Subgroups — #32154250

Order 18176.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22601478	C₂	1	1	yes	maximal, characteristic, complemented
2	#22600126	C₂	1	2	yes	maximal, complemented
3	#22590602	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590462	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590460	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590411	C₂	1	2	yes	maximal, complemented
7	#22590308	C₂	1	2	yes	maximal, complemented
8	#22590304	C₂	1	2	yes	maximal, complemented
9	#22590221	C₂	1	1	yes	maximal, characteristic, complemented
10	#22590175	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#3417	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986807	—	2	2	no	—
13	#17986801	C₂²	1	2	yes	—
14	#17986436	—	2	2	no	—
15	#17986434	C₂²	1	2	yes	—
16	#17986432	—	2	1	no	—
17	C₂×#14556071	C₂²	1	2	yes	complemented
18	C₂×D₁₁₃₆	C₂²	1	1	yes	characteristic
19	D₁₆×D₇₁	—	2	1	no	—
20	#17986415	C₂²	1	1	yes	characteristic
21	#17986384	C₂²	1	2	yes	complemented
22	#17986271	C₂²	1	2	yes	—
23	#17986269	C₂²	1	2	yes	complemented
24	#17986264	—	2	2	no	—
25	#17986175	—	2	2	no	—
26	#17986172	—	2	2	no	—
27	#17986116	C₂²	1	2	yes	complemented
28	#17986114	C₂²	1	1	yes	characteristic
29	#17985829	C₂²	1	1	yes	characteristic, complemented
30	#17985819	C₂²	1	1	yes	characteristic, complemented
31	#17985815	C₂²	1	2	yes	complemented
32	#17985813	C₂²	1	2	yes	complemented
33	#17985806	C₂²	1	2	yes	complemented
34	#17985784	C₂²	1	1	yes	characteristic
35	#17985783	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic, complemented
37	#578×C₇₁	C₂²	1	2	yes	complemented
38	#510×C₇₁	—	2	2	no	—
39	#510×C₇₁	C₂²	1	2	yes	complemented
40	#508×C₇₁	—	2	1	no	—
41	#508×C₇₁	C₂²	1	1	yes	characteristic
42	C₂×Q₃₂×C₇₁	—	2	1	no	—
43	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic, complemented
44	#504×C₇₁	C₂²	1	1	yes	characteristic, complemented
45	#14556174	—	2	2	no	—
46	#14556159	—	2	2	no	—
47	#14556159	—	4	2	no	—
48	#14556155	—	4	2	no	—
49	#14556153	—	2	2	no	—
50	#14556153	—	2	2	no	—
51	#14556153	—	4	2	no	—
52	#14556151	—	2	2	no	—
53	#14556148	—	4	2	no	—
54	#14556148	D₄	1	1	yes	characteristic
55	C₂×#11765852	—	2	2	no	—
56	C₂×#11765852	—	4	2	no	—
57	C₂×D₅₆₈	C₂³	1	2	yes	—
58	D₈×D₇₁	—	2	2	no	—
59	#14556135	—	2	2	no	—
60	#14556134	C₂³	1	1	yes	characteristic
61	#14556117	—	2	2	no	—
62	#14556071	—	2	2	no	—
63	#14556071	—	4	2	no	—
64	D₁₁₃₆	—	2	1	no	—
65	D₁₁₃₆	—	2	1	no	—
66	#14556069	—	2	2	no	—
67	#14556068	C₂³	1	1	yes	characteristic
68	#14556067	—	2	1	no	—
69	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₁₆×D₇₁	—	2	1	no	—
