Subgroups — #33077837

Order 18688.

626 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622828	C₂	1	2	yes	maximal
2	#22622324	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609528	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609438	C₂	1	2	yes	maximal, complemented
6	#22609330	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609296	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609148	C₂	1	2	yes	maximal, complemented
9	#22609099	C₂	1	2	yes	maximal, complemented
10	#22608829	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
12	C₂²×Dic₂₉₂	—	2	1	no	—
13	#17989294	C₂²	1	2	yes	—
14	D₄×Dic₁₄₆	C₂²	1	2	yes	—
15	#17988995	—	2	2	no	—
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988985	C₂²	1	2	yes	—
18	#17988967	C₂²	1	1	yes	characteristic
19	#17988938	C₂²	1	2	yes	—
20	#17988923	—	2	2	no	—
21	#17988889	—	2	2	no	—
22	#17988883	C₂²	1	2	yes	—
23	#17988877	C₂²	1	2	yes	—
24	#17988875	—	2	2	no	—
25	#17988867	C₂²	1	2	yes	—
26	#17988828	—	2	1	no	—
27	#17988828	—	2	1	no	—
28	#17988798	C₂²	1	2	yes	—
29	#17988745	—	2	2	no	—
30	#17988717	C₂²	1	2	yes	—
31	#17988699	—	2	2	no	—
32	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
33	C₂×#14556525	—	2	1	no	—
34	#17988340	C₂²	1	1	yes	characteristic
35	#17988335	C₂²	1	2	yes	—
36	#17988334	C₂²	1	2	yes	—
37	#17988319	C₂²	1	2	yes	—
38	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
41	#335×C₇₃	C₂²	1	1	yes	characteristic
42	D₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂²×Dic₁₄₆	—	2	2	no	—
46	C₂×Dic₂₉₂	—	2	2	no	—
47	C₂×Dic₂₉₂	—	2	1	no	—
48	C₂×Dic₂₉₂	—	2	1	no	—
49	C₂×Dic₂₉₂	—	4	4	no	—
50	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
51	#14556591	C₂³	1	2	yes	—
52	#14556591	C₂³	1	2	yes	—
53	#14556590	—	2	2	no	—
54	#14556582	—	2	2	no	—
55	#14556570	—	2	2	no	—
56	#14556570	—	2	2	no	—
57	#14556568	—	4	2	no	—
58	#14556568	—	4	2	no	—
59	C₄×Dic₁₄₆	C₂³	1	2	yes	—
60	D₄×Dic₇₃	—	2	2	no	—
61	D₄×Dic₇₃	—	2	2	no	—
62	C₂×C₄⋊Dic₇₃	—	2	2	no	—
63	#14556525	—	2	2	no	—
64	#14556525	—	2	1	no	—
65	#14556525	—	2	1	no	—
66	#14556525	D₄	1	1	yes	characteristic
67	#14556524	—	2	2	no	—
68	#14556523	—	2	2	no	—
69	#14556523	—	2	2	no	—
70	#14556522	—	2	2	no	—
71	C₄⋊#10596831	C₂³	1	2	yes	—
72	#14556520	—	2	2	no	—
73	#14556516	—	2	2	no	—
74	#14556516	—	2	2	no	—
75	#14556516	—	4	2	no	—
76	#14556516	D₄	1	2	yes	—
77	#14556512	—	2	2	no	—
78	#14556512	—	2	2	no	—
79	#14556501	—	2	2	no	—
80	C₂×D₈×C₇₃	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₇₃	—	2	1	no	—
82	C₂²×C₈×C₇₃	—	2	1	no	—
83	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
84	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
85	#107×C₇₃	D₄	1	1	yes	characteristic
86	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
88	#102×C₇₃	D₄	1	1	yes	characteristic
89	#102×C₇₃	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₇₃	—	2	1	no	—
91	C₂²⋊C₈×C₇₃	—	2	1	no	—
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₄₆	—	2	2	no	—
