Subgroups — #32152192

Order 18176.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598852	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17986116	C₂	1	1	yes	maximal, characteristic
3	#22590514	C₂	1	4	yes	maximal, complemented
4	#22590509	C₂	1	4	yes	maximal, complemented
5	#22590113	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590112	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17985819	C₂	1	1	yes	maximal, characteristic, complemented
8	#22590076	C₂	1	1	yes	maximal, characteristic
9	C₇₁×#2145	C₂	1	1	yes	maximal, characteristic
10	C₂×#14556135	—	2	1	no	—
11	#17986552	—	2	1	no	—
12	C₂×#14556068	C₂²	1	1	yes	characteristic
13	#17986403	—	2	1	no	—
14	#17986400	—	2	1	no	—
15	#17986399	C₂²	1	2	yes	complemented
16	#17986396	C₂²	1	2	yes	complemented
17	#17986116	—	2	1	no	—
18	#17986116	—	2	1	no	—
19	#17986116	C₂²	1	2	yes	—
20	#17986116	C₂²	1	2	yes	—
21	C₂×#14556064	C₂²	1	1	yes	characteristic
22	#17986037	C₂²	1	1	yes	characteristic
23	#17986035	—	2	1	no	—
24	C₂×#14556055	—	2	1	no	—
25	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
26	#17985822	C₂²	1	4	yes	complemented
27	#17985821	C₂²	1	4	yes	complemented
28	#17985819	C₂²	1	2	yes	complemented
29	#17985819	C₂²	1	2	yes	complemented
30	#17985809	C₂²	1	4	yes	—
31	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
33	#427×C₇₁	C₂²	1	1	yes	characteristic
34	#360×C₇₁	C₂²	1	4	yes	—
35	#14556135	—	2	2	no	—
36	#14556135	—	2	2	no	—
37	#14556135	—	4	1	no	—
38	#14556135	—	4	1	no	—
39	C₂×#11765849	—	2	1	no	—
40	C₂×#11765849	—	4	1	no	—
41	C₂²×C₇₁⋊C₈	—	4	1	no	—
42	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
43	#14556127	—	2	2	no	—
44	#14556119	—	2	2	no	—
45	#14556105	—	4	2	no	—
46	C₄×Dic₁₄₂	—	4	2	no	—
47	#14556087	—	2	1	no	—
48	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
49	#14556068	—	2	1	no	—
50	#14556068	C₂³	1	2	yes	—
51	C₂×C₇₁⋊C₁₆	—	2	1	no	—
52	#14556064	—	2	1	no	—
53	#14556064	C₂³	1	2	yes	—
54	#14556062	—	2	1	no	—
55	#14556061	—	2	2	no	—
56	#14556060	C₂³	1	2	yes	—
57	#14556059	—	2	2	no	—
58	#14556055	—	2	2	no	—
59	#14556055	—	2	2	no	—
60	C₈×Dic₇₁	D₄	1	1	yes	characteristic
61	C₈×Dic₇₁	D₄	1	1	yes	characteristic
62	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
63	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
64	C₂×#41×C₇₁	—	2	1	no	—
65	#135×C₇₁	—	2	1	no	—
66	#135×C₇₁	—	2	1	no	—
67	#135×C₇₁	Q₈	1	2	yes	complemented
68	#135×C₇₁	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₇₁	Q₈	1	1	yes	characteristic, complemented
70	C₂×SD₁₆×C₇₁	—	2	1	no	—
71	C₂×D₈×C₇₁	Q₈	1	1	yes	characteristic, complemented
72	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
73	#117×C₇₁	—	2	1	no	—
74	M₃₂×C₇₁	—	2	1	no	—
75	M₃₂×C₇₁	C₂³	1	2	yes	—
76	C₂×C₁₆×C₇₁	—	2	1	no	—
77	#107×C₇₁	—	2	1	no	—
78	#106×C₇₁	C₂³	1	2	yes	—
79	C₂×Dic₁₄₂	—	4	2	no	—
80	C₂²×Dic₇₁	—	2	1	no	—
81	#11765849	—	2	2	no	—
82	#11765849	—	4	1	no	—
83	#11765849	—	4	2	no	—
84	#11765849	—	4	2	no	—
