Subgroups — #33077299

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622280	C₂	1	1	yes	maximal, characteristic
3	#22621579	C₂	1	2	yes	maximal, complemented
4	#22609549	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609420	C₂	1	2	yes	maximal, complemented
6	#22609326	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609321	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609183	C₂	1	2	yes	maximal, complemented
9	#22609169	C₂	1	2	yes	maximal, complemented
10	C₈×#11924017	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1589	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556624	—	2	2	no	—
13	D₄×Dic₁₄₆	C₂²	1	2	yes	—
14	#17989242	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988992	C₂²	1	2	yes	—
18	#17988991	—	2	1	no	—
19	#17988967	C₂²	1	2	yes	—
20	#17988899	—	2	2	no	—
21	#17988867	C₂²	1	2	yes	—
22	#17988867	C₂²	1	2	yes	—
23	#17988865	—	2	2	no	—
24	#17988845	—	2	2	no	—
25	#17988820	—	2	1	no	—
26	#17988798	C₂²	1	2	yes	—
27	#17988793	C₂²	1	2	yes	—
28	#17988751	—	2	2	no	—
29	#17988745	—	2	2	no	—
30	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
31	C₄×#11924017	C₂²	1	1	yes	characteristic
32	D₈×Dic₇₃	—	2	1	no	—
33	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
34	#107×D₇₃	—	2	1	no	—
35	#17988338	C₂²	1	1	yes	characteristic
36	#17988334	C₂²	1	2	yes	—
37	#17988334	C₂²	1	2	yes	—
38	#17988319	C₂²	1	1	yes	characteristic
39	#17988316	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
41	#436×C₇₃	C₂²	1	2	yes	—
42	#335×C₇₃	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₇₃	C₂²	1	2	yes	—
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924028	—	2	2	no	—
46	C₂²×Dic₁₄₆	—	2	2	no	—
47	#14556624	—	4	8	no	—
48	C₂×Dic₂₉₂	—	2	2	no	—
49	C₂×Dic₂₉₂	—	2	2	no	—
50	C₂×Dic₂₉₂	—	2	2	no	—
51	C₂×#11924017	—	2	2	no	—
52	C₂×#11924017	D₄	1	1	yes	characteristic
53	C₂×C₈×D₇₃	—	2	1	no	—
54	#14556591	C₂³	1	2	yes	—
55	#14556591	C₂³	1	2	yes	—
56	#14556590	—	2	2	no	—
57	#14556573	—	4	2	no	—
58	#14556570	—	2	2	no	—
59	#14556570	—	2	2	no	—
60	#14556568	—	4	2	no	—
61	#14556562	—	2	2	no	—
62	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
63	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
64	D₄×Dic₇₃	—	2	2	no	—
65	C₄⋊C₄×D₇₃	—	2	2	no	—
66	#14556525	—	2	1	no	—
67	#14556525	—	2	1	no	—
68	#14556525	—	2	1	no	—
69	#14556525	—	2	1	no	—
70	#14556525	—	4	2	no	—
71	#14556524	—	2	2	no	—
72	#14556524	—	2	2	no	—
73	#14556523	D₄	1	1	yes	characteristic
74	C₄⋊#10596831	C₂³	1	1	yes	characteristic
75	C₄⋊#10596831	C₂³	1	1	yes	characteristic
76	#14556520	—	2	1	no	—
77	#14556516	—	2	2	no	—
78	#14556516	—	2	2	no	—
79	#14556516	D₄	1	1	yes	characteristic
80	#14556513	D₄	1	1	yes	characteristic
81	#14556512	—	2	2	no	—
82	#14556512	—	2	2	no	—
83	#14556503	—	2	1	no	—
84	C₈×Dic₇₃	—	2	1	no	—
85	C₂×D₈×C₇₃	D₄	1	1	yes	characteristic
86	C₂×M₁₆×C₇₃	—	2	2	no	—
87	C₄×D₄×C₇₃	C₂³	1	2	yes	—
88	#107×C₇₃	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
90	#102×C₇₃	D₄	1	2	yes	—
