Subgroups — #34025243

Order 18688.

674 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	#22626426	C₂	1	2	yes	maximal
2	#22626115	C₂	1	2	yes	maximal
3	#22625995	C₂	1	4	yes	maximal
4	#22625992	C₂	1	4	yes	maximal
5	#22625853	C₂	1	1	yes	maximal, characteristic
6	#22625649	C₂	1	1	yes	maximal, characteristic
7	#500×D₇₃	C₂	1	1	yes	maximal, characteristic
8	#17989546	C₂²	1	1	yes	characteristic
9	(C₂×#64408)⋊Q₈	—	2	4	no	—
10	#17989538	C₂²	1	4	yes	—
11	#17989536	—	2	4	no	—
12	#17989531	C₂²	1	2	yes	—
13	C₄⋊C₄×#64408	C₂²	1	4	yes	—
14	#17989523	C₂²	1	1	yes	characteristic
15	#17989461	C₂²	1	2	yes	—
16	#17989461	C₂²	1	2	yes	—
17	#17989460	C₂²	1	8	yes	—
18	#17989452	C₂²	1	2	yes	—
19	#17989452	C₂²	1	2	yes	—
20	C₄⋊Q₈×D₇₃	C₂²	1	2	yes	—
21	#17988992	C₄	1	1	yes	characteristic, complemented
22	#17988991	C₄	1	2	yes	complemented
23	#17988991	C₄	1	2	yes	complemented
24	#17988990	C₄	1	2	yes	complemented
25	#107×D₇₃	C₂²	1	2	yes	—
26	#107×D₇₃	C₂²	1	2	yes	—
27	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
28	#500×C₇₃	C₄	1	1	yes	characteristic, complemented
29	C₂×#11924036	C₂³	1	2	yes	—
30	C₂×C₄⋊#64408	—	2	4	no	—
31	C₂×C₄⋊#64408	C₂³	1	4	yes	—
32	C₂×C₄×#64408	—	2	4	no	—
33	C₂×C₄×#64408	C₂³	1	2	yes	—
34	#14556646	—	2	1	no	—
35	#14556646	—	2	1	no	—
36	(C₂×#64408)⋊C₄	—	2	4	no	—
37	(C₂×#64408)⋊C₄	—	2	4	no	—
38	(C₂×#64408)⋊C₄	—	2	2	no	—
39	(C₂×#64408)⋊C₄	—	2	2	no	—
40	(C₂×#64408)⋊C₄	D₄	1	2	yes	—
41	(C₂×#64408)⋊C₄	D₄	1	2	yes	—
42	C₂×Q₈×D₇₃	—	2	2	no	—
43	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
45	#14556591	C₂×C₄	1	2	yes	—
46	#14556590	—	2	2	no	—
47	#14556590	C₂×C₄	1	4	yes	—
48	#14556589	—	2	2	no	—
49	C₄⋊C₄×D₇₃	—	2	2	no	—
50	C₄⋊C₄×D₇₃	C₂³	1	4	yes	—
51	C₄²×D₇₃	C₂³	1	1	yes	characteristic
52	#14556525	C₂×C₄	1	2	yes	—
53	#14556525	C₂×C₄	1	2	yes	—
54	#14556524	C₂×C₄	1	4	yes	—
55	#14556524	C₂×C₄	1	4	yes	—
56	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
57	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
59	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
60	#107×C₇₃	C₂×C₄	1	2	yes	—
61	#107×C₇₃	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×#64408	—	2	4	no	—
64	C₂²×#64408	C₂×D₄	1	4	yes	—
65	#11924036	—	2	2	no	—
66	#11924036	—	2	2	no	—
67	C₂×#10596833	—	2	2	no	—
68	C₄⋊#64408	—	2	4	no	—
69	C₄⋊#64408	—	2	4	no	—
70	C₄⋊#64408	—	8	4	no	—
71	C₄×#64408	—	2	2	no	—
72	C₄×#64408	—	2	2	no	—
73	C₄×#64408	—	8	4	no	—
74	C₂×Dic₁₄₆	—	2	2	no	—
75	C₂×Dic₁₄₆	—	2	4	no	—
76	Q₈×D₇₃	—	8	4	no	—
77	C₂×C₄×D₇₃	—	2	4	no	—
78	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
81	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
82	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
83	C₈×D₇₃	—	2	2	no	—
84	C₈×D₇₃	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
87	#11924007	—	2	4	no	—
88	#11924007	—	2	4	no	—
89	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₄×Dic₇₃	#41	1	1	yes	characteristic
