Subgroups — #32150867

Order 18176.

642 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601473	C₂	1	1	yes	maximal, characteristic
2	#22597252	C₂	1	2	yes	maximal, complemented
3	#22590226	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590225	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590224	C₂	1	2	yes	maximal, complemented
6	#22590222	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590218	C₂	1	2	yes	maximal, complemented
8	#22590133	C₂	1	1	yes	maximal, characteristic, complemented
9	#359×D₇₁	C₂	1	2	yes	maximal, complemented
10	#22589935	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#2178	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₇₁	C₂²	1	2	yes	—
13	#17986764	—	2	2	no	—
14	#17986441	C₄	1	2	yes	complemented
15	#17986438	C₄	1	2	yes	complemented
16	#17986436	C₄	1	2	yes	complemented
17	#17986433	C₄	1	2	yes	complemented
18	#17986431	C₂²	1	1	yes	characteristic
19	#17986429	C₂²	1	2	yes	—
20	C₂×#14556067	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
22	#17986119	—	2	2	no	—
23	#17986118	—	2	2	no	—
24	#17986117	C₂²	1	2	yes	complemented
25	#17986116	C₂²	1	2	yes	complemented
26	#17986115	C₂²	1	2	yes	complemented
27	#108×D₇₁	C₂²	1	2	yes	—
28	#17986014	—	2	2	no	—
29	#17985953	C₂²	1	1	yes	characteristic
30	#17985829	C₂²	1	1	yes	characteristic, complemented
31	#17985828	C₂²	1	2	yes	complemented
32	#17985807	C₂²	1	2	yes	complemented
33	#17985806	C₂²	1	2	yes	complemented
34	#17985805	C₂²	1	2	yes	complemented
35	#17985775	C₂²	1	1	yes	characteristic, complemented
36	#17985774	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic, complemented
38	#508×C₇₁	C₂²	1	1	yes	characteristic
39	#443×C₇₁	C₂²	1	2	yes	complemented
40	#368×C₇₁	C₂²	1	1	yes	characteristic, complemented
41	#359×C₇₁	C₂²	1	2	yes	complemented
42	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic, complemented
43	#41×D₇₁	—	2	2	no	—
44	#14556151	C₂×C₄	1	2	yes	—
45	#14556150	C₂³	1	2	yes	—
46	#14556149	—	2	2	no	—
47	#14556148	—	2	2	no	—
48	#14556147	C₂×C₄	1	2	yes	—
49	Q₁₆×D₇₁	C₂×C₄	1	2	yes	—
50	SD₁₆×D₇₁	—	2	2	no	—
51	D₈×D₇₁	C₂×C₄	1	2	yes	—
52	#14556136	—	2	2	no	—
53	#14556136	—	4	2	no	—
54	#14556135	—	2	2	no	—
55	#14556134	—	2	2	no	—
56	#14556134	—	4	2	no	—
57	C₂×#11765848	—	4	2	no	—
58	C₂×#11765848	D₄	1	1	yes	characteristic
59	M₁₆×D₇₁	—	2	2	no	—
60	M₁₆×D₇₁	—	4	2	no	—
61	C₂×C₈×D₇₁	C₂³	1	1	yes	characteristic
62	#14556085	D₄	1	1	yes	characteristic
63	Dic₅₆₈	C₂×C₄	1	1	yes	characteristic
64	#14556074	C₂×C₄	1	2	yes	—
65	#14556073	C₂×C₄	1	2	yes	—
66	#14556072	C₂×C₄	1	2	yes	—
