Subgroups — #32042582

Order 18176.

620 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	C₂×#17986271	C₂	1	1	yes	maximal, characteristic
2	#22589808	C₂	1	2	yes	maximal, complemented
3	#22589618	C₂	1	2	yes	maximal
4	#22589256	C₂	1	1	yes	maximal, characteristic, complemented
5	#22589130	C₂	1	1	yes	maximal, characteristic, complemented
6	#22588834	C₂	1	2	yes	maximal
7	#22588463	C₂	1	1	yes	maximal, characteristic, complemented
8	#22588454	C₂	1	2	yes	maximal, complemented
9	#22588391	C₂	1	1	yes	maximal, characteristic
10	#22588141	C₂	1	1	yes	maximal, characteristic
11	C₇₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556159	C₂²	1	2	yes	—
13	C₂×#14556134	C₂²	1	1	yes	characteristic
14	#17986271	—	2	2	no	—
15	#17986271	—	2	2	no	—
16	#17986047	C₂²	1	2	yes	—
17	C₂×#14556064	C₂²	1	1	yes	characteristic
18	C₂×#14556064	C₂²	1	1	yes	characteristic
19	#17986017	C₂²	1	2	yes	—
20	#17986008	C₂²	1	2	yes	—
21	#17985981	C₂²	1	1	yes	characteristic
22	C₂×#14556047	C₂²	1	2	yes	—
23	C₂×#14556047	C₂²	1	2	yes	—
24	#17985972	C₂²	1	1	yes	characteristic
25	#17985954	C₂²	1	1	yes	characteristic
26	#17985771	C₂²	1	1	yes	characteristic
27	#17985769	C₂²	1	1	yes	characteristic
28	#17985769	C₂²	1	2	yes	—
29	#17985769	C₂²	1	2	yes	—
30	#17985769	C₂²	1	2	yes	—
31	#17985755	C₂²	1	2	yes	—
32	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
33	#519×C₇₁	C₂²	1	2	yes	—
34	#441×C₇₁	—	2	2	no	—
35	C₄×Q₁₆×C₇₁	—	2	2	no	—
36	#427×C₇₁	C₂²	1	1	yes	characteristic
37	#419×C₇₁	C₂²	1	2	yes	—
38	#405×C₇₁	C₂²	1	1	yes	characteristic
39	C₂×#11765865	C₂³	1	1	yes	characteristic
40	#14556159	—	2	4	no	—
41	#14556159	—	4	2	no	—
42	C₂×#11765855	—	2	2	no	—
43	C₂×#11765852	—	2	2	no	—
44	#14556134	—	2	1	no	—
45	#14556134	—	2	1	no	—
46	#14556134	—	2	1	no	—
47	#14556134	—	2	1	no	—
48	C₂×#11765849	C₂³	1	1	yes	characteristic
49	C₂×#11765849	C₂³	1	1	yes	characteristic
50	C₂×#11765849	C₂³	1	2	yes	—
51	C₂×#11765849	C₂³	1	2	yes	—
52	C₂×#11765848	—	2	1	no	—
53	C₂×C₈×D₇₁	—	2	1	no	—
54	#14556087	C₂³	1	1	yes	characteristic
55	#14556064	—	2	1	no	—
56	#14556064	—	2	1	no	—
57	#14556064	—	2	1	no	—
58	#14556064	—	2	1	no	—
59	#14556063	—	4	2	no	—
60	#14556056	—	4	2	no	—
61	#14556052	—	4	2	no	—
62	#14556047	—	2	4	no	—
63	#14556047	—	4	2	no	—
64	#14556046	—	2	1	no	—
65	#14556044	—	2	1	no	—
66	#14556042	—	2	2	no	—
67	#14556038	—	2	1	no	—
68	C₈×Dic₇₁	—	2	1	no	—
