Subgroups — #32154263

Order 18176.

636 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601491	C₂	1	2	yes	maximal, complemented
2	#22600145	C₂	1	1	yes	maximal, characteristic, complemented
3	#22590592	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590463	C₂	1	2	yes	maximal, complemented
5	#22590421	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590418	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590318	C₂	1	2	yes	maximal, complemented
8	#22590316	C₂	1	2	yes	maximal, complemented
9	#22590219	C₂	1	1	yes	maximal, characteristic, complemented
10	#22590177	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#3293	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986816	C₂²	1	2	yes	—
13	#17986813	—	2	1	no	—
14	#17986440	—	2	2	no	—
15	#17986440	C₂²	1	2	yes	—
16	#17986439	—	2	2	no	—
17	#17986439	C₂²	1	2	yes	—
18	#17986429	—	2	2	no	—
19	#17986429	C₂²	1	1	yes	characteristic
20	C₂×#14556074	—	2	2	no	—
21	C₂×#14556074	C₂²	1	2	yes	—
22	C₂×#14556074	C₂²	1	2	yes	—
23	#17986416	C₂²	1	2	yes	—
24	#17986379	C₂²	1	1	yes	characteristic
25	#17986275	—	2	1	no	—
26	#17986274	C₂²	1	2	yes	—
27	#17986183	—	2	2	no	—
28	#17986117	C₂²	1	1	yes	characteristic
29	#17985828	C₂²	1	1	yes	characteristic
30	#17985820	C₂²	1	1	yes	characteristic
31	#17985815	C₂²	1	2	yes	—
32	#17985814	C₂²	1	2	yes	—
33	#17985806	C₂²	1	1	yes	characteristic
34	#17985805	C₂²	1	1	yes	characteristic
35	#17985783	C₂²	1	2	yes	—
36	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
37	#578×C₇₁	—	2	1	no	—
38	#578×C₇₁	C₂²	1	2	yes	—
39	#494×C₇₁	C₂²	1	1	yes	characteristic
40	#473×C₇₁	—	2	1	no	—
41	#473×C₇₁	C₂²	1	2	yes	—
42	#456×C₇₁	—	2	2	no	—
43	#443×C₇₁	C₂²	1	1	yes	characteristic
44	#443×C₇₁	C₂²	1	1	yes	characteristic
45	#14556175	—	2	2	no	—
46	#14556161	—	2	2	no	—
47	#14556161	—	4	2	no	—
48	#14556158	—	2	2	no	—
49	#14556157	—	4	2	no	—
50	#14556150	—	2	2	no	—
51	#14556150	C₂³	1	1	yes	characteristic
52	#14556147	—	4	1	no	—
53	C₂×Dic₂₈₄	—	2	2	no	—
54	C₂×Dic₂₈₄	—	2	1	no	—
55	C₂×Dic₂₈₄	C₂³	1	2	yes	—
56	Q₁₆×D₇₁	—	4	1	no	—
57	#14556134	—	2	1	no	—
58	#14556128	—	2	1	no	—
59	#14556074	—	2	2	no	—
60	#14556074	—	2	2	no	—
61	#14556074	—	2	2	no	—
62	#14556074	—	2	2	no	—
63	#14556074	—	4	2	no	—
64	#14556074	D₄	1	1	yes	characteristic
65	#14556072	—	2	2	no	—
66	#14556072	D₄	1	1	yes	characteristic
67	#14556071	—	2	2	no	—
68	#14556071	D₄	1	1	yes	characteristic
69	#14556069	D₄	1	1	yes	characteristic
70	#14556068	—	2	2	no	—
71	#14556068	C₂³	1	2	yes	—
72	C₂×C₇₁⋊C₁₆	—	2	2	no	—