67	#14556071	C₂×C₄	1	2	yes	—
68	D₁₁₃₆	C₂×C₄	1	1	yes	characteristic
69	#14556069	C₂×C₄	1	2	yes	—
70	#14556067	C₂×C₄	1	2	yes	—
71	#14556067	C₂×C₄	1	2	yes	—
72	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₁₆×D₇₁	—	2	2	no	—
75	#14556064	C₂³	1	2	yes	—
76	#14556063	—	2	2	no	—
77	#14556056	—	2	2	no	—
78	#14556055	—	2	2	no	—
79	#14556054	—	2	2	no	—
80	#14556039	D₄	1	1	yes	characteristic
81	#14556038	D₄	1	1	yes	characteristic
82	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
83	#135×C₇₁	C₂³	1	2	yes	—
84	#131×C₇₁	—	2	2	no	—
85	Q₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
86	SD₃₂×C₇₁	C₂×C₄	1	2	yes	—
87	D₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
88	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
90	#108×C₇₁	C₂³	1	2	yes	—
91	#104×C₇₁	—	2	2	no	—
92	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
93	#11765866	—	2	2	no	—
94	#11765865	—	2	2	no	—
95	#11765865	—	4	2	no	—
96	#11765864	—	2	2	no	—
97	#11765864	—	4	2	no	—
98	Q₈×D₇₁	—	2	2	no	—
99	D₄×D₇₁	—	2	2	no	—
100	D₄×D₇₁	—	4	2	no	—
101	C₂×C₄×D₇₁	—	4	2	no	—
102	C₂×C₄×D₇₁	C₂×D₄	1	1	yes	characteristic
103	Dic₂₈₄	C₂²×C₄	1	2	yes	—
104	#11765855	—	2	2	no	—
105	#11765854	—	2	2	no	—
106	#11765853	—	2	2	no	—
107	#11765852	—	2	2	no	—
108	D₅₆₈	C₂²×C₄	1	2	yes	—
109	#11765850	—	2	2	no	—
110	#11765849	—	2	2	no	—
111	#11765849	—	4	2	no	—
112	#11765848	—	4	2	no	—
113	#11765848	—	4	2	no	—
114	#11765848	—	4	2	no	—
115	#11765848	D₈	1	2	yes	—
116	#11765848	D₈	1	2	yes	—
117	C₂×C₇₁⋊C₈	—	4	2	no	—
118	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
120	C₈×D₇₁	—	4	2	no	—
121	C₈×D₇₁	—	4	2	no	—
122	C₈×D₇₁	—	4	2	no	—
123	C₈×D₇₁	C₂²×C₄	1	2	yes	—
124	C₈×D₇₁	#41	1	2	yes	—
125	Dic₇₁⋊C₄	—	2	1	no	—
126	(C₂×D₇₁)⋊C₄	—	2	1	no	—
127	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
128	C₇₁⋊C₁₆	—	2	1	no	—
129	C₇₁⋊C₁₆	C₂²×C₄	1	2	yes	—
130	#41×C₇₁	—	2	2	no	—
131	Q₁₆×C₇₁	C₂²×C₄	1	2	yes	—
132	SD₁₆×C₇₁	—	2	2	no	—
133	D₈×C₇₁	C₂²×C₄	1	2	yes	—
134	M₁₆×C₇₁	—	2	2	no	—
135	M₁₆×C₇₁	—	4	2	no	—
136	C₂×C₈×C₇₁	—	4	2	no	—
137	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
139	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
140	C₁₆×C₇₁	—	2	1	no	—
141	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
142	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
143	C₂²×D₇₁	—	4	2	no	—
144	C₂²×D₇₁	#135	1	1	yes	characteristic
145	Q₈×C₇₁	—	2	2	no	—
146	D₄×C₇₁	—	2	2	no	—
147	D₄×C₇₁	—	4	2	no	—
148	C₂×C₄×C₇₁	—	2	1	no	—
149	C₂×C₄×C₇₁	—	4	2	no	—
150	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₇₁	—	4	2	no	—
152	C₂²⋊D₇₁	—	4	2	no	—
