Subgroups — #34018985

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626877	C₂	1	2	yes	maximal, complemented
2	#22625580	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625577	C₂	1	1	yes	maximal, characteristic, complemented
4	#22625566	C₂	1	2	yes	maximal, complemented
5	#22625529	C₂	1	1	yes	maximal, characteristic, complemented
6	#22625526	C₂	1	1	yes	maximal, characteristic, complemented
7	#22625516	C₂	1	2	yes	maximal, complemented
8	#22625486	C₂	1	1	yes	maximal, characteristic, complemented
9	#22625289	C₂	1	2	yes	maximal, complemented
10	#22625135	C₂	1	1	yes	maximal, characteristic, complemented
11	#22620503	C₂	1	1	yes	maximal, characteristic
12	#17989637	C₂²	1	2	yes	—
13	#17989636	C₂²	1	2	yes	—
14	#17989621	—	2	2	no	—
15	#17989621	C₂²	1	2	yes	—
16	#17989617	—	2	2	no	—
17	#17989615	C₂²	1	2	yes	—
18	#17989612	—	2	2	no	—
19	#17989605	C₂²	1	1	yes	characteristic
20	#17989605	C₂²	1	1	yes	characteristic
21	#17989594	—	2	2	no	—
22	#17989591	C₂²	1	2	yes	—
23	#17989448	C₂²	1	2	yes	—
24	#17989448	C₂²	1	1	yes	characteristic
25	#17989448	C₂²	1	1	yes	characteristic
26	#17989437	C₂²	1	2	yes	—
27	#17989437	C₂²	1	2	yes	—
28	#17989435	C₂²	1	2	yes	—
29	#17989435	C₂²	1	2	yes	—
30	#17989428	C₂²	1	1	yes	characteristic
31	#17989428	C₂²	1	1	yes	characteristic
32	C₈×#10596833	C₂²	1	1	yes	characteristic
33	C₈×#10596833	C₂²	1	1	yes	characteristic
34	#17989122	C₂²	1	2	yes	—
35	#17988971	C₄	1	1	yes	characteristic
36	#17988944	C₄	1	2	yes	—
37	#17988829	C₄	1	1	yes	characteristic
38	#17988824	C₄	1	1	yes	characteristic
39	#17988817	C₄	1	2	yes	—
40	#17988681	C₂²	1	1	yes	characteristic
41	#17988674	C₂²	1	2	yes	—
42	#17988669	C₂²	1	1	yes	characteristic
43	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
44	#499×C₇₃	C₄	1	1	yes	characteristic
45	C₂×#11924038	—	2	2	no	—
46	C₂×#11924038	—	2	2	no	—
47	C₂×#11924036	—	2	2	no	—
48	#14556658	—	2	2	no	—
49	#14556658	—	2	1	no	—
50	#14556658	D₄	1	2	yes	—
51	C₄⋊#10596833	—	2	2	no	—
52	C₄⋊#10596833	—	2	2	no	—
53	C₄⋊#10596833	C₂³	1	2	yes	—
54	C₄⋊#10596833	C₂³	1	2	yes	—
55	#14556648	—	2	2	no	—
56	#14556648	—	2	2	no	—
57	#14556648	—	2	1	no	—
58	#14556648	D₄	1	2	yes	—
59	#14556646	—	2	2	no	—
60	C₄×#10596833	—	2	2	no	—
61	C₄×#10596833	C₂³	1	1	yes	characteristic
62	C₄×#10596833	C₂³	1	1	yes	characteristic
63	C₄×#10596833	C₂³	1	1	yes	characteristic
64	C₄×#10596833	C₂³	1	1	yes	characteristic
65	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
66	C₂×#11924017	C₂×C₄	1	2	yes	—
67	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
68	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
70	#14556586	—	2	2	no	—
71	#14556579	C₂×C₄	1	2	yes	—
72	#14556576	—	2	2	no	—
73	#14556571	C₂×C₄	1	2	yes	—
74	#14556566	C₂×C₄	1	2	yes	—
75	#14556556	C₂³	1	2	yes	—
76	#14556553	—	2	2	no	—
77	#14556553	C₂³	1	2	yes	—
78	#14556549	—	2	2	no	—
