Subgroups — #34025253

Order 18688.

578 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626724	C₂	1	2	yes	maximal
2	#22626119	C₂	1	2	yes	maximal
3	#22625999	C₂	1	4	yes	maximal
4	#22625996	C₂	1	4	yes	maximal
5	#22625856	C₂	1	1	yes	maximal, characteristic
6	#22625649	C₂	1	1	yes	maximal, characteristic
7	#22620442	C₂	1	1	yes	maximal, characteristic
8	#17989639	—	2	4	no	—
9	#17989638	—	2	4	no	—
10	#17989594	C₂²	1	4	yes	—
11	#17989592	C₂²	1	2	yes	—
12	#17989552	C₂²	1	4	yes	—
13	#17989546	C₂²	1	1	yes	characteristic
14	#17989523	C₂²	1	1	yes	characteristic
15	#17989463	C₂²	1	2	yes	—
16	#17989463	C₂²	1	2	yes	—
17	#17989457	C₂²	1	8	yes	—
18	#17989452	C₂²	1	2	yes	—
19	#17989452	C₂²	1	2	yes	—
20	#17989142	C₂²	1	2	yes	—
21	#17988992	C₄	1	1	yes	characteristic, complemented
22	#17988988	C₄	1	2	yes	complemented
23	#17988988	C₄	1	2	yes	complemented
24	#17988983	C₄	1	2	yes	complemented
25	#17988624	C₂²	1	2	yes	—
26	#17988624	C₂²	1	2	yes	—
27	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
28	#500×C₇₃	C₄	1	1	yes	characteristic, complemented
29	C₂×#11924038	C₂³	1	4	yes	—
30	C₂×#11924036	C₂³	1	2	yes	—
31	C₂×C₄×#64408	C₂³	1	2	yes	—
32	#14556658	—	2	4	no	—
33	#14556658	—	2	2	no	—
34	#14556658	—	2	2	no	—
35	C₄⋊#10596833	—	2	4	no	—
36	#14556646	—	2	4	no	—
37	#14556646	D₄	1	2	yes	—
38	#14556646	D₄	1	2	yes	—
39	C₄×#10596833	—	2	4	no	—
40	(C₂×#64408)⋊C₄	—	2	1	no	—
41	(C₂×#64408)⋊C₄	—	2	1	no	—
42	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
43	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
44	#14556591	C₂×C₄	1	2	yes	—
45	#14556582	C₂×C₄	1	4	yes	—
46	#14556571	—	2	2	no	—
47	#14556567	—	2	2	no	—
48	Q₈×Dic₇₃	—	2	2	no	—
49	#14556556	—	2	2	no	—
50	#14556549	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	#14556522	C₂×C₄	1	4	yes	—
55	#14556522	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	2	no	—
64	#11924038	—	2	4	no	—
65	#11924038	—	2	4	no	—
66	#11924036	—	2	2	no	—
67	#11924036	—	2	2	no	—
68	C₂×#10596833	—	2	4	no	—
69	C₂×#10596833	C₂×D₄	1	4	yes	—
70	C₄×#64408	—	2	2	no	—
71	C₄×#64408	—	2	2	no	—
72	C₂×Dic₁₄₆	—	2	2	no	—
73	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
74	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
75	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
76	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
77	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
78	C₈×D₇₃	—	2	2	no	—
79	C₈×D₇₃	—	2	2	no	—
80	C₄⋊Dic₇₃	—	2	2	no	—
81	C₄⋊Dic₇₃	—	2	4	no	—
82	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
83	#11924007	—	2	4	no	—
84	#11924005	—	2	4	no	—
85	C₄×Dic₇₃	—	2	4	no	—
86	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