96	C₂×Dic₁₄₆	—	2	2	no	—
97	C₂×Dic₁₄₆	—	4	4	no	—
98	C₂×Dic₁₄₆	—	4	2	no	—
99	C₂×Dic₁₄₆	—	8	2	no	—
100	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
101	C₂²×Dic₇₃	—	4	2	no	—
102	C₂²×Dic₇₃	—	4	2	no	—
103	Dic₂₉₂	—	2	2	no	—
104	Dic₂₉₂	—	4	4	no	—
105	Dic₂₉₂	—	4	2	no	—
106	Dic₂₉₂	—	8	1	no	—
107	C₂×#10596831	—	2	2	no	—
108	C₄⋊Dic₇₃	—	2	2	no	—
109	C₄⋊Dic₇₃	—	2	2	no	—
110	C₄⋊Dic₇₃	—	2	2	no	—
111	C₄⋊Dic₇₃	—	2	2	no	—
112	C₄⋊Dic₇₃	—	4	2	no	—
113	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
114	#11924007	—	2	2	no	—
115	#11924007	—	4	2	no	—
116	#11924007	—	4	2	no	—
117	C₂²⋊Dic₇₃	—	4	2	no	—
118	C₂²⋊Dic₇₃	—	4	2	no	—
119	C₄×Dic₇₃	—	2	2	no	—
120	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₇₃	—	2	1	no	—
123	C₂²×C₄×C₇₃	—	2	1	no	—
124	D₈×C₇₃	Q₁₆	1	4	yes	complemented
125	C₂×C₈×C₇₃	—	2	1	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	—	2	1	no	—
128	C₂×C₈×C₇₃	—	4	2	no	—
129	C₂×C₈×C₇₃	—	4	2	no	—
130	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₇₃	—	2	1	no	—
137	C₂²⋊C₄×C₇₃	—	2	1	no	—
138	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
139	C₂³×C₇₃	—	2	1	no	—
140	C₂³×C₇₃	—	2	1	no	—
141	D₄×C₇₃	—	2	1	no	—
142	D₄×C₇₃	—	2	1	no	—
143	D₄×C₇₃	C₂×Q₁₆	1	2	yes	—
144	D₄×C₇₃	C₂×Q₁₆	1	2	yes	complemented
145	C₂×C₄×C₇₃	—	2	1	no	—
146	C₂×C₄×C₇₃	—	2	1	no	—
147	C₂×C₄×C₇₃	—	4	1	no	—
148	C₂×C₄×C₇₃	—	4	1	no	—
149	C₂×C₄×C₇₃	—	4	1	no	—
150	C₂×C₄×C₇₃	—	4	1	no	—
151	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×Dic₇₃	—	2	2	no	—
155	C₂×Dic₇₃	—	2	2	no	—
156	C₂×Dic₇₃	—	4	2	no	—
157	C₂×Dic₇₃	—	4	2	no	—
158	C₂×Dic₇₃	—	8	4	no	—
159	C₂×Dic₇₃	—	16	2	no	—
160	Dic₁₄₆	—	2	2	no	—
161	Dic₁₄₆	—	2	2	no	—
162	Dic₁₄₆	—	4	2	no	—
163	Dic₁₄₆	—	4	2	no	—
164	Dic₁₄₆	—	8	4	no	—
165	Dic₁₄₆	—	8	2	no	—
166	C₈×C₇₃	—	2	1	no	—
167	C₈×C₇₃	—	2	1	no	—
168	C₈×C₇₃	—	4	1	no	—
169	C₈×C₇₃	—	4	1	no	—
170	C₈×C₇₃	—	4	1	no	—
171	C₈×C₇₃	C₂×Q₁₆	1	2	yes	complemented
172	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
173	#10596831	—	4	2	no	—
174	C₂²×C₇₃	—	2	2	no	—
175	C₂²×C₇₃	—	2	2	no	—
176	C₂²×C₇₃	—	4	1	no	—
177	C₂²×C₇₃	—	4	1	no	—
178	C₂²×C₇₃	—	4	1	no	—
179	C₂²×C₇₃	—	4	1	no	—
180	C₂²×C₇₃	D₄²	1	1	yes	characteristic
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	4	1	no	—
183	C₄×C₇₃	—	4	1	no	—
184	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
185	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₇₃	#577	1	1	yes	characteristic
187	C₄×C₇₃	#578	1	1	yes	characteristic
188	Dic₇₃	—	4	2	no	—
189	Dic₇₃	—	4	2	no	—
190	Dic₇₃	—	8	2	no	—
191	Dic₇₃	—	8	2	no	—
192	#19926	—	73	1	no	maximal, Sylow
193	C₂×C₇₃	—	4	2	no	—
194	C₂×C₇₃	—	4	2	no	—
195	C₂×C₇₃	#3272	1	1	yes	characteristic
196	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
197	C₂×C₇₃	#3290	1	1	yes	characteristic
198	#3384	—	73	2	no	—
199	#3338	—	73	1	no	—
200	D₄×Q₁₆	—	73	1	no	—
201	#1707	—	73	1	no	—
202	#1692	—	73	2	no	—
203	#1675	—	73	1	no	—
204	#1669	—	73	1	no	—
205	#1644	—	73	2	no	—
206	#1637	—	73	2	no	—
207	#1591	—	73	1	no	—
