Subgroups — #31997822

Order 18176.

608 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22596767	C₂	1	1	yes	maximal, characteristic
2	#22588488	C₂	1	2	yes	maximal, complemented
3	#22588464	C₂	1	1	yes	maximal, characteristic, complemented
4	#22588442	C₂	1	2	yes	maximal
5	#22588387	C₂	1	1	yes	maximal, characteristic
6	#22588383	C₂	1	1	yes	maximal, characteristic
7	#22588227	C₂	1	1	yes	maximal, characteristic, complemented
8	#22588226	C₂	1	1	yes	maximal, characteristic, complemented
9	#22588139	C₂	1	2	yes	maximal
10	#22588094	C₂	1	2	yes	maximal, complemented
11	C₇₁×#1833	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556136	C₂²	1	1	yes	characteristic
13	C₂×#14556134	C₂²	1	1	yes	characteristic
14	#17986467	C₂²	1	1	yes	characteristic
15	C₂×#14556064	C₂²	1	1	yes	characteristic
16	#17986037	C₂²	1	1	yes	characteristic
17	#17985981	C₂²	1	2	yes	—
18	#17985972	C₂²	1	2	yes	—
19	#17985957	C₂²	1	2	yes	—
20	#17985956	C₄	1	2	yes	complemented
21	#17985955	C₂²	1	2	yes	—
22	#17985955	C₂²	1	1	yes	characteristic
23	#17985953	C₄	1	2	yes	complemented
24	M₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
25	C₄×#11765848	C₄	1	2	yes	complemented
26	C₂×#14556038	C₂²	1	1	yes	characteristic
27	#97×D₇₁	C₄	1	2	yes	complemented
28	#17985770	C₂²	1	2	yes	—
29	#17985760	C₂²	1	2	yes	—
30	#17985758	C₂²	1	2	yes	—
31	C₇₁⋊#338	C₂²	1	2	yes	—
32	#17985755	C₂²	1	2	yes	—
33	#17985755	C₂²	1	2	yes	—
34	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
35	#427×C₇₁	C₂²	1	1	yes	characteristic
36	#405×C₇₁	C₂²	1	1	yes	characteristic
37	#344×C₇₁	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₇₁	C₂²	1	2	yes	—
39	C₂×#11765865	C₂³	1	1	yes	characteristic
40	#14556136	—	2	2	no	—
41	#14556136	—	2	2	no	—
42	#14556134	—	2	1	no	—
43	#14556134	—	2	1	no	—
44	#14556134	—	2	1	no	—
45	#14556134	—	2	1	no	—
46	C₂×#11765849	C₂³	1	2	yes	—
47	C₂×#11765849	C₂³	1	1	yes	characteristic
48	C₂×#11765848	C₂×C₄	1	2	yes	—
49	C₂×#11765848	C₂×C₄	1	2	yes	—
50	C₂²×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
51	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
52	C₂×C₈×D₇₁	C₂×C₄	1	2	yes	—
53	#14556093	—	2	1	no	—
54	#14556090	—	2	1	no	—
55	(C₂²⋊D₇₁)⋊C₄	—	2	2	no	—
56	#14556087	C₂³	1	2	yes	—
57	#14556085	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₇₁	C₂³	1	1	yes	characteristic
59	C₄²×D₇₁	C₂×C₄	1	2	yes	—
60	#14556064	—	2	1	no	—
61	#14556064	—	2	1	no	—
62	#14556062	—	2	1	no	—
63	#14556060	—	2	1	no	—