153	C₂²⋊D₇₁	—	4	2	no	—
154	C₂×Dic₇₁	—	2	1	no	—
155	C₂×Dic₇₁	—	4	2	no	—
156	C₂×Dic₇₁	#135	1	1	yes	characteristic
157	D₂₈₄	—	2	2	no	—
158	D₂₈₄	—	4	2	no	—
159	C₄×D₇₁	—	4	2	no	—
160	C₄×D₇₁	—	4	2	no	—
161	C₄×D₇₁	—	4	2	no	—
162	C₄×D₇₁	C₄×D₄	1	2	yes	—
163	C₄×D₇₁	C₂×D₈	1	2	yes	—
164	Dic₁₄₂	—	2	2	no	—
165	Dic₁₄₂	—	4	2	no	—
166	C₈×C₇₁	—	4	2	no	—
167	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
168	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
169	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
170	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
171	C₇₁⋊C₈	—	4	2	no	—
172	C₇₁⋊C₈	C₄×D₄	1	2	yes	—
173	C₇₁⋊C₈	C₂×D₈	1	2	yes	—
174	C₂²×C₇₁	—	4	2	no	—
175	C₂²×C₇₁	C₂×#135	1	1	yes	characteristic
176	C₂×D₇₁	—	4	1	no	—
177	C₂×D₇₁	—	4	2	no	—
178	C₂×D₇₁	—	16	2	no	—
179	C₂×D₇₁	C₄×D₈	1	2	yes	—
180	C₄×C₇₁	—	4	1	no	—
181	C₄×C₇₁	—	4	2	no	—
182	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
184	Dic₇₁	—	4	1	no	—
185	Dic₇₁	—	4	2	no	—
186	Dic₇₁	C₄×D₈	1	2	yes	—
187	#33464	—	71	1	no	maximal, Sylow
188	C₂×C₇₁	—	2	1	no	—
189	C₂×C₇₁	—	8	2	no	—
190	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
191	D₇₁	—	2	2	no	—
192	D₇₁	—	8	2	no	—
193	#3414	—	71	1	no	—
194	#2957	—	71	2	no	—
195	#2179	—	71	1	no	—
196	#2179	—	71	2	no	—
197	#2179	—	71	1	no	—
198	#2178	—	71	2	no	—
199	#2178	—	71	1	no	—
200	#2178	D₇₁	1	1	yes	characteristic, complemented
201	#2161	—	71	1	no	—
202	C₂×#359	—	71	2	no	—
203	#2108	—	71	1	no	—
204	C₇₁	#33464	1	1	yes	characteristic, complemented, Sylow
205	C₂×#135	—	71	2	no	—
206	#568	—	142	2	no	—
207	#510	—	71	2	no	—
208	#510	—	71	2	no	—
209	#509	—	71	2	no	—
210	#509	—	71	2	no	—
211	#508	—	71	2	no	—
212	#508	—	71	1	no	—
213	#508	C₂×D₇₁	1	1	yes	characteristic
214	C₂×M₃₂	—	71	1	no	—
215	C₂²×C₁₆	—	71	1	no	—
216	#444	—	142	2	no	—
217	#444	—	142	2	no	—
218	#443	—	71	2	no	—
219	#443	—	71	2	no	—
220	#443	—	71	2	no	—
221	#443	C₂×D₇₁	1	2	yes	complemented
222	C₂×#108	—	71	2	no	—
223	#421	—	142	2	no	—
224	#405	—	71	1	no	—
225	#368	—	71	1	no	—
226	#368	—	71	2	no	—
227	#368	C₂×D₇₁	1	1	yes	characteristic, complemented
228	#359	—	71	2	no	—
229	#359	—	71	2	no	—
230	#359	—	71	2	no	—
231	#359	C₂×D₇₁	1	2	yes	complemented
232	#346	—	71	1	no	—
233	#346	—	71	1	no	—
234	C₄×C₁₆	—	71	1	no	—
235	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic, complemented
236	C₂×#41	—	142	2	no	—
237	#135	—	71	2	no	—
238	#135	—	71	2	no	—
239	#135	—	71	2	no	—
240	#135	—	142	2	no	—
241	#135	—	142	2	no	—
242	#135	C₂²×D₇₁	1	2	yes	—
243	C₂×Q₁₆	—	71	2	no	—
244	C₂×SD₁₆	—	142	2	no	—
245	C₂×D₈	—	71	2	no	—
