Subgroups — #32042577

Order 18176.

656 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986269	C₂	1	1	yes	maximal, characteristic
2	#22597193	C₂	1	1	yes	maximal, characteristic, complemented
3	#22589803	C₂	1	2	yes	maximal, complemented
4	#22589613	C₂	1	2	yes	maximal, complemented
5	#22589250	C₂	1	1	yes	maximal, characteristic, complemented
6	#22589130	C₂	1	1	yes	maximal, characteristic, complemented
7	#22588830	C₂	1	2	yes	maximal, complemented
8	#22588464	C₂	1	1	yes	maximal, characteristic, complemented
9	#22588450	C₂	1	2	yes	maximal, complemented
10	#22588394	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556153	C₂²	1	2	yes	—
13	C₂²×D₅₆₈	C₂²	1	1	yes	characteristic
14	#17986473	C₂²	1	2	yes	—
15	#17986269	—	2	2	no	—
16	#17986269	—	2	2	no	—
17	#17986112	—	2	2	no	—
18	C₄×D₅₆₈	—	2	2	no	—
19	#17986047	C₂²	1	2	yes	—
20	C₂×#14556064	C₂²	1	1	yes	characteristic
21	C₂×#14556064	C₂²	1	1	yes	characteristic
22	#17986037	C₂²	1	1	yes	characteristic
23	#17986016	C₂²	1	2	yes	—
24	#17986005	C₂²	1	2	yes	—
25	#17986003	C₂²	1	2	yes	—
26	C₂×#14556044	C₂²	1	2	yes	—
27	C₂×#14556044	C₂²	1	2	yes	—
28	#17985771	C₂²	1	2	yes	—
29	#17985770	C₂²	1	2	yes	—
30	#17985769	C₂²	1	2	yes	—
31	#17985755	C₂²	1	2	yes	—
32	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
33	#413×C₇₁	C₂²	1	1	yes	characteristic
34	#410×C₇₁	C₂²	1	1	yes	characteristic
35	#405×C₇₁	C₂²	1	1	yes	characteristic
36	#405×C₇₁	C₂²	1	1	yes	characteristic
37	#344×C₇₁	C₂²	1	1	yes	characteristic
38	#344×C₇₁	C₂²	1	1	yes	characteristic
39	C₂²×D₂₈₄	C₂³	1	2	yes	—
40	#14556153	—	2	4	no	—
41	#14556153	—	4	2	no	—
42	C₂×#11765852	—	2	2	no	—
43	C₂×D₅₆₈	—	2	1	no	—
44	C₂×D₅₆₈	—	2	1	no	—
45	C₂×D₅₆₈	—	2	1	no	—
46	C₂×D₅₆₈	—	2	1	no	—
47	C₂×D₅₆₈	D₄	1	2	yes	—
48	C₂×D₅₆₈	D₄	1	2	yes	—
49	C₂×#11765850	—	2	2	no	—
50	C₂×#11765849	C₂³	1	2	yes	—
51	C₂×#11765849	C₂³	1	2	yes	—
52	C₄×D₂₈₄	—	2	4	no	—
53	#14556090	—	4	2	no	—
54	#14556087	C₂³	1	2	yes	—
55	C₂²⋊C₄×D₇₁	—	2	4	no	—
56	#14556064	—	2	1	no	—
57	#14556064	—	2	1	no	—
58	#14556064	—	2	1	no	—
59	#14556064	—	2	1	no	—
60	#14556063	—	4	2	no	—
61	#14556062	—	2	1	no	—
62	#14556060	—	2	1	no	—
63	#14556055	—	4	2	no	—
64	#14556050	—	2	4	no	—
65	#14556048	—	4	2	no	—
66	#14556044	—	2	4	no	—
67	#14556044	—	4	2	no	—
68	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
69	#131×C₇₁	—	2	1	no	—
70	#131×C₇₁	—	2	1	no	—
71	#131×C₇₁	—	2	1	no	—
72	#131×C₇₁	—	2	1	no	—
73	C₂×M₁₆×C₇₁	—	2	1	no	—
74	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₇₁	—	2	1	no	—