69	C₂²×Q₈×C₇₁	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₇₁	—	2	1	no	—
71	C₂×Q₁₆×C₇₁	—	2	1	no	—
72	C₂×Q₁₆×C₇₁	—	2	1	no	—
73	C₂×Q₁₆×C₇₁	—	2	1	no	—
74	C₂×Q₁₆×C₇₁	D₄	1	2	yes	—
75	C₂×Q₁₆×C₇₁	D₄	1	2	yes	—
76	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₇₁	—	2	4	no	—
79	C₄×Q₈×C₇₁	—	4	2	no	—
80	#117×C₇₁	—	2	4	no	—
81	#117×C₇₁	C₂³	1	2	yes	—
82	#117×C₇₁	C₂³	1	1	yes	characteristic
83	#107×C₇₁	—	2	1	no	—
84	#106×C₇₁	—	2	1	no	—
85	#103×C₇₁	—	2	4	no	—
86	#97×C₇₁	—	2	1	no	—
87	C₄×C₈×C₇₁	—	2	1	no	—
88	#11765865	—	2	1	no	—
89	#11765865	—	2	1	no	—
90	#11765865	—	2	1	no	—
91	#11765865	—	2	1	no	—
92	C₂×Dic₁₄₂	#41	1	1	yes	characteristic
93	C₂×D₂₈₄	#41	1	1	yes	characteristic
94	C₂×C₂²⋊D₇₁	—	2	1	no	—
95	C₂×C₄×D₇₁	—	2	1	no	—
96	#11765855	—	4	4	no	—
97	#11765852	—	4	4	no	—
98	#11765849	—	2	1	no	—
99	#11765849	—	2	2	no	—
100	#11765849	—	2	1	no	—
101	#11765849	—	2	2	no	—
102	#11765849	—	2	1	no	—
103	#11765849	—	2	1	no	—
104	#11765849	—	2	2	no	—
105	#11765849	—	2	2	no	—
106	#11765848	—	4	1	no	—
107	#11765848	—	4	1	no	—
108	C₂×C₇₁⋊C₈	—	2	1	no	—
109	C₂×C₇₁⋊C₈	—	2	1	no	—
110	C₂×C₇₁⋊C₈	—	2	2	no	—
111	C₂×C₇₁⋊C₈	—	2	2	no	—
112	C₈×D₇₁	—	4	1	no	—
113	C₈×D₇₁	—	4	1	no	—
114	C₄⋊Dic₇₁	—	2	1	no	—
115	C₂²⋊Dic₇₁	—	2	1	no	—
116	C₄×Dic₇₁	—	2	1	no	—
117	C₂×Q₈×C₇₁	—	2	2	no	—
118	C₂×Q₈×C₇₁	—	2	2	no	—
119	C₂×Q₈×C₇₁	—	2	2	no	—
120	C₂×Q₈×C₇₁	—	2	2	no	—
121	C₂×Q₈×C₇₁	—	2	2	no	—
122	C₂×Q₈×C₇₁	C₂×D₄	1	4	yes	—
123	C₂²×C₄×C₇₁	—	2	2	no	—
124	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₇₁	—	2	2	no	—
126	Q₁₆×C₇₁	—	2	2	no	—
127	Q₁₆×C₇₁	—	2	2	no	—
128	Q₁₆×C₇₁	—	2	2	no	—
129	M₁₆×C₇₁	—	2	1	no	—
130	M₁₆×C₇₁	—	2	1	no	—
131	C₂×C₈×C₇₁	—	2	1	no	—
132	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
135	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
136	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
137	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
138	C₄⋊C₄×C₇₁	—	2	1	no	—
139	C₄⋊C₄×C₇₁	—	2	2	no	—
140	C₄⋊C₄×C₇₁	—	4	4	no	—
141	C₂²⋊C₄×C₇₁	—	2	2	no	—
142	C₂²⋊C₄×C₇₁	#41	1	2	yes	—
143	C₄²×C₇₁	—	2	1	no	—
144	C₄²×C₇₁	—	2	2	no	—
145	C₄²×C₇₁	—	4	4	no	—
146	C₂³×C₇₁	C₂×#41	1	1	yes	characteristic
147	C₂²×D₇₁	—	2	1	no	—
148	Q₈×C₇₁	—	2	4	no	—
149	Q₈×C₇₁	—	2	4	no	—
150	Q₈×C₇₁	—	4	2	no	—
151	Q₈×C₇₁	—	4	2	no	—
152	C₂×C₄×C₇₁	—	2	2	no	—
153	C₂×C₄×C₇₁	—	2	4	no	—
154	C₂×C₄×C₇₁	—	4	2	no	—
155	C₂×C₄×C₇₁	—	4	2	no	—
156	C₂×C₄×C₇₁	—	4	2	no	—
