Subgroups — #33077877

Order 18688.

570 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622840	C₂	1	2	yes	maximal
2	Q₈×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22622323	C₂	1	1	yes	maximal, characteristic
4	#22609548	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609458	C₂	1	2	yes	maximal, complemented
6	#22609331	C₂	1	1	yes	maximal, characteristic
7	#22609070	C₂	1	2	yes	maximal, complemented
8	#22609066	C₂	1	2	yes	maximal
9	#22608827	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924015	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1676	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₄₆	C₂²	1	2	yes	—
13	#17989298	C₂²	1	2	yes	—
14	#17988995	—	2	2	no	—
15	#17988992	—	2	1	no	—
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988990	C₂²	1	2	yes	—
18	#17988966	C₂²	1	2	yes	—
19	#17988924	—	2	2	no	—
20	#17988892	—	2	2	no	—
21	#17988883	—	2	2	no	—
22	#17988883	C₂²	1	2	yes	—
23	#17988879	C₂²	1	2	yes	—
24	#17988878	C₂²	1	2	yes	—
25	#17988866	—	2	2	no	—
26	#17988829	—	2	1	no	—
27	#17988714	C₂²	1	2	yes	—
28	#17988690	—	2	2	no	—
29	C₄×Dic₂₉₂	—	2	1	no	—
30	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
31	C₄×#11924015	C₂²	1	1	yes	characteristic
32	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
33	#107×D₇₃	—	2	1	no	—
34	#17988337	C₂²	1	1	yes	characteristic
35	#17988335	C₂²	1	2	yes	—
36	Q₈⋊#10596831	C₂²	1	1	yes	characteristic
37	#17988317	C₂²	1	2	yes	—
38	#17988314	C₂²	1	1	yes	characteristic
39	C₈×#10596831	C₂²	1	1	yes	characteristic
40	#500×C₇₃	C₂²	1	1	yes	characteristic
41	#477×C₇₃	C₂²	1	2	yes	—
42	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
43	#335×C₇₃	C₂²	1	1	yes	characteristic
44	#333×C₇₃	C₂²	1	2	yes	—
45	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
46	C₂×#11924018	—	4	2	no	—
47	C₂×#11924015	D₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	—	2	1	no	—
49	#14556591	—	2	2	no	—
50	#14556591	C₂³	1	2	yes	—
51	#14556590	—	2	2	no	—
52	#14556590	—	4	2	no	—
53	#14556582	—	2	2	no	—
54	#14556571	—	2	2	no	—
55	#14556567	—	4	2	no	—
56	Q₈×Dic₇₃	—	2	2	no	—
57	C₄×Dic₁₄₆	—	2	2	no	—
58	C₄×Dic₁₄₆	C₂³	1	2	yes	—
59	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
60	#14556553	—	2	2	no	—
61	C₄⋊C₄×D₇₃	—	2	2	no	—
62	#14556525	—	2	2	no	—
63	#14556525	—	2	1	no	—
64	#14556525	—	2	1	no	—
65	#14556525	D₄	1	1	yes	characteristic
66	#14556524	—	2	2	no	—
67	#14556524	—	2	2	no	—
68	#14556524	—	2	2	no	—
69	#14556523	—	2	2	no	—
70	#14556522	D₄	1	1	yes	characteristic
71	C₄⋊#10596831	C₂³	1	1	yes	characteristic
72	#14556520	—	2	2	no	—
73	#14556516	—	2	2	no	—
74	#14556516	D₄	1	2	yes	—
75	#14556515	D₄	1	1	yes	characteristic
76	#14556514	—	2	2	no	—
77	#14556513	—	2	2	no	—
78	#14556511	D₄	1	1	yes	characteristic
79	#14556503	—	2	1	no	—
80	#14556501	—	2	2	no	—
81	C₄×#10596831	C₂³	1	1	yes	characteristic