71	#14556064	C₂³	1	2	yes	—
72	#14556063	D₄	1	1	yes	characteristic
73	#14556055	—	2	2	no	—
74	#14556055	D₄	1	1	yes	characteristic
75	#14556054	D₄	1	1	yes	characteristic
76	#14556047	—	2	2	no	—
77	#14556044	—	2	2	no	—
78	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
79	2₋¹⁺⁴×C₇₁	—	2	2	no	—
80	#137×C₇₁	—	2	2	no	—
81	#137×C₇₁	—	4	2	no	—
82	#135×C₇₁	—	2	2	no	—
83	#135×C₇₁	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₇₁	—	2	2	no	—
85	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
86	#131×C₇₁	C₂³	1	1	yes	characteristic
87	Q₃₂×C₇₁	—	2	2	no	—
88	Q₃₂×C₇₁	—	2	1	no	—
89	Q₃₂×C₇₁	—	2	1	no	—
90	Q₃₂×C₇₁	D₄	1	1	yes	characteristic
91	SD₃₂×C₇₁	—	2	2	no	—
92	SD₃₂×C₇₁	D₄	1	2	yes	—
93	D₁₆×C₇₁	D₄	1	1	yes	characteristic
94	M₃₂×C₇₁	—	2	1	no	—
95	M₃₂×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₇₁	—	2	1	no	—
97	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
98	#11765866	—	4	2	no	—
99	#11765865	—	4	2	no	—
100	#11765865	C₂×D₄	1	1	yes	characteristic
101	C₂×D₂₈₄	—	2	2	no	—
102	C₂×D₂₈₄	—	4	2	no	—
103	D₄×D₇₁	—	4	2	no	—
104	D₄×D₇₁	—	8	2	no	—
105	#11765855	—	4	2	no	—
106	#11765855	D₈	1	1	yes	characteristic
107	#11765853	D₈	1	1	yes	characteristic
108	#11765852	—	4	2	no	—
109	#11765852	—	4	2	no	—
110	#11765852	—	4	2	no	—
111	#11765852	—	4	2	no	—
112	#11765852	—	4	2	no	—
113	#11765852	D₈	1	1	yes	characteristic
114	D₅₆₈	—	2	2	no	—
115	D₅₆₈	—	2	2	no	—
116	#11765850	—	4	2	no	—
117	#11765850	—	4	2	no	—
118	#11765850	D₈	1	1	yes	characteristic
119	#11765849	—	2	2	no	—
120	#11765849	—	4	2	no	—
121	#11765849	C₂×D₄	1	1	yes	characteristic
122	#11765848	—	2	1	no	—
123	C₂×C₇₁⋊C₈	—	4	2	no	—
124	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₈×D₇₁	—	2	1	no	—
126	(C₂×D₇₁)⋊C₄	—	8	2	no	—
127	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
128	C₇₁⋊C₁₆	—	2	1	no	—
129	C₇₁⋊C₁₆	—	2	1	no	—
130	#41×C₇₁	—	2	2	no	—
131	#41×C₇₁	—	4	2	no	—
132	#41×C₇₁	—	4	2	no	—
133	#41×C₇₁	—	8	2	no	—
134	#41×C₇₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₇₁	—	2	2	no	—
136	C₂×Q₈×C₇₁	—	4	2	no	—
137	C₂×Q₈×C₇₁	—	4	2	no	—
138	Q₁₆×C₇₁	—	2	2	no	—
139	Q₁₆×C₇₁	—	2	2	no	—
140	Q₁₆×C₇₁	—	4	2	no	—
141	Q₁₆×C₇₁	C₂×D₄	1	2	yes	—
142	SD₁₆×C₇₁	—	2	2	no	—
143	SD₁₆×C₇₁	—	4	2	no	—
144	D₈×C₇₁	C₂×D₄	1	2	yes	—
145	M₁₆×C₇₁	—	2	1	no	—
146	M₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₇₁	—	2	1	no	—
148	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
149	C₁₆×C₇₁	—	2	1	no	—
150	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
151	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
152	C₂²×D₇₁	—	4	2	no	—
153	C₂²×D₇₁	—	8	2	no	—