85	C₂×C₇₁⋊C₈	—	4	1	no	—
86	C₂×C₇₁⋊C₈	—	4	2	no	—
87	C₂×C₇₁⋊C₈	—	4	2	no	—
88	C₂×C₇₁⋊C₈	SD₁₆	1	1	yes	characteristic
89	C₂×C₇₁⋊C₈	SD₁₆	1	1	yes	characteristic
90	C₂×C₇₁⋊C₈	SD₁₆	1	1	yes	characteristic
91	C₂×C₇₁⋊C₈	SD₁₆	1	1	yes	characteristic
92	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
94	C₄⋊Dic₇₁	—	2	2	no	—
95	C₄⋊Dic₇₁	—	4	2	no	—
96	Dic₇₁⋊C₄	—	4	2	no	—
97	Dic₇₁⋊C₄	—	4	2	no	—
98	C₂²⋊Dic₇₁	—	4	1	no	—
99	C₄×Dic₇₁	—	2	1	no	—
100	C₄×Dic₇₁	—	4	1	no	—
101	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
102	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
103	C₇₁⋊C₁₆	—	2	2	no	—
104	#41×C₇₁	—	2	2	no	—
105	#41×C₇₁	—	2	2	no	—
106	#41×C₇₁	—	4	1	no	—
107	#41×C₇₁	—	4	1	no	—
108	C₂×Q₈×C₇₁	—	2	1	no	—
109	C₂×D₄×C₇₁	—	2	1	no	—
110	C₂×D₄×C₇₁	—	4	1	no	—
111	C₂²×C₄×C₇₁	—	4	1	no	—
112	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₇₁	—	2	1	no	—
114	Q₁₆×C₇₁	C₂×Q₈	1	2	yes	—
115	SD₁₆×C₇₁	—	2	2	no	—
116	SD₁₆×C₇₁	—	2	2	no	—
117	D₈×C₇₁	—	2	1	no	—
118	D₈×C₇₁	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
122	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
123	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
124	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
125	C₄⋊C₄×C₇₁	—	2	2	no	—
126	C₂²⋊C₄×C₇₁	—	4	1	no	—
127	C₄²×C₇₁	—	4	1	no	—
128	C₁₆×C₇₁	—	2	2	no	—
129	C₂³×C₇₁	—	4	1	no	—
130	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	Q₈×C₇₁	—	2	2	no	—
132	Q₈×C₇₁	—	4	1	no	—
133	D₄×C₇₁	—	2	2	no	—
134	D₄×C₇₁	—	4	2	no	—
135	D₄×C₇₁	—	4	2	no	—
136	D₄×C₇₁	—	4	1	no	—
137	C₂×C₄×C₇₁	—	4	1	no	—
138	C₂×C₄×C₇₁	—	4	2	no	—
139	C₂×C₄×C₇₁	—	4	2	no	—
140	C₂×C₄×C₇₁	—	4	2	no	—
141	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
142	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
143	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
145	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₇₁	—	2	1	no	—
148	C₂×Dic₇₁	—	2	1	no	—
149	C₂×Dic₇₁	—	4	2	no	—
150	C₂×Dic₇₁	—	4	2	no	—
151	Dic₁₄₂	—	8	4	no	—
152	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₇₁	C₂²×Q₈	1	1	yes	characteristic
155	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
156	C₇₁⋊C₈	—	4	2	no	—
157	C₇₁⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
158	C₇₁⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
159	C₇₁⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
160	C₇₁⋊C₈	C₂×SD₁₆	1	1	yes	characteristic
161	C₂²×C₇₁	—	4	1	no	—
162	C₂²×C₇₁	—	4	2	no	—
163	C₂²×C₇₁	—	8	2	no	—
164	C₂²×C₇₁	#476	1	1	yes	characteristic
165	C₂²×C₇₁	#476	1	1	yes	characteristic
166	C₂²×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₄×C₇₁	—	4	2	no	—
168	C₄×C₇₁	—	8	2	no	—
169	C₄×C₇₁	#476	1	1	yes	characteristic
170	C₄×C₇₁	#476	1	1	yes	characteristic
171	C₄×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
172	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
173	Dic₇₁	—	2	2	no	—
174	Dic₇₁	—	4	1	no	—
175	Dic₇₁	—	8	2	no	—
176	#33637	—	71	1	no	maximal, Sylow
