Subgroups — #34019619

Order 18688.

586 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	#22626727	C₂	1	2	yes	maximal
2	#22625570	C₂	1	2	yes	maximal, complemented
3	#22625543	C₂	1	4	yes	maximal, complemented
4	#22625519	C₂	1	2	yes	maximal, complemented
5	#22625476	C₂	1	1	yes	maximal, characteristic
6	#22625241	C₂	1	2	yes	maximal
7	C₈×#11924038	C₂	1	1	yes	maximal, characteristic
8	#22620518	C₂	1	1	yes	maximal, characteristic
9	#17989638	—	2	4	no	—
10	#17989637	C₂²	1	4	yes	—
11	#17989636	C₂²	1	4	yes	—
12	#17989606	C₂²	1	1	yes	characteristic
13	#17989594	—	2	2	no	—
14	#17989588	C₂²	1	2	yes	—
15	#17989552	—	2	2	no	—
16	C₄×#11924038	C₂²	1	1	yes	characteristic
17	#17989451	C₂²	1	2	yes	—
18	#17989450	C₂²	1	1	yes	characteristic
19	#17989450	C₂²	1	1	yes	characteristic
20	#17989437	C₂²	1	4	yes	—
21	#17989437	C₂²	1	4	yes	—
22	#17989429	C₂²	1	2	yes	—
23	C₈×#10596833	C₂²	1	2	yes	—
24	#17989142	C₂²	1	2	yes	—
25	#17988967	C₄	1	1	yes	characteristic
26	#17988939	C₄	1	2	yes	—
27	#17988829	C₄	1	2	yes	—
28	#17988829	C₄	1	2	yes	—
29	#17988681	C₂²	1	2	yes	—
30	#17988681	C₂²	1	2	yes	—
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#500×C₇₃	C₄	1	1	yes	characteristic
33	C₂×#11924038	D₄	1	1	yes	characteristic
34	C₂×#11924038	D₄	1	1	yes	characteristic
35	C₂×#11924036	—	2	2	no	—
36	#14556658	—	2	4	no	—
37	#14556658	—	2	2	no	—
38	C₄⋊#10596833	—	2	4	no	—
39	C₄⋊#10596833	—	2	1	no	—
40	C₄⋊#10596833	C₂³	1	2	yes	—
41	C₄⋊#10596833	C₂³	1	2	yes	—
42	#14556648	—	2	1	no	—
43	#14556646	—	2	4	no	—
44	#14556646	D₄	1	2	yes	—
45	C₄×#10596833	—	2	4	no	—
46	C₄×#10596833	C₂³	1	2	yes	—
47	C₄×#10596833	C₂³	1	2	yes	—
48	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
49	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
52	#14556591	C₂×C₄	1	2	yes	—
53	#14556582	—	2	2	no	—
54	#14556571	C₂×C₄	1	4	yes	—
55	#14556567	—	2	2	no	—
56	Q₈×Dic₇₃	—	2	2	no	—
57	#14556556	C₂³	1	4	yes	—
58	#14556549	—	2	2	no	—
59	C₄²×D₇₃	C₂³	1	1	yes	characteristic
60	#14556514	C₂×C₄	1	4	yes	—
61	#14556514	C₂×C₄	1	4	yes	—
62	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
63	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
64	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
66	#107×C₇₃	C₂×C₄	1	2	yes	—
67	#107×C₇₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	#11924038	—	2	2	no	—
70	#11924038	Q₁₆	1	4	yes	complemented
71	#11924036	—	4	4	no	—
72	C₂×#10596833	—	2	2	no	—
73	C₂×#10596833	—	2	2	no	—
74	C₂×#10596833	—	2	2	no	—
75	C₂×#10596833	C₂×D₄	1	2	yes	—
76	C₂×#10596833	C₂×D₄	1	2	yes	—
77	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
81	Dic₂₉₂	—	2	4	no	—
82	Dic₂₉₂	—	4	2	no	—
83	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
84	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
85	C₈×D₇₃	—	2	2	no	—
86	C₈×D₇₃	—	4	1	no	—
87	C₄⋊Dic₇₃	—	2	2	no	—
88	C₄⋊Dic₇₃	—	2	2	no	—
89	C₄⋊Dic₇₃	—	2	4	no	—
90	#11924007	—	2	4	no	—