82	C₄×#10596831	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
84	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
85	#107×C₇₃	—	2	1	no	—
86	#107×C₇₃	—	2	1	no	—
87	#107×C₇₃	—	2	2	no	—
88	C₄⋊C₈×C₇₃	—	2	1	no	—
89	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
90	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
91	#103×C₇₃	—	2	2	no	—
92	C₄×C₈×C₇₃	—	2	1	no	—
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₄₆	—	2	2	no	—
95	C₂×Dic₁₄₆	—	4	2	no	—
96	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
97	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₇₃	—	2	1	no	—
99	C₂×C₄×D₇₃	—	4	2	no	—
100	Dic₂₉₂	—	2	2	no	—
101	#11924018	—	8	4	no	—
102	#11924015	Q₁₆	1	4	yes	complemented
103	C₂×#10596831	—	2	1	no	—
104	C₂×#10596831	—	2	1	no	—
105	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
106	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
107	C₈×D₇₃	—	4	2	no	—
108	C₄⋊Dic₇₃	—	2	2	no	—
109	C₄⋊Dic₇₃	—	2	2	no	—
110	C₄⋊Dic₇₃	—	4	2	no	—
111	C₄⋊Dic₇₃	—	4	2	no	—
112	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
113	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
114	#11924007	—	2	2	no	—
115	#11924007	—	4	2	no	—
116	#11924007	—	4	2	no	—
117	#11924005	—	2	1	no	—
118	#11924005	—	4	2	no	—
119	C₄×Dic₇₃	—	2	2	no	—
120	C₄×Dic₇₃	—	4	2	no	—
121	C₂×Q₈×C₇₃	—	2	2	no	—
122	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₃	—	2	1	no	—
124	C₂×C₈×C₇₃	—	2	2	no	—
125	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄×C₇₃	—	2	1	no	—
128	C₄⋊C₄×C₇₃	—	2	2	no	—
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₇₃	—	2	1	no	—
132	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
133	C₂²×D₇₃	—	2	1	no	—
134	Q₈×C₇₃	—	2	1	no	—
135	Q₈×C₇₃	—	8	2	no	—
136	Q₈×C₇₃	C₂×Q₁₆	1	2	yes	—
137	C₂×C₄×C₇₃	—	2	1	no	—
138	C₂×C₄×C₇₃	—	2	1	no	—
139	C₂×C₄×C₇₃	—	4	2	no	—
140	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×Dic₇₃	—	2	2	no	—
144	C₂×Dic₇₃	—	2	2	no	—
145	C₂×Dic₇₃	—	2	1	no	—
146	C₂×Dic₇₃	—	4	2	no	—
147	D₂₉₂	—	2	1	no	—
148	D₂₉₂	C₂×Q₁₆	1	2	yes	—
149	C₄×D₇₃	—	4	1	no	—
150	C₄×D₇₃	—	4	1	no	—
151	C₄×D₇₃	—	16	2	no	—
152	Dic₁₄₆	—	2	2	no	—
153	Dic₁₄₆	—	2	2	no	—
154	Dic₁₄₆	—	8	2	no	—
155	Dic₁₄₆	—	16	2	no	—
156	C₈×C₇₃	—	2	1	no	—
157	C₈×C₇₃	—	4	2	no	—
158	C₈×C₇₃	—	4	1	no	—
159	C₈×C₇₃	#137	1	2	yes	—
160	#10596831	—	2	2	no	—
161	#10596831	—	2	1	no	—
162	#10596831	—	4	1	no	—
163	#10596831	C₂×Q₁₆	1	2	yes	—
164	C₂²×C₇₃	D₄²	1	1	yes	characteristic
165	C₂×D₇₃	—	2	2	no	—
166	C₂×D₇₃	—	4	1	no	—
167	C₂×D₇₃	—	4	1	no	—
168	C₄×C₇₃	—	2	2	no	—
169	C₄×C₇₃	—	2	1	no	—
170	C₄×C₇₃	—	4	1	no	—
171	C₄×C₇₃	—	8	2	no	—
172	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
173	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
174	C₄×C₇₃	#578	1	1	yes	characteristic
175	C₄×C₇₃	#578	1	1	yes	characteristic
176	Dic₇₃	—	4	2	no	—
177	Dic₇₃	—	4	1	no	—
178	Dic₇₃	—	4	2	no	—
179	Dic₇₃	—	8	2	no	—