88	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂×Q₈×C₇₃	—	2	2	no	—
90	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₇₃	—	2	2	no	—
93	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
94	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×D₇₃	C₂×#41	1	1	yes	characteristic
96	C₂×#64408	—	4	2	no	—
97	C₂×#64408	—	4	2	no	—
98	C₂×#64408	—	4	2	no	—
99	Q₈×C₇₃	—	8	2	no	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
104	C₂×Dic₇₃	—	2	4	no	—
105	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×D₇₃	—	4	1	no	—
109	C₄×D₇₃	—	4	1	no	—
110	C₄×D₇₃	2₋¹⁺⁴	1	2	yes	—
111	C₄×D₇₃	2₋¹⁺⁴	1	2	yes	—
112	Dic₁₄₆	—	8	2	no	—
113	#10596833	—	2	4	no	—
114	#10596833	—	4	4	no	—
115	C₈×C₇₃	—	2	1	no	—
116	C₈×C₇₃	—	2	1	no	—
117	#10596831	—	2	1	no	—
118	#10596831	—	2	1	no	—
119	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂×D₇₃	—	2	1	no	—
121	C₂×D₇₃	—	2	1	no	—
122	C₂×D₇₃	#541	1	2	yes	—
123	#64408	—	8	2	no	—
124	C₄×C₇₃	—	2	1	no	—
125	C₄×C₇₃	—	2	1	no	—
126	C₄×C₇₃	—	4	4	no	—
127	C₄×C₇₃	#519	1	1	yes	characteristic
128	C₄×C₇₃	#519	1	1	yes	characteristic
129	Dic₇₃	—	2	1	no	—
130	Dic₇₃	—	2	1	no	—
131	Dic₇₃	—	4	4	no	—
132	Dic₇₃	#536	1	2	yes	—
133	#22258	—	73	1	no	maximal, Sylow
134	C₂×C₇₃	#2327	1	1	yes	characteristic
135	C₂×C₇₃	#2957	1	1	yes	characteristic
136	C₂×C₇₃	#2957	1	1	yes	characteristic
137	D₇₃	—	2	2	no	—
138	#3160	—	73	1	no	—
139	#2961	—	73	2	no	—
140	#1995	—	73	2	no	—
141	#1922	—	73	4	no	—
142	#1922	—	73	4	no	—
143	#1837	—	73	1	no	—
144	#1746	—	73	1	no	—
145	C₇₃	#22258	1	1	yes	characteristic, complemented, Sylow
146	#533	—	73	2	no	—
147	#500	—	73	1	no	—
148	#500	#64408	1	1	yes	characteristic, complemented
149	#499	—	73	2	no	—
150	#499	—	73	2	no	—
151	#498	—	73	2	no	—
152	#446	—	146	4	no	—
153	#446	—	146	4	no	—
154	#427	—	73	2	no	—
155	#427	—	73	2	no	—
156	#424	—	73	4	no	—
157	#423	—	73	2	no	—
158	#405	—	73	4	no	—
159	C₂×#97	—	73	1	no	—
160	C₂×C₄×C₈	—	73	1	no	—
161	#376	—	73	1	no	—
162	#341	—	73	2	no	—
163	#341	—	73	2	no	—
164	#339	—	73	8	no	—
165	#336	—	73	2	no	—
166	#336	—	73	2	no	—
167	C₂×M₁₆	—	73	4	no	—
168	C₂²×C₈	—	73	2	no	—
169	C₂²×C₈	—	73	1	no	—
170	C₂²×C₈	—	73	1	no	—
171	C₄⋊Q₈	—	73	2	no	—
172	C₄⋊Q₈	C₂×#64408	1	2	yes	—
173	#125	—	73	4	no	—
174	C₂²⋊Q₈	—	146	2	no	—
175	C₂²⋊Q₈	—	146	2	no	—
176	C₄×Q₈	—	146	2	no	—
177	C₄×Q₈	—	146	2	no	—
178	#117	—	73	4	no	—
179	C₂×C₄²	—	73	1	no	—
180	C₂×C₄²	—	73	2	no	—
181	#107	—	73	2	no	—
182	#107	—	73	2	no	—
183	#107	C₂×#64408	1	2	yes	—
184	#107	C₂×#64408	1	2	yes	—
185	#106	—	73	4	no	—
186	#106	—	73	4	no	—
187	C₄⋊C₈	—	146	2	no	—
188	C₄⋊C₈	—	146	2	no	—
189	C₄⋊C₈	—	146	4	no	—
190	C₄⋊C₈	—	146	4	no	—