79	C₄²×D₇₃	C₂³	1	1	yes	characteristic
80	#14556514	C₂×C₄	1	2	yes	—
81	#14556514	C₂×C₄	1	2	yes	—
82	#14556511	C₂×C₄	1	2	yes	—
83	#14556511	C₂×C₄	1	2	yes	—
84	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
85	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
86	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
87	#125×C₇₃	C₂×C₄	1	2	yes	—
88	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
89	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
90	#106×C₇₃	C₂×C₄	1	2	yes	—
91	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
92	#11924038	—	8	2	no	—
93	#11924038	—	8	2	no	—
94	#11924036	—	8	2	no	—
95	C₂×#10596833	—	2	2	no	—
96	C₂×#10596833	—	2	2	no	—
97	C₂×#10596833	—	2	1	no	—
98	C₂×#10596833	—	2	1	no	—
99	C₂×#10596833	C₂×D₄	1	2	yes	—
100	C₂×#10596833	C₂×D₄	1	2	yes	—
101	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
102	C₂×D₂₉₂	—	2	2	no	—
103	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
104	C₂×C₄×D₇₃	—	2	2	no	—
105	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
108	Dic₂₉₂	—	2	2	no	—
109	#11924017	—	2	4	no	—
110	D₅₈₄	—	2	2	no	—
111	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
112	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
113	C₈×D₇₃	—	2	2	no	—
114	C₈×D₇₃	—	2	2	no	—
115	C₄⋊Dic₇₃	—	2	2	no	—
116	#11924007	—	2	2	no	—
117	#11924005	—	2	2	no	—
118	#11924005	—	2	2	no	—
119	#11924005	—	2	2	no	—
120	#11924005	—	2	2	no	—
121	#11924005	—	2	2	no	—
122	C₄×Dic₇₃	—	2	2	no	—
123	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₇₃	#41	1	1	yes	characteristic
125	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
128	C₄⋊C₄×C₇₃	—	2	2	no	—
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
131	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
132	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₂²×D₇₃	—	2	2	no	—
134	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₃	—	2	2	no	—
136	C₂×C₄×C₇₃	—	2	2	no	—
137	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×Dic₇₃	—	2	2	no	—
141	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
144	D₂₉₂	—	2	2	no	—
145	D₂₉₂	—	2	2	no	—
146	D₂₉₂	—	8	2	no	—
147	C₄×D₇₃	—	2	1	no	—
148	C₄×D₇₃	—	2	1	no	—
149	C₄×D₇₃	—	2	2	no	—
150	C₄×D₇₃	—	4	1	no	—
151	C₄×D₇₃	—	4	2	no	—
152	C₄×D₇₃	—	4	2	no	—
153	Dic₁₄₆	—	2	2	no	—
154	Dic₁₄₆	—	2	2	no	—
155	#10596833	—	2	2	no	—
156	#10596833	—	2	2	no	—
157	#10596833	—	4	1	no	—
158	#10596833	—	4	1	no	—
159	#10596833	—	4	2	no	—
160	#10596833	—	4	2	no	—
161	C₈×C₇₃	#131	1	2	yes	—
162	C₈×C₇₃	#131	1	2	yes	—
163	#10596831	—	2	1	no	—
164	#10596831	—	2	1	no	—
165	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂×D₇₃	—	2	1	no	—
167	C₂×D₇₃	—	2	1	no	—
168	C₂×D₇₃	—	2	1	no	—
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₂×D₇₃	—	4	2	no	—
172	C₄×C₇₃	—	2	1	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	4	2	no	—
175	C₄×C₇₃	—	4	2	no	—