154	Q₈×C₇₁	—	2	2	no	—
155	Q₈×C₇₁	—	4	2	no	—
156	Q₈×C₇₁	—	4	2	no	—
157	Q₈×C₇₁	—	8	2	no	—
158	Q₈×C₇₁	C₂×D₈	1	1	yes	characteristic
159	D₄×C₇₁	—	2	1	no	—
160	D₄×C₇₁	—	2	2	no	—
161	D₄×C₇₁	—	4	2	no	—
162	D₄×C₇₁	—	8	2	no	—
163	D₄×C₇₁	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₇₁	—	2	1	no	—
165	C₂×C₄×C₇₁	—	4	2	no	—
166	C₂×C₄×C₇₁	—	4	2	no	—
167	C₂×C₄×C₇₁	—	8	2	no	—
168	C₂×C₄×C₇₁	—	8	2	no	—
169	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
170	C₂²⋊D₇₁	—	2	1	no	—
171	C₂²⋊D₇₁	—	8	2	no	—
172	C₂×Dic₇₁	—	2	1	no	—
173	D₂₈₄	—	4	2	no	—
174	D₂₈₄	—	4	2	no	—
175	D₂₈₄	—	8	2	no	—
176	D₂₈₄	C₂×D₈	1	1	yes	characteristic
177	C₄×D₇₁	—	2	1	no	—
178	C₄×D₇₁	—	8	2	no	—
179	Dic₁₄₂	C₂×D₈	1	1	yes	characteristic
180	C₈×C₇₁	—	2	1	no	—
181	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
183	C₇₁⋊C₈	—	2	1	no	—
184	C₇₁⋊C₈	—	4	2	no	—
185	C₇₁⋊C₈	C₂×D₈	1	1	yes	characteristic
186	C₇₁⋊C₈	C₂×D₈	1	1	yes	characteristic
187	C₂²×C₇₁	—	2	1	no	—
188	C₂²×C₇₁	—	4	2	no	—
189	C₂²×C₇₁	#576	1	1	yes	characteristic
190	C₂×D₇₁	—	2	1	no	—
191	C₂×D₇₁	—	8	2	no	—
192	C₂×D₇₁	—	16	2	no	—
193	C₂×D₇₁	—	32	2	no	—
194	C₄×C₇₁	—	2	1	no	—
195	C₄×C₇₁	—	4	2	no	—
196	C₄×C₇₁	—	8	2	no	—
197	C₄×C₇₁	D₄²	1	1	yes	characteristic
198	C₄×C₇₁	C₂²×D₈	1	1	yes	characteristic
199	Dic₇₁	—	2	1	no	—
200	Dic₇₁	—	4	1	no	—
201	#33859	—	71	1	no	maximal, Sylow
202	C₂×C₇₁	—	2	1	no	—
203	C₂×C₇₁	—	4	1	no	—
204	C₂×C₇₁	—	8	2	no	—
205	C₂×C₇₁	D₄×D₈	1	1	yes	characteristic
206	D₇₁	—	4	1	no	—
207	D₇₁	—	16	2	no	—
208	#3417	D₇₁	1	1	yes	characteristic, complemented
209	#3415	—	71	1	no	—
210	#3290	—	71	2	no	—
211	#2249	—	71	1	no	—
212	#2222	—	71	1	no	—
213	#2221	—	71	1	no	—
214	#2212	—	71	2	no	—
215	#2194	—	71	2	no	—
216	#2193	—	71	2	no	—
217	#2178	—	71	1	no	—
218	#2170	—	71	1	no	—
219	C₇₁	#33859	1	1	yes	characteristic, complemented, Sylow
220	#578	C₂×D₇₁	1	2	yes	complemented
221	#577	—	142	2	no	—
222	#576	—	71	2	no	—
223	#510	—	2	2	no	—
224	#510	C₂×D₇₁	1	2	yes	complemented
225	#509	—	71	2	no	—
226	#509	—	142	2	no	—
227	#508	—	2	1	no	—
228	#508	—	142	1	no	—
229	#508	C₂×D₇₁	1	1	yes	characteristic
230	C₂×Q₃₂	—	2	1	no	—
231	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic, complemented
232	C₂×SD₃₂	—	71	2	no	—
233	C₂×D₁₆	—	71	1	no	—
234	C₂×D₁₆	—	142	1	no	—
235	#504	—	71	1	no	—
236	#504	C₂×D₇₁	1	1	yes	characteristic, complemented
237	#495	—	71	2	no	—
238	#473	—	71	2	no	—
239	#472	—	71	2	no	—
240	#471	—	142	2	no	—
241	#455	—	142	2	no	—
242	#454	—	142	2	no	—
243	#443	—	71	1	no	—
244	#443	—	71	2	no	—
245	#368	—	71	1	no	—
246	#364	—	71	1	no	—
247	#362	—	71	2	no	—
248	#361	—	71	2	no	—
249	#359	—	71	2	no	—
