Subgroups — #34019759

Order 18688.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626842	C₂	1	2	yes	maximal
2	#22625578	C₂	1	2	yes	maximal, complemented
3	#22625544	C₂	1	4	yes	maximal, complemented
4	#22625524	C₂	1	2	yes	maximal, complemented
5	#22625479	C₂	1	1	yes	maximal, characteristic
6	#22625254	C₂	1	2	yes	maximal
7	#22625141	C₂	1	1	yes	maximal, characteristic
8	#22620515	C₂	1	1	yes	maximal, characteristic
9	#17989638	C₂²	1	4	yes	—
10	#17989637	—	2	2	no	—
11	#17989636	—	2	2	no	—
12	Q₈×#10596833	C₂²	1	4	yes	—
13	#17989612	—	2	4	no	—
14	#17989606	C₂²	1	2	yes	—
15	#17989606	C₂²	1	1	yes	characteristic
16	C₄×#11924038	C₂²	1	1	yes	characteristic
17	#17989451	C₂²	1	2	yes	—
18	#17989448	C₂²	1	1	yes	characteristic
19	#17989448	C₂²	1	1	yes	characteristic
20	Q₈⋊#10596833	C₂²	1	4	yes	—
21	Q₈⋊#10596833	C₂²	1	4	yes	—
22	#17989429	C₂²	1	2	yes	—
23	#17989428	C₂²	1	2	yes	—
24	#17989142	C₂²	1	2	yes	—
25	#17988992	C₄	1	1	yes	characteristic
26	#17988937	C₄	1	2	yes	—
27	#17988825	C₄	1	2	yes	—
28	#17988825	C₄	1	2	yes	—
29	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
30	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#494×C₇₃	C₄	1	1	yes	characteristic
33	C₂×#11924038	—	2	1	no	—
34	#14556658	—	2	4	no	—
35	C₄⋊#10596833	—	2	4	no	—
36	C₄⋊#10596833	D₄	1	2	yes	—
37	C₄⋊#10596833	C₂³	1	2	yes	—
38	C₄⋊#10596833	C₂³	1	2	yes	—
39	#14556648	—	2	2	no	—
40	#14556648	D₄	1	2	yes	—
41	#14556646	—	2	4	no	—
42	#14556646	—	2	1	no	—
43	C₄×#10596833	—	2	4	no	—
44	C₄×#10596833	—	2	2	no	—
45	C₄×#10596833	C₂³	1	2	yes	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
49	#14556591	C₂×C₄	1	2	yes	—
50	#14556582	—	2	2	no	—
51	#14556571	—	2	2	no	—
52	#14556567	C₂×C₄	1	4	yes	—
53	Q₈×Dic₇₃	C₂³	1	4	yes	—
54	#14556556	—	2	2	no	—
55	#14556549	—	2	2	no	—
56	C₄²×D₇₃	C₂³	1	1	yes	characteristic
57	#14556525	C₂×C₄	1	2	yes	—
58	#14556525	C₂×C₄	1	2	yes	—
59	#14556515	C₂×C₄	1	4	yes	—
60	#14556515	C₂×C₄	1	4	yes	—
61	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
62	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
63	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
65	C₂×Q₁₆×C₇₃	C₂×C₄	1	2	yes	—
66	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
68	#11924038	—	4	2	no	—
69	C₂×#10596833	—	2	2	no	—
70	C₂×#10596833	—	2	2	no	—
71	C₂×#10596833	—	2	2	no	—
72	C₂×#10596833	C₂×D₄	1	4	yes	—
73	C₂×Dic₁₄₆	—	2	2	no	—
74	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
77	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
78	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
79	C₈×D₇₃	—	2	2	no	—
80	C₈×D₇₃	—	4	1	no	—
81	C₄⋊Dic₇₃	—	2	2	no	—
82	C₄⋊Dic₇₃	—	2	4	no	—
83	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