91	C₄×Dic₇₃	#41	1	1	yes	characteristic
92	C₂×Q₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₇₃	—	2	2	no	—
96	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
97	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×D₇₃	C₂²×D₄	1	1	yes	characteristic
99	C₂×#64408	—	2	4	no	—
100	C₂×#64408	—	2	4	no	—
101	C₂×#64408	—	2	4	no	—
102	C₂×#64408	—	4	4	no	—
103	C₂×#64408	—	4	4	no	—
104	C₂×#64408	—	4	4	no	—
105	Q₈×C₇₃	—	8	2	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₇₃	—	2	4	no	—
111	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
112	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
113	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
114	C₄×D₇₃	—	2	2	no	—
115	C₄×D₇₃	—	2	2	no	—
116	C₄×D₇₃	—	4	4	no	—
117	C₄×D₇₃	—	4	4	no	—
118	C₄×D₇₃	2₋¹⁺⁴	1	2	yes	—
119	C₄×D₇₃	2₋¹⁺⁴	1	2	yes	—
120	Dic₁₄₆	—	8	4	no	—
121	Dic₁₄₆	—	8	2	no	—
122	#10596833	—	4	2	no	—
123	C₈×C₇₃	—	2	1	no	—
124	C₈×C₇₃	—	2	1	no	—
125	#10596831	—	2	1	no	—
126	#10596831	—	2	1	no	—
127	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂×D₇₃	#536	1	2	yes	—
129	C₂×D₇₃	#576	1	2	yes	—
130	C₂×D₇₃	#578	1	2	yes	—
131	#64408	—	4	4	no	—
132	#64408	—	8	4	no	—
133	C₄×C₇₃	—	2	1	no	—
134	C₄×C₇₃	—	2	1	no	—
135	C₄×C₇₃	—	4	4	no	—
136	C₄×C₇₃	#519	1	1	yes	characteristic
137	C₄×C₇₃	#519	1	1	yes	characteristic
138	Dic₇₃	—	2	1	no	—
139	Dic₇₃	—	2	1	no	—
140	Dic₇₃	—	4	4	no	—
141	Dic₇₃	#541	1	2	yes	—
142	#22254	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	#2327	1	1	yes	characteristic
144	C₂×C₇₃	#2948	1	1	yes	characteristic
145	C₂×C₇₃	#2950	1	1	yes	characteristic
146	D₇₃	#3243	1	4	yes	—
147	C₂×#500	—	73	1	no	—
148	#2331	—	73	2	no	—
149	#1993	—	73	2	no	—
150	#1921	—	73	4	no	—
151	#1918	—	73	4	no	—
152	#1835	—	73	1	no	—
153	#1746	—	73	1	no	—
154	C₇₃	#22254	1	1	yes	characteristic, complemented, Sylow
155	C₂×C₄⋊Q₈	—	73	2	no	—
156	#500	—	73	2	no	—
157	#500	—	73	2	no	—
158	#500	—	73	2	no	—
159	#500	—	73	1	no	—
160	#500	#64408	1	1	yes	characteristic, complemented
161	C₂×#107	—	73	2	no	—
162	C₂×#107	—	73	2	no	—
163	C₂×#97	—	73	1	no	—
164	C₂×C₄×C₈	—	73	1	no	—
165	#391	—	146	4	no	—
166	#389	—	73	4	no	—
167	#387	—	146	4	no	—
168	#382	—	73	2	no	—
169	C₄×C₄⋊C₄	—	73	4	no	—
170	#376	—	73	1	no	—
171	#340	—	73	2	no	—
172	#340	—	73	2	no	—
173	#340	—	73	8	no	—
174	#336	—	73	2	no	—
175	#336	—	73	2	no	—
176	C₂²×Q₈	—	146	2	no	—
177	C₂²×C₈	—	73	1	no	—
178	C₂²×C₈	—	73	1	no	—
179	C₂²×C₈	—	73	2	no	—
180	C₄⋊Q₈	—	73	4	no	—
181	C₄⋊Q₈	—	73	2	no	—
182	C₄⋊Q₈	—	146	2	no	—
183	C₄⋊Q₈	—	146	2	no	—
184	C₄⋊Q₈	C₂×#64408	1	2	yes	—
185	C₂×C₄⋊C₄	—	73	4	no	—
186	C₂×C₄⋊C₄	—	73	4	no	—
187	C₂×C₄⋊C₄	—	146	2	no	—
188	C₂×C₄⋊C₄	—	146	4	no	—
189	C₂×C₄²	—	73	1	no	—
190	C₂×C₄²	—	73	2	no	—
191	C₂×C₄²	—	146	4	no	—
192	#107	—	73	2	no	—
193	#107	—	73	2	no	—
194	#107	—	73	4	no	—
195	#107	—	73	4	no	—
196	#107	C₂×#64408	1	2	yes	—
197	#107	C₂×#64408	1	2	yes	—
198	#97	—	146	1	no	—
199	#97	—	146	1	no	—