64	#14556046	—	2	2	no	—
65	#14556044	—	2	2	no	—
66	#14556042	—	2	4	no	—
67	#14556039	—	2	2	no	—
68	#14556039	C₂×C₄	1	2	yes	—
69	#14556038	—	2	2	no	—
70	#14556038	C₂×C₄	1	1	yes	characteristic
71	#14556038	C₂×C₄	1	2	yes	—
72	#14556038	C₂×C₄	1	1	yes	characteristic
73	#14556038	C₂×C₄	1	1	yes	characteristic
74	#14556038	C₂×C₄	1	1	yes	characteristic
75	C₄×C₇₁⋊C₈	—	2	2	no	—
76	C₄×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
77	C₈×Dic₇₁	—	2	2	no	—
78	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
79	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
80	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
82	#117×C₇₁	C₂³	1	2	yes	—
83	#117×C₇₁	C₂³	1	1	yes	characteristic
84	#108×C₇₁	—	2	1	no	—
85	#108×C₇₁	—	2	1	no	—
86	#107×C₇₁	—	2	1	no	—
87	#106×C₇₁	—	2	1	no	—
88	#97×C₇₁	C₂×C₄	1	2	yes	—
89	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
90	#11765865	—	2	1	no	—
91	#11765865	—	2	1	no	—
92	#11765865	—	2	1	no	—
93	#11765865	—	2	1	no	—
94	C₂×Dic₁₄₂	#41	1	1	yes	characteristic
95	C₂×D₂₈₄	#41	1	1	yes	characteristic
96	C₂×C₂²⋊D₇₁	#41	1	2	yes	—
97	C₂²×Dic₇₁	—	2	1	no	—
98	C₂×C₄×D₇₁	—	2	2	no	—
99	C₂×C₄×D₇₁	C₂²×C₄	1	2	yes	—
100	#11765849	—	2	1	no	—
101	#11765849	—	2	1	no	—
102	#11765849	—	2	2	no	—
103	#11765849	—	2	2	no	—
104	#11765848	—	2	2	no	—
105	#11765848	—	2	2	no	—
106	#11765848	—	2	4	no	—
107	C₂×C₇₁⋊C₈	—	2	2	no	—
108	C₂×C₇₁⋊C₈	C₂²×C₄	1	2	yes	—
109	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₇₁⋊C₈	#41	1	2	yes	—
112	C₂×C₇₁⋊C₈	#41	1	2	yes	—
113	C₈×D₇₁	—	2	2	no	—
114	C₈×D₇₁	—	2	2	no	—
115	C₈×D₇₁	—	2	4	no	—
116	C₄⋊Dic₇₁	—	2	2	no	—
117	Dic₇₁⋊C₄	—	2	2	no	—
118	C₂²⋊Dic₇₁	—	2	2	no	—
119	(C₂×D₇₁)⋊C₄	—	2	2	no	—
120	C₄×Dic₇₁	—	2	1	no	—
121	C₄×Dic₇₁	—	2	2	no	—
122	C₄×Dic₇₁	—	2	2	no	—
123	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
126	M₁₆×C₇₁	—	2	1	no	—
127	M₁₆×C₇₁	—	2	1	no	—
128	M₁₆×C₇₁	—	2	2	no	—
129	M₁₆×C₇₁	—	2	2	no	—
130	C₂×C₈×C₇₁	—	2	2	no	—
131	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
132	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₇₁	C₂²×C₄	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	C₂²⋊C₄×C₇₁	—	2	1	no	—
141	C₂²⋊C₄×C₇₁	—	2	2	no	—
142	C₄²×C₇₁	—	2	2	no	—
143	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
144	C₂³×C₇₁	C₂×#41	1	1	yes	characteristic
145	C₂²×D₇₁	#117	1	2	yes	—
146	C₂×C₄×C₇₁	—	2	2	no	—
147	C₂×C₄×C₇₁	—	4	2	no	—
148	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
154	C₂²⋊D₇₁	—	2	2	no	—
155	C₂²⋊D₇₁	—	2	2	no	—