177	C₂×C₇₁	—	2	1	no	—
178	C₂×C₇₁	—	2	1	no	—
179	C₂×C₇₁	—	2	1	no	—
180	C₂×C₇₁	—	8	2	no	—
181	C₂×C₇₁	C₂×#476	1	1	yes	characteristic
182	#3160	—	71	1	no	—
183	C₂×#443	—	71	1	no	—
184	#2231	—	71	4	no	—
185	#2230	—	71	4	no	—
186	#2155	—	71	1	no	—
187	#2155	—	71	1	no	—
188	C₂×#364	—	71	1	no	—
189	#2145	—	71	1	no	—
190	#2145	D₇₁	1	1	yes	characteristic
191	C₇₁	#33637	1	1	yes	characteristic, complemented, Sylow
192	C₂×#135	C₂×D₇₁	1	1	yes	characteristic
193	C₂×#131	—	142	1	no	—
194	#533	—	142	1	no	—
195	C₂×M₃₂	—	71	1	no	—
196	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
197	#500	—	142	1	no	—
198	#499	—	71	2	no	—
199	#499	—	142	1	no	—
200	#498	—	71	2	no	—
201	#443	—	71	2	no	—
202	#443	—	71	2	no	—
203	#443	—	142	1	no	—
204	#443	—	142	1	no	—
205	C₂×#108	—	71	1	no	—
206	#427	—	71	1	no	—
207	#427	—	142	1	no	—
208	#427	C₂×D₇₁	1	1	yes	characteristic
209	C₂×#104	—	142	1	no	—
210	C₂×C₄×C₈	—	71	1	no	—
211	#365	—	71	4	no	—
212	#365	—	71	4	no	—
213	#364	—	71	2	no	—
214	#364	—	71	2	no	—
215	#360	—	71	4	no	—
216	#360	C₂×D₇₁	1	4	yes	—
217	C₂×#41	—	2	1	no	—
218	#135	—	2	1	no	—
219	#135	—	2	1	no	—
220	#135	Dic₁₄₂	1	2	yes	complemented
221	#135	C₂²×D₇₁	1	2	yes	—
222	C₂×Q₁₆	Dic₁₄₂	1	1	yes	characteristic, complemented
223	C₂×SD₁₆	—	2	1	no	—
224	C₂×D₈	Dic₁₄₂	1	1	yes	characteristic, complemented
225	#131	—	284	1	no	—
226	#131	—	284	1	no	—
227	#131	—	142	2	no	—
228	#131	—	142	2	no	—
229	C₂×M₁₆	—	284	1	no	—
230	C₂×M₁₆	—	142	1	no	—
231	C₂²×C₈	—	284	1	no	—
232	C₂²×C₈	—	71	1	no	—
233	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
234	C₄⋊Q₈	—	142	2	no	—
235	#125	—	142	2	no	—
236	C₂²⋊Q₈	—	284	2	no	—
237	C₄×Q₈	—	284	2	no	—
238	#117	—	2	1	no	—
239	#117	—	142	1	no	—
240	C₂×C₄²	—	71	1	no	—
241	M₃₂	—	2	1	no	—
242	M₃₂	—	71	2	no	—
243	M₃₂	—	142	1	no	—
244	M₃₂	C₂²×D₇₁	1	2	yes	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	142	1	no	—
247	#108	—	71	2	no	—
248	#108	—	142	1	no	—
249	#107	—	2	1	no	—
250	#107	—	142	2	no	—
251	#107	—	142	1	no	—
252	#106	—	71	2	no	—
253	#106	—	142	2	no	—
254	#106	C₂²×D₇₁	1	2	yes	—
255	#104	—	142	2	no	—
256	#104	—	142	2	no	—
257	C₄×C₈	—	71	1	no	—
258	C₄×C₈	—	71	1	no	—
259	C₄×C₈	—	71	1	no	—
260	C₄×C₈	—	71	1	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	#41	—	4	1	no	—
264	#41	—	4	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	284	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	142	1	no	—
271	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
272	Q₁₆	—	2	1	no	—
273	Q₁₆	C₂×Dic₁₄₂	1	2	yes	—
274	SD₁₆	—	2	2	no	—
275	SD₁₆	—	2	2	no	—
276	D₈	—	2	1	no	—
277	D₈	C₂×Dic₁₄₂	1	2	yes	—
278	M₁₆	—	284	1	no	—
279	M₁₆	—	284	2	no	—
280	M₁₆	—	284	2	no	—
281	M₁₆	—	142	2	no	—
282	C₂×C₈	—	284	1	no	—
283	C₂×C₈	—	284	2	no	—
284	C₂×C₈	—	284	2	no	—
285	C₂×C₈	—	71	1	no	—
286	C₂×C₈	—	71	1	no	—
287	C₂×C₈	—	71	1	no	—
288	C₂×C₈	—	71	1	no	—
289	C₂×C₈	—	71	1	no	—
290	C₂×C₈	—	71	1	no	—
291	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