208	C₈×D₈	D₇₃	1	1	yes	characteristic
209	C₇₃	#19926	1	1	yes	characteristic, complemented, Sylow
210	C₂²×Q₁₆	—	146	1	no	—
211	#554	—	73	2	no	—
212	D₄×Q₈	—	73	2	no	—
213	#501	—	146	2	no	—
214	#500	—	73	1	no	—
215	#498	—	73	2	no	—
216	#494	—	73	1	no	—
217	#487	—	73	2	no	—
218	#484	—	146	2	no	—
219	#478	—	146	2	no	—
220	#477	—	73	2	no	—
221	#476	—	73	2	no	—
222	#476	—	146	2	no	—
223	#474	—	73	2	no	—
224	#467	—	146	1	no	—
225	#467	—	146	1	no	—
226	#462	—	73	2	no	—
227	#451	—	146	2	no	—
228	#446	—	73	2	no	—
229	#442	—	146	2	no	—
230	C₄×Q₁₆	—	73	1	no	—
231	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
232	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
233	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
234	C₂×#107	—	146	1	no	—
235	#335	—	73	1	no	—
236	#335	C₂×D₇₃	1	1	yes	characteristic
237	#333	—	73	2	no	—
238	#332	—	73	2	no	—
239	Q₈⋊C₈	—	73	2	no	—
240	D₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
241	D₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
242	C₈²	C₂×D₇₃	1	1	yes	characteristic
243	C₂²×Q₈	—	146	2	no	—
244	C₂×Q₁₆	—	292	4	no	—
245	C₂×Q₁₆	—	73	1	no	—
246	C₂×Q₁₆	—	146	2	no	—
247	C₂×Q₁₆	—	146	1	no	—
248	C₂×Q₁₆	—	146	1	no	—
249	C₂×D₈	D₂₉₂	1	1	yes	characteristic
250	C₂²×C₈	—	2	1	no	—
251	C₂²×C₈	—	2	1	no	—
252	C₄⋊Q₈	—	73	2	no	—
253	C₄⋊Q₈	—	73	2	no	—
254	C₄⋊Q₈	—	146	2	no	—
255	#125	—	146	2	no	—
256	C₂²⋊Q₈	—	292	2	no	—
257	C₂²⋊Q₈	—	292	2	no	—
258	C₂²⋊Q₈	—	146	2	no	—
259	C₂²⋊Q₈	—	146	2	no	—
260	C₄×Q₈	—	73	2	no	—
261	C₄×D₄	—	146	2	no	—
262	C₄×D₄	—	146	2	no	—
263	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
264	C₄×D₄	C₂²×D₇₃	1	1	yes	characteristic
265	C₂×C₄⋊C₄	—	146	2	no	—
266	#107	—	73	1	no	—
267	#107	—	146	2	no	—
268	#107	—	146	1	no	—
269	#107	—	146	1	no	—
270	#107	—	146	2	no	—
271	#107	D₂₉₂	1	1	yes	characteristic
272	#106	—	146	2	no	—
273	#106	—	146	2	no	—
274	#106	—	146	2	no	—
275	C₄⋊C₈	—	73	2	no	—
276	C₄⋊C₈	—	146	2	no	—
277	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
278	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
279	#103	—	292	2	no	—
280	#103	—	73	2	no	—
281	#103	—	146	2	no	—
282	#103	—	146	2	no	—
283	#102	—	146	2	no	—
284	#102	—	146	2	no	—
285	#102	D₂₉₂	1	1	yes	characteristic
286	#102	D₂₉₂	1	1	yes	characteristic
287	C₂²⋊C₈	—	2	1	no	—
288	C₂²⋊C₈	—	2	1	no	—
289	#97	—	146	2	no	—
290	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
293	C₂×Q₈	—	292	2	no	—
294	C₂×Q₈	—	292	4	no	—
295	C₂×Q₈	—	584	2	no	—
296	C₂×Q₈	—	73	2	no	—
297	C₂×Q₈	—	146	2	no	—
298	C₂×Q₈	—	146	2	no	—
299	C₂×D₄	C₂×D₂₉₂	1	1	yes	characteristic
300	C₂×D₄	C₂×D₂₉₂	1	1	yes	characteristic
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	292	2	no	—
304	C₂²×C₄	—	292	2	no	—
305	Q₁₆	—	292	4	no	—
306	Q₁₆	—	292	2	no	—
307	Q₁₆	—	584	1	no	—
308	Q₁₆	—	146	2	no	—
309	D₈	Dic₂₉₂	1	4	yes	complemented
310	C₂×C₈	—	2	1	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	4	2	no	—
314	C₂×C₈	—	4	2	no	—
315	C₂×C₈	—	146	2	no	—