157	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²⋊D₇₁	—	4	1	no	—
164	C₂²⋊D₇₁	—	4	1	no	—
165	C₂×Dic₇₁	—	2	1	no	—
166	C₂×Dic₇₁	—	4	1	no	—
167	D₂₈₄	—	2	1	no	—
168	D₂₈₄	—	2	1	no	—
169	C₄×D₇₁	—	4	1	no	—
170	C₄×D₇₁	—	4	1	no	—
171	Dic₁₄₂	—	2	1	no	—
172	Dic₁₄₂	—	2	1	no	—
173	C₈×C₇₁	—	4	1	no	—
174	C₈×C₇₁	#123	1	1	yes	characteristic
175	C₈×C₇₁	#123	1	1	yes	characteristic
176	C₈×C₇₁	#123	1	1	yes	characteristic
177	C₈×C₇₁	#123	1	1	yes	characteristic
178	C₇₁⋊C₈	—	4	1	no	—
179	C₇₁⋊C₈	—	4	2	no	—
180	C₇₁⋊C₈	—	4	2	no	—
181	C₇₁⋊C₈	—	4	1	no	—
182	C₂²×C₇₁	—	4	1	no	—
183	C₂²×C₇₁	#529	1	1	yes	characteristic
184	C₂²×C₇₁	#546	1	1	yes	characteristic
185	C₂²×C₇₁	#555	1	1	yes	characteristic
186	C₂×D₇₁	—	4	1	no	—
187	C₂×D₇₁	—	8	1	no	—
188	C₄×C₇₁	—	4	2	no	—
189	C₄×C₇₁	—	4	4	no	—
190	C₄×C₇₁	—	8	2	no	—
191	C₄×C₇₁	C₂×#123	1	1	yes	characteristic
192	C₄×C₇₁	#540	1	1	yes	characteristic
193	C₄×C₇₁	#546	1	1	yes	characteristic
194	C₄×C₇₁	#553	1	1	yes	characteristic
195	Dic₇₁	—	4	1	no	—
196	Dic₇₁	—	8	1	no	—
197	#24989	—	71	1	no	maximal, Sylow
198	C₂×C₇₁	—	2	1	no	—
199	C₂×C₇₁	—	2	1	no	—
200	C₂×C₇₁	—	2	1	no	—
201	C₂×C₇₁	#2420	1	1	yes	characteristic
202	D₇₁	—	8	1	no	—
203	C₂×#473	—	71	1	no	—
204	#2950	D₇₁	1	1	yes	characteristic, complemented
205	#2082	—	71	2	no	—
206	#2049	—	71	2	no	—
207	#1979	—	71	1	no	—
208	#1953	—	71	1	no	—
209	#1905	—	71	2	no	—
210	#1833	—	71	1	no	—
211	#1830	—	71	2	no	—
212	#1814	—	71	1	no	—
213	#1757	—	71	1	no	—
214	C₇₁	#24989	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	71	2	no	—
216	C₂²×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
217	C₂×#131	—	71	1	no	—
218	#519	C₂×D₇₁	1	2	yes	—
219	#473	—	142	2	no	—
220	#473	—	142	2	no	—
221	#441	—	2	2	no	—
222	C₄×Q₁₆	—	2	2	no	—
223	#430	—	71	2	no	—
224	C₂×#108	—	71	1	no	—
225	C₂×#108	—	71	1	no	—
226	#427	C₂×D₇₁	1	1	yes	characteristic
227	#421	—	71	2	no	—
228	#419	—	71	2	no	—
229	#419	C₂×D₇₁	1	2	yes	—
230	#413	—	71	1	no	—
231	C₂×#101	—	71	2	no	—
232	C₂×#101	—	71	2	no	—
233	#410	—	71	1	no	—
234	#405	—	71	1	no	—
235	#405	C₂×D₇₁	1	1	yes	characteristic
236	#344	—	71	1	no	—
237	#344	—	71	2	no	—
238	#344	—	71	2	no	—
239	#344	—	71	1	no	—
240	#344	—	71	2	no	—
241	H(3,ℤ/4)	—	71	2	no	—
242	C₂×#41	—	71	1	no	—
243	C₂²×Q₈	C₂²×D₇₁	1	2	yes	—
244	#137	—	284	2	no	—
245	#137	—	142	4	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	2	1	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	142	2	no	—