78	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
79	#117×C₇₁	C₂³	1	1	yes	characteristic
80	#117×C₇₁	C₂³	1	1	yes	characteristic
81	#101×C₇₁	—	2	1	no	—
82	#100×C₇₁	—	2	1	no	—
83	#99×C₇₁	—	2	2	no	—
84	#97×C₇₁	—	2	1	no	—
85	#97×C₇₁	—	2	1	no	—
86	C₄×C₈×C₇₁	—	2	1	no	—
87	C₄×C₈×C₇₁	—	2	1	no	—
88	C₂³×D₇₁	—	2	2	no	—
89	C₂×D₂₈₄	—	2	2	no	—
90	C₂×D₂₈₄	—	2	2	no	—
91	C₂×D₂₈₄	—	2	2	no	—
92	C₂×D₂₈₄	—	2	2	no	—
93	C₂×D₂₈₄	—	2	2	no	—
94	C₂×D₂₈₄	C₂×D₄	1	4	yes	—
95	C₂×C₄×D₇₁	—	4	4	no	—
96	#11765852	—	4	4	no	—
97	D₅₆₈	—	2	2	no	—
98	D₅₆₈	—	2	2	no	—
99	D₅₆₈	—	2	2	no	—
100	D₅₆₈	—	2	2	no	—
101	#11765850	—	4	4	no	—
102	#11765849	—	2	2	no	—
103	#11765849	—	2	2	no	—
104	#11765849	—	2	2	no	—
105	#11765849	—	2	2	no	—
106	C₂×C₇₁⋊C₈	—	2	2	no	—
107	C₂×C₇₁⋊C₈	—	2	2	no	—
108	C₄⋊Dic₇₁	—	2	2	no	—
109	C₂²⋊Dic₇₁	—	2	2	no	—
110	(C₂×D₇₁)⋊C₄	—	4	4	no	—
111	C₄×Dic₇₁	—	2	2	no	—
112	#41×C₇₁	—	2	1	no	—
113	#41×C₇₁	—	2	1	no	—
114	#41×C₇₁	—	2	1	no	—
115	#41×C₇₁	—	2	1	no	—
116	C₂×Q₈×C₇₁	#41	1	1	yes	characteristic
117	C₂×D₄×C₇₁	—	2	1	no	—
118	C₂×D₄×C₇₁	#41	1	1	yes	characteristic
119	C₂²×C₄×C₇₁	—	2	1	no	—
120	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
121	M₁₆×C₇₁	—	2	1	no	—
122	M₁₆×C₇₁	—	2	1	no	—
123	M₁₆×C₇₁	—	2	1	no	—
124	M₁₆×C₇₁	—	2	1	no	—
125	M₁₆×C₇₁	—	2	1	no	—
126	M₁₆×C₇₁	—	2	1	no	—
127	M₁₆×C₇₁	—	4	1	no	—
128	M₁₆×C₇₁	—	4	1	no	—
129	C₂×C₈×C₇₁	—	2	1	no	—
130	C₂×C₈×C₇₁	—	2	1	no	—
131	C₂×C₈×C₇₁	—	2	1	no	—
132	C₂×C₈×C₇₁	—	4	1	no	—
133	C₂×C₈×C₇₁	—	4	1	no	—
134	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
137	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
138	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
139	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
140	C₄⋊C₄×C₇₁	—	2	1	no	—
141	C₄⋊C₄×C₇₁	—	2	1	no	—
142	C₂²⋊C₄×C₇₁	—	2	1	no	—
143	C₂²⋊C₄×C₇₁	#41	1	2	yes	—
144	C₄²×C₇₁	—	2	1	no	—
145	C₄²×C₇₁	—	2	1	no	—
146	C₂³×C₇₁	—	2	1	no	—
147	C₂³×C₇₁	C₂×#41	1	1	yes	characteristic
148	C₂²×D₇₁	—	2	4	no	—
149	C₂²×D₇₁	—	4	2	no	—
150	C₂²×D₇₁	—	4	2	no	—
151	C₂²×D₇₁	—	16	2	no	—
152	Q₈×C₇₁	—	2	1	no	—
153	Q₈×C₇₁	—	2	1	no	—
154	D₄×C₇₁	—	2	1	no	—
155	D₄×C₇₁	—	2	1	no	—
156	D₄×C₇₁	—	4	1	no	—
157	D₄×C₇₁	—	4	1	no	—
158	C₂×C₄×C₇₁	—	2	1	no	—
159	C₂×C₄×C₇₁	—	2	2	no	—
160	C₂×C₄×C₇₁	—	4	1	no	—
161	C₂×C₄×C₇₁	—	4	1	no	—
162	C₂×C₄×C₇₁	—	4	1	no	—
163	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
166	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×Dic₇₁	—	4	2	no	—
170	D₂₈₄	—	2	4	no	—
171	D₂₈₄	—	2	4	no	—
172	D₂₈₄	—	4	2	no	—
173	D₂₈₄	—	4	2	no	—