91	C₂²⋊C₈×C₇₃	—	2	2	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	#11924028	—	4	4	no	—
96	#11924028	—	8	2	no	—
97	C₂×Dic₁₄₆	—	2	2	no	—
98	C₂×Dic₁₄₆	—	2	2	no	—
99	C₂×Dic₁₄₆	—	4	4	no	—
100	C₂×Dic₁₄₆	—	4	2	no	—
101	C₂×Dic₁₄₆	—	8	2	no	—
102	C₂×Dic₁₄₆	C₂×D₄	1	1	yes	characteristic
103	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
104	C₂×C₂²⋊D₇₃	—	4	2	no	—
105	C₂²×Dic₇₃	—	4	2	no	—
106	C₂×C₄×D₇₃	—	2	1	no	—
107	C₂×C₄×D₇₃	—	4	2	no	—
108	Dic₂₉₂	—	4	4	no	—
109	Dic₂₉₂	—	4	4	no	—
110	Dic₂₉₂	—	8	2	no	—
111	#11924017	—	4	4	no	—
112	#11924017	D₈	1	4	yes	complemented
113	C₂×#10596831	—	2	1	no	—
114	C₂×#10596831	—	2	1	no	—
115	C₈×D₇₃	—	4	2	no	—
116	C₄⋊Dic₇₃	—	2	2	no	—
117	C₄⋊Dic₇₃	—	2	2	no	—
118	C₄⋊Dic₇₃	—	2	2	no	—
119	C₄⋊Dic₇₃	—	2	2	no	—
120	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
121	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
122	#11924007	—	2	1	no	—
123	#11924007	—	4	2	no	—
124	#11924007	—	4	2	no	—
125	C₂²⋊Dic₇₃	—	4	2	no	—
126	C₂²⋊Dic₇₃	—	4	2	no	—
127	#11924005	—	2	1	no	—
128	C₄×Dic₇₃	—	2	1	no	—
129	C₂×D₄×C₇₃	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₇₃	—	2	2	no	—
131	D₈×C₇₃	—	2	2	no	—
132	M₁₆×C₇₃	—	4	4	no	—
133	C₂×C₈×C₇₃	—	2	1	no	—
134	C₂×C₈×C₇₃	—	2	2	no	—
135	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
140	C₂²⋊C₄×C₇₃	—	2	2	no	—
141	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
142	C₂³×C₇₃	—	2	2	no	—
143	C₂²×D₇₃	—	2	1	no	—
144	D₄×C₇₃	—	2	2	no	—
145	D₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	—	4	2	no	—
148	C₂×C₄×C₇₃	—	4	2	no	—
149	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂²⋊D₇₃	—	8	4	no	—
153	C₂×Dic₇₃	—	2	1	no	—
154	C₂×Dic₇₃	—	2	1	no	—
155	C₂×Dic₇₃	—	2	1	no	—
156	C₂×Dic₇₃	—	4	2	no	—
157	C₂×Dic₇₃	—	4	2	no	—
158	C₂×Dic₇₃	—	8	4	no	—
159	D₂₉₂	—	2	1	no	—
160	D₂₉₂	C₂×D₈	1	2	yes	—
161	C₄×D₇₃	—	4	1	no	—
162	C₄×D₇₃	—	4	1	no	—
163	C₄×D₇₃	—	8	4	no	—
164	Dic₁₄₆	—	2	1	no	—
165	Dic₁₄₆	—	4	2	no	—
166	Dic₁₄₆	—	4	2	no	—
167	Dic₁₄₆	—	4	2	no	—
168	Dic₁₄₆	—	4	2	no	—
169	Dic₁₄₆	—	8	4	no	—
170	Dic₁₄₆	C₂×D₈	1	2	yes	—
171	C₈×C₇₃	—	2	2	no	—
172	C₈×C₇₃	—	2	1	no	—
173	C₈×C₇₃	—	2	1	no	—
174	C₈×C₇₃	—	4	2	no	—
175	C₈×C₇₃	C₂×D₈	1	2	yes	—
176	C₈×C₇₃	#137	1	2	yes	complemented
177	#10596831	—	4	1	no	—
178	#10596831	—	4	1	no	—
179	C₂²×C₇₃	—	4	2	no	—
180	C₂²×C₇₃	—	4	2	no	—
181	C₂²×C₇₃	—	4	2	no	—
182	C₂²×C₇₃	D₄²	1	1	yes	characteristic
183	C₂×D₇₃	—	2	2	no	—
184	C₂×D₇₃	—	4	1	no	—
185	C₂×D₇₃	—	4	1	no	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	—	4	2	no	—
188	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
189	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
190	C₄×C₇₃	#576	1	1	yes	characteristic
191	C₄×C₇₃	#578	1	1	yes	characteristic
192	Dic₇₃	—	2	2	no	—
193	Dic₇₃	—	4	1	no	—
194	Dic₇₃	—	4	1	no	—
195	Dic₇₃	—	8	2	no	—
196	Dic₇₃	—	8	2	no	—
197	#19887	—	73	1	no	maximal, Sylow
198	C₂×C₇₃	—	8	2	no	—
199	C₂×C₇₃	#3276	1	1	yes	characteristic
200	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
201	C₂×C₇₃	#3293	1	1	yes	characteristic