91	#11924005	—	2	4	no	—
92	C₄×Dic₇₃	—	2	4	no	—
93	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
94	C₄×Dic₇₃	#41	1	1	yes	characteristic
95	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
96	C₂×Q₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₃	—	2	2	no	—
100	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
101	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
103	Q₈×C₇₃	—	8	2	no	—
104	C₂×C₄×C₇₃	—	2	4	no	—
105	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
108	C₂×Dic₇₃	—	2	4	no	—
109	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
110	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
112	C₄×D₇₃	—	2	1	no	—
113	C₄×D₇₃	—	2	1	no	—
114	C₄×D₇₃	—	4	1	no	—
115	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
116	C₄×D₇₃	#135	1	2	yes	—
117	Dic₁₄₆	—	2	4	no	—
118	Dic₁₄₆	—	2	4	no	—
119	#10596833	—	2	2	no	—
120	#10596833	—	4	2	no	—
121	#10596833	—	4	2	no	—
122	#10596833	—	4	2	no	—
123	#10596833	C₂×Q₁₆	1	4	yes	—
124	C₈×C₇₃	—	2	1	no	—
125	C₈×C₇₃	#131	1	2	yes	—
126	#10596831	—	2	1	no	—
127	#10596831	—	2	1	no	—
128	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂×D₇₃	—	2	1	no	—
130	C₂×D₇₃	—	2	1	no	—
131	C₂×D₇₃	C₄×Q₁₆	1	2	yes	—
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	—	4	4	no	—
134	C₄×C₇₃	C₄×Q₁₆	1	2	yes	—
135	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
136	C₄×C₇₃	#568	1	1	yes	characteristic
137	Dic₇₃	—	2	1	no	—
138	Dic₇₃	—	2	1	no	—
139	Dic₇₃	—	4	4	no	—
140	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
141	Dic₇₃	C₂×#135	1	1	yes	characteristic
142	#18068	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	#2935	1	1	yes	characteristic
144	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
145	C₂×C₇₃	#2957	1	1	yes	characteristic
146	D₇₃	—	2	2	no	—
147	#3170	—	73	1	no	—
148	#2962	—	73	2	no	—
149	#1591	—	73	2	no	—
150	#1584	—	73	4	no	—
151	C₈×Q₁₆	—	73	2	no	—
152	C₂×#335	—	73	1	no	—
153	#1481	—	73	2	no	—
154	C₈×M₁₆	—	73	1	no	—
155	C₇₃	#18068	1	1	yes	characteristic, complemented, Sylow
156	#533	—	73	2	no	—
157	#500	#64408	1	1	yes	characteristic
158	#494	—	73	1	no	—
159	#487	—	73	2	no	—
160	#467	—	73	2	no	—
161	#467	—	73	2	no	—
162	#446	—	73	4	no	—
163	#446	—	146	4	no	—
164	C₈×Q₈	—	73	4	no	—
165	C₄×Q₁₆	—	73	2	no	—
166	C₄×Q₁₆	—	73	2	no	—
167	#431	—	73	1	no	—
168	#424	—	146	2	no	—
169	C₂×C₄⋊C₈	—	73	2	no	—
170	#405	—	146	2	no	—
171	C₄×M₁₆	—	73	1	no	—
172	C₂×C₄×C₈	—	73	1	no	—
173	#335	—	73	1	no	—
174	#335	—	73	2	no	—
175	#335	—	73	1	no	—
176	Q₈⋊C₈	—	73	4	no	—
177	Q₈⋊C₈	—	73	4	no	—
178	#322	—	73	2	no	—
179	C₈²	—	73	2	no	—
180	C₂×Q₁₆	—	73	2	no	—
181	C₂×Q₁₆	—	73	2	no	—
182	C₂×M₁₆	—	73	1	no	—
183	C₂×M₁₆	—	73	1	no	—
184	C₂²×C₈	—	73	1	no	—
185	C₂²×C₈	—	73	1	no	—
186	C₂²×C₈	—	146	2	no	—
187	C₄⋊Q₈	—	73	2	no	—
188	C₄⋊Q₈	C₂×#64408	1	2	yes	—
189	#125	—	146	2	no	—
190	C₂²⋊Q₈	—	73	4	no	—
191	C₂²⋊Q₈	—	146	2	no	—
192	C₄×Q₈	—	73	4	no	—
193	C₄×Q₈	—	146	2	no	—