250	#350	—	71	1	no	—
251	#350	—	71	1	no	—
252	C₄×C₁₆	—	71	1	no	—
253	2₋¹⁺⁴	—	2	2	no	—
254	H(5,𝔽₂)	—	142	2	no	—
255	#137	—	2	2	no	—
256	#137	—	4	2	no	—
257	#137	—	284	2	no	—
258	#137	—	142	2	no	—
259	AGL(1,ℤ/8)	—	284	2	no	—
260	AGL(1,ℤ/8)	—	284	2	no	—
261	AGL(1,ℤ/8)	—	142	2	no	—
262	AGL(1,ℤ/8)	—	142	2	no	—
263	#135	—	2	2	no	—
264	#135	—	284	2	no	—
265	#135	—	71	1	no	—
266	#135	—	142	2	no	—
267	#135	C₂²×D₇₁	1	2	yes	—
268	C₂×Q₁₆	—	2	2	no	—
269	C₂×Q₁₆	C₂²×D₇₁	1	2	yes	—
270	C₂×SD₁₆	—	284	2	no	—
271	C₂×SD₁₆	—	142	2	no	—
272	C₂×D₈	—	71	2	no	—
273	C₂×D₈	—	142	2	no	—
274	#131	—	71	1	no	—
275	#131	—	142	2	no	—
276	#131	C₂²×D₇₁	1	1	yes	characteristic
277	#124	—	142	2	no	—
278	Q₃₂	—	2	2	no	—
279	Q₃₂	—	2	1	no	—
280	Q₃₂	—	2	1	no	—
281	Q₃₂	C₂²⋊D₇₁	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	284	2	no	—
284	SD₃₂	—	142	2	no	—
285	SD₃₂	C₂²⋊D₇₁	1	2	yes	—
286	D₁₆	—	142	2	no	—
287	D₁₆	—	142	1	no	—
288	D₁₆	—	142	1	no	—
289	D₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	71	1	no	—
292	M₃₂	—	142	1	no	—
293	M₃₂	C₂²×D₇₁	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	71	1	no	—
296	C₂×C₁₆	—	142	1	no	—
297	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
298	#108	—	71	1	no	—
299	#108	—	71	2	no	—
300	#104	—	71	1	no	—
301	#104	—	71	1	no	—
302	#104	—	142	2	no	—
303	#101	—	142	2	no	—
304	#100	—	142	2	no	—
305	C₄×C₈	—	71	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	8	2	no	—
310	#41	—	284	2	no	—
311	#41	—	284	2	no	—
312	#41	—	71	1	no	—
313	#41	D₄×D₇₁	1	1	yes	characteristic
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	4	2	no	—
316	C₂×Q₈	—	4	2	no	—
317	C₂×D₄	—	284	2	no	—
318	C₂×D₄	—	284	2	no	—
319	C₂×D₄	—	568	2	no	—
320	C₂×D₄	—	142	2	no	—
321	Q₁₆	—	2	2	no	—
322	Q₁₆	—	2	2	no	—
323	Q₁₆	—	4	2	no	—
324	Q₁₆	—	284	2	no	—
325	Q₁₆	—	71	1	no	—
326	Q₁₆	C₂×C₂²⋊D₇₁	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	284	2	no	—
330	SD₁₆	—	284	2	no	—
331	SD₁₆	—	284	2	no	—
332	SD₁₆	—	284	2	no	—
333	SD₁₆	—	284	2	no	—
334	SD₁₆	—	71	1	no	—
335	SD₁₆	—	71	1	no	—
336	D₈	—	284	2	no	—
337	D₈	—	284	2	no	—
338	D₈	—	71	1	no	—
339	D₈	—	142	2	no	—
340	D₈	—	142	2	no	—
341	D₈	C₂×C₂²⋊D₇₁	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	284	2	no	—
344	M₁₆	—	71	1	no	—
345	M₁₆	—	142	2	no	—
346	M₁₆	—	142	1	no	—
347	M₁₆	D₄×D₇₁	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	284	2	no	—
350	C₂×C₈	—	71	1	no	—
351	C₂×C₈	—	142	1	no	—
352	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
353	C₂²⋊C₄	—	568	2	no	—
354	C₄²	—	71	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	142	1	no	—
357	C₁₆	—	142	1	no	—
