Subgroups — #31912341

Order 18176.

554 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556064	C₂	1	1	yes	maximal, characteristic
3	#22586433	C₂	1	4	yes	maximal, complemented
4	C₂²×#108×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11765849	—	2	1	no	—
6	C₂×#14556068	—	2	2	no	—
7	C₂×#14556068	—	2	2	no	—
8	C₂×#14556068	C₄	1	2	yes	—
9	C₂×#14556068	C₄	1	2	yes	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	C₂×#14556064	—	2	1	no	—
12	C₂×#14556064	—	2	2	no	—
13	C₂×#14556064	C₄	1	4	yes	—
14	C₂×#14556064	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₇₁	—	2	1	no	—
16	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×#108×C₇₁	—	2	1	no	—
18	C₂×#108×C₇₁	—	2	2	no	—
19	C₂×#108×C₇₁	C₄	1	4	yes	—
20	C₂×#108×C₇₁	C₂²	1	2	yes	—
21	C₂×#11765849	—	2	4	no	—
22	C₂×#11765849	—	2	2	no	—
23	C₂×#11765849	—	4	1	no	—
24	C₂×#11765849	—	4	2	no	—
25	C₂²×C₇₁⋊C₈	—	4	1	no	—
26	#14556068	—	2	4	no	—
27	#14556068	—	2	4	no	—
28	C₂×C₇₁⋊C₁₆	—	2	2	no	—
29	C₂×C₇₁⋊C₁₆	—	2	2	no	—
30	C₂×C₇₁⋊C₁₆	C₂×C₄	1	4	yes	—
31	#14556064	—	2	4	no	—
32	#14556064	—	2	2	no	—
33	#14556064	C₂×C₄	1	4	yes	—
34	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₇₁	—	2	4	no	—
36	C₂×M₁₆×C₇₁	—	2	2	no	—
37	C₂×M₁₆×C₇₁	—	4	1	no	—
38	C₂×M₁₆×C₇₁	—	4	2	no	—
39	C₂²×C₈×C₇₁	—	4	1	no	—
40	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	D₄	1	2	yes	—
46	C₂²×C₈×C₇₁	D₄	1	2	yes	—
47	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
48	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
49	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
50	#108×C₇₁	—	2	4	no	—
51	#108×C₇₁	—	2	2	no	—
52	#108×C₇₁	C₂×C₄	1	4	yes	—
53	#11765849	—	2	4	no	—
54	#11765849	—	4	4	no	—
55	#11765849	—	4	2	no	—
56	C₂×C₇₁⋊C₈	—	4	4	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₇₁⋊C₁₆	—	2	4	no	—
59	C₂⁴×C₇₁	Q₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
70	M₁₆×C₇₁	—	2	4	no	—
71	M₁₆×C₇₁	—	4	4	no	—
72	M₁₆×C₇₁	—	4	2	no	—
73	C₂×C₈×C₇₁	—	4	4	no	—
74	C₂×C₈×C₇₁	—	4	2	no	—
75	C₂×C₈×C₇₁	C₄²	1	2	yes	—
76	C₂×C₈×C₇₁	C₄²	1	2	yes	—
77	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
84	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
86	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
93	C₂³×C₇₁	—	4	2	no	—
94	C₂³×C₇₁	#103	1	2	yes	—
95	C₂³×C₇₁	#103	1	2	yes	—
96	C₂³×C₇₁	#107	1	1	yes	characteristic
97	C₂³×C₇₁	#107	1	1	yes	characteristic
98	C₂³×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₇₁	#95	1	2	yes	—
100	C₂×C₄×C₇₁	#95	1	2	yes	—
101	C₂×C₄×C₇₁	#102	1	2	yes	—
102	C₂×C₄×C₇₁	#102	1	2	yes	—
103	C₂×C₄×C₇₁	#102	1	2	yes	—
104	C₂×C₄×C₇₁	#102	1	2	yes	—
105	C₂×C₄×C₇₁	#103	1	2	yes	—
106	C₂×C₄×C₇₁	#103	1	2	yes	—
107	C₂×C₄×C₇₁	#106	1	2	yes	—
108	C₂×C₄×C₇₁	#106	1	2	yes	—
109	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₈×C₇₁	—	4	4	no	—
118	C₈×C₇₁	#95	1	2	yes	—