316	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
317	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
318	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
319	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
320	C₄⋊C₄	—	292	2	no	—
321	C₄⋊C₄	—	292	2	no	—
322	C₄⋊C₄	—	292	2	no	—
323	C₄⋊C₄	—	73	2	no	—
324	C₄⋊C₄	—	146	2	no	—
325	C₄⋊C₄	—	146	2	no	—
326	C₄⋊C₄	—	146	2	no	—
327	C₄⋊C₄	—	146	2	no	—
328	C₄⋊C₄	—	146	2	no	—
329	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
330	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
331	C₂²⋊C₄	—	2	1	no	—
332	C₂²⋊C₄	—	2	1	no	—
333	C₂²⋊C₄	—	292	2	no	—
334	C₂²⋊C₄	—	292	2	no	—
335	C₄²	—	146	2	no	—
336	C₄²	C₂³×D₇₃	1	1	yes	characteristic
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	Q₈	—	292	2	no	—
340	Q₈	—	292	2	no	—
341	Q₈	—	584	2	no	—
342	Q₈	—	584	4	no	—
343	Q₈	—	146	2	no	—
344	Q₈	—	146	2	no	—
345	D₄	—	2	1	no	—
346	D₄	—	2	1	no	—
347	D₄	C₂×Dic₂₉₂	1	2	yes	—
348	D₄	C₂×Dic₂₉₂	1	2	yes	complemented
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	292	2	no	—
356	C₂×C₄	—	292	2	no	—
357	C₂×C₄	—	584	4	no	—
358	C₂×C₄	—	1168	2	no	—
359	C₂×C₄	—	146	2	no	—
360	C₂×C₄	—	146	2	no	—
361	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
362	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
363	C₂×C₄	#14556637	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	4	1	no	—
367	C₈	—	4	1	no	—
368	C₈	—	4	1	no	—
369	C₈	—	292	2	no	—
370	C₈	C₂×Dic₂₉₂	1	2	yes	complemented
371	C₈	#14556615	1	2	yes	—
372	C₂²	—	2	2	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	4	1	no	—
378	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	1	no	—
382	C₄	—	292	2	no	—
383	C₄	—	292	2	no	—
384	C₄	—	584	2	no	—
385	C₄	—	584	2	no	—
386	C₄	C₂²×Dic₂₉₂	1	1	yes	characteristic
387	C₄	C₂×#14556615	1	1	yes	characteristic
388	C₄	#17989394	1	1	yes	characteristic
389	C₄	#17989404	1	1	yes	characteristic
390	C₂	—	4	2	no	—
391	C₂	—	4	2	no	—
392	C₂	#22621532	1	1	yes	characteristic
393	C₂	D₄×Dic₂₉₂	1	1	yes	characteristic
394	C₂	#22621718	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×D₈×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
prime radical	C₂×D₈×C₇₃	D₄
hypercenter	C₈×D₈	D₇₃
hypocenter	C₇₃	#19926
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
2. #22622324	C₂	semidirect
3. D₄×Dic₂₉₂	C₂	semidirect
4. #22609528	C₂	semidirect
5. #22609438	C₂	semidirect
6. #22609330	C₂	semidirect
7. #22609296	C₂	semidirect
8. #22609148	C₂	semidirect
9. #22609099	C₂	semidirect
10. #22608829	C₂	semidirect
124. D₈×C₇₃	Q₁₆	semidirect
144. D₄×C₇₃	C₂×Q₁₆	semidirect
171. C₈×C₇₃	C₂×Q₁₆	semidirect
209. C₇₃	#19926	semidirect
309. D₈	Dic₂₉₂	semidirect
348. D₄	C₂×Dic₂₉₂	semidirect
370. C₈	C₂×Dic₂₉₂	semidirect

Maximal subgroups

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

subgroup	index
#22622828	2
#22622324	2
D₄×Dic₂₉₂	2
#22609528	2
#22609438	2
#22609330	2
#22609296	2
#22609148	2
#22609099	2
#22608829	2
C₈×D₈×C₇₃	2
#19926	73