251	C₂×Q₁₆	C₂²⋊D₇₁	1	2	yes	—
252	C₂×Q₁₆	C₂²⋊D₇₁	1	2	yes	—
253	C₂×SD₁₆	—	142	2	no	—
254	#131	—	142	1	no	—
255	#131	—	142	1	no	—
256	#131	—	142	1	no	—
257	#131	—	142	1	no	—
258	C₂×M₁₆	—	71	1	no	—
259	C₂×M₁₆	—	71	2	no	—
260	C₂×M₁₆	—	71	1	no	—
261	C₂×M₁₆	—	71	2	no	—
262	C₂×M₁₆	—	142	1	no	—
263	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
264	C₂²×C₈	—	142	1	no	—
265	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	4	2	no	—
268	#117	—	2	4	no	—
269	#117	—	71	1	no	—
270	#117	C₂²×D₇₁	1	2	yes	—
271	#117	C₂²×D₇₁	1	1	yes	characteristic
272	#108	—	284	2	no	—
273	#108	—	142	1	no	—
274	#108	—	142	1	no	—
275	#108	—	142	1	no	—
276	#108	—	142	1	no	—
277	#107	—	2	1	no	—
278	#106	—	2	1	no	—
279	#104	—	284	2	no	—
280	#103	—	2	4	no	—
281	#103	—	284	2	no	—
282	#101	—	284	2	no	—
283	#101	—	142	1	no	—
284	#101	—	142	4	no	—
285	#100	—	142	1	no	—
286	#99	—	142	2	no	—
287	#97	—	2	1	no	—
288	#97	—	142	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	142	1	no	—
291	#41	—	142	1	no	—
292	#41	—	142	1	no	—
293	#41	—	142	1	no	—
294	#41	—	142	1	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	71	1	no	—
301	C₂×Q₈	C₂×C₂²⋊D₇₁	1	4	yes	—
302	C₂×D₄	—	71	1	no	—
303	C₂×D₄	—	142	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	142	1	no	—
306	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
307	Q₁₆	—	2	2	no	—
308	Q₁₆	—	2	2	no	—
309	Q₁₆	—	2	2	no	—
310	Q₁₆	—	2	2	no	—
311	Q₁₆	—	284	4	no	—
312	SD₁₆	—	284	4	no	—
313	M₁₆	—	2	1	no	—
314	M₁₆	—	2	1	no	—
315	M₁₆	—	284	1	no	—
316	M₁₆	—	284	1	no	—
317	M₁₆	—	142	1	no	—
318	M₁₆	—	142	2	no	—
319	M₁₆	—	142	1	no	—
320	M₁₆	—	142	2	no	—
321	M₁₆	—	142	1	no	—
322	M₁₆	—	142	1	no	—
323	M₁₆	—	142	2	no	—
324	M₁₆	—	142	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	284	1	no	—
327	C₂×C₈	—	284	1	no	—
328	C₂×C₈	—	142	2	no	—
329	C₂×C₈	—	142	1	no	—
330	C₂×C₈	—	142	1	no	—
331	C₂×C₈	—	142	2	no	—
332	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
333	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
334	C₂×C₈	#11765865	1	1	yes	characteristic
335	C₂×C₈	#11765865	1	1	yes	characteristic
336	C₂×C₈	#11765865	1	1	yes	characteristic
337	C₂×C₈	#11765865	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₄⋊C₄	—	142	1	no	—
342	C₂²⋊C₄	—	2	2	no	—
343	C₂²⋊C₄	—	142	1	no	—
344	C₂²⋊C₄	#11765864	1	2	yes	—