176	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
177	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
178	Dic₇₃	—	2	1	no	—
179	Dic₇₃	—	2	1	no	—
180	Dic₇₃	—	4	2	no	—
181	Dic₇₃	#577	1	2	yes	—
182	#17974	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	#2931	1	1	yes	characteristic
184	C₂×C₇₃	#2952	1	1	yes	characteristic
185	C₂×C₇₃	#2957	1	1	yes	characteristic
186	D₇₃	—	4	1	no	—
187	D₇₃	—	8	2	no	—
188	#3168	—	73	1	no	—
189	#2993	—	73	2	no	—
190	#1594	—	73	1	no	—
191	#1593	—	73	1	no	—
192	#1590	—	73	2	no	—
193	#1580	—	73	1	no	—
194	#1579	—	73	1	no	—
195	C₈×SD₁₆	—	73	2	no	—
196	#1566	—	73	1	no	—
197	#1495	—	73	2	no	—
198	#1450	—	73	1	no	—
199	C₇₃	#17974	1	1	yes	characteristic, complemented, Sylow
200	#529	—	73	2	no	—
201	#499	#64408	1	1	yes	characteristic
202	#495	—	73	1	no	—
203	#488	—	73	2	no	—
204	#467	—	73	1	no	—
205	#466	—	73	1	no	—
206	#465	—	73	2	no	—
207	#446	—	73	2	no	—
208	C₈×Q₈	—	73	2	no	—
209	C₄×Q₁₆	—	73	1	no	—
210	C₄×SD₁₆	—	73	2	no	—
211	C₄×D₈	—	73	1	no	—
212	#436	—	73	2	no	—
213	#436	—	146	2	no	—
214	#435	—	146	2	no	—
215	C₈×D₄	—	73	2	no	—
216	#433	—	146	2	no	—
217	#431	—	73	1	no	—
218	#431	—	73	1	no	—
219	#424	—	146	2	no	—
220	#423	—	73	2	no	—
221	C₂×C₄×C₈	—	73	1	no	—
222	#334	—	73	2	no	—
223	#334	—	73	1	no	—
224	#334	—	73	1	no	—
225	Q₈⋊C₈	—	73	2	no	—
226	Q₈⋊C₈	—	73	2	no	—
227	D₄⋊C₈	—	73	2	no	—
228	D₄⋊C₈	—	73	2	no	—
229	#322	—	73	1	no	—
230	#322	—	73	1	no	—
231	C₈²	—	73	1	no	—
232	C₈²	—	73	1	no	—
233	C₂×Q₁₆	—	73	1	no	—
234	C₂×SD₁₆	—	73	2	no	—
235	C₂×D₈	—	73	1	no	—
236	C₂×M₁₆	—	146	2	no	—
237	C₂×M₁₆	—	146	2	no	—
238	C₂²×C₈	—	73	1	no	—
239	C₂²×C₈	—	73	1	no	—
240	C₂²×C₈	—	146	2	no	—
241	#126	—	146	2	no	—
242	#125	C₂×#64408	1	2	yes	—
243	#124	—	73	2	no	—
244	#123	—	146	2	no	—
245	C₂²⋊Q₈	—	73	2	no	—
246	#121	—	73	2	no	—
247	C₄×Q₈	—	73	2	no	—
248	C₄×D₄	—	73	2	no	—
249	C₄×D₄	—	146	2	no	—
250	#117	—	146	2	no	—
251	C₂×C₄²	—	73	1	no	—
252	#107	C₂×#64408	1	1	yes	characteristic
253	#107	C₂×#64408	1	1	yes	characteristic
254	#106	C₂×#64408	1	2	yes	—
255	C₄⋊C₈	—	73	2	no	—
256	C₄⋊C₈	—	73	2	no	—
257	C₄⋊C₈	—	73	2	no	—
258	C₄⋊C₈	—	146	2	no	—
259	C₄⋊C₈	—	146	1	no	—
260	C₄⋊C₈	—	146	2	no	—
261	C₄⋊C₈	—	146	2	no	—
262	#103	—	73	2	no	—
263	#103	—	73	2	no	—
264	#102	—	73	2	no	—
265	#102	—	73	2	no	—
266	C₂²⋊C₈	—	73	2	no	—
267	C₂²⋊C₈	—	146	2	no	—
268	C₂²⋊C₈	—	146	2	no	—
269	C₂²⋊C₈	—	146	1	no	—
270	#97	—	146	2	no	—
271	C₄×C₈	—	73	1	no	—
272	C₄×C₈	—	73	1	no	—
273	C₄×C₈	—	73	1	no	—
274	C₄×C₈	—	73	1	no	—
275	C₄×C₈	—	73	1	no	—
276	C₄×C₈	—	73	1	no	—
277	C₄×C₈	—	73	1	no	—
278	C₄×C₈	—	146	2	no	—
279	C₄×C₈	C₂×#64408	1	1	yes	characteristic
280	C₂×Q₈	—	73	2	no	—
281	C₂×D₄	—	73	2	no	—
282	C₂×D₄	—	146	2	no	—
283	C₂²×C₄	—	73	1	no	—
284	C₂²×C₄	—	73	1	no	—
285	C₂²×C₄	—	73	1	no	—
286	C₂²×C₄	—	146	2	no	—
287	Q₁₆	—	146	2	no	—
288	SD₁₆	—	146	4	no	—
289	D₈	—	146	2	no	—