156	C₂×Dic₇₁	—	2	1	no	—
157	C₂×Dic₇₁	—	2	1	no	—
158	C₂×Dic₇₁	—	4	2	no	—
159	C₂×Dic₇₁	—	8	1	no	—
160	C₂×Dic₇₁	#117	1	2	yes	—
161	D₂₈₄	—	2	1	no	—
162	D₂₈₄	—	2	1	no	—
163	C₄×D₇₁	—	2	2	no	—
164	C₄×D₇₁	—	2	2	no	—
165	C₄×D₇₁	—	8	2	no	—
166	Dic₁₄₂	—	2	1	no	—
167	Dic₁₄₂	—	2	1	no	—
168	C₈×C₇₁	—	2	2	no	—
169	C₈×C₇₁	—	4	2	no	—
170	C₈×C₇₁	#117	1	1	yes	characteristic
171	C₈×C₇₁	#117	1	1	yes	characteristic
172	C₈×C₇₁	#117	1	1	yes	characteristic
173	C₈×C₇₁	#117	1	1	yes	characteristic
174	C₇₁⋊C₈	—	2	2	no	—
175	C₇₁⋊C₈	—	4	2	no	—
176	C₇₁⋊C₈	#117	1	2	yes	—
177	C₇₁⋊C₈	#117	1	2	yes	—
178	C₂²×C₇₁	—	4	1	no	—
179	C₂²×C₇₁	C₂×#117	1	1	yes	characteristic
180	C₂²×C₇₁	#529	1	1	yes	characteristic
181	C₂²×C₇₁	#529	1	1	yes	characteristic
182	C₂×D₇₁	—	2	2	no	—
183	C₂×D₇₁	—	4	2	no	—
184	C₄×C₇₁	—	4	2	no	—
185	C₄×C₇₁	—	8	2	no	—
186	C₄×C₇₁	C₂×#117	1	1	yes	characteristic
187	C₄×C₇₁	C₂×#117	1	1	yes	characteristic
188	C₄×C₇₁	#529	1	1	yes	characteristic
189	C₄×C₇₁	#529	1	1	yes	characteristic
190	Dic₇₁	—	2	2	no	—
191	Dic₇₁	—	4	1	no	—
192	Dic₇₁	—	4	1	no	—
193	Dic₇₁	—	8	2	no	—
194	#20788	—	71	1	no	maximal, Sylow
195	C₂×C₇₁	—	2	1	no	—
196	C₂×C₇₁	—	2	1	no	—
197	C₂×C₇₁	—	2	1	no	—
198	C₂×C₇₁	#2283	1	1	yes	characteristic
199	D₇₁	—	4	2	no	—
200	#2875	—	71	1	no	—
201	#1841	—	71	2	no	—
202	#1833	—	71	1	no	—
203	#1833	D₇₁	1	1	yes	characteristic, complemented
204	#1827	—	71	2	no	—
205	#1813	—	71	1	no	—
206	#1812	—	71	1	no	—
207	#1773	—	71	1	no	—
208	#1773	—	71	1	no	—
209	#1757	—	71	2	no	—
210	#1747	—	71	2	no	—
211	C₇₁	#20788	1	1	yes	characteristic, complemented, Sylow
212	C₂×#131	—	71	1	no	—
213	C₂×#131	—	71	1	no	—
214	C₄×#41	—	71	1	no	—
215	C₂×#108	—	71	1	no	—
216	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
217	#427	—	71	1	no	—
218	#427	C₂×D₇₁	1	1	yes	characteristic
219	#413	—	71	2	no	—
220	#410	—	71	2	no	—
221	#405	—	71	1	no	—
222	#405	—	71	2	no	—
223	#405	—	71	2	no	—
224	#405	—	71	2	no	—
225	#405	—	71	2	no	—
226	#405	C₂×D₇₁	1	1	yes	characteristic
227	C₄×M₁₆	—	71	1	no	—
228	C₄×M₁₆	—	71	2	no	—
229	C₂×#97	—	71	1	no	—
230	C₂×#97	—	71	2	no	—
231	#344	—	71	2	no	—
232	#344	C₂×D₇₁	1	2	yes	—
233	#339	—	71	2	no	—
234	#339	—	71	2	no	—
235	#338	—	71	2	no	—
236	H(3,ℤ/4)	—	71	2	no	—
237	H(3,ℤ/4)	—	71	2	no	—
238	H(3,ℤ/4)	C₂×D₇₁	1	2	yes	—
239	C₂×#41	—	71	1	no	—
240	#131	—	142	2	no	—
241	#131	—	142	2	no	—
242	#131	—	142	1	no	—
243	#131	—	142	1	no	—
244	#131	—	142	1	no	—
245	#131	—	142	1	no	—
246	C₂×M₁₆	—	71	1	no	—