202	D₇₃	—	4	2	no	—
203	Q₈×D₈	—	73	1	no	—
204	#3334	—	73	1	no	—
205	#3276	—	73	2	no	—
206	#1713	—	73	1	no	—
207	#1689	—	73	2	no	—
208	#1674	—	73	1	no	—
209	#1673	—	73	1	no	—
210	#1649	—	73	2	no	—
211	#1647	—	73	2	no	—
212	#1589	D₇₃	1	1	yes	characteristic, complemented
213	C₈×SD₁₆	—	73	1	no	—
214	C₇₃	#19887	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	146	2	no	—
216	D₄×Q₈	—	73	2	no	—
217	#547	—	73	2	no	—
218	#500	—	73	2	no	—
219	#500	—	73	1	no	—
220	#500	—	73	1	no	—
221	#500	—	146	1	no	—
222	#494	—	73	2	no	—
223	#480	—	146	2	no	—
224	#474	—	73	2	no	—
225	#474	—	73	2	no	—
226	#474	—	146	2	no	—
227	#470	—	146	2	no	—
228	#466	—	146	1	no	—
229	#462	—	73	2	no	—
230	#461	—	73	2	no	—
231	#452	—	146	2	no	—
232	#451	—	146	2	no	—
233	C₈×Q₈	—	73	1	no	—
234	C₄×SD₁₆	—	73	1	no	—
235	C₄×D₈	—	146	1	no	—
236	C₄×D₈	C₂×D₇₃	1	1	yes	characteristic
237	#436	C₂×D₇₃	1	2	yes	—
238	C₈×D₄	—	73	1	no	—
239	C₂×#107	—	146	1	no	—
240	#335	C₂×D₇₃	1	1	yes	characteristic
241	#334	—	73	1	no	—
242	#332	—	73	2	no	—
243	#332	—	73	2	no	—
244	Q₈⋊C₈	—	73	1	no	—
245	D₄⋊C₈	—	73	1	no	—
246	D₄⋊C₈	C₂×D₇₃	1	2	yes	—
247	C₈²	C₂×D₇₃	1	1	yes	characteristic
248	C₂×#41	—	146	2	no	—
249	C₂²×Q₈	—	146	2	no	—
250	#137	—	292	8	no	—
251	C₂×Q₁₆	—	146	2	no	—
252	C₂×Q₁₆	—	146	2	no	—
253	C₂×Q₁₆	—	146	2	no	—
254	C₂×SD₁₆	—	73	1	no	—
255	C₂×SD₁₆	—	146	2	no	—
256	C₂×D₈	D₂₉₂	1	1	yes	characteristic
257	C₂×M₁₆	—	2	2	no	—
258	C₂²×C₈	—	146	1	no	—
259	C₄⋊Q₈	—	73	2	no	—
260	C₄⋊Q₈	—	73	2	no	—
261	C₄⋊Q₈	—	146	2	no	—
262	#123	—	292	2	no	—
263	C₂²⋊Q₈	—	292	2	no	—
264	C₂²⋊Q₈	—	146	2	no	—
265	C₂²⋊Q₈	—	146	2	no	—
266	#121	—	146	2	no	—
267	C₄×Q₈	—	73	1	no	—
268	C₄×D₄	—	73	1	no	—
269	C₄×D₄	—	146	2	no	—
270	C₄×D₄	C₂²×D₇₃	1	2	yes	—
271	C₂×C₄⋊C₄	—	146	2	no	—
272	#107	—	292	2	no	—
273	#107	—	146	1	no	—
274	#107	—	146	1	no	—
275	#107	—	146	1	no	—
276	#107	—	146	1	no	—
277	#107	—	146	2	no	—
278	#107	—	146	2	no	—
279	#107	D₂₉₂	1	1	yes	characteristic
280	#106	—	73	1	no	—
281	C₄⋊C₈	—	73	1	no	—
282	C₄⋊C₈	—	73	1	no	—
283	C₄⋊C₈	—	146	1	no	—
284	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
285	#103	—	73	1	no	—
286	#103	—	146	2	no	—
287	#103	—	146	2	no	—
288	#102	—	73	1	no	—
289	#102	—	146	2	no	—
290	#102	—	146	2	no	—
291	#102	D₂₉₂	1	2	yes	—
292	C₂²⋊C₈	—	2	2	no	—
293	C₂²⋊C₈	—	146	1	no	—
294	C₄×C₈	—	146	1	no	—
295	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
298	#41	—	292	4	no	—
299	#41	—	584	2	no	—
300	C₂×Q₈	—	292	2	no	—
301	C₂×Q₈	—	292	4	no	—
302	C₂×Q₈	—	584	2	no	—
303	C₂×Q₈	—	73	1	no	—
304	C₂×Q₈	—	146	2	no	—
305	C₂×Q₈	—	146	2	no	—
306	C₂×D₄	—	292	2	no	—
307	C₂×D₄	—	73	1	no	—
308	C₂×D₄	C₂×D₂₉₂	1	2	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	292	2	no	—
311	C₂²×C₄	—	292	2	no	—
312	C₂²×C₄	—	146	1	no	—
313	Q₁₆	—	292	4	no	—
314	Q₁₆	—	292	4	no	—
315	Q₁₆	—	584	2	no	—
316	SD₁₆	—	292	4	no	—
317	SD₁₆	—	73	4	no	—
318	D₈	—	2	2	no	—
319	M₁₆	—	4	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	292	2	no	—
323	C₂×C₈	—	146	1	no	—
324	C₂×C₈	—	146	1	no	—
