Subgroups — #32017421

Order 18176.

764 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589277	C₂	1	2	yes	maximal
2	#22588899	C₂	1	4	yes	maximal
3	#22588483	C₂	1	2	yes	maximal
4	#22588457	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₄×C₇₁⋊C₈	C₂	1	2	yes	maximal
6	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#22588086	C₂	1	2	yes	maximal
8	C₄²×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×C₇₁⋊C₈	C₂²	1	4	yes	—
13	#17985754	C₂²	1	1	yes	characteristic
14	#17985754	C₂²	1	1	yes	characteristic
15	#17985754	C₂²	1	2	yes	—
16	#17985754	C₂²	1	2	yes	—
17	#17985754	C₂²	1	4	yes	—
18	#17985754	C₂²	1	4	yes	—
19	#17985754	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₇₁	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₇₁	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₇₁	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₇₁	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
29	C₄³×C₇₁	C₂²	1	1	yes	characteristic
30	C₄³×C₇₁	C₂²	1	2	yes	—
31	C₂²×C₇₁⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₇₁⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₇₁⋊C₈	C₂³	1	2	yes	—
34	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
35	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
36	C₄×C₇₁⋊C₈	—	2	8	no	—
37	C₄×C₇₁⋊C₈	—	2	4	no	—
38	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₄×Q₈×C₇₁	—	2	4	no	—
40	C₄×Q₈×C₇₁	—	2	2	no	—
41	C₄×Q₈×C₇₁	—	2	2	no	—
42	C₄×Q₈×C₇₁	C₈	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
55	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
59	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
62	#95×C₇₁	C₂×C₄	1	2	yes	—
63	#95×C₇₁	C₂×C₄	1	2	yes	—
64	#95×C₇₁	C₂×C₄	1	2	yes	—
65	C₂×C₇₁⋊C₈	—	2	2	no	—
66	C₂×C₇₁⋊C₈	—	2	2	no	—
67	C₂×C₇₁⋊C₈	—	2	4	no	—
68	C₂×C₇₁⋊C₈	—	2	2	no	—
69	C₂×C₇₁⋊C₈	—	2	2	no	—
70	C₂×C₇₁⋊C₈	—	2	4	no	—
71	C₂×C₇₁⋊C₈	—	2	2	no	—
72	C₂×C₇₁⋊C₈	—	2	4	no	—
73	C₂×C₇₁⋊C₈	#41	1	4	yes	—
74	C₂×Q₈×C₇₁	—	2	2	no	—
75	C₂×Q₈×C₇₁	—	2	2	no	—
76	C₂×Q₈×C₇₁	C₂×C₈	1	4	yes	—
77	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₇₁	—	2	4	no	—
89	C₄⋊C₄×C₇₁	—	2	2	no	—
90	C₄⋊C₄×C₇₁	—	2	4	no	—
91	C₄⋊C₄×C₇₁	C₂×C₈	1	8	yes	—
92	C₄⋊C₄×C₇₁	C₂×C₈	1	4	yes	—
93	C₄⋊C₄×C₇₁	M₁₆	1	4	yes	—
94	C₄²×C₇₁	—	2	4	no	—
95	C₄²×C₇₁	—	2	2	no	—
96	C₄²×C₇₁	—	2	2	no	—
97	C₄²×C₇₁	—	2	4	no	—
98	C₄²×C₇₁	—	2	2	no	—
99	C₄²×C₇₁	—	2	2	no	—
100	C₄²×C₇₁	—	2	4	no	—
101	C₄²×C₇₁	—	2	2	no	—
102	C₄²×C₇₁	—	2	2	no	—
103	C₄²×C₇₁	—	4	1	no	—
104	C₄²×C₇₁	—	4	2	no	—
105	C₄²×C₇₁	C₂×C₈	1	8	yes	—
106	C₄²×C₇₁	C₂×C₈	1	4	yes	—
107	C₄²×C₇₁	#41	1	2	yes	—
108	C₄²×C₇₁	#41	1	2	yes	—
109	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
110	Q₈×C₇₁	—	2	8	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	2	2	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	2	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	2	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	—	2	2	no	—
122	C₂×C₄×C₇₁	—	2	1	no	—
123	C₂×C₄×C₇₁	—	2	1	no	—
124	C₂×C₄×C₇₁	—	2	2	no	—
125	C₂×C₄×C₇₁	—	2	2	no	—
126	C₂×C₄×C₇₁	—	2	1	no	—
127	C₂×C₄×C₇₁	—	2	1	no	—
128	C₂×C₄×C₇₁	—	2	2	no	—
129	C₂×C₄×C₇₁	—	2	2	no	—
130	C₂×C₄×C₇₁	—	2	1	no	—
131	C₂×C₄×C₇₁	#117	1	2	yes	—
132	C₂×C₄×C₇₁	#117	1	2	yes	—
133	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
