Subgroups — #32976093

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
2	#22612338	C₂	1	1	yes	maximal, characteristic
3	#22607815	C₂	1	4	yes	maximal, complemented
4	C₂×#432×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#10596831	—	2	1	no	—
6	C₂×#14556550	C₄	1	2	yes	—
7	C₂×C₄⋊#10596831	—	2	1	no	—
8	C₂×C₄⋊#10596831	—	2	2	no	—
9	C₂×C₄⋊#10596831	C₄	1	4	yes	complemented
10	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
11	#17988457	C₂²	1	2	yes	—
12	#17988453	C₄	1	2	yes	—
13	#17988447	—	2	1	no	—
14	#17988447	—	2	1	no	—
15	C₄×C₄⋊Dic₇₃	—	2	1	no	—
16	C₄×C₄⋊Dic₇₃	—	2	1	no	—
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	4	no	—
24	C₂²×#10596831	—	2	2	no	—
25	C₂²×#10596831	—	4	4	no	—
26	#14556550	—	4	4	no	—
27	C₂×C₄⋊Dic₇₃	—	2	2	no	—
28	C₂×C₄⋊Dic₇₃	—	2	2	no	—
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
31	C₂×C₄×Dic₇₃	—	2	2	no	—
32	C₄⋊#10596831	—	2	4	no	—
33	C₄⋊#10596831	—	2	2	no	—
34	C₄⋊#10596831	C₂×C₄	1	4	yes	—
35	#14556498	—	2	2	no	—
36	#14556498	—	2	2	no	—
37	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₃	—	2	2	no	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	D₄	1	2	yes	—
47	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
48	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₇₃	—	2	2	no	—
50	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
51	C₂²⋊C₈×C₇₃	—	2	4	no	—
52	#97×C₇₃	—	2	4	no	—
53	C₂²×Dic₇₃	—	2	4	no	—
54	C₂×#10596831	—	2	4	no	—
55	C₂×#10596831	—	4	8	no	—
56	C₂×#10596831	—	4	4	no	—
57	C₄⋊Dic₇₃	—	4	4	no	—
58	C₄⋊Dic₇₃	—	4	2	no	—
59	C₄⋊Dic₇₃	—	4	2	no	—
60	C₄⋊Dic₇₃	—	4	4	no	—
61	C₂²⋊Dic₇₃	—	4	4	no	—
62	C₄×Dic₇₃	—	4	4	no	—
63	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	—	2	2	no	—
65	C₂²×C₄×C₇₃	—	2	2	no	—
66	C₂²×C₄×C₇₃	—	2	4	no	—
67	C₂²×C₄×C₇₃	—	2	2	no	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
70	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₃	—	2	4	no	—
82	C₂×C₈×C₇₃	—	4	4	no	—
83	C₄²×C₇₃	C₄²	1	2	yes	—
84	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
85	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
86	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
88	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
89	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
93	C₂³×C₇₃	—	2	2	no	—
94	C₂³×C₇₃	—	2	2	no	—
95	C₂³×C₇₃	#95	1	2	yes	—
96	C₂³×C₇₃	#100	1	2	yes	—
97	C₂³×C₇₃	#101	1	2	yes	—
98	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	—	2	4	no	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	2	1	no	—
102	C₂×C₄×C₇₃	—	2	1	no	—
103	C₂×C₄×C₇₃	—	2	1	no	—
104	C₂×C₄×C₇₃	—	2	1	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	2	no	—
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	4	no	—
111	C₂×C₄×C₇₃	—	2	4	no	—
112	C₂×C₄×C₇₃	#95	1	2	yes	—
113	C₂×C₄×C₇₃	#95	1	2	yes	—
114	C₂×C₄×C₇₃	#95	1	2	yes	—
115	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	#135	1	4	yes	—
122	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇₃	#137	1	2	yes	—
124	C₂×Dic₇₃	—	4	4	no	—
125	C₂×Dic₇₃	—	4	4	no	—
126	C₂×Dic₇₃	—	4	4	no	—
127	C₈×C₇₃	—	4	4	no	—
128	#10596831	—	4	4	no	—
129	C₂²×C₇₃	—	2	4	no	—
130	C₂²×C₇₃	—	2	4	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	#343	1	4	yes	—
134	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
135	C₂²×C₇₃	C₂×#100	1	1	yes	characteristic
136	C₂²×C₇₃	C₂×#101	1	1	yes	characteristic
137	C₂²×C₇₃	#416	1	2	yes	—
138	C₂²×C₇₃	#417	1	2	yes	—
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	2	2	no	—
142	C₄×C₇₃	—	2	1	no	—
143	C₄×C₇₃	—	2	1	no	—
144	C₄×C₇₃	#416	1	2	yes	—
145	C₄×C₇₃	#418	1	1	yes	characteristic
146	C₄×C₇₃	#419	1	1	yes	characteristic
147	C₄×C₇₃	#427	1	2	yes	—
148	C₄×C₇₃	#428	1	2	yes	—
149	Dic₇₃	—	8	4	no	—
150	#8044	—	73	1	no	maximal, Sylow
151	C₂×C₇₃	—	2	4	no	—
152	C₂×C₇₃	#1551	1	2	yes	—
153	C₂×C₇₃	#1552	1	2	yes	—
154	C₂×C₇₃	#1735	1	1	yes	characteristic
155	C₂×C₇₃	#1736	1	1	yes	characteristic
156	C₂×C₇₃	C₂×#343	1	1	yes	characteristic
157	C₂×#432	D₇₃	1	1	yes	characteristic
158	C₂²×C₄⋊C₈	—	73	1	no	—
159	#2285	—	73	1	no	—
160	#1296	—	73	4	no	—
161	C₇₃	#8044	1	1	yes	characteristic, complemented, Sylow
162	C₂³×C₈	—	146	1	no	—
163	C₂×#117	—	73	2	no	—
164	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
165	#432	—	2	2	no	—