174	C₄×D₇₁	—	16	4	no	—
175	C₈×C₇₁	—	4	1	no	—
176	C₈×C₇₁	—	4	1	no	—
177	C₈×C₇₁	—	4	1	no	—
178	C₈×C₇₁	#123	1	1	yes	characteristic
179	C₈×C₇₁	#123	1	1	yes	characteristic
180	C₈×C₇₁	#123	1	1	yes	characteristic
181	C₈×C₇₁	#123	1	1	yes	characteristic
182	C₇₁⋊C₈	—	4	2	no	—
183	C₇₁⋊C₈	—	4	2	no	—
184	C₂²×C₇₁	—	4	1	no	—
185	C₂²×C₇₁	—	4	1	no	—
186	C₂²×C₇₁	—	8	1	no	—
187	C₂²×C₇₁	#529	1	1	yes	characteristic
188	C₂²×C₇₁	#546	1	1	yes	characteristic
189	C₂²×C₇₁	#555	1	1	yes	characteristic
190	C₂×D₇₁	—	4	4	no	—
191	C₂×D₇₁	—	8	4	no	—
192	C₂×D₇₁	—	16	2	no	—
193	C₂×D₇₁	—	16	2	no	—
194	C₄×C₇₁	—	4	2	no	—
195	C₄×C₇₁	—	4	1	no	—
196	C₄×C₇₁	—	8	1	no	—
197	C₄×C₇₁	C₂×#123	1	1	yes	characteristic
198	C₄×C₇₁	#540	1	1	yes	characteristic
199	C₄×C₇₁	#546	1	1	yes	characteristic
200	C₄×C₇₁	#553	1	1	yes	characteristic
201	Dic₇₁	—	8	2	no	—
202	#24988	—	71	1	no	maximal, Sylow
203	C₂×C₇₁	—	2	1	no	—
204	C₂×C₇₁	—	2	1	no	—
205	C₂×C₇₁	—	2	1	no	—
206	C₂×C₇₁	—	8	1	no	—
207	C₂×C₇₁	#2420	1	1	yes	characteristic
208	D₇₁	—	8	4	no	—
209	C₂×#472	—	71	1	no	—
210	#2948	—	71	1	no	—
211	#2081	—	71	2	no	—
212	#2048	—	71	2	no	—
213	#1978	—	71	1	no	—
214	#1953	—	71	1	no	—
215	#1904	—	71	2	no	—
216	#1833	—	71	1	no	—
217	#1829	—	71	2	no	—
218	#1814	—	71	1	no	—
219	#1757	D₇₁	1	1	yes	characteristic, complemented
220	C₇₁	#24988	1	1	yes	characteristic, complemented, Sylow
221	C₂×AGL(1,ℤ/8)	—	71	2	no	—
222	C₂²×D₈	—	71	1	no	—
223	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
224	#518	—	71	2	no	—
225	#472	—	142	2	no	—
226	#472	—	142	2	no	—
227	#442	—	142	2	no	—
228	C₄×D₈	—	142	2	no	—
229	#430	—	71	2	no	—
230	C₂×#108	—	71	1	no	—
231	C₂×#108	—	71	1	no	—
232	#427	—	71	1	no	—
233	#421	—	71	2	no	—
234	#418	—	71	2	no	—
235	#418	—	71	2	no	—
236	#413	C₂×D₇₁	1	1	yes	characteristic
237	C₂×#100	—	71	2	no	—
238	C₂×#100	—	71	2	no	—
239	#410	C₂×D₇₁	1	1	yes	characteristic
240	#405	C₂×D₇₁	1	1	yes	characteristic
241	#405	C₂×D₇₁	1	1	yes	characteristic
242	#344	—	71	2	no	—
243	#344	—	71	2	no	—
244	#344	—	71	2	no	—
245	#344	C₂×D₇₁	1	1	yes	characteristic
246	#344	C₂×D₇₁	1	1	yes	characteristic
247	H(3,ℤ/4)	—	71	2	no	—
248	C₂×#41	C₂²×D₇₁	1	1	yes	characteristic
249	C₂²×D₄	—	71	2	no	—
250	AGL(1,ℤ/8)	—	284	2	no	—
251	AGL(1,ℤ/8)	—	142	4	no	—
252	C₂×SD₁₆	—	142	2	no	—
253	C₂×D₈	—	71	2	no	—
254	C₂×D₈	—	71	2	no	—
255	C₂×D₈	—	142	2	no	—
256	C₂×D₈	—	142	1	no	—
257	C₂×D₈	—	142	1	no	—
258	C₂×D₈	—	142	1	no	—
259	C₂×D₈	—	142	1	no	—
260	#131	—	2	1	no	—
261	#131	—	2	1	no	—
262	#131	—	2	1	no	—
263	#131	—	2	1	no	—