73	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
75	#14556064	C₂³	1	1	yes	characteristic
76	#14556056	—	2	1	no	—
77	#14556046	—	2	2	no	—
78	2₋¹⁺⁴×C₇₁	—	2	2	no	—
79	#137×C₇₁	—	2	2	no	—
80	#137×C₇₁	—	2	2	no	—
81	#137×C₇₁	—	4	2	no	—
82	#137×C₇₁	—	4	2	no	—
83	#135×C₇₁	—	2	2	no	—
84	#135×C₇₁	—	4	1	no	—
85	#135×C₇₁	D₄	1	1	yes	characteristic
86	#135×C₇₁	C₂³	1	1	yes	characteristic
87	C₂×Q₁₆×C₇₁	—	2	2	no	—
88	C₂×Q₁₆×C₇₁	—	2	1	no	—
89	C₂×Q₁₆×C₇₁	—	4	1	no	—
90	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
91	#131×C₇₁	—	2	1	no	—
92	#131×C₇₁	C₂³	1	2	yes	—
93	C₄⋊Q₈×C₇₁	—	2	1	no	—
94	#108×C₇₁	D₄	1	1	yes	characteristic
95	#108×C₇₁	C₂³	1	1	yes	characteristic
96	#104×C₇₁	—	2	1	no	—
97	#104×C₇₁	D₄	1	2	yes	—
98	#101×C₇₁	—	2	2	no	—
99	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
100	#11765865	—	2	1	no	—
101	#11765865	—	4	2	no	—
102	#11765864	—	4	2	no	—
103	#11765864	—	8	2	no	—
104	C₂×Dic₁₄₂	—	2	2	no	—
105	C₂×Dic₁₄₂	—	4	2	no	—
106	Q₈×D₇₁	—	4	2	no	—
107	Dic₂₈₄	—	2	2	no	—
108	Dic₂₈₄	—	2	2	no	—
109	Dic₂₈₄	—	4	2	no	—
110	Dic₂₈₄	—	4	1	no	—
111	Dic₂₈₄	—	4	1	no	—
112	Dic₂₈₄	C₂×D₄	1	1	yes	characteristic
113	#11765855	—	4	2	no	—
114	#11765854	—	2	1	no	—
115	#11765854	—	4	2	no	—
116	#11765853	—	4	2	no	—
117	D₅₆₈	C₂×D₄	1	1	yes	characteristic
118	#11765849	—	2	1	no	—
119	#11765848	—	4	1	no	—
120	C₈×D₇₁	—	4	1	no	—
121	C₄⋊Dic₇₁	—	8	1	no	—
122	C₇₁⋊C₁₆	—	2	2	no	—
123	C₇₁⋊C₁₆	—	2	1	no	—
124	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
125	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
126	#41×C₇₁	—	2	1	no	—
127	#41×C₇₁	—	4	2	no	—
128	#41×C₇₁	—	4	2	no	—
129	#41×C₇₁	—	8	2	no	—
130	#41×C₇₁	C₂×D₄	1	2	yes	—
131	C₂×Q₈×C₇₁	—	2	2	no	—
132	C₂×Q₈×C₇₁	—	4	2	no	—
133	C₂×Q₈×C₇₁	—	4	2	no	—
134	Q₁₆×C₇₁	—	2	2	no	—
135	Q₁₆×C₇₁	—	2	2	no	—
136	Q₁₆×C₇₁	—	4	2	no	—
137	Q₁₆×C₇₁	—	4	2	no	—
138	Q₁₆×C₇₁	—	4	1	no	—
139	Q₁₆×C₇₁	—	4	1	no	—
140	Q₁₆×C₇₁	D₈	1	1	yes	characteristic
141	Q₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
142	SD₁₆×C₇₁	—	2	1	no	—
143	SD₁₆×C₇₁	—	4	2	no	—
144	SD₁₆×C₇₁	—	4	2	no	—
145	SD₁₆×C₇₁	D₈	1	2	yes	—
146	D₈×C₇₁	D₈	1	1	yes	characteristic
147	D₈×C₇₁	C₂×D₄	1	1	yes	characteristic
148	M₁₆×C₇₁	—	2	2	no	—
149	M₁₆×C₇₁	—	2	1	no	—
150	M₁₆×C₇₁	—	4	1	no	—
151	M₁₆×C₇₁	C₂×D₄	1	2	yes	—
152	C₂×C₈×C₇₁	—	2	2	no	—
153	C₂×C₈×C₇₁	—	4	1	no	—
154	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
155	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
156	C₄⋊C₄×C₇₁	—	8	1	no	—
157	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