246	#131	—	2	2	no	—
247	#131	—	284	2	no	—
248	#131	—	284	2	no	—
249	#131	—	142	2	no	—
250	#131	—	142	2	no	—
251	#131	—	142	2	no	—
252	C₂×M₁₆	—	284	2	no	—
253	C₂×M₁₆	—	284	2	no	—
254	C₂×M₁₆	—	71	1	no	—
255	C₂×M₁₆	—	142	2	no	—
256	C₂²×C₈	—	71	1	no	—
257	#117	—	71	1	no	—
258	Q₃₂	—	71	2	no	—
259	Q₃₂	—	71	1	no	—
260	Q₃₂	C₄×D₇₁	1	1	yes	characteristic
261	SD₃₂	—	71	2	no	—
262	SD₃₂	—	71	2	no	—
263	SD₃₂	—	71	2	no	—
264	SD₃₂	C₄×D₇₁	1	2	yes	—
265	D₁₆	—	71	2	no	—
266	D₁₆	—	71	1	no	—
267	D₁₆	C₄×D₇₁	1	1	yes	characteristic
268	M₃₂	—	71	2	no	—
269	M₃₂	—	71	2	no	—
270	C₂×C₁₆	—	71	1	no	—
271	C₂×C₁₆	—	71	1	no	—
272	C₂×C₁₆	—	142	2	no	—
273	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
274	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
275	#108	—	71	2	no	—
276	#108	—	142	2	no	—
277	#108	C₂²×D₇₁	1	2	yes	—
278	#104	—	2	2	no	—
279	#104	—	142	2	no	—
280	#104	—	142	2	no	—
281	#104	—	142	2	no	—
282	#97	—	71	1	no	—
283	#97	—	71	1	no	—
284	C₄×C₈	—	71	1	no	—
285	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
286	#41	—	2	2	no	—
287	#41	—	284	2	no	—
288	#41	—	284	2	no	—
289	#41	—	142	2	no	—
290	#41	—	142	2	no	—
291	#41	—	142	2	no	—
292	C₂×Q₈	—	142	2	no	—
293	C₂×D₄	—	284	2	no	—
294	C₂×D₄	—	142	2	no	—
295	C₂²×C₄	—	284	2	no	—
296	C₂²×C₄	—	71	1	no	—
297	Q₁₆	—	71	2	no	—
298	Q₁₆	—	142	2	no	—
299	Q₁₆	C₂×C₄×D₇₁	1	2	yes	—
300	SD₁₆	—	2	2	no	—
301	SD₁₆	—	142	2	no	—
302	SD₁₆	—	142	2	no	—
303	SD₁₆	—	142	2	no	—
304	D₈	—	71	2	no	—
305	D₈	—	142	2	no	—
306	D₈	C₂×C₄×D₇₁	1	2	yes	—
307	M₁₆	—	2	2	no	—
308	M₁₆	—	4	2	no	—
309	M₁₆	—	284	2	no	—
310	M₁₆	—	284	2	no	—
311	M₁₆	—	284	2	no	—
312	M₁₆	—	284	2	no	—
313	M₁₆	—	71	2	no	—
314	M₁₆	—	71	2	no	—
315	M₁₆	—	142	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	284	2	no	—
318	C₂×C₈	—	284	2	no	—
319	C₂×C₈	—	284	2	no	—
320	C₂×C₈	—	284	2	no	—
321	C₂×C₈	—	71	1	no	—
322	C₂×C₈	—	71	1	no	—
323	C₂×C₈	—	71	2	no	—
324	C₂×C₈	—	71	2	no	—
325	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
326	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
327	C₄⋊C₄	—	142	1	no	—
328	C₂²⋊C₄	—	142	1	no	—
329	C₄²	—	71	1	no	—
330	C₄²	D₄×D₇₁	1	1	yes	characteristic
331	C₁₆	—	2	1	no	—
332	C₁₆	—	71	2	no	—
333	C₁₆	—	142	1	no	—
334	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
335	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
336	C₂³	—	284	2	no	—
337	C₂³	—	71	1	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	284	2	no	—
340	Q₈	—	142	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	284	2	no	—
344	D₄	—	284	2	no	—
345	D₄	—	284	2	no	—
346	D₄	—	284	2	no	—