247	C₂×M₁₆	—	71	2	no	—
248	C₂×M₁₆	—	71	2	no	—
249	C₂×M₁₆	—	71	2	no	—
250	C₂×M₁₆	C₂²×D₇₁	1	2	yes	—
251	C₂×M₁₆	C₂²×D₇₁	1	1	yes	characteristic
252	C₂²×C₈	—	71	1	no	—
253	C₂²×C₈	—	71	2	no	—
254	C₂²×C₈	—	71	2	no	—
255	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
256	C₄×Q₈	—	142	1	no	—
257	C₄×D₄	—	142	2	no	—
258	C₄×D₄	—	142	1	no	—
259	#117	—	71	2	no	—
260	#117	—	71	2	no	—
261	#117	C₂²×D₇₁	1	2	yes	—
262	#117	C₂²×D₇₁	1	1	yes	characteristic
263	C₂×C₄²	—	71	1	no	—
264	C₂×C₄²	—	71	2	no	—
265	#108	—	2	1	no	—
266	#108	—	2	1	no	—
267	#108	—	142	1	no	—
268	#108	—	142	1	no	—
269	#107	—	2	1	no	—
270	#107	—	142	1	no	—
271	#106	—	2	1	no	—
272	#106	—	142	1	no	—
273	#101	—	142	2	no	—
274	#100	—	142	2	no	—
275	#99	—	142	4	no	—
276	#97	—	71	1	no	—
277	#97	—	71	1	no	—
278	#97	—	71	2	no	—
279	#97	—	71	1	no	—
280	#97	—	71	1	no	—
281	#97	—	71	2	no	—
282	#97	—	142	2	no	—
283	#97	—	142	2	no	—
284	#97	C₄×D₇₁	1	2	yes	—
285	C₄×C₈	—	71	2	no	—
286	C₄×C₈	—	71	2	no	—
287	C₄×C₈	—	142	2	no	—
288	C₄×C₈	—	142	2	no	—
289	C₄×C₈	C₄×D₇₁	1	2	yes	—
290	#41	—	142	1	no	—
291	#41	—	142	1	no	—
292	#41	—	142	1	no	—
293	#41	—	142	1	no	—
294	C₂×Q₈	—	71	1	no	—
295	C₂×D₄	—	71	2	no	—
296	C₂×D₄	—	71	1	no	—
297	C₂²×C₄	—	71	2	no	—
298	C₂²×C₄	—	142	2	no	—
299	C₂²×C₄	—	142	1	no	—
300	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
301	M₁₆	—	2	1	no	—
302	M₁₆	—	2	1	no	—
303	M₁₆	—	2	2	no	—
304	M₁₆	—	2	2	no	—
305	M₁₆	—	142	4	no	—
306	M₁₆	—	142	2	no	—
307	M₁₆	—	142	2	no	—
308	M₁₆	—	142	1	no	—
309	M₁₆	—	142	1	no	—
310	M₁₆	—	142	2	no	—
311	M₁₆	—	142	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	71	1	no	—
314	C₂×C₈	—	71	1	no	—
315	C₂×C₈	—	71	2	no	—
316	C₂×C₈	—	71	2	no	—
317	C₂×C₈	—	71	2	no	—
318	C₂×C₈	—	142	4	no	—
319	C₂×C₈	—	142	2	no	—
320	C₂×C₈	—	142	2	no	—
321	C₂×C₈	—	142	2	no	—
322	C₂×C₈	C₂×C₄×D₇₁	1	2	yes	—
323	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
325	C₂×C₈	#11765864	1	1	yes	characteristic
326	C₂×C₈	#11765864	1	1	yes	characteristic
327	C₂×C₈	#11765866	1	1	yes	characteristic
328	C₂×C₈	#11765866	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	142	2	no	—
332	C₄⋊C₄	—	142	2	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	2	2	no	—
335	C₂²⋊C₄	—	142	2	no	—
336	C₂²⋊C₄	—	142	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	71	1	no	—
339	C₄²	—	71	1	no	—
340	C₄²	—	142	2	no	—
341	C₄²	—	142	2	no	—
342	C₄²	—	142	1	no	—
343	C₄²	C₂×C₄×D₇₁	1	2	yes	—
344	C₂³	—	71	2	no	—
345	C₂³	#41×D₇₁	1	1	yes	characteristic