158	Q₈×C₇₁	—	2	1	no	—
159	Q₈×C₇₁	—	4	2	no	—
160	Q₈×C₇₁	—	4	2	no	—
161	Q₈×C₇₁	—	8	2	no	—
162	Q₈×C₇₁	C₂×D₈	1	2	yes	—
163	D₄×C₇₁	—	2	2	no	—
164	D₄×C₇₁	—	2	1	no	—
165	D₄×C₇₁	—	4	1	no	—
166	D₄×C₇₁	—	8	2	no	—
167	D₄×C₇₁	C₂×D₈	1	2	yes	—
168	C₂×C₄×C₇₁	—	2	1	no	—
169	C₂×C₄×C₇₁	—	2	2	no	—
170	C₂×C₄×C₇₁	—	4	1	no	—
171	C₂×C₄×C₇₁	—	4	2	no	—
172	C₂×C₄×C₇₁	—	8	2	no	—
173	C₂×C₄×C₇₁	—	8	2	no	—
174	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
175	C₂²⋊D₇₁	—	4	1	no	—
176	C₂²⋊D₇₁	—	8	2	no	—
177	C₂×Dic₇₁	—	4	2	no	—
178	C₂×Dic₇₁	—	8	2	no	—
179	D₂₈₄	—	2	1	no	—
180	C₄×D₇₁	—	4	1	no	—
181	C₄×D₇₁	—	8	2	no	—
182	Dic₁₄₂	—	2	1	no	—
183	Dic₁₄₂	—	4	2	no	—
184	Dic₁₄₂	—	4	2	no	—
185	Dic₁₄₂	—	8	2	no	—
186	C₈×C₇₁	—	2	2	no	—
187	C₈×C₇₁	—	4	1	no	—
188	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
189	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
190	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
191	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
192	C₇₁⋊C₈	—	4	1	no	—
193	C₂²×C₇₁	—	2	2	no	—
194	C₂²×C₇₁	—	4	1	no	—
195	C₂²×C₇₁	#576	1	1	yes	characteristic
196	C₂×D₇₁	—	4	1	no	—
197	C₄×C₇₁	—	2	2	no	—
198	C₄×C₇₁	—	2	2	no	—
199	C₄×C₇₁	—	4	1	no	—
200	C₄×C₇₁	—	8	2	no	—
201	C₄×C₇₁	D₄²	1	1	yes	characteristic
202	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
203	Dic₇₁	—	4	1	no	—
204	Dic₇₁	—	8	2	no	—
205	#33860	—	71	1	no	maximal, Sylow
206	C₂×C₇₁	—	2	1	no	—
207	C₂×C₇₁	—	4	2	no	—
208	C₂×C₇₁	—	8	1	no	—
209	C₂×C₇₁	D₄×D₈	1	1	yes	characteristic
210	D₇₁	—	8	1	no	—
211	#3417	—	71	2	no	—
212	#3293	—	71	1	no	—
213	#3293	D₇₁	1	1	yes	characteristic, complemented
214	#2247	—	71	1	no	—
215	#2222	—	71	2	no	—
216	#2214	—	71	1	no	—
217	#2214	—	71	1	no	—
218	#2195	—	71	2	no	—
219	#2195	—	71	2	no	—
220	#2178	—	71	1	no	—
221	#2170	—	71	1	no	—
222	C₇₁	#33860	1	1	yes	characteristic, complemented, Sylow
223	#578	—	2	1	no	—
224	#578	—	71	2	no	—
225	#578	—	142	1	no	—
226	#578	C₂×D₇₁	1	2	yes	—
227	#510	—	71	2	no	—
228	#510	—	71	2	no	—
229	#510	—	142	2	no	—
230	#510	—	142	2	no	—
231	#508	—	71	1	no	—
232	#508	—	142	2	no	—
233	C₂×Q₃₂	—	71	2	no	—
234	C₂×Q₃₂	—	71	2	no	—
235	C₂×Q₃₂	—	142	2	no	—
236	#504	—	71	2	no	—
237	#494	—	71	1	no	—
238	#494	C₂×D₇₁	1	1	yes	characteristic
239	#473	—	2	1	no	—
240	#473	—	71	2	no	—
241	#473	—	142	1	no	—
242	#473	C₂×D₇₁	1	2	yes	—
243	#456	—	2	2	no	—
244	#456	—	142	2	no	—
245	#443	—	71	1	no	—
246	#443	C₂×D₇₁	1	1	yes	characteristic
247	#443	C₂×D₇₁	1	1	yes	characteristic
248	#368	—	71	1	no	—
249	#364	—	71	1	no	—