191	#97	—	73	2	no	—
192	#97	—	73	2	no	—
193	#97	—	146	4	no	—
194	C₄×C₈	—	73	1	no	—
195	C₄×C₈	—	73	1	no	—
196	C₄×C₈	—	73	1	no	—
197	C₄×C₈	—	146	4	no	—
198	C₄×C₈	C₂×#64408	1	1	yes	characteristic
199	#95	—	146	1	no	—
200	#95	—	146	1	no	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	146	2	no	—
203	C₂²×C₄	—	73	1	no	—
204	C₂²×C₄	—	73	1	no	—
205	C₂²×C₄	—	73	1	no	—
206	C₂²×C₄	—	146	2	no	—
207	M₁₆	—	146	4	no	—
208	M₁₆	—	146	4	no	—
209	C₂×C₈	—	73	4	no	—
210	C₂×C₈	—	73	1	no	—
211	C₂×C₈	—	73	1	no	—
212	C₂×C₈	—	146	2	no	—
213	C₂×C₈	—	146	2	no	—
214	C₂×C₈	—	146	4	no	—
215	C₂×C₈	—	146	2	no	—
216	C₂×C₈	—	146	2	no	—
217	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
218	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	73	4	no	—
221	C₄⋊C₄	—	146	4	no	—
222	C₄⋊C₄	—	146	4	no	—
223	C₄⋊C₄	—	146	2	no	—
224	C₄⋊C₄	C₂²×#64408	1	4	yes	—
225	C₂²⋊C₄	—	146	4	no	—
226	C₄²	—	73	1	no	—
227	C₄²	—	73	1	no	—
228	C₄²	—	73	1	no	—
229	C₄²	—	146	4	no	—
230	C₄²	—	146	2	no	—
231	C₄²	—	146	2	no	—
232	C₄²	C₂²×#64408	1	1	yes	characteristic
233	C₂³	—	73	1	no	—
234	Q₈	—	8	2	no	—
235	Q₈	—	584	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	292	1	no	—
238	C₂×C₄	—	292	1	no	—
239	C₂×C₄	—	292	2	no	—
240	C₂×C₄	—	292	2	no	—
241	C₂×C₄	—	292	2	no	—
242	C₂×C₄	—	73	1	no	—
243	C₂×C₄	—	73	2	no	—
244	C₂×C₄	—	73	2	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	146	4	no	—
248	C₂×C₄	D₄×#64408	1	1	yes	characteristic
249	C₂×C₄	D₄×#64408	1	1	yes	characteristic
250	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
251	C₈	—	2	1	no	—
252	C₈	—	2	1	no	—
253	C₈	—	292	4	no	—
254	C₈	—	146	4	no	—
255	C₈	—	146	1	no	—
256	C₈	—	146	1	no	—
257	C₂²	—	73	2	no	—
258	C₂²	—	146	1	no	—
259	C₂²	—	146	1	no	—
260	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	4	no	—
264	C₄	—	292	4	no	—
265	C₄	—	584	2	no	—
266	C₄	—	73	2	no	—
267	C₄	—	146	1	no	—
268	C₄	—	146	1	no	—
269	C₄	#17989655	1	1	yes	characteristic
270	C₄	#17989655	1	1	yes	characteristic
271	C₂	—	146	2	no	—
272	C₂	#22626422	1	1	yes	characteristic
273	C₂	#22626707	1	1	yes	characteristic
274	C₂	#22626711	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	C₂×#41
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	#500	#64408
hypocenter	C₇₃	#22258
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. #17988988	C₄	semidirect
23. #17988988	C₄	semidirect
24. #17988983	C₄	semidirect
28. #500×C₇₃	C₄	semidirect
145. C₇₃	#22258	semidirect
148. #500	#64408	semidirect

Maximal subgroups

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

subgroup	index
#22626724	2
#22626119	2
#22625999	2
#22625996	2
#22625856	2
#22625649	2
#22620442	2
#22258	73