346	Q₈	—	142	1	no	—
347	Q₈	—	142	1	no	—
348	D₄	—	142	2	no	—
349	D₄	—	142	2	no	—
350	D₄	—	142	1	no	—
351	D₄	—	142	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	284	2	no	—
355	C₂×C₄	—	568	2	no	—
356	C₂×C₄	—	568	1	no	—
357	C₂×C₄	—	71	2	no	—
358	C₂×C₄	—	142	1	no	—
359	C₂×C₄	—	142	1	no	—
360	C₂×C₄	—	142	2	no	—
361	C₂×C₄	—	142	2	no	—
362	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
363	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
364	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
365	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
366	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
367	C₂×C₄	C₂×#11765866	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	284	2	no	—
371	C₈	—	71	2	no	—
372	C₈	—	71	2	no	—
373	C₈	—	142	2	no	—
374	C₈	#14556086	1	1	yes	characteristic
375	C₈	#14556086	1	1	yes	characteristic
376	C₈	#14556086	1	1	yes	characteristic
377	C₈	#14556086	1	1	yes	characteristic
378	C₂²	—	4	1	no	—
379	C₂²	—	284	2	no	—
380	C₂²	—	142	2	no	—
381	C₂²	#117×D₇₁	1	1	yes	characteristic
382	C₂²	#17986528	1	1	yes	characteristic
383	C₂²	#17986534	1	1	yes	characteristic
384	C₄	—	4	2	no	—
385	C₄	—	8	2	no	—
386	C₄	—	284	1	no	—
387	C₄	—	284	1	no	—
388	C₄	—	568	2	no	—
389	C₄	—	142	2	no	—
390	C₄	#117×D₇₁	1	1	yes	characteristic
391	C₄	C₂×#14556086	1	1	yes	characteristic
392	C₄	#17986528	1	1	yes	characteristic
393	C₄	#17986534	1	1	yes	characteristic
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	2	1	no	—
397	C₂	—	284	2	no	—
398	C₂	#22590733	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₁
Fitting	C₇₁×#1833	C₂
socle	C₂×C₇₁	#2283
prime radical	C₂×#11765865	C₂³
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	#1833	D₇₁
hypocenter	C₇₁	#20788
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1833

Complements

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

N	H ≅ G/N	type
2. #22588488	C₂	semidirect
3. #22588464	C₂	semidirect
7. #22588227	C₂	semidirect
8. #22588226	C₂	semidirect
10. #22588094	C₂	semidirect
11. C₇₁×#1833	C₂	semidirect
20. #17985956	C₄	semidirect
23. #17985953	C₄	semidirect
25. C₄×#11765848	C₄	semidirect
27. #97×D₇₁	C₄	semidirect
203. #1833	D₇₁	semidirect
211. C₇₁	#20788	semidirect

Maximal subgroups

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

subgroup	index
#22596767	2
#22588488	2
#22588464	2
#22588442	2
#22588387	2
#22588383	2
#22588227	2
#22588226	2
#22588139	2
#22588094	2
C₇₁×#1833	2
#20788	71