Subgroups — #31890287

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588320	C₂	1	2	yes	maximal
2	C₂×#17985750	C₂	1	2	yes	maximal, complemented
3	#22586175	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	4	yes	—
6	C₂×C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
7	#17985754	—	2	2	no	—
8	#17985750	—	2	4	no	—
9	C₂³×C₈×C₇₁	C₄	1	1	yes	characteristic
10	C₂³×C₈×C₇₁	C₄	1	1	yes	characteristic
11	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₇₁	—	2	4	no	—
13	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	—	2	4	no	—
18	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
19	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₇₁⋊C₈)	—	4	4	no	—
21	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₇₁	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₇₁	—	2	2	no	—
25	C₂²×C₈×C₇₁	—	2	2	no	—
26	C₂²×C₈×C₇₁	—	2	2	no	—
27	C₂²×C₈×C₇₁	—	2	2	no	—
28	C₂²×C₈×C₇₁	—	2	2	no	—
29	C₂²×C₈×C₇₁	C₈	1	4	yes	complemented
30	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	—	2	2	no	—
35	C₂×C₄²×C₇₁	—	2	2	no	—
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₇₁	—	2	2	no	—
41	C₄×C₈×C₇₁	—	2	2	no	—
42	C₄×C₈×C₇₁	—	2	2	no	—
43	C₄×C₈×C₇₁	D₄	1	4	yes	—
44	C₂×C₇₁⋊C₈	—	4	4	no	—
45	C₂×C₇₁⋊C₈	—	4	4	no	—
46	C₂×C₇₁⋊C₈	—	4	4	no	—
47	C₂⁴×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₇₁	—	2	2	no	—
49	C₂²×C₄×C₇₁	—	2	2	no	—
50	C₂²×C₄×C₇₁	—	2	1	no	—
51	C₂²×C₄×C₇₁	—	2	1	no	—
52	C₂²×C₄×C₇₁	—	2	2	no	—
53	C₂²×C₄×C₇₁	—	2	2	no	—
54	C₂²×C₄×C₇₁	—	2	1	no	—
55	C₂²×C₄×C₇₁	—	2	1	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₇₁	M₁₆	1	2	yes	—
62	C₂²×C₄×C₇₁	SD₁₆	1	4	yes	—
63	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₇₁	—	2	8	no	—
67	C₂×C₈×C₇₁	—	2	1	no	—
68	C₂×C₈×C₇₁	—	2	1	no	—
69	C₂×C₈×C₇₁	—	2	1	no	—
70	C₂×C₈×C₇₁	—	2	1	no	—
71	C₂×C₈×C₇₁	—	2	1	no	—
72	C₂×C₈×C₇₁	—	2	1	no	—
73	C₂×C₈×C₇₁	—	2	8	no	—
74	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
75	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
76	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
79	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
80	C₄²×C₇₁	—	2	1	no	—
81	C₄²×C₇₁	—	2	1	no	—
82	C₄²×C₇₁	—	2	2	no	—
83	C₄²×C₇₁	—	2	2	no	—
84	C₄²×C₇₁	C₂×D₄	1	2	yes	—
85	C₄²×C₇₁	#41	1	2	yes	—
86	C₂³×C₇₁	—	2	1	no	—
87	C₂³×C₇₁	—	2	1	no	—
88	C₂³×C₇₁	—	2	1	no	—
89	C₂³×C₇₁	—	2	1	no	—
90	C₂³×C₇₁	C₄⋊C₈	1	2	yes	—
91	C₂³×C₇₁	#106	1	2	yes	—
92	C₂³×C₇₁	#108	1	2	yes	—
93	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	—	2	8	no	—
95	C₂×C₄×C₇₁	—	2	1	no	—
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	1	no	—
98	C₂×C₄×C₇₁	—	2	1	no	—
99	C₂×C₄×C₇₁	—	2	1	no	—
100	C₂×C₄×C₇₁	—	2	1	no	—
101	C₂×C₄×C₇₁	—	2	1	no	—
102	C₂×C₄×C₇₁	—	2	1	no	—
103	C₂×C₄×C₇₁	—	2	2	no	—
104	C₂×C₄×C₇₁	—	2	2	no	—
105	C₂×C₄×C₇₁	—	2	2	no	—
106	C₂×C₄×C₇₁	—	2	2	no	—
107	C₂×C₄×C₇₁	—	2	8	no	—
108	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
109	C₂×C₄×C₇₁	#106	1	2	yes	—
110	C₂×C₄×C₇₁	#108	1	2	yes	—
111	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#121	1	2	yes	—