325	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
326	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
327	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
328	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
329	C₄⋊C₄	—	292	2	no	—
330	C₄⋊C₄	—	292	2	no	—
331	C₄⋊C₄	—	73	1	no	—
332	C₄⋊C₄	—	73	1	no	—
333	C₄⋊C₄	—	146	2	no	—
334	C₄⋊C₄	—	146	2	no	—
335	C₄⋊C₄	—	146	2	no	—
336	C₄⋊C₄	—	146	2	no	—
337	C₄⋊C₄	—	146	1	no	—
338	C₄⋊C₄	C₂×D₂₉₂	1	2	yes	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	292	2	no	—
341	C₂²⋊C₄	—	292	2	no	—
342	C₂²⋊C₄	—	146	1	no	—
343	C₄²	—	146	1	no	—
344	C₄²	C₂³×D₇₃	1	1	yes	characteristic
345	C₂³	—	2	2	no	—
346	C₂³	—	146	1	no	—
347	Q₈	—	292	2	no	—
348	Q₈	—	292	2	no	—
349	Q₈	—	292	2	no	—
350	Q₈	—	292	2	no	—
351	Q₈	—	584	4	no	—
352	Q₈	—	73	2	no	—
353	Q₈	—	146	1	no	—
354	D₄	—	2	2	no	—
355	D₄	—	2	2	no	—
356	D₄	—	584	4	no	—
357	D₄	—	73	2	no	—
358	D₄	—	146	1	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	292	1	no	—
363	C₂×C₄	—	292	1	no	—
364	C₂×C₄	—	292	2	no	—
365	C₂×C₄	—	292	2	no	—
366	C₂×C₄	—	584	4	no	—
367	C₂×C₄	—	584	4	no	—
368	C₂×C₄	—	146	1	no	—
369	C₂×C₄	—	146	1	no	—
370	C₂×C₄	—	146	1	no	—
371	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
373	C₂×C₄	#14556637	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	4	2	no	—
378	C₈	—	292	1	no	—
379	C₈	—	292	1	no	—
380	C₈	D₈×D₇₃	1	2	yes	—
381	C₈	#14556624	1	2	yes	complemented
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	292	1	no	—
386	C₂²	—	292	1	no	—
387	C₂²	—	146	2	no	—
388	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	292	1	no	—
392	C₄	—	292	1	no	—
393	C₄	—	584	2	no	—
394	C₄	—	584	2	no	—
395	C₄	—	146	2	no	—
396	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
397	C₄	C₂×#14556624	1	1	yes	characteristic
398	C₄	#17989389	1	1	yes	characteristic
399	C₄	#17989404	1	1	yes	characteristic
400	C₂	—	8	2	no	—
401	C₂	—	292	2	no	—
402	C₂	#22621579	1	1	yes	characteristic
403	C₂	D₈×D₂₉₂	1	1	yes	characteristic
404	C₂	#22621736	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1589	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#22622280	C₂
hypercenter	#1589	D₇₃
hypocenter	C₇₃	#19887
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1589

Complements

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

N	H ≅ G/N	type
1. D₈×Dic₁₄₆	C₂	semidirect
3. #22621579	C₂	semidirect
4. #22609549	C₂	semidirect
5. #22609420	C₂	semidirect
6. #22609326	C₂	semidirect
7. #22609321	C₂	semidirect
8. #22609183	C₂	semidirect
9. #22609169	C₂	semidirect
10. C₈×#11924017	C₂	semidirect
11. C₇₃×#1589	C₂	semidirect
112. #11924017	D₈	semidirect
176. C₈×C₇₃	#137	semidirect
212. #1589	D₇₃	semidirect
214. C₇₃	#19887	semidirect
381. C₈	#14556624	semidirect

Maximal subgroups

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

subgroup	index
D₈×Dic₁₄₆	2
#22622280	2
#22621579	2
#22609549	2
#22609420	2
#22609326	2
#22609321	2
#22609183	2
#22609169	2
C₈×#11924017	2
C₇₃×#1589	2
#19887	73