84	#11924007	—	2	4	no	—
85	#11924005	—	2	4	no	—
86	C₄×Dic₇₃	—	2	4	no	—
87	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
88	C₄×Dic₇₃	#41	1	1	yes	characteristic
89	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
90	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
91	Q₁₆×C₇₃	—	2	4	no	—
92	Q₁₆×C₇₃	—	4	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	—	2	2	no	—
97	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₇₃	—	2	4	no	—
100	Q₈×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₇₃	—	2	4	no	—
106	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
109	C₄×D₇₃	—	2	1	no	—
110	C₄×D₇₃	—	2	1	no	—
111	C₄×D₇₃	—	2	2	no	—
112	C₄×D₇₃	—	2	2	no	—
113	Dic₁₄₆	—	4	4	no	—
114	#10596833	—	2	4	no	—
115	#10596833	—	4	2	no	—
116	#10596833	—	4	2	no	—
117	#10596833	—	4	2	no	—
118	C₈×C₇₃	—	2	1	no	—
119	C₈×C₇₃	#131	1	2	yes	—
120	#10596831	—	2	1	no	—
121	#10596831	—	2	1	no	—
122	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
123	C₂×D₇₃	—	2	1	no	—
124	C₂×D₇₃	—	2	1	no	—
125	C₂×D₇₃	—	2	1	no	—
126	C₄×C₇₃	—	2	1	no	—
127	C₄×C₇₃	—	4	4	no	—
128	C₄×C₇₃	#444	1	2	yes	—
129	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
130	C₄×C₇₃	#568	1	1	yes	characteristic
131	Dic₇₃	—	2	1	no	—
132	Dic₇₃	—	2	1	no	—
133	Dic₇₃	—	4	4	no	—
134	Dic₇₃	#578	1	2	yes	—
135	#18090	—	73	1	no	maximal, Sylow
136	C₂×C₇₃	#2935	1	1	yes	characteristic
137	C₂×C₇₃	#2950	1	1	yes	characteristic
138	C₂×C₇₃	C₂×#444	1	1	yes	characteristic
139	D₇₃	—	4	1	no	—
140	#3170	—	73	1	no	—
141	#2986	—	73	2	no	—
142	#1593	—	73	2	no	—
143	#1584	—	73	4	no	—
144	#1579	—	73	2	no	—
145	#1564	—	73	1	no	—
146	#1485	—	73	2	no	—
147	#1451	—	73	1	no	—
148	C₇₃	#18090	1	1	yes	characteristic, complemented, Sylow
149	#533	—	73	2	no	—
150	#500	—	73	1	no	—
151	#494	#64408	1	1	yes	characteristic
152	#487	—	73	2	no	—
153	#467	—	73	2	no	—
154	#467	—	73	2	no	—
155	#446	—	73	4	no	—
156	#446	—	146	2	no	—
157	C₈×Q₈	—	73	4	no	—
158	C₈×Q₈	—	146	2	no	—
159	C₄×Q₁₆	—	73	2	no	—
160	C₄×Q₁₆	—	73	2	no	—
161	#433	—	146	4	no	—
162	#431	—	73	1	no	—
163	#431	—	73	2	no	—
164	C₄×M₁₆	—	73	1	no	—
165	C₂×C₄×C₈	—	73	1	no	—
166	#335	—	73	2	no	—
167	#334	—	73	1	no	—
168	#334	—	73	1	no	—
169	Q₈⋊C₈	—	73	4	no	—
170	Q₈⋊C₈	—	73	4	no	—
171	#322	—	73	2	no	—
172	#322	—	73	2	no	—
173	C₂×Q₁₆	C₂×#64408	1	2	yes	—
174	C₂×Q₁₆	C₂×#64408	1	2	yes	—
175	C₂×M₁₆	—	146	1	no	—
176	C₂²×C₈	—	73	1	no	—
177	C₂²×C₈	—	73	1	no	—
178	C₄⋊Q₈	—	73	2	no	—
179	C₄⋊Q₈	C₂×#64408	1	2	yes	—
180	#125	—	146	2	no	—
181	C₂²⋊Q₈	—	73	4	no	—
182	C₂²⋊Q₈	—	146	2	no	—
183	C₄×Q₈	—	73	4	no	—
184	C₄×Q₈	—	146	2	no	—
185	#117	—	146	2	no	—
186	C₂×C₄²	—	73	1	no	—
187	#107	—	73	2	no	—
188	#107	—	73	2	no	—
189	C₄⋊C₈	—	73	2	no	—
190	C₄⋊C₈	—	73	2	no	—
191	C₄⋊C₈	—	73	2	no	—