292	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
293	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
294	C₂×C₈	#11765864	1	1	yes	characteristic
295	C₂×C₈	#11765864	1	1	yes	characteristic
296	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	284	2	no	—
299	C₄⋊C₄	—	284	2	no	—
300	C₄⋊C₄	—	284	2	no	—
301	C₄⋊C₄	—	142	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	284	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	284	1	no	—
306	C₄²	—	71	1	no	—
307	C₄²	—	71	1	no	—
308	C₄²	—	142	1	no	—
309	C₁₆	—	2	2	no	—
310	C₁₆	—	142	2	no	—
311	C₂³	—	4	1	no	—
312	C₂³	#14556152	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	4	1	no	—
315	Q₈	—	568	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	284	2	no	—
325	C₂×C₄	—	284	2	no	—
326	C₂×C₄	—	142	1	no	—
327	C₂×C₄	—	142	1	no	—
328	C₂×C₄	#14556105	1	1	yes	characteristic
329	C₂×C₄	#14556105	1	1	yes	characteristic
330	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
331	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
332	C₂×C₄	#14556152	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
334	C₈	—	284	2	no	—
335	C₈	—	71	1	no	—
336	C₈	—	71	1	no	—
337	C₈	—	71	1	no	—
338	C₈	—	71	1	no	—
339	C₈	#14556105	1	1	yes	characteristic
340	C₈	#14556105	1	1	yes	characteristic
341	C₈	C₂²×Dic₁₄₂	1	1	yes	characteristic
342	C₈	C₂×#11765864	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	#17986287	1	1	yes	characteristic
347	C₂²	#17986287	1	1	yes	characteristic
348	C₂²	C₂×#14556152	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	8	2	no	—
351	C₄	—	284	1	no	—
352	C₄	—	568	2	no	—
353	C₄	—	142	2	no	—
354	C₄	#17986287	1	1	yes	characteristic
355	C₄	#17986287	1	1	yes	characteristic
356	C₄	C₂×#14556105	1	1	yes	characteristic
357	C₄	C₂×SD₁₆×D₇₁	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₂×#17986287	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17986287
commutator, cosocle	C₂×C₈×C₇₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2145	C₂
socle	C₂×C₇₁	C₂×#476
prime radical	C₂×#135×C₇₁	C₂²
hypercenter	#2145	D₇₁
hypocenter	C₇₁	#33637
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2145

Complements

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

N	H ≅ G/N	type
1. #22598852	C₂	semidirect
3. #22590514	C₂	semidirect
4. #22590509	C₂	semidirect
5. #22590113	C₂	semidirect
6. #22590112	C₂	semidirect
7. C₂×#17985819	C₂	semidirect
15. #17986399	C₂²	semidirect
16. #17986396	C₂²	semidirect
26. #17985822	C₂²	semidirect
27. #17985821	C₂²	semidirect
28. #17985819	C₂²	semidirect
29. #17985819	C₂²	semidirect
67. #135×C₇₁	Q₈	semidirect
69. C₂×Q₁₆×C₇₁	Q₈	semidirect
71. C₂×D₈×C₇₁	Q₈	semidirect
191. C₇₁	#33637	semidirect
220. #135	Dic₁₄₂	semidirect
222. C₂×Q₁₆	Dic₁₄₂	semidirect
224. C₂×D₈	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22598852	2
C₂×#17986116	2
#22590514	2
#22590509	2
#22590113	2
#22590112	2
C₂×#17985819	2
#22590076	2
C₇₁×#2145	2
#33637	71