Subgroups — #32975551

Order 18688.

638 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	C₂×#17988640	C₂	1	1	yes	maximal, characteristic
2	#22612338	C₂	1	1	yes	maximal, characteristic
3	#22607770	C₂	1	4	yes	maximal, complemented
4	C₂×#432×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556550	—	2	1	no	—
6	#17988640	—	2	2	no	—
7	#17988640	C₄	1	4	yes	complemented
8	C₂×C₄⋊#10596831	—	2	1	no	—
9	C₂×C₂²⋊#10596831	—	2	1	no	—
10	C₂×C₄×#10596831	C₂²	1	2	yes	—
11	#17988457	C₄	1	2	yes	—
12	#17988453	—	2	1	no	—
13	#17988447	—	2	1	no	—
14	#17988447	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₇₃	—	2	1	no	—
16	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	#432×C₇₃	—	2	2	no	—
19	#432×C₇₃	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₇₃	—	2	1	no	—
22	C₂×#97×C₇₃	—	2	1	no	—
23	C₂²×#10596831	—	2	2	no	—
24	#14556550	—	4	4	no	—
25	C₂×C₄⋊Dic₇₃	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	2	no	—
27	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
29	C₂×C₄×Dic₇₃	—	2	2	no	—
30	C₄⋊#10596831	—	2	4	no	—
31	C₂²⋊#10596831	—	2	4	no	—
32	C₄×#10596831	—	2	2	no	—
33	C₄×#10596831	C₂×C₄	1	4	yes	—
34	#14556498	—	2	2	no	—
35	#14556498	—	2	2	no	—
36	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₇₃	—	2	2	no	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	D₄	1	2	yes	—
46	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₃	—	2	2	no	—
49	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
50	C₂²⋊C₈×C₇₃	—	2	4	no	—
51	#97×C₇₃	—	2	4	no	—
52	C₂²×Dic₇₃	—	2	4	no	—
53	C₂×#10596831	—	2	4	no	—
54	C₂×#10596831	—	4	4	no	—
55	C₄⋊Dic₇₃	—	2	8	no	—
56	C₄⋊Dic₇₃	—	4	4	no	—
57	C₄⋊Dic₇₃	—	4	4	no	—
58	C₂²⋊Dic₇₃	—	4	4	no	—
59	C₄×Dic₇₃	—	4	4	no	—
60	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	—	2	4	no	—
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	—	2	2	no	—
64	C₂²×C₄×C₇₃	—	4	2	no	—
65	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
66	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₃	—	2	4	no	—
78	C₂×C₈×C₇₃	—	4	2	no	—
79	C₂×C₈×C₇₃	—	4	2	no	—
80	C₄²×C₇₃	C₄²	1	2	yes	—
81	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
90	C₂³×C₇₃	—	2	2	no	—
91	C₂³×C₇₃	—	2	1	no	—
92	C₂³×C₇₃	—	2	1	no	—
93	C₂³×C₇₃	—	2	2	no	—
94	C₂³×C₇₃	#95	1	2	yes	—
95	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	2	1	no	—
98	C₂×C₄×C₇₃	—	2	1	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	2	no	—
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	—	2	4	no	—
103	C₂×C₄×C₇₃	—	2	4	no	—
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	#95	1	2	yes	—
108	C₂×C₄×C₇₃	#95	1	2	yes	—
109	C₂×C₄×C₇₃	#95	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	#131	1	4	yes	—
117	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₇₃	#137	1	2	yes	—
119	C₂×Dic₇₃	—	4	4	no	—
120	C₂×Dic₇₃	—	4	4	no	—
121	C₂×Dic₇₃	—	4	4	no	—
122	C₈×C₇₃	—	4	4	no	—
123	#10596831	—	4	4	no	—
124	C₂²×C₇₃	—	2	2	no	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	—	2	2	no	—
127	C₂²×C₇₃	—	4	2	no	—
128	C₂²×C₇₃	—	4	2	no	—
129	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
130	C₂²×C₇₃	#408	1	2	yes	—
131	C₂²×C₇₃	#418	1	1	yes	characteristic
132	C₂²×C₇₃	#419	1	1	yes	characteristic
133	C₂²×C₇₃	#421	1	2	yes	—
134	C₄×C₇₃	—	2	2	no	—
135	C₄×C₇₃	—	2	1	no	—
136	C₄×C₇₃	—	2	2	no	—
137	C₄×C₇₃	—	2	1	no	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	#405	1	2	yes	—
140	C₄×C₇₃	#417	1	2	yes	—
141	C₄×C₇₃	#424	1	2	yes	—
142	C₄×C₇₃	#428	1	2	yes	—
143	Dic₇₃	—	8	4	no	—
144	#7900	—	73	1	no	maximal, Sylow
145	C₂×C₇₃	—	4	2	no	—
146	C₂×C₇₃	#1496	1	2	yes	—
147	C₂×C₇₃	#1497	1	2	yes	—
148	C₂×C₇₃	#1729	1	1	yes	characteristic
149	C₂×C₇₃	#1733	1	1	yes	characteristic
150	C₂×C₇₃	#1736	1	1	yes	characteristic
151	C₂×#432	D₇₃	1	1	yes	characteristic
152	C₂×#431	—	73	1	no	—
153	#2285	—	73	1	no	—
154	#1279	—	73	4	no	—
155	C₇₃	#7900	1	1	yes	characteristic, complemented, Sylow
156	C₂×#117	—	146	1	no	—
157	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
158	#432	—	2	2	no	—
159	#432	Dic₇₃	1	4	yes	complemented
160	#431	—	73	4	no	—