192	C₄⋊C₈	—	146	4	no	—
193	C₄⋊C₈	—	146	4	no	—
194	#103	—	73	4	no	—
195	#103	—	73	4	no	—
196	C₂²⋊C₈	—	73	2	no	—
197	C₂²⋊C₈	—	146	2	no	—
198	#97	—	146	1	no	—
199	#97	—	146	4	no	—
200	C₄×C₈	—	73	2	no	—
201	C₄×C₈	—	73	1	no	—
202	C₄×C₈	—	73	1	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	1	no	—
205	C₄×C₈	—	146	2	no	—
206	C₄×C₈	—	146	4	no	—
207	C₄×C₈	C₂×#64408	1	1	yes	characteristic
208	C₂×Q₈	—	146	2	no	—
209	C₂×Q₈	C₂²×#64408	1	4	yes	—
210	C₂²×C₄	—	73	1	no	—
211	C₂²×C₄	—	73	1	no	—
212	C₂²×C₄	—	73	1	no	—
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	2	no	—
215	M₁₆	—	292	2	no	—
216	C₂×C₈	—	292	1	no	—
217	C₂×C₈	—	73	4	no	—
218	C₂×C₈	—	73	1	no	—
219	C₂×C₈	—	73	1	no	—
220	C₂×C₈	—	146	2	no	—
221	C₂×C₈	—	146	2	no	—
222	C₂×C₈	—	146	2	no	—
223	C₂×C₈	—	146	2	no	—
224	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
225	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	73	4	no	—
229	C₄⋊C₄	—	146	4	no	—
230	C₄⋊C₄	—	146	4	no	—
231	C₄⋊C₄	—	146	2	no	—
232	C₂²⋊C₄	—	146	4	no	—
233	C₄²	—	73	1	no	—
234	C₄²	—	73	1	no	—
235	C₄²	—	73	1	no	—
236	C₄²	—	146	4	no	—
237	C₄²	C₂²×#64408	1	1	yes	characteristic
238	C₂³	—	73	1	no	—
239	Q₈	—	2	4	no	—
240	Q₈	—	2	4	no	—
241	Q₈	—	292	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	73	1	no	—
244	C₂×C₄	—	73	1	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	146	2	no	—
247	C₂×C₄	—	146	4	no	—
248	C₂×C₄	—	146	2	no	—
249	C₂×C₄	—	146	1	no	—
250	C₂×C₄	—	146	1	no	—
251	C₂×C₄	D₄×#64408	1	1	yes	characteristic
252	C₂×C₄	D₄×#64408	1	1	yes	characteristic
253	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	292	2	no	—
256	C₈	—	292	2	no	—
257	C₈	—	292	2	no	—
258	C₈	—	146	4	no	—
259	C₈	—	146	1	no	—
260	C₈	—	146	1	no	—
261	C₈	#14556678	1	2	yes	—
262	C₂²	—	146	1	no	—
263	C₂²	—	146	1	no	—
264	C₂²	—	146	1	no	—
265	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	4	4	no	—
268	C₄	—	292	4	no	—
269	C₄	—	73	2	no	—
270	C₄	—	146	1	no	—
271	C₄	—	146	1	no	—
272	C₄	#17989633	1	2	yes	—
273	C₄	C₂×#14556678	1	1	yes	characteristic
274	C₄	#17989665	1	1	yes	characteristic
275	C₂	—	292	1	no	—
276	C₂	#22626674	1	1	yes	characteristic
277	C₂	#22626692	1	1	yes	characteristic
278	C₂	C₂×#17989633	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#494×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#494	#64408
hypocenter	C₇₃	#18090
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #22625578	C₂	semidirect
3. #22625544	C₂	semidirect
4. #22625524	C₂	semidirect
148. C₇₃	#18090	semidirect

Maximal subgroups

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

subgroup	index
#22626842	2
#22625578	2
#22625544	2
#22625524	2
#22625479	2
#22625254	2
#22625141	2
#22620515	2
#18090	73