250	#362	—	71	2	no	—
251	#362	—	71	2	no	—
252	#359	—	71	1	no	—
253	#359	—	71	1	no	—
254	#350	—	71	2	no	—
255	C₄×C₁₆	—	71	1	no	—
256	2₋¹⁺⁴	—	2	2	no	—
257	2₋¹⁺⁴	—	142	2	no	—
258	#137	—	2	2	no	—
259	#137	—	2	2	no	—
260	#137	—	4	2	no	—
261	#137	—	4	2	no	—
262	#137	—	284	2	no	—
263	#137	—	284	2	no	—
264	#137	—	142	2	no	—
265	#137	—	142	2	no	—
266	#135	—	2	2	no	—
267	#135	—	4	1	no	—
268	#135	—	284	1	no	—
269	#135	—	71	1	no	—
270	#135	—	142	2	no	—
271	#135	C₂²⋊D₇₁	1	1	yes	characteristic
272	#135	C₂²×D₇₁	1	1	yes	characteristic
273	C₂×Q₁₆	—	2	2	no	—
274	C₂×Q₁₆	—	2	1	no	—
275	C₂×Q₁₆	—	4	1	no	—
276	C₂×Q₁₆	—	284	1	no	—
277	C₂×Q₁₆	—	71	2	no	—
278	C₂×Q₁₆	—	142	2	no	—
279	C₂×Q₁₆	—	142	1	no	—
280	C₂×Q₁₆	C₂²×D₇₁	1	2	yes	—
281	#131	—	2	1	no	—
282	#131	—	142	1	no	—
283	#131	C₂²×D₇₁	1	2	yes	—
284	C₄⋊Q₈	—	2	1	no	—
285	C₄⋊Q₈	—	142	1	no	—
286	Q₃₂	—	284	2	no	—
287	Q₃₂	—	71	1	no	—
288	Q₃₂	—	142	2	no	—
289	Q₃₂	—	142	2	no	—
290	Q₃₂	—	142	2	no	—
291	Q₃₂	—	142	2	no	—
292	SD₃₂	—	71	1	no	—
293	SD₃₂	—	71	1	no	—
294	SD₃₂	—	142	2	no	—
295	SD₃₂	—	142	2	no	—
296	D₁₆	—	71	1	no	—
297	M₃₂	—	71	2	no	—
298	M₃₂	—	142	2	no	—
299	C₂×C₁₆	—	71	1	no	—
300	C₂×C₁₆	—	71	1	no	—
301	C₂×C₁₆	—	142	2	no	—
302	#108	—	71	1	no	—
303	#108	C₂²⋊D₇₁	1	1	yes	characteristic
304	#108	C₂²×D₇₁	1	1	yes	characteristic
305	#104	—	2	1	no	—
306	#104	—	142	1	no	—
307	#104	C₂²⋊D₇₁	1	2	yes	—
308	#101	—	2	2	no	—
309	#101	—	142	2	no	—
310	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
311	#41	—	2	1	no	—
312	#41	—	4	2	no	—
313	#41	—	4	2	no	—
314	#41	—	8	2	no	—
315	#41	—	284	2	no	—
316	#41	—	284	2	no	—
317	#41	—	568	2	no	—
318	#41	—	142	1	no	—
319	#41	C₂×C₂²⋊D₇₁	1	2	yes	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×Q₈	—	284	2	no	—
324	C₂×Q₈	—	284	2	no	—
325	C₂×Q₈	—	142	2	no	—
326	Q₁₆	—	2	2	no	—
327	Q₁₆	—	2	2	no	—
328	Q₁₆	—	4	2	no	—
329	Q₁₆	—	4	2	no	—
330	Q₁₆	—	4	1	no	—
331	Q₁₆	—	4	1	no	—
332	Q₁₆	—	284	2	no	—
333	Q₁₆	—	284	2	no	—
334	Q₁₆	—	284	1	no	—
335	Q₁₆	—	284	1	no	—
336	Q₁₆	—	71	1	no	—
337	Q₁₆	—	142	2	no	—
338	Q₁₆	—	142	2	no	—
339	Q₁₆	#11765850	1	1	yes	characteristic
340	Q₁₆	D₄×D₇₁	1	1	yes	characteristic
341	SD₁₆	—	2	1	no	—
342	SD₁₆	—	4	2	no	—
343	SD₁₆	—	4	2	no	—
344	SD₁₆	—	284	2	no	—
345	SD₁₆	—	284	2	no	—
346	SD₁₆	—	142	1	no	—
347	SD₁₆	#11765850	1	2	yes	—
348	D₈	—	71	1	no	—
349	D₈	#11765850	1	1	yes	characteristic
350	D₈	D₄×D₇₁	1	1	yes	characteristic
351	M₁₆	—	2	2	no	—
352	M₁₆	—	2	1	no	—
353	M₁₆	—	4	1	no	—
354	M₁₆	—	284	1	no	—
355	M₁₆	—	142	1	no	—