161	#431	—	146	2	no	—
162	C₂×C₄⋊C₈	—	146	1	no	—
163	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
164	C₂×C₂²⋊C₈	—	2	1	no	—
165	C₂×C₂²⋊C₈	—	146	1	no	—
166	C₂×#97	—	2	1	no	—
167	C₂×C₄×C₈	—	73	2	no	—
168	#384	—	73	2	no	—
169	#382	—	146	1	no	—
170	#380	—	73	2	no	—
171	#380	—	146	1	no	—
172	C₄×C₄⋊C₄	—	73	2	no	—
173	C₄×C₄⋊C₄	—	146	1	no	—
174	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
175	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	146	2	no	—
179	#117	—	292	4	no	—
180	C₂×C₄⋊C₄	—	73	4	no	—
181	C₂×C₄⋊C₄	—	146	2	no	—
182	C₂×C₄⋊C₄	—	146	2	no	—
183	C₂×C₂²⋊C₄	—	146	2	no	—
184	C₂×C₄²	—	146	2	no	—
185	C₂×C₄²	Dic₁₄₆	1	2	yes	—
186	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
187	C₂×C₄²	C₄×D₇₃	1	2	yes	—
188	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
190	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
192	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	—	146	4	no	—
195	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
196	C₂²⋊C₈	—	2	4	no	—
197	C₂²⋊C₈	—	146	4	no	—
198	#97	—	2	4	no	—
199	C₄×C₈	—	73	4	no	—
200	C₄×C₈	—	146	2	no	—
201	#95	—	146	2	no	—
202	#95	—	146	2	no	—
203	C₂⁴	#11924005	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	146	4	no	—
209	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
210	C₂²×C₄	#11924005	1	1	yes	characteristic
211	C₂²×C₄	#11924005	1	1	yes	characteristic
212	C₂²×C₄	#11924005	1	1	yes	characteristic
213	C₂²×C₄	#11924005	1	2	yes	—
214	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
215	C₂²×C₄	#11924007	1	2	yes	—
216	C₂²×C₄	#11924007	1	2	yes	—
217	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
218	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	292	4	no	—
225	C₂×C₈	—	146	4	no	—
226	C₄⋊C₄	—	292	4	no	—
227	C₄⋊C₄	—	292	4	no	—
228	C₄⋊C₄	—	146	8	no	—
229	C₂²⋊C₄	—	292	4	no	—
230	C₄²	—	292	4	no	—
231	C₄²	C₄×Dic₇₃	1	2	yes	—
232	C₄²	#11924005	1	2	yes	—
233	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
234	C₄²	#11924007	1	2	yes	—
235	C₄²	#11924007	1	2	yes	—
236	C₄²	C₄⋊Dic₇₃	1	2	yes	—
237	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
238	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
240	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
241	C₂³	—	2	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	#14556498	1	2	yes	—
246	C₂³	C₂×#11924005	1	1	yes	characteristic
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	292	4	no	—
259	C₂×C₄	—	292	4	no	—
260	C₂×C₄	—	292	4	no	—
261	C₂×C₄	#14556498	1	2	yes	—
262	C₂×C₄	#14556498	1	2	yes	—
263	C₂×C₄	#14556498	1	2	yes	—
264	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
265	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
266	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
268	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
269	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
270	C₂×C₄	#14556599	1	4	yes	—
271	C₂×C₄	#14556616	1	2	yes	—
272	C₂×C₄	#14556621	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	292	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	C₂×#14556498	1	1	yes	characteristic
281	C₂²	#17988555	1	2	yes	—
282	C₂²	#17988591	1	1	yes	characteristic
283	C₂²	#17988595	1	1	yes	characteristic
284	C₂²	#17988603	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	584	4	no	—
291	C₄	#17988545	1	2	yes	—
292	C₄	#17988588	1	2	yes	—
293	C₄	#17988615	1	2	yes	—
294	C₄	#17988630	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	#22608399	1	2	yes	—
297	C₂	#22608403	1	2	yes	—
298	C₂	#22609629	1	1	yes	characteristic
299	C₂	#22609642	1	1	yes	characteristic
300	C₂	#22609652	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#432×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#432	D₇₃
hypocenter	C₇₃	#7900
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #22607770	C₂	semidirect
7. #17988640	C₄	semidirect
19. #432×C₇₃	C₄	semidirect
155. C₇₃	#7900	semidirect
159. #432	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988640	2
#22612338	2
#22607770	2
C₂×#432×C₇₃	2
#7900	73