347	D₄	—	142	2	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	284	2	no	—
351	C₂×C₄	—	284	2	no	—
352	C₂×C₄	—	284	2	no	—
353	C₂×C₄	—	284	2	no	—
354	C₂×C₄	—	71	1	no	—
355	C₂×C₄	—	71	2	no	—
356	C₂×C₄	—	71	2	no	—
357	C₂×C₄	—	142	1	no	—
358	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
359	C₈	—	4	2	no	—
360	C₈	—	284	2	no	—
361	C₈	—	71	2	no	—
362	C₈	—	71	2	no	—
363	C₈	D₈×D₇₁	1	1	yes	characteristic
364	C₈	D₈×D₇₁	1	1	yes	characteristic
365	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
366	C₈	#41×D₇₁	1	1	yes	characteristic
367	C₂²	—	4	2	no	—
368	C₂²	—	284	1	no	—
369	C₂²	—	284	2	no	—
370	C₂²	—	71	2	no	—
371	C₂²	—	1136	2	no	—
372	C₂²	#135×D₇₁	1	1	yes	characteristic
373	C₄	—	4	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	284	1	no	—
376	C₄	—	284	2	no	—
377	C₄	—	71	2	no	—
378	C₄	C₄×D₄×D₇₁	1	1	yes	characteristic
379	C₄	C₂×D₈×D₇₁	1	1	yes	characteristic
380	C₂	—	2	1	no	—
381	C₂	—	8	2	no	—
382	C₂	—	568	2	no	—
383	C₂	—	142	2	no	—
384	C₂	C₄×D₈×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₇₁
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2178	C₂
socle	C₂×C₇₁	C₂×C₄×D₈
prime radical	#22601473	C₂
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2178	D₇₁
hypocenter	C₇₁	#33464
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

Complements

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

N	H ≅ G/N	type
2. #22597252	C₂	semidirect
3. #22590226	C₂	semidirect
4. #22590225	C₂	semidirect
5. #22590224	C₂	semidirect
6. #22590222	C₂	semidirect
7. #22590218	C₂	semidirect
8. #22590133	C₂	semidirect
9. #359×D₇₁	C₂	semidirect
10. #22589935	C₂	semidirect
11. C₇₁×#2178	C₂	semidirect
14. #17986441	C₄	semidirect
15. #17986438	C₄	semidirect
16. #17986436	C₄	semidirect
17. #17986433	C₄	semidirect
24. #17986117	C₂²	semidirect
25. #17986116	C₂²	semidirect
26. #17986115	C₂²	semidirect
30. #17985829	C₂²	semidirect
31. #17985828	C₂²	semidirect
32. #17985807	C₂²	semidirect
33. #17985806	C₂²	semidirect
34. #17985805	C₂²	semidirect
35. #17985775	C₂²	semidirect
36. #17985774	C₂²	semidirect
37. C₁₆×Dic₇₁	C₂²	semidirect
39. #443×C₇₁	C₂²	semidirect
40. #368×C₇₁	C₂²	semidirect
41. #359×C₇₁	C₂²	semidirect
42. C₄×C₁₆×C₇₁	C₂²	semidirect
200. #2178	D₇₁	semidirect
204. C₇₁	#33464	semidirect
221. #443	C₂×D₇₁	semidirect
227. #368	C₂×D₇₁	semidirect
231. #359	C₂×D₇₁	semidirect
235. C₄×C₁₆	C₂×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22601473	2
#22597252	2
#22590226	2
#22590225	2
#22590224	2
#22590222	2
#22590218	2
#22590133	2
#359×D₇₁	2
#22589935	2
C₇₁×#2178	2
#33464	71