194	#117	—	146	2	no	—
195	C₂×C₄²	—	73	1	no	—
196	#107	C₂×#64408	1	2	yes	—
197	#107	C₂×#64408	1	2	yes	—
198	C₄⋊C₈	—	73	2	no	—
199	C₄⋊C₈	—	73	2	no	—
200	C₄⋊C₈	—	146	2	no	—
201	C₄⋊C₈	—	146	1	no	—
202	C₄⋊C₈	—	146	4	no	—
203	C₄⋊C₈	—	146	4	no	—
204	#103	—	73	4	no	—
205	#103	—	73	4	no	—
206	C₂²⋊C₈	—	146	1	no	—
207	#97	—	73	2	no	—
208	#97	—	146	4	no	—
209	C₄×C₈	—	73	2	no	—
210	C₄×C₈	—	73	1	no	—
211	C₄×C₈	—	73	1	no	—
212	C₄×C₈	—	73	2	no	—
213	C₄×C₈	—	73	1	no	—
214	C₄×C₈	—	146	4	no	—
215	C₄×C₈	C₂×#64408	1	1	yes	characteristic
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	73	4	no	—
218	C₂²×C₄	—	73	1	no	—
219	C₂²×C₄	—	73	1	no	—
220	C₂²×C₄	—	73	1	no	—
221	Q₁₆	—	292	2	no	—
222	Q₁₆	—	146	4	no	—
223	M₁₆	—	73	4	no	—
224	M₁₆	—	146	2	no	—
225	C₂×C₈	—	292	1	no	—
226	C₂×C₈	—	292	4	no	—
227	C₂×C₈	—	73	2	no	—
228	C₂×C₈	—	73	2	no	—
229	C₂×C₈	—	73	1	no	—
230	C₂×C₈	—	73	1	no	—
231	C₂×C₈	—	146	2	no	—
232	C₂×C₈	—	146	2	no	—
233	C₂×C₈	—	146	2	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
236	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	146	4	no	—
239	C₄⋊C₄	—	146	4	no	—
240	C₄⋊C₄	—	146	2	no	—
241	C₄⋊C₄	—	146	2	no	—
242	C₄⋊C₄	C₂²×#64408	1	4	yes	—
243	C₂²⋊C₄	—	146	4	no	—
244	C₄²	—	73	1	no	—
245	C₄²	—	73	1	no	—
246	C₄²	—	73	1	no	—
247	C₄²	—	146	4	no	—
248	C₄²	C₂²×#64408	1	1	yes	characteristic
249	C₂³	—	73	1	no	—
250	Q₈	—	8	2	no	—
251	Q₈	—	146	4	no	—
252	Q₈	—	146	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	292	1	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	4	no	—
261	C₂×C₄	—	146	1	no	—
262	C₂×C₄	—	146	1	no	—
263	C₂×C₄	D₄×#64408	1	1	yes	characteristic
264	C₂×C₄	D₄×#64408	1	1	yes	characteristic
265	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	292	2	no	—
268	C₈	—	292	2	no	—
269	C₈	—	292	2	no	—
270	C₈	—	73	4	no	—
271	C₈	—	146	2	no	—
272	C₈	—	146	1	no	—
273	C₈	—	146	1	no	—
274	C₈	#14556679	1	2	yes	—
275	C₂²	—	73	2	no	—
276	C₂²	—	146	1	no	—
277	C₂²	—	146	1	no	—
278	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	4	4	no	—
281	C₄	—	292	4	no	—
282	C₄	—	73	1	no	—
283	C₄	—	73	1	no	—
284	C₄	—	146	1	no	—
285	C₄	—	146	1	no	—
286	C₄	Q₁₆×#64408	1	2	yes	—
287	C₄	C₂×#14556679	1	1	yes	characteristic
288	C₄	#17989667	1	1	yes	characteristic
289	C₂	—	146	2	no	—
290	C₂	#22626677	1	1	yes	characteristic
291	C₂	C₂×Q₁₆×#64408	1	1	yes	characteristic
292	C₂	#22626707	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	C₄×D₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#500	#64408
hypocenter	C₇₃	#18068
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
2. #22625570	C₂	semidirect
3. #22625543	C₂	semidirect
4. #22625519	C₂	semidirect
70. #11924038	Q₁₆	semidirect
155. C₇₃	#18068	semidirect

Maximal subgroups

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

subgroup	index
#22626727	2
#22625570	2
#22625543	2
#22625519	2
#22625476	2
#22625241	2
C₈×#11924038	2
#22620518	2
#18068	73