114	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
120	C₂×C₄×C₇₁	#135	1	2	yes	—
121	C₈×C₇₁	—	2	2	no	—
122	C₈×C₇₁	—	2	2	no	—
123	C₈×C₇₁	—	2	2	no	—
124	C₈×C₇₁	C₂²⋊C₈	1	4	yes	complemented
125	C₇₁⋊C₈	—	8	4	no	—
126	C₂²×C₇₁	—	2	2	no	—
127	C₂²×C₇₁	—	2	2	no	—
128	C₂²×C₇₁	—	2	2	no	—
129	C₂²×C₇₁	—	2	2	no	—
130	C₂²×C₇₁	—	2	2	no	—
131	C₂²×C₇₁	—	2	2	no	—
132	C₂²×C₇₁	#334	1	4	yes	complemented
133	C₂²×C₇₁	#380	1	1	yes	characteristic
134	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
135	C₂²×C₇₁	#423	1	1	yes	characteristic
136	C₂²×C₇₁	C₂×#106	1	1	yes	characteristic
137	C₂²×C₇₁	#427	1	1	yes	characteristic
138	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
139	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
140	C₄×C₇₁	—	2	2	no	—
141	C₄×C₇₁	—	2	2	no	—
142	C₄×C₇₁	—	2	2	no	—
143	C₄×C₇₁	—	2	2	no	—
144	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₇₁	#407	1	1	yes	characteristic
146	C₄×C₇₁	#431	1	1	yes	characteristic
147	C₄×C₇₁	#432	1	1	yes	characteristic
148	C₄×C₇₁	#465	1	4	yes	—
149	#8595	—	71	1	no	maximal, Sylow
150	C₂×C₇₁	—	2	4	no	—
151	C₂×C₇₁	C₂×#334	1	1	yes	characteristic
152	C₂×C₇₁	#1564	1	1	yes	characteristic
153	C₂×C₇₁	#1566	1	1	yes	characteristic
154	C₂×C₇₁	#1568	1	1	yes	characteristic
155	C₂×C₇₁	#1797	1	1	yes	characteristic
156	C₂×C₇₁	#1807	1	1	yes	characteristic
157	C₂×C₇₁	#1810	1	1	yes	characteristic
158	C₂²×C₄×C₈	D₇₁	1	1	yes	characteristic
159	#1797	—	71	2	no	—
160	C₂×#334	—	71	2	no	—
161	#1277	—	71	2	no	—
162	C₇₁	#8595	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₈	Dic₇₁	1	1	yes	characteristic
164	C₂³×C₈	Dic₇₁	1	1	yes	characteristic
165	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic
166	C₂×C₄⋊C₈	—	71	4	no	—
167	C₂×C₂²⋊C₈	—	142	2	no	—
168	C₂×C₄×C₈	—	2	4	no	—
169	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
170	C₂×C₄×C₈	Dic₇₁	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
173	#336	—	142	2	no	—
174	#334	—	142	4	no	—
175	C₂³×C₄	Dic₁₄₂	1	1	yes	characteristic
176	C₂³×C₄	D₂₈₄	1	1	yes	characteristic
177	C₂³×C₄	C₂×Dic₇₁	1	1	yes	characteristic
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	142	4	no	—
184	C₂²×C₈	C₇₁⋊C₈	1	4	yes	complemented
185	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
186	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
187	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
188	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	Dic₁₄₂	1	1	yes	characteristic
192	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
193	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
194	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
195	C₄⋊C₈	—	142	4	no	—
196	C₄⋊C₈	—	142	4	no	—
197	C₂²⋊C₈	—	284	4	no	—
198	C₄×C₈	—	2	2	no	—
199	C₄×C₈	—	2	2	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	C₂²⋊D₇₁	1	4	yes	—
202	C₂⁴	C₄⋊Dic₇₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
213	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
214	C₂²×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₇₁⋊C₈	1	2	yes	—
216	C₂²×C₄	#11765849	1	2	yes	—