345	C₄²	—	2	1	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	4	4	no	—
348	C₄²	—	142	1	no	—
349	C₂³	—	142	1	no	—
350	C₂³	C₂×#11765864	1	1	yes	characteristic
351	Q₈	—	2	4	no	—
352	Q₈	—	2	4	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	142	1	no	—
356	Q₈	—	142	1	no	—
357	D₄	—	284	1	no	—
358	D₄	—	284	1	no	—
359	D₄	—	142	1	no	—
360	D₄	—	142	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	284	1	no	—
367	C₂×C₄	—	284	1	no	—
368	C₂×C₄	—	284	1	no	—
369	C₂×C₄	—	142	1	no	—
370	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
371	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
372	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
373	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
374	C₂×C₄	#14556173	1	1	yes	characteristic
375	C₂×C₄	#14556174	1	1	yes	characteristic
376	C₈	—	4	1	no	—
377	C₈	—	284	2	no	—
378	C₈	—	284	2	no	—
379	C₈	—	284	1	no	—
380	C₈	—	284	1	no	—
381	C₈	#14556111	1	1	yes	characteristic
382	C₈	#14556111	1	1	yes	characteristic
383	C₈	#14556111	1	1	yes	characteristic
384	C₈	#14556111	1	1	yes	characteristic
385	C₂²	—	4	1	no	—
386	C₂²	—	284	1	no	—
387	C₂²	—	568	1	no	—
388	C₂²	#17986531	1	1	yes	characteristic
389	C₂²	#17986643	1	1	yes	characteristic
390	C₂²	#17986714	1	1	yes	characteristic
391	C₄	—	4	2	no	—
392	C₄	—	4	4	no	—
393	C₄	—	8	2	no	—
394	C₄	—	284	1	no	—
395	C₄	—	568	1	no	—
396	C₄	C₂×#14556111	1	1	yes	characteristic
397	C₄	#17986600	1	1	yes	characteristic
398	C₄	#17986637	1	1	yes	characteristic
399	C₄	#17986699	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	568	1	no	—
404	C₂	#22591826	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22591826
commutator, cosocle	C₂²×C₄×C₇₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₇₁×#2950	C₂
socle	C₂×C₇₁	#2420
prime radical	C₂×#11765865	C₂³
hypercenter	#2950	D₇₁
hypocenter	C₇₁	#24989
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2950

Complements

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

N	H ≅ G/N	type
2. #22589808	C₂	semidirect
4. #22589256	C₂	semidirect
5. #22589130	C₂	semidirect
7. #22588463	C₂	semidirect
8. #22588454	C₂	semidirect
11. C₇₁×#2950	C₂	semidirect
204. #2950	D₇₁	semidirect
214. C₇₁	#24989	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986271	2
#22589808	2
#22589618	2
#22589256	2
#22589130	2
#22588834	2
#22588463	2
#22588454	2
#22588391	2
#22588141	2
C₇₁×#2950	2
#24989	71