166	#432	Dic₇₃	1	4	yes	complemented
167	C₂×C₄⋊C₈	—	73	4	no	—
168	C₂×C₄⋊C₈	—	73	2	no	—
169	C₂×C₄⋊C₈	—	146	1	no	—
170	C₂×C₄⋊C₈	—	146	2	no	—
171	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
172	C₂×C₂²⋊C₈	—	2	1	no	—
173	C₂×#97	—	2	1	no	—
174	#384	—	73	2	no	—
175	#382	—	73	2	no	—
176	#380	—	146	1	no	—
177	#380	—	146	1	no	—
178	C₄×C₄⋊C₄	—	146	1	no	—
179	C₄×C₄⋊C₄	—	146	1	no	—
180	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
181	C₂³×C₄	D₂₉₂	1	1	yes	characteristic
182	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	292	4	no	—
185	C₂²×C₈	—	146	4	no	—
186	C₂²×C₈	—	146	2	no	—
187	#117	—	292	4	no	—
188	C₂×C₄⋊C₄	—	73	4	no	—
189	C₂×C₄⋊C₄	—	146	2	no	—
190	C₂×C₄⋊C₄	—	146	2	no	—
191	C₂×C₂²⋊C₄	—	146	2	no	—
192	C₂×C₄²	—	146	2	no	—
193	C₂×C₄²	Dic₁₄₆	1	2	yes	—
194	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
195	C₂×C₄²	C₄×D₇₃	1	2	yes	—
196	C₂×C₄²	D₂₉₂	1	1	yes	characteristic
197	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
198	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
199	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
200	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	73	4	no	—
203	C₄⋊C₈	—	146	4	no	—
204	C₄⋊C₈	—	146	2	no	—
205	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
206	C₂²⋊C₈	—	2	4	no	—
207	#97	—	2	4	no	—
208	#95	—	146	2	no	—
209	#95	—	146	2	no	—
210	C₂⁴	#11924005	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	146	4	no	—
217	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
218	C₂²×C₄	#11924005	1	1	yes	characteristic
219	C₂²×C₄	#11924005	1	1	yes	characteristic
220	C₂²×C₄	#11924005	1	1	yes	characteristic
221	C₂²×C₄	#11924005	1	2	yes	—
222	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
223	C₂²×C₄	#11924007	1	2	yes	—
224	C₂²×C₄	#11924007	1	2	yes	—
225	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	292	8	no	—
232	C₂×C₈	—	292	4	no	—
233	C₂×C₈	—	146	4	no	—
234	C₄⋊C₄	—	292	4	no	—
235	C₄⋊C₄	—	292	4	no	—
236	C₄⋊C₄	—	292	2	no	—
237	C₄⋊C₄	—	292	2	no	—
238	C₂²⋊C₄	—	292	4	no	—
239	C₄²	—	292	4	no	—
240	C₄²	C₄×Dic₇₃	1	2	yes	—
241	C₄²	#11924005	1	2	yes	—
242	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
243	C₄²	#11924007	1	2	yes	—
244	C₄²	#11924007	1	2	yes	—
245	C₄²	C₄⋊Dic₇₃	1	2	yes	—
246	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
247	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
248	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
249	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	#14556498	1	2	yes	—
253	C₂³	#14556507	1	2	yes	—
254	C₂³	#14556509	1	2	yes	—
255	C₂³	C₂×#11924005	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	292	4	no	—
270	C₂×C₄	—	292	4	no	—
271	C₂×C₄	—	292	4	no	—
272	C₂×C₄	#14556498	1	2	yes	—
273	C₂×C₄	#14556498	1	2	yes	—
274	C₂×C₄	#14556498	1	2	yes	—
275	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
276	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
277	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
279	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
281	C₂×C₄	#14556611	1	4	yes	—
282	C₂×C₄	#14556619	1	2	yes	—
283	C₂×C₄	#14556624	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	292	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	#17988355	1	4	yes	—
291	C₂²	C₂×#14556498	1	1	yes	characteristic
292	C₂²	C₂×#14556507	1	1	yes	characteristic
293	C₂²	C₂×#14556509	1	1	yes	characteristic
294	C₂²	#17988581	1	2	yes	—
295	C₂²	#17988589	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	584	4	no	—
302	C₄	#17988583	1	2	yes	—
303	C₄	#17988593	1	1	yes	characteristic
304	C₄	#17988597	1	1	yes	characteristic
305	C₄	#17988623	1	2	yes	—
306	C₄	#17988629	1	2	yes	—
307	C₂	—	2	4	no	—
308	C₂	#22608627	1	2	yes	—
309	C₂	#22608632	1	2	yes	—
310	C₂	#22609647	1	1	yes	characteristic
311	C₂	#22609651	1	1	yes	characteristic
312	C₂	C₂×#17988355	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11924005
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	#8044
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #22607815	C₂	semidirect
9. C₂×C₄⋊#10596831	C₄	semidirect
19. #432×C₇₃	C₄	semidirect
161. C₇₃	#8044	semidirect
166. #432	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊#10596831	2
#22612338	2
#22607815	2
C₂×#432×C₇₃	2
#8044	73