180	#19930	—	73	1	no	maximal, Sylow
181	C₂×C₇₃	#3276	1	1	yes	characteristic
182	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
183	C₂×C₇₃	#3292	1	1	yes	characteristic
184	D₇₃	—	4	2	no	—
185	#3385	—	73	2	no	—
186	Q₈×Q₁₆	—	73	1	no	—
187	#3338	—	73	1	no	—
188	#1713	—	73	1	no	—
189	#1696	—	73	2	no	—
190	#1676	D₇₃	1	1	yes	characteristic, complemented
191	#1675	—	73	1	no	—
192	#1633	—	73	2	no	—
193	#1632	—	73	2	no	—
194	#1591	—	73	1	no	—
195	C₈×SD₁₆	—	73	1	no	—
196	C₇₃	#19930	1	1	yes	characteristic, complemented, Sylow
197	Q₈²	—	73	2	no	—
198	#554	—	73	2	no	—
199	#501	—	146	2	no	—
200	#500	—	73	2	no	—
201	#500	—	73	1	no	—
202	#500	—	146	1	no	—
203	#500	C₂×D₇₃	1	1	yes	characteristic
204	#494	—	73	2	no	—
205	#484	—	146	2	no	—
206	#479	—	146	2	no	—
207	#477	—	73	2	no	—
208	#477	—	73	2	no	—
209	#477	—	146	2	no	—
210	#477	C₂×D₇₃	1	2	yes	—
211	#476	—	73	2	no	—
212	#474	—	146	2	no	—
213	#467	—	146	1	no	—
214	#446	—	73	2	no	—
215	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
216	#440	—	146	2	no	—
217	C₄×Q₁₆	—	73	1	no	—
218	C₄×Q₁₆	—	146	1	no	—
219	C₄×SD₁₆	—	73	1	no	—
220	C₈×D₄	—	73	1	no	—
221	C₂×#107	—	146	1	no	—
222	#335	C₂×D₇₃	1	1	yes	characteristic
223	#334	—	73	1	no	—
224	#333	—	73	2	no	—
225	#333	C₂×D₇₃	1	2	yes	—
226	Q₈⋊C₈	—	73	1	no	—
227	Q₈⋊C₈	—	73	2	no	—
228	D₄⋊C₈	—	73	1	no	—
229	C₈²	—	73	1	no	—
230	C₂×Q₁₆	—	292	2	no	—
231	C₂×Q₁₆	—	73	1	no	—
232	C₂×SD₁₆	—	73	1	no	—
233	C₂²×C₈	—	146	1	no	—
234	C₄⋊Q₈	—	292	2	no	—
235	C₄⋊Q₈	—	73	2	no	—
236	C₄⋊Q₈	—	146	2	no	—
237	C₄⋊Q₈	—	146	2	no	—
238	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
239	#125	—	146	2	no	—
240	C₂²⋊Q₈	—	292	2	no	—
241	C₂²⋊Q₈	—	146	2	no	—
242	C₄×Q₈	—	73	2	no	—
243	C₄×Q₈	—	146	2	no	—
244	C₄×Q₈	—	146	2	no	—
245	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
246	C₄×D₄	—	73	1	no	—
247	C₄×D₄	—	146	2	no	—
248	C₂×C₄⋊C₄	—	146	2	no	—
249	#107	—	2	1	no	—
250	#107	—	2	1	no	—
251	#107	—	2	2	no	—
252	#107	—	73	1	no	—
253	#107	—	146	2	no	—
254	#107	—	146	2	no	—
255	#107	—	146	2	no	—
256	#107	—	146	2	no	—
257	#107	—	146	1	no	—
258	#107	—	146	1	no	—
259	#106	—	73	1	no	—
260	#106	—	146	2	no	—
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	73	1	no	—
263	C₄⋊C₈	—	146	2	no	—
264	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
265	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
266	#103	—	2	2	no	—
267	#103	—	73	1	no	—
268	#103	—	73	2	no	—
269	#103	—	146	2	no	—
270	#103	—	146	2	no	—
271	#102	—	73	1	no	—
272	#102	—	146	2	no	—
273	C₂²⋊C₈	—	146	1	no	—
274	#97	—	146	2	no	—
275	C₄×C₈	—	2	1	no	—
276	C₄×C₈	—	73	1	no	—
277	C₄×C₈	—	73	1	no	—
278	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	292	2	no	—
281	C₂×Q₈	—	73	2	no	—
282	C₂×Q₈	—	146	2	no	—
283	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
284	C₂×D₄	—	73	1	no	—
285	C₂²×C₄	—	292	2	no	—
286	C₂²×C₄	—	146	1	no	—