264	C₂×M₁₆	—	2	1	no	—
265	C₂×M₁₆	—	71	2	no	—
266	C₂×M₁₆	—	71	2	no	—
267	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
268	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
269	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
270	C₂²×C₈	—	2	1	no	—
271	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
272	C₄×D₄	—	284	2	no	—
273	C₄×D₄	—	142	4	no	—
274	#117	—	71	2	no	—
275	#117	C₂²×D₇₁	1	1	yes	characteristic
276	#117	C₂²×D₇₁	1	1	yes	characteristic
277	C₂×C₂²⋊C₄	—	142	4	no	—
278	#108	—	284	2	no	—
279	#108	—	142	1	no	—
280	#108	—	142	1	no	—
281	#108	—	142	1	no	—
282	#108	—	142	1	no	—
283	#107	—	142	1	no	—
284	#106	—	142	1	no	—
285	#104	—	284	2	no	—
286	#102	—	284	2	no	—
287	#102	—	142	4	no	—
288	#101	—	2	1	no	—
289	#100	—	2	1	no	—
290	#100	—	284	2	no	—
291	#100	—	142	4	no	—
292	#99	—	2	2	no	—
293	#97	—	2	1	no	—
294	#97	—	2	1	no	—
295	C₄×C₈	—	2	1	no	—
296	C₄×C₈	—	2	1	no	—
297	C₂⁴	—	142	2	no	—
298	#41	—	2	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	—	2	1	no	—
302	C₂×Q₈	#11765865	1	1	yes	characteristic
303	C₂×D₄	—	2	1	no	—
304	C₂×D₄	—	71	4	no	—
305	C₂×D₄	—	142	2	no	—
306	C₂×D₄	—	142	2	no	—
307	C₂×D₄	—	142	2	no	—
308	C₂×D₄	—	142	2	no	—
309	C₂×D₄	—	142	2	no	—
310	C₂×D₄	#11765865	1	1	yes	characteristic
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	284	4	no	—
313	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
314	SD₁₆	—	284	4	no	—
315	D₈	—	284	4	no	—
316	D₈	—	142	2	no	—
317	D₈	—	142	2	no	—
318	D₈	—	142	2	no	—
319	D₈	—	142	2	no	—
320	M₁₆	—	2	1	no	—
321	M₁₆	—	2	1	no	—
322	M₁₆	—	2	1	no	—
323	M₁₆	—	2	1	no	—
324	M₁₆	—	2	1	no	—
325	M₁₆	—	2	1	no	—
326	M₁₆	—	4	1	no	—
327	M₁₆	—	4	1	no	—
328	M₁₆	—	142	2	no	—
329	M₁₆	—	142	2	no	—
330	M₁₆	—	142	2	no	—
331	M₁₆	—	142	2	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	1	no	—
336	C₂×C₈	—	4	1	no	—
337	C₂×C₈	—	142	2	no	—
338	C₂×C₈	—	142	2	no	—
339	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
340	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
341	C₂×C₈	#11765864	1	1	yes	characteristic
342	C₂×C₈	#11765864	1	1	yes	characteristic
343	C₂×C₈	#11765865	1	1	yes	characteristic
344	C₂×C₈	#11765865	1	1	yes	characteristic
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	142	2	no	—
348	C₂²⋊C₄	—	2	1	no	—
349	C₂²⋊C₄	—	284	4	no	—
350	C₂²⋊C₄	—	142	2	no	—
351	C₂²⋊C₄	#11765865	1	2	yes	—
352	C₄²	—	2	1	no	—
353	C₄²	—	2	1	no	—
354	C₄²	—	142	2	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	284	2	no	—
357	C₂³	—	284	2	no	—
358	C₂³	—	1136	2	no	—