135	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₇₁	C₂²×C₈	1	4	yes	—
137	C₂×C₄×C₇₁	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
139	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
140	C₂×C₄×C₇₁	#131	1	2	yes	—
141	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
142	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
145	C₇₁⋊C₈	—	2	4	no	—
146	C₇₁⋊C₈	—	4	2	no	—
147	C₇₁⋊C₈	—	4	4	no	—
148	C₂²×C₇₁	C₂×#117	1	1	yes	characteristic
149	C₂²×C₇₁	C₄×#41	1	1	yes	characteristic
150	C₂²×C₇₁	#519	1	1	yes	characteristic
151	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
152	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
153	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
154	C₂²×C₇₁	#568	1	1	yes	characteristic
155	C₄×C₇₁	—	2	4	no	—
156	C₄×C₇₁	—	2	2	no	—
157	C₄×C₇₁	—	2	4	no	—
158	C₄×C₇₁	—	2	2	no	—
159	C₄×C₇₁	—	2	4	no	—
160	C₄×C₇₁	—	2	2	no	—
161	C₄×C₇₁	—	2	2	no	—
162	C₄×C₇₁	—	4	2	no	—
163	C₄×C₇₁	—	4	1	no	—
164	C₄×C₇₁	#431	1	4	yes	—
165	C₄×C₇₁	#552	1	2	yes	—
166	C₄×C₇₁	#556	1	2	yes	—
167	#22828	—	71	1	no	maximal, Sylow
168	C₂×C₇₁	#2356	1	1	yes	characteristic
169	C₂×C₇₁	C₂×#431	1	1	yes	characteristic
170	C₂×C₇₁	#2925	1	1	yes	characteristic
171	C₂×C₇₁	C₈×#41	1	1	yes	characteristic
172	C₂×C₇₁	#2970	1	1	yes	characteristic
173	C₂×C₇₁	#2982	1	1	yes	characteristic
174	C₂×C₇₁	#2984	1	1	yes	characteristic
175	C₄²×Q₈	D₇₁	1	1	yes	characteristic
176	#1984	—	71	2	no	—
177	#1916	—	71	4	no	—
178	#1839	—	71	2	no	—
179	#1831	—	71	1	no	—
180	C₈×C₄⋊C₄	—	71	2	no	—
181	C₄×C₄⋊C₈	—	71	1	no	—
182	#1744	—	71	2	no	—
183	C₇₁	#22828	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
185	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
186	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
187	C₂×C₄⋊C₈	—	71	2	no	—
188	C₂×C₄⋊C₈	—	71	2	no	—
189	C₂×C₄×C₈	—	71	4	no	—
190	C₂×C₄×C₈	—	71	2	no	—
191	C₄×C₄⋊C₄	Dic₇₁	1	1	yes	characteristic
192	C₄×C₄⋊C₄	Dic₇₁	1	2	yes	—
193	C₄×C₄⋊C₄	Dic₇₁	1	2	yes	—
194	C₄×C₄⋊C₄	Dic₇₁	1	1	yes	characteristic
195	C₄×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
196	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
197	C₄³	C₂×D₇₁	1	1	yes	characteristic
198	C₄³	C₂×D₇₁	1	2	yes	—
199	#336	—	71	2	no	—
200	#336	—	71	2	no	—
201	#336	—	71	1	no	—
202	#336	—	71	4	no	—
203	#336	—	71	4	no	—
204	#336	—	71	4	no	—
205	#336	—	71	1	no	—
206	C₂²×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
207	C₂²×C₈	—	71	2	no	—
208	C₂²×C₈	—	71	4	no	—
209	C₂²×C₈	—	71	2	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	2	no	—
213	C₄×Q₈	C₇₁⋊C₈	1	8	yes	complemented
214	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
217	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
218	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
219	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
220	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
221	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
222	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
223	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
224	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
226	C₂×C₄²	C₂×Dic₇₁	1	2	yes	—
227	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