356	M₁₆	C₂×C₂²⋊D₇₁	1	2	yes	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	4	1	no	—
359	C₂×C₈	—	284	1	no	—
360	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
361	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
362	C₄⋊C₄	—	8	1	no	—
363	C₄⋊C₄	—	568	1	no	—
364	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
365	C₁₆	—	71	1	no	—
366	C₁₆	—	71	1	no	—
367	C₁₆	—	142	2	no	—
368	C₁₆	—	142	1	no	—
369	Q₈	—	2	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	8	2	no	—
373	Q₈	—	284	2	no	—
374	Q₈	—	284	2	no	—
375	Q₈	—	568	2	no	—
376	Q₈	—	142	1	no	—
377	Q₈	C₂×#11765850	1	2	yes	—
378	D₄	—	2	2	no	—
379	D₄	—	2	1	no	—
380	D₄	—	4	1	no	—
381	D₄	—	8	2	no	—
382	D₄	—	284	1	no	—
383	D₄	—	568	2	no	—
384	D₄	—	142	1	no	—
385	D₄	C₂×#11765850	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	8	2	no	—
391	C₂×C₄	—	8	2	no	—
392	C₂×C₄	—	284	1	no	—
393	C₂×C₄	—	284	2	no	—
394	C₂×C₄	—	568	2	no	—
395	C₂×C₄	—	568	2	no	—
396	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	4	1	no	—
399	C₈	—	284	1	no	—
400	C₈	C₂×#11765850	1	1	yes	characteristic
401	C₈	C₂×#11765850	1	1	yes	characteristic
402	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
403	C₈	#14556173	1	1	yes	characteristic
404	C₂²	—	2	2	no	—
405	C₂²	—	4	1	no	—
406	C₂²	—	284	1	no	—
407	C₂²	#17986802	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	4	1	no	—
411	C₄	—	8	2	no	—
412	C₄	—	284	1	no	—
413	C₄	—	568	2	no	—
414	C₄	D₄×C₂²⋊D₇₁	1	1	yes	characteristic
415	C₄	C₂²×#11765850	1	1	yes	characteristic
416	C₂	—	2	1	no	—
417	C₂	—	4	2	no	—
418	C₂	—	8	1	no	—
419	C₂	—	568	1	no	—
420	C₂	D₄×#11765850	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11765850
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3293	C₂
socle	C₂×C₇₁	D₄×D₈
hypercenter	#3293	D₇₁
hypocenter	C₇₁	#33860
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #3293

Complements

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

N	H ≅ G/N	type
1. #22601491	C₂	semidirect
2. #22600145	C₂	semidirect
3. #22590592	C₂	semidirect
4. #22590463	C₂	semidirect
5. #22590421	C₂	semidirect
6. #22590418	C₂	semidirect
7. #22590318	C₂	semidirect
8. #22590316	C₂	semidirect
9. #22590219	C₂	semidirect
10. #22590177	C₂	semidirect
11. C₇₁×#3293	C₂	semidirect
213. #3293	D₇₁	semidirect
222. C₇₁	#33860	semidirect

Maximal subgroups

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

subgroup	index
#22601491	2
#22600145	2
#22590592	2
#22590463	2
#22590421	2
#22590418	2
#22590318	2
#22590316	2
#22590219	2
#22590177	2
C₇₁×#3293	2
#33860	71