Subgroups — #32975548

Order 18688.

638 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
2	#22612329	C₂	1	1	yes	maximal, characteristic
3	#22607771	C₂	1	4	yes	maximal, complemented
4	C₂×#431×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	—	2	1	no	—
6	#17988611	—	2	2	no	—
7	#17988611	C₄	1	4	yes	complemented
8	C₂×C₄⋊#10596831	—	2	1	no	—
9	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
10	#17988455	—	2	1	no	—
11	#17988453	C₄	1	2	yes	—
12	#17988450	—	2	1	no	—
13	#17988447	C₄	1	2	yes	—
14	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
15	C₄×C₂²⋊Dic₇₃	—	2	1	no	—
16	#17988433	—	2	1	no	—
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	#431×C₇₃	—	2	2	no	—
19	#431×C₇₃	C₄	1	4	yes	complemented
20	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
21	C₂×C₂²⋊C₈×C₇₃	—	2	1	no	—
22	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
23	C₂×#11924012	—	2	4	no	—
24	C₂×#11924012	—	4	4	no	—
25	C₂²×#10596831	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
27	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
28	C₂×C₄×Dic₇₃	—	2	2	no	—
29	C₄⋊#10596831	—	2	4	no	—
30	C₄⋊#10596831	—	2	2	no	—
31	C₄⋊#10596831	C₂×C₄	1	4	yes	—
32	#14556498	—	2	2	no	—
33	#14556498	—	2	2	no	—
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	4	no	—
49	C₂²⋊C₈×C₇₃	—	2	4	no	—
50	C₄×C₈×C₇₃	—	2	2	no	—
51	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
52	C₂²×Dic₇₃	—	2	4	no	—
53	#11924012	—	4	8	no	—
54	C₂×#10596831	—	2	4	no	—
55	C₂×#10596831	—	4	4	no	—
56	C₄⋊Dic₇₃	—	2	8	no	—
57	C₂²⋊Dic₇₃	—	4	4	no	—
58	C₄×Dic₇₃	—	4	2	no	—
59	C₄×Dic₇₃	—	4	2	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	4	no	—
79	C₄²×C₇₃	C₄²	1	2	yes	—
80	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
89	C₂³×C₇₃	—	2	2	no	—
90	C₂³×C₇₃	—	2	2	no	—
91	C₂³×C₇₃	—	2	2	no	—
92	C₂³×C₇₃	#95	1	2	yes	—
93	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	—	2	2	no	—
97	C₂×C₄×C₇₃	—	2	2	no	—
98	C₂×C₄×C₇₃	—	2	4	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	—	4	2	no	—
104	C₂×C₄×C₇₃	—	4	2	no	—
105	C₂×C₄×C₇₃	#95	1	2	yes	—
106	C₂×C₄×C₇₃	#95	1	2	yes	—
107	C₂×C₄×C₇₃	#95	1	2	yes	—
108	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₃	#131	1	4	yes	—
115	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₇₃	#137	1	2	yes	—
117	C₂×Dic₇₃	—	4	4	no	—
118	C₂×Dic₇₃	—	4	4	no	—
119	C₂×Dic₇₃	—	4	4	no	—
120	C₈×C₇₃	—	4	4	no	—
121	#10596831	—	4	4	no	—
122	C₂²×C₇₃	—	2	2	no	—
123	C₂²×C₇₃	—	2	2	no	—
124	C₂²×C₇₃	—	2	2	no	—
125	C₂²×C₇₃	—	4	4	no	—
126	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
127	C₂²×C₇₃	#408	1	2	yes	—
128	C₂²×C₇₃	#417	1	2	yes	—
129	C₂²×C₇₃	#421	1	2	yes	—
130	C₄×C₇₃	—	2	2	no	—
131	C₄×C₇₃	—	2	1	no	—
132	C₄×C₇₃	—	2	2	no	—
133	C₄×C₇₃	—	2	1	no	—
134	C₄×C₇₃	—	2	2	no	—
135	C₄×C₇₃	#405	1	2	yes	—
136	C₄×C₇₃	#418	1	1	yes	characteristic
137	C₄×C₇₃	#419	1	1	yes	characteristic
138	C₄×C₇₃	#424	1	2	yes	—
139	C₄×C₇₃	#428	1	2	yes	—
140	Dic₇₃	—	8	4	no	—
141	#7899	—	73	1	no	maximal, Sylow
142	C₂×C₇₃	—	4	2	no	—
143	C₂×C₇₃	#1495	1	4	yes	—
144	C₂×C₇₃	#1729	1	1	yes	characteristic
145	C₂×C₇₃	#1733	1	1	yes	characteristic
146	C₂×C₇₃	#1736	1	1	yes	characteristic
147	C₂×#431	D₇₃	1	1	yes	characteristic
148	C₂×#423	—	73	1	no	—
149	#2283	—	73	1	no	—
150	#1279	—	73	4	no	—
151	C₇₃	#7899	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	—	146	1	no	—
153	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
154	#431	—	2	2	no	—
155	#431	Dic₇₃	1	4	yes	complemented
156	#423	—	73	4	no	—
157	#423	—	146	2	no	—
158	C₂×C₄⋊C₈	—	2	1	no	—
159	C₂×C₄⋊C₈	—	73	2	no	—
160	C₂×C₄⋊C₈	—	146	1	no	—
161	C₂×C₂²⋊C₈	—	2	1	no	—
162	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
163	#383	—	146	1	no	—
164	#382	—	73	2	no	—
165	#381	—	146	1	no	—