287	Q₁₆	—	584	4	no	—
288	Q₁₆	—	146	2	no	—
289	SD₁₆	—	73	4	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	292	2	no	—
293	C₂×C₈	—	73	1	no	—
294	C₂×C₈	—	73	1	no	—
295	C₂×C₈	—	146	1	no	—
296	C₂×C₈	—	146	1	no	—
297	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
298	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	292	2	no	—
303	C₄⋊C₄	—	292	2	no	—
304	C₄⋊C₄	—	292	2	no	—
305	C₄⋊C₄	—	292	2	no	—
306	C₄⋊C₄	—	73	1	no	—
307	C₄⋊C₄	—	73	2	no	—
308	C₄⋊C₄	—	146	2	no	—
309	C₄⋊C₄	—	146	2	no	—
310	C₄⋊C₄	—	146	2	no	—
311	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
312	C₂²⋊C₄	—	292	2	no	—
313	C₂²⋊C₄	—	146	1	no	—
314	C₄²	—	2	1	no	—
315	C₄²	—	292	2	no	—
316	C₄²	—	146	2	no	—
317	C₄²	C₂³×D₇₃	1	1	yes	characteristic
318	C₂³	—	146	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	8	2	no	—
321	Q₈	—	584	2	no	—
322	Q₈	—	1168	2	no	—
323	Q₈	—	146	2	no	—
324	Q₈	—	146	2	no	—
325	Q₈	Q₁₆×D₇₃	1	2	yes	—
326	D₄	—	73	2	no	—
327	D₄	—	146	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	292	1	no	—
332	C₂×C₄	—	292	1	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	1168	2	no	—
335	C₂×C₄	—	146	2	no	—
336	C₂×C₄	—	146	2	no	—
337	C₂×C₄	—	146	1	no	—
338	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
339	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
340	C₂×C₄	#14556637	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	4	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	292	1	no	—
345	C₈	—	73	2	no	—
346	C₈	—	146	2	no	—
347	C₈	—	146	1	no	—
348	C₈	#14556623	1	2	yes	—
349	C₂²	—	292	1	no	—
350	C₂²	—	292	1	no	—
351	C₂²	—	146	2	no	—
352	C₂²	#17989221	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	292	2	no	—
358	C₄	—	292	1	no	—
359	C₄	—	292	2	no	—
360	C₄	—	584	2	no	—
361	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
362	C₄	C₂×#14556623	1	1	yes	characteristic
363	C₄	#17989401	1	1	yes	characteristic
364	C₄	#17989403	1	1	yes	characteristic
365	C₂	—	292	2	no	—
366	C₂	#22621572	1	1	yes	characteristic
367	C₂	#22621680	1	1	yes	characteristic
368	C₂	#22621729	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17989221
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1676	C₂
socle	C₂²×C₇₃	D₄²
prime radical	C₂×D₂₉₂	C₂×D₄
hypercenter	#1676	D₇₃
hypocenter	C₇₃	#19930
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1676

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₂₉₂	C₂	semidirect
4. #22609548	C₂	semidirect
5. #22609458	C₂	semidirect
7. #22609070	C₂	semidirect
9. #22608827	C₂	semidirect
11. C₇₃×#1676	C₂	semidirect
102. #11924015	Q₁₆	semidirect
190. #1676	D₇₃	semidirect
196. C₇₃	#19930	semidirect

Maximal subgroups

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

subgroup	index
#22622840	2
Q₈×Dic₂₉₂	2
#22622323	2
#22609548	2
#22609458	2
#22609331	2
#22609070	2
#22609066	2
#22608827	2
C₈×#11924015	2
C₇₃×#1676	2
#19930	73