290	M₁₆	—	584	2	no	—
291	M₁₆	—	584	2	no	—
292	C₂×C₈	—	584	2	no	—
293	C₂×C₈	—	73	2	no	—
294	C₂×C₈	—	73	2	no	—
295	C₂×C₈	—	73	1	no	—
296	C₂×C₈	—	73	1	no	—
297	C₂×C₈	—	146	1	no	—
298	C₂×C₈	—	146	1	no	—
299	C₂×C₈	—	146	2	no	—
300	C₂×C₈	—	146	2	no	—
301	C₂×C₈	—	146	2	no	—
302	C₂×C₈	—	146	2	no	—
303	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
304	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	146	2	no	—
308	C₄⋊C₄	—	146	2	no	—
309	C₄⋊C₄	C₂²×#64408	1	2	yes	—
310	C₄⋊C₄	C₂²×#64408	1	2	yes	—
311	C₂²⋊C₄	—	146	2	no	—
312	C₂²⋊C₄	—	146	2	no	—
313	C₂²⋊C₄	—	146	2	no	—
314	C₂²⋊C₄	—	146	2	no	—
315	C₂²⋊C₄	—	146	2	no	—
316	C₄²	—	73	1	no	—
317	C₄²	—	73	1	no	—
318	C₄²	—	73	1	no	—
319	C₄²	—	146	2	no	—
320	C₄²	C₂²×#64408	1	1	yes	characteristic
321	C₂³	—	73	1	no	—
322	C₂³	—	146	2	no	—
323	Q₈	—	146	2	no	—
324	Q₈	—	146	2	no	—
325	D₄	—	584	2	no	—
326	D₄	—	146	2	no	—
327	D₄	—	146	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	292	1	no	—
331	C₂×C₄	—	292	2	no	—
332	C₂×C₄	—	292	2	no	—
333	C₂×C₄	—	73	1	no	—
334	C₂×C₄	—	73	1	no	—
335	C₂×C₄	—	73	1	no	—
336	C₂×C₄	—	146	2	no	—
337	C₂×C₄	—	146	2	no	—
338	C₂×C₄	—	146	1	no	—
339	C₂×C₄	—	146	1	no	—
340	C₂×C₄	D₄×#64408	1	1	yes	characteristic
341	C₂×C₄	D₄×#64408	1	1	yes	characteristic
342	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
343	C₈	—	292	2	no	—
344	C₈	—	292	2	no	—
345	C₈	—	292	1	no	—
346	C₈	—	292	1	no	—
347	C₈	—	146	2	no	—
348	C₈	—	146	2	no	—
349	C₈	—	146	1	no	—
350	C₈	—	146	1	no	—
351	C₈	#14556679	1	2	yes	—
352	C₈	#14556679	1	2	yes	—
353	C₂²	—	292	2	no	—
354	C₂²	—	292	2	no	—
355	C₂²	—	292	2	no	—
356	C₂²	—	146	1	no	—
357	C₂²	—	146	1	no	—
358	C₂²	—	146	1	no	—
359	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	2	no	—
364	C₄	—	292	2	no	—
365	C₄	—	73	2	no	—
366	C₄	—	146	1	no	—
367	C₄	—	146	1	no	—
368	C₄	C₂×#14556679	1	1	yes	characteristic
369	C₄	C₂×#14556679	1	1	yes	characteristic
370	C₂	—	292	1	no	—
371	C₂	—	584	2	no	—
372	C₂	#22626656	1	1	yes	characteristic
373	C₂	#22626694	1	1	yes	characteristic
374	C₂	#22626710	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#499×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17988824	C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#499	#64408
hypocenter	C₇₃	#17974
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #22626877	C₂	semidirect
2. #22625580	C₂	semidirect
3. #22625577	C₂	semidirect
4. #22625566	C₂	semidirect
5. #22625529	C₂	semidirect
6. #22625526	C₂	semidirect
7. #22625516	C₂	semidirect
8. #22625486	C₂	semidirect
9. #22625289	C₂	semidirect
10. #22625135	C₂	semidirect
199. C₇₃	#17974	semidirect

Maximal subgroups

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

subgroup	index
#22626877	2
#22625580	2
#22625577	2
#22625566	2
#22625529	2
#22625526	2
#22625516	2
#22625486	2
#22625289	2
#22625135	2
#22620503	2
#17974	73