217	C₂²×C₄	#11765854	1	4	yes	—
218	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
220	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
221	C₂×C₈	—	2	8	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	284	4	no	—
230	C₂×C₈	—	284	4	no	—
231	C₂×C₈	—	284	4	no	—
232	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
233	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
234	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
235	C₂×C₈	C₂×C₇₁⋊C₈	1	2	yes	—
236	C₂×C₈	#11765849	1	2	yes	—
237	C₂×C₈	#11765849	1	2	yes	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	C₂×C₂²⋊D₇₁	1	2	yes	—
243	C₄²	#11765865	1	2	yes	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
249	C₂³	#14556060	1	2	yes	—
250	C₂³	#14556064	1	2	yes	—
251	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	2	yes	—
267	C₂×C₄	#14556060	1	2	yes	—
268	C₂×C₄	#14556064	1	2	yes	—
269	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
270	C₂×C₄	#14556087	1	1	yes	characteristic
271	C₂×C₄	#14556102	1	2	yes	—
272	C₂×C₄	#14556107	1	2	yes	—
273	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
274	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
275	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
276	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
277	C₂×C₄	C₂×#11765854	1	2	yes	—
278	C₂×C₄	#14556150	1	2	yes	—
279	C₈	—	2	2	no	—
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	568	4	no	—
283	C₈	C₂²⋊(C₇₁⋊C₈)	1	4	yes	complemented
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	#17985750	1	4	yes	complemented
291	C₂²	#17985859	1	1	yes	characteristic
292	C₂²	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
293	C₂²	#17986023	1	1	yes	characteristic
294	C₂²	C₂×#14556060	1	1	yes	characteristic
295	C₂²	#17986037	1	1	yes	characteristic
296	C₂²	C₂×#14556064	1	1	yes	characteristic
297	C₂²	C₂×#14556064	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
303	C₄	#17985963	1	1	yes	characteristic
304	C₄	#17986052	1	1	yes	characteristic
305	C₄	#17986056	1	1	yes	characteristic
306	C₄	#17986228	1	4	yes	—
307	C₂	—	2	4	no	—
308	C₂	C₂×#17985750	1	1	yes	characteristic
309	C₂	#22587088	1	1	yes	characteristic
310	C₂	#22587096	1	1	yes	characteristic
311	C₂	#22587103	1	1	yes	characteristic
312	C₂	#22588320	1	1	yes	characteristic
313	C₂	#22588365	1	1	yes	characteristic
314	C₂	#22588378	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂²×C₄×C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₁	C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂²×C₄×C₈	D₇₁
hypocenter	C₇₁	#8595
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17985750	C₂	semidirect
3. #22586175	C₂	semidirect
29. C₂²×C₈×C₇₁	C₈	semidirect
124. C₈×C₇₁	C₂²⋊C₈	semidirect
132. C₂²×C₇₁	#334	semidirect
162. C₇₁	#8595	semidirect
184. C₂²×C₈	C₇₁⋊C₈	semidirect
283. C₈	C₂²⋊(C₇₁⋊C₈)	semidirect
290. C₂²	#17985750	semidirect

Maximal subgroups

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

subgroup	index
#22588320	2
C₂×#17985750	2
#22586175	2
C₂²×C₄×C₈×C₇₁	2
#8595	71