200	C₄×C₈	—	73	1	no	—
201	C₄×C₈	—	73	1	no	—
202	C₄×C₈	—	73	1	no	—
203	C₄×C₈	C₂×#64408	1	1	yes	characteristic
204	#95	—	73	2	no	—
205	#95	—	73	2	no	—
206	#95	—	146	4	no	—
207	#95	—	146	2	no	—
208	#95	—	146	2	no	—
209	#95	—	146	4	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	584	4	no	—
212	C₂×Q₈	—	146	4	no	—
213	C₂×Q₈	—	146	2	no	—
214	C₂²×C₄	—	73	1	no	—
215	C₂²×C₄	—	73	1	no	—
216	C₂²×C₄	—	73	1	no	—
217	C₂²×C₄	—	73	4	no	—
218	C₂²×C₄	—	146	4	no	—
219	C₂²×C₄	—	146	4	no	—
220	C₂×C₈	—	73	1	no	—
221	C₂×C₈	—	73	1	no	—
222	C₂×C₈	—	146	2	no	—
223	C₂×C₈	—	146	2	no	—
224	C₂×C₈	—	146	2	no	—
225	C₂×C₈	—	146	2	no	—
226	C₂×C₈	—	146	2	no	—
227	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
228	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	584	4	no	—
231	C₄⋊C₄	—	73	4	no	—
232	C₄⋊C₄	—	146	4	no	—
233	C₄⋊C₄	—	146	4	no	—
234	C₄⋊C₄	—	146	4	no	—
235	C₄⋊C₄	—	146	4	no	—
236	C₄⋊C₄	—	146	2	no	—
237	C₄⋊C₄	C₂²×#64408	1	4	yes	—
238	C₄²	—	584	4	no	—
239	C₄²	—	73	1	no	—
240	C₄²	—	73	1	no	—
241	C₄²	—	73	1	no	—
242	C₄²	—	146	2	no	—
243	C₄²	—	146	2	no	—
244	C₄²	C₂²×#64408	1	1	yes	characteristic
245	C₂³	—	73	1	no	—
246	Q₈	—	8	2	no	—
247	Q₈	—	584	2	no	—
248	Q₈	—	584	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	292	4	no	—
251	C₂×C₄	—	292	4	no	—
252	C₂×C₄	—	292	4	no	—
253	C₂×C₄	—	292	4	no	—
254	C₂×C₄	—	292	4	no	—
255	C₂×C₄	—	73	1	no	—
256	C₂×C₄	—	73	2	no	—
257	C₂×C₄	—	73	2	no	—
258	C₂×C₄	—	73	1	no	—
259	C₂×C₄	—	73	1	no	—
260	C₂×C₄	—	146	2	no	—
261	C₂×C₄	—	146	2	no	—
262	C₂×C₄	—	146	4	no	—
263	C₂×C₄	—	146	4	no	—
264	C₂×C₄	—	146	4	no	—
265	C₂×C₄	—	146	4	no	—
266	C₂×C₄	D₄×#64408	1	1	yes	characteristic
267	C₂×C₄	D₄×#64408	1	1	yes	characteristic
268	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	292	2	no	—
272	C₈	—	146	1	no	—
273	C₈	—	146	1	no	—
274	C₂²	—	73	2	no	—
275	C₂²	—	73	2	no	—
276	C₂²	—	73	2	no	—
277	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	—	292	4	no	—
282	C₄	—	292	4	no	—
283	C₄	—	584	4	no	—
284	C₄	—	73	2	no	—
285	C₄	—	146	1	no	—
286	C₄	—	146	1	no	—
287	C₄	#17989655	1	1	yes	characteristic
288	C₄	#17989655	1	1	yes	characteristic
289	C₂	—	73	4	no	—
290	C₂	#22626422	1	1	yes	characteristic
291	C₂	#22626690	1	1	yes	characteristic
292	C₂	#22626692	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#500×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₂²×D₄
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	#500	#64408
hypocenter	C₇₃	#22254
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

Complements

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

N	H ≅ G/N	type
21. #17988992	C₄	semidirect
22. #17988991	C₄	semidirect
23. #17988991	C₄	semidirect
24. #17988990	C₄	semidirect
28. #500×C₇₃	C₄	semidirect
154. C₇₃	#22254	semidirect
160. #500	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626426	2
#22626115	2
#22625995	2
#22625992	2
#22625853	2
#22625649	2
#500×D₇₃	2
#22254	73