359	C₂³	—	142	4	no	—
360	C₂³	C₂×#11765865	1	1	yes	characteristic
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	D₄	—	2	1	no	—
364	D₄	—	2	1	no	—
365	D₄	—	4	1	no	—
366	D₄	—	4	1	no	—
367	D₄	—	284	2	no	—
368	D₄	—	284	2	no	—
369	D₄	—	142	4	no	—
370	D₄	—	142	4	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	—	284	2	no	—
377	C₂×C₄	—	1136	4	no	—
378	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
379	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
380	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
381	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
382	C₂×C₄	#14556174	1	1	yes	characteristic
383	C₂×C₄	#14556174	1	1	yes	characteristic
384	C₈	—	4	1	no	—
385	C₈	—	4	1	no	—
386	C₈	—	4	1	no	—
387	C₈	—	284	2	no	—
388	C₈	—	284	2	no	—
389	C₈	#14556110	1	1	yes	characteristic
390	C₈	#14556110	1	1	yes	characteristic
391	C₈	#14556110	1	1	yes	characteristic
392	C₈	#14556110	1	1	yes	characteristic
393	C₂²	—	4	1	no	—
394	C₂²	—	4	1	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	284	4	no	—
397	C₂²	—	568	4	no	—
398	C₂²	—	1136	2	no	—
399	C₂²	—	1136	2	no	—
400	C₂²	#17986533	1	1	yes	characteristic
401	C₂²	#17986645	1	1	yes	characteristic
402	C₂²	#17986714	1	1	yes	characteristic
403	C₄	—	4	2	no	—
404	C₄	—	4	1	no	—
405	C₄	—	8	1	no	—
406	C₄	—	568	2	no	—
407	C₄	C₂×#14556110	1	1	yes	characteristic
408	C₄	#17986604	1	1	yes	characteristic
409	C₄	#17986645	1	1	yes	characteristic
410	C₄	#17986695	1	1	yes	characteristic
411	C₂	—	2	1	no	—
412	C₂	—	2	1	no	—
413	C₂	—	2	1	no	—
414	C₂	—	8	1	no	—
415	C₂	—	568	4	no	—
416	C₂	#22591832	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22591832
commutator, cosocle	C₂²×C₄×C₇₁	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₇₁×#1757	C₂
socle	C₂×C₇₁	#2420
prime radical	C₂×#17986269	C₂
hypercenter	#1757	D₇₁
hypocenter	C₇₁	#24988
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757

Complements

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

N	H ≅ G/N	type
2. #22597193	C₂	semidirect
3. #22589803	C₂	semidirect
4. #22589613	C₂	semidirect
5. #22589250	C₂	semidirect
6. #22589130	C₂	semidirect
7. #22588830	C₂	semidirect
8. #22588464	C₂	semidirect
9. #22588450	C₂	semidirect
10. #22588394	C₂	semidirect
11. C₇₁×#1757	C₂	semidirect
219. #1757	D₇₁	semidirect
220. C₇₁	#24988	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986269	2
#22597193	2
#22589803	2
#22589613	2
#22589250	2
#22589130	2
#22588830	2
#22588464	2
#22588450	2
#22588394	2
C₇₁×#1757	2
#24988	71