166	#380	—	73	2	no	—
167	C₄×C₄⋊C₄	—	73	2	no	—
168	C₄×C₂²⋊C₄	—	146	1	no	—
169	#376	—	146	1	no	—
170	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
171	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
172	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
173	C₂×M₁₆	—	292	4	no	—
174	C₂×M₁₆	—	146	4	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	146	2	no	—
177	C₂×C₄⋊C₄	—	73	4	no	—
178	C₂×C₂²⋊C₄	—	146	2	no	—
179	C₂×C₄²	—	146	2	no	—
180	C₂×C₄²	Dic₁₄₆	1	2	yes	—
181	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₇₃	1	2	yes	—
183	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
185	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
186	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	73	4	no	—
190	C₄⋊C₈	—	146	4	no	—
191	C₄⋊C₈	—	146	2	no	—
192	C₂²⋊C₈	—	2	4	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
195	#95	—	146	2	no	—
196	#95	—	146	2	no	—
197	#95	—	146	2	no	—
198	#95	—	146	2	no	—
199	C₂⁴	#11924005	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	146	4	no	—
205	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
206	C₂²×C₄	#11924005	1	1	yes	characteristic
207	C₂²×C₄	#11924005	1	1	yes	characteristic
208	C₂²×C₄	#11924005	1	1	yes	characteristic
209	C₂²×C₄	#11924005	1	2	yes	—
210	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
211	C₂²×C₄	#11924007	1	2	yes	—
212	C₂²×C₄	#11924007	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
217	M₁₆	—	292	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	4	no	—
220	C₂×C₈	—	292	4	no	—
221	C₂×C₈	—	146	4	no	—
222	C₄⋊C₄	—	146	8	no	—
223	C₂²⋊C₄	—	292	4	no	—
224	C₄²	—	292	2	no	—
225	C₄²	—	292	2	no	—
226	C₄²	C₄×Dic₇₃	1	2	yes	—
227	C₄²	#11924005	1	2	yes	—
228	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
229	C₄²	#11924007	1	2	yes	—
230	C₄²	#11924007	1	2	yes	—
231	C₄²	C₄⋊Dic₇₃	1	2	yes	—
232	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
233	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
234	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
235	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	#14556498	1	2	yes	—
240	C₂³	C₂×#11924005	1	1	yes	characteristic
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	292	4	no	—
253	C₂×C₄	—	292	4	no	—
254	C₂×C₄	—	292	4	no	—
255	C₂×C₄	#14556498	1	2	yes	—
256	C₂×C₄	#14556498	1	2	yes	—
257	C₂×C₄	#14556498	1	2	yes	—
258	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
259	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
261	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
262	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
264	C₂×C₄	#14556597	1	4	yes	—
265	C₂×C₄	#14556619	1	2	yes	—
266	C₂×C₄	#14556624	1	2	yes	—
267	C₈	—	4	4	no	—
268	C₈	—	292	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂×#14556498	1	1	yes	characteristic
274	C₂²	#17988553	1	2	yes	—
275	C₂²	#17988589	1	2	yes	—
276	C₂²	#17988605	1	2	yes	—
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	2	no	—
282	C₄	—	584	4	no	—
283	C₄	#17988542	1	2	yes	—
284	C₄	#17988593	1	1	yes	characteristic
285	C₄	#17988597	1	1	yes	characteristic
286	C₄	#17988613	1	2	yes	—
287	C₄	#17988629	1	2	yes	—
288	C₂	—	4	2	no	—
289	C₂	#22608397	1	4	yes	—
290	C₂	#22609627	1	1	yes	characteristic
291	C₂	#22609641	1	1	yes	characteristic
292	C₂	#22609651	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₂×#431×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	C₂×#431	D₇₃
hypocenter	C₇₃	#7899
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22607771	C₂	semidirect
7. #17988611	C₄	semidirect
19. #431×C₇₃	C₄	semidirect
151. C₇₃	#7899	semidirect
155. #431	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988611	2
#22612329	2
#22607771	2
C₂×#431×C₇₃	2
#7899	73