229	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
230	C₂×C₄²	C₂²×D₇₁	1	2	yes	—
231	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
233	C₄⋊C₈	—	142	4	no	—
234	C₄⋊C₈	—	142	4	no	—
235	C₄×C₈	—	142	8	no	—
236	C₄×C₈	—	142	4	no	—
237	#95	C₂×Dic₇₁	1	2	yes	—
238	#95	C₂×Dic₇₁	1	2	yes	—
239	#95	C₂×Dic₇₁	1	2	yes	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	C₂×C₇₁⋊C₈	1	4	yes	—
243	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
245	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
248	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₇₁	1	2	yes	—
251	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
253	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
254	C₂×C₈	—	71	4	no	—
255	C₂×C₈	—	142	4	no	—
256	C₂×C₈	—	142	2	no	—
257	C₂×C₈	—	142	2	no	—
258	C₂×C₈	—	142	4	no	—
259	C₂×C₈	—	142	2	no	—
260	C₂×C₈	—	142	4	no	—
261	C₂×C₈	—	142	2	no	—
262	C₂×C₈	—	142	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	C₂×C₇₁⋊C₈	1	8	yes	—
267	C₄⋊C₄	C₂×C₇₁⋊C₈	1	4	yes	—
268	C₄⋊C₄	#11765849	1	4	yes	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	2	no	—
280	C₄²	C₂×C₇₁⋊C₈	1	8	yes	—
281	C₄²	C₂×C₇₁⋊C₈	1	4	yes	—
282	C₄²	#11765865	1	2	yes	—
283	C₄²	#11765865	1	2	yes	—
284	C₂³	C₂³×Dic₇₁	1	1	yes	characteristic
285	Q₈	—	2	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	#14556087	1	2	yes	—
307	C₂×C₄	#14556087	1	2	yes	—
308	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
309	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
310	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
311	C₂×C₄	C₂²×C₇₁⋊C₈	1	4	yes	—
312	C₂×C₄	C₂²×C₇₁⋊C₈	1	2	yes	—
313	C₂×C₄	C₂×#11765849	1	4	yes	—
314	C₂×C₄	C₂×#11765849	1	2	yes	—
315	C₂×C₄	#14556135	1	2	yes	—
316	C₂×C₄	#41×D₇₁	1	2	yes	—
317	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
318	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
319	C₂×C₄	#14556176	1	2	yes	—
320	C₈	—	284	2	no	—
321	C₈	—	284	4	no	—
322	C₈	—	142	4	no	—
323	C₂²	C₂×#14556087	1	1	yes	characteristic
324	C₂²	#41×Dic₇₁	1	1	yes	characteristic
325	C₂²	#17986477	1	1	yes	characteristic
326	C₂²	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
327	C₂²	C₂²×#11765849	1	1	yes	characteristic
328	C₂²	C₂×#14556135	1	1	yes	characteristic
329	C₂²	#17986766	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	4	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	1	no	—
339	C₄	#17986052	1	4	yes	—
340	C₄	#17986690	1	2	yes	—
341	C₄	#17986721	1	2	yes	—
342	C₂	#22591305	1	1	yes	characteristic
343	C₂	C₂×#17986052	1	1	yes	characteristic
344	C₂	#22597006	1	1	yes	characteristic
345	C₂	#41×C₇₁⋊C₈	1	1	yes	characteristic
346	C₂	#22597354	1	1	yes	characteristic
347	C₂	#22597421	1	1	yes	characteristic
348	C₂	#22597429	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₁
commutator	C₂²×C₇₁	C₂³×C₈
Fitting	C₄²×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂³×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₄²×Q₈	D₇₁
hypocenter	C₇₁	#22828
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
42. C₄×Q₈×C₇₁	C₈	semidirect
183. C₇₁	#22828	semidirect
213. C₄×Q₈	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22589277	2
#22588899	2
#22588483	2
#22588457	2
C₄⋊C₄×C₇₁⋊C₈	2
C₄×C₄⋊(C₇₁⋊C₈)	2
#22588086	2
C₄²×Q₈×C₇₁	2
#22828	71