2	no	—
186	#137	—	292	16	no	—
187	C₂×Q₁₆	—	146	8	no	—
188	C₂×SD₁₆	—	146	8	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×M₁₆	—	73	2	no	—
191	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
192	C₄⋊Q₈	—	73	4	no	—
193	C₄⋊Q₈	—	146	2	no	—
194	C₄⋊Q₈	—	146	2	no	—
195	C₄⋊Q₈	—	146	2	no	—
196	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
197	#127	C₂²×D₇₃	1	2	yes	—
198	#124	—	2	2	no	—
199	#121	—	2	2	no	—
200	#121	—	2	2	no	—
201	C₄×D₄	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	146	2	no	—
204	C₂×C₄⋊C₄	—	146	2	no	—
205	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
206	#107	—	2	2	no	—
207	#107	—	146	8	no	—
208	#107	—	146	4	no	—
209	#107	—	146	4	no	—
210	C₄⋊C₈	—	73	2	no	—
211	#102	—	2	4	no	—
212	#102	—	146	8	no	—
213	C₂²⋊C₈	—	73	2	no	—
214	#97	—	73	2	no	—
215	C₄×C₈	—	73	2	no	—
216	C₄×C₈	—	73	2	no	—
217	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
218	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
220	#41	—	4	8	no	—
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	292	2	no	—
223	C₂×Q₈	—	292	8	no	—
224	C₂×Q₈	—	146	4	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	292	2	no	—
231	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
234	Q₁₆	—	292	16	no	—
235	SD₁₆	—	292	16	no	—
236	D₈	—	4	4	no	—
237	M₁₆	—	73	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	73	4	no	—
240	C₂×C₈	—	73	4	no	—
241	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
242	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	292	2	no	—
246	C₄⋊C₄	—	292	2	no	—
247	C₄⋊C₄	—	146	4	no	—
248	C₄⋊C₄	—	146	4	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	D₄×D₇₃	1	1	yes	characteristic
252	C₄²	D₄×D₇₃	1	1	yes	characteristic
253	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
254	C₄²	C₂³×D₇₃	1	1	yes	characteristic
255	C₂³	—	4	2	no	—
256	C₂³	#14556587	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	292	8	no	—
259	Q₈	—	584	4	no	—
260	D₄	—	4	4	no	—
261	D₄	—	4	4	no	—
262	D₄	—	4	4	no	—
263	D₄	—	8	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	4	no	—
268	C₂×C₄	—	292	4	no	—
269	C₂×C₄	—	584	4	no	—
270	C₂×C₄	#14556587	1	1	yes	characteristic
271	C₂×C₄	#14556587	1	1	yes	characteristic
272	C₂×C₄	#14556587	1	1	yes	characteristic
273	C₂×C₄	D₈×D₇₃	1	4	yes	—
274	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
275	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	73	8	no	—
279	C₈	—	146	4	no	—
280	C₈	#14556621	1	4	yes	complemented
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	#17988962	1	2	yes	—
287	C₂²	C₂×#14556587	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	8	2	no	—
291	C₄	—	584	4	no	—
292	C₄	#17988962	1	2	yes	—
293	C₄	C₂×D₈×D₇₃	1	2	yes	—
294	C₄	C₂×#14556621	1	2	yes	—
295	C₄	#17989401	1	2	yes	—
296	C₂	—	2	2	no	—
297	C₂	—	16	2	no	—
298	C₂	C₂×#17988962	1	1	yes	characteristic
299	C₂	#22620391	1	1	yes	characteristic
300	C₂	#22620426	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556587
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting, prime radical	C₇₃×#3169	C₂
socle	C₂²×C₇₃	C₂×#127
hypercenter	#3169	D₇₃
hypocenter	C₇₃	#19630
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #22621049	C₂	semidirect
2. C₂×#17988992	C₂	semidirect
3. #22620391	C₂	semidirect
4. #22609604	C₂	semidirect
5. #22609548	C₂	semidirect
6. #22609520	C₂	semidirect
7. C₈×#11924012	C₂	semidirect
75. #11924012	D₈	semidirect
127. C₈×C₇₃	#137	semidirect
159. C₇₃	#19630	semidirect
280. C₈	#14556621	semidirect

Maximal subgroups

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

subgroup	index
#22621049	2
C₂×#17988992	2
#22620391	2
#22609604	2
#22609548	2
#22609520	2
C₈×#11924012	2
C₇₃×#3169	2
#19630	73