358	C₁₆	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
360	C₂³	—	284	2	no	—
361	C₂³	—	568	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	8	2	no	—
366	Q₈	—	71	1	no	—
367	Q₈	D₈×D₇₁	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	8	2	no	—
372	D₄	—	284	2	no	—
373	D₄	—	284	2	no	—
374	D₄	—	568	2	no	—
375	D₄	—	568	2	no	—
376	D₄	—	71	1	no	—
377	D₄	—	142	1	no	—
378	D₄	D₈×D₇₁	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	8	2	no	—
383	C₂×C₄	—	8	2	no	—
384	C₂×C₄	—	568	2	no	—
385	C₂×C₄	—	142	1	no	—
386	C₂×C₄	—	142	1	no	—
387	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	284	2	no	—
390	C₈	—	71	1	no	—
391	C₈	—	71	1	no	—
392	C₈	—	142	1	no	—
393	C₈	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
394	C₈	#14556173	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	568	2	no	—
398	C₂²	—	1136	2	no	—
399	C₂²	—	142	1	no	—
400	C₂²	—	2272	2	no	—
401	C₂²	#17986800	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	2	no	—
405	C₄	—	284	1	no	—
406	C₄	—	142	1	no	—
407	C₄	D₄×C₂²⋊D₇₁	1	1	yes	characteristic
408	C₄	C₂×D₈×D₇₁	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	284	1	no	—
413	C₂	—	1136	2	no	—
414	C₂	D₈×C₂²⋊D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₇₁
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#3417	C₂
socle	C₂×C₇₁	D₄×D₈
hypercenter	#3417	D₇₁
hypocenter	C₇₁	#33859
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3417

Complements

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

N	H ≅ G/N	type
1. #22601478	C₂	semidirect
2. #22600126	C₂	semidirect
3. #22590602	C₂	semidirect
4. #22590462	C₂	semidirect
5. #22590460	C₂	semidirect
6. #22590411	C₂	semidirect
7. #22590308	C₂	semidirect
8. #22590304	C₂	semidirect
9. #22590221	C₂	semidirect
10. #22590175	C₂	semidirect
11. C₇₁×#3417	C₂	semidirect
17. C₂×#14556071	C₂²	semidirect
21. #17986384	C₂²	semidirect
23. #17986269	C₂²	semidirect
27. #17986116	C₂²	semidirect
29. #17985829	C₂²	semidirect
30. #17985819	C₂²	semidirect
31. #17985815	C₂²	semidirect
32. #17985813	C₂²	semidirect
33. #17985806	C₂²	semidirect
35. #17985783	C₂²	semidirect
36. C₁₆×Dic₇₁	C₂²	semidirect
37. #578×C₇₁	C₂²	semidirect
39. #510×C₇₁	C₂²	semidirect
43. C₂×Q₃₂×C₇₁	C₂²	semidirect
44. #504×C₇₁	C₂²	semidirect
208. #3417	D₇₁	semidirect
219. C₇₁	#33859	semidirect
220. #578	C₂×D₇₁	semidirect
224. #510	C₂×D₇₁	semidirect
231. C₂×Q₃₂	C₂×D₇₁	semidirect
236. #504	C₂×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22601478	2
#22600126	2
#22590602	2
#22590462	2
#22590460	2
#22590411	2
#22590308	2
#22590304	2
#22590221	2
#22590175	2
C₇₁×#3417	2
#33859	71