119	C₈×C₇₁	#95	1	2	yes	—
120	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
121	C₈×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₈×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₇₁⋊C₈	—	4	4	no	—
125	C₂²×C₇₁	—	2	2	no	—
126	C₂²×C₇₁	—	2	2	no	—
127	C₂²×C₇₁	—	2	2	no	—
128	C₂²×C₇₁	—	4	2	no	—
129	C₂²×C₇₁	—	4	2	no	—
130	C₂²×C₇₁	#340	1	2	yes	—
131	C₂²×C₇₁	#340	1	2	yes	—
132	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
133	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
134	C₂²×C₇₁	C₂×#107	1	1	yes	characteristic
135	C₄×C₇₁	#340	1	2	yes	—
136	C₄×C₇₁	#340	1	2	yes	—
137	C₄×C₇₁	C₂×#95	1	1	yes	characteristic
138	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
139	C₄×C₇₁	C₂×#102	1	1	yes	characteristic
140	C₄×C₇₁	C₂×#106	1	1	yes	characteristic
141	#12559	—	71	1	no	maximal, Sylow
142	C₂×C₇₁	—	2	2	no	—
143	C₂×C₇₁	—	2	2	no	—
144	C₂×C₇₁	—	2	1	no	—
145	C₂×C₇₁	—	2	1	no	—
146	C₂×C₇₁	C₂×#340	1	1	yes	characteristic
147	C₂×C₇₁	#2161	1	2	yes	—
148	C₂²×M₃₂	—	71	1	no	—
149	C₂²×#108	—	71	1	no	—
150	C₂²×#108	D₇₁	1	1	yes	characteristic
151	#1381	—	71	4	no	—
152	C₇₁	#12559	1	1	yes	characteristic, complemented, Sylow
153	C₂²×M₁₆	—	2	1	no	—
154	C₂²×M₁₆	—	142	1	no	—
155	C₂³×C₈	C₂×D₇₁	1	1	yes	characteristic
156	C₂×M₃₂	—	71	2	no	—
157	C₂×M₃₂	—	71	2	no	—
158	C₂×M₃₂	—	142	2	no	—
159	C₂×M₃₂	—	142	2	no	—
160	C₂²×C₁₆	—	71	2	no	—
161	C₂×#108	—	2	1	no	—
162	C₂×#108	—	2	2	no	—
163	C₂×#108	—	71	4	no	—
164	C₂×#108	—	71	2	no	—
165	C₂×#108	—	142	2	no	—
166	C₂×#108	—	142	1	no	—
167	C₂×#108	Dic₇₁	1	4	yes	—
168	C₂×#108	C₂×D₇₁	1	2	yes	—
169	C₂³×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	4	1	no	—
173	C₂×M₁₆	—	4	2	no	—
174	C₂×M₁₆	—	284	1	no	—
175	C₂×M₁₆	—	284	2	no	—
176	C₂×M₁₆	—	142	4	no	—
177	C₂×M₁₆	—	142	2	no	—
178	C₂²×C₈	—	4	1	no	—
179	C₂²×C₈	—	284	1	no	—
180	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
181	C₂²×C₈	Dic₁₄₂	1	1	yes	characteristic
182	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
183	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
184	C₂²×C₈	C₄×D₇₁	1	2	yes	—
185	C₂²×C₈	D₂₈₄	1	2	yes	—
186	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₇₁	1	2	yes	—
189	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
190	M₃₂	—	142	4	no	—
191	M₃₂	—	142	4	no	—
192	C₂×C₁₆	—	71	4	no	—
193	C₂×C₁₆	—	142	2	no	—
194	C₂×C₁₆	—	142	2	no	—
195	#108	—	2	4	no	—
196	#108	—	2	2	no	—
197	#108	—	71	4	no	—
198	#108	—	142	4	no	—
199	#108	—	142	2	no	—
200	#108	C₂×Dic₇₁	1	4	yes	—
201	C₂⁴	#11765855	1	1	yes	characteristic
202	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
203	C₂²×C₄	C₂²⋊Dic₇₁	1	2	yes	—
204	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
205	C₂²×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
206	C₂²×C₄	#11765850	1	1	yes	characteristic
207	C₂²×C₄	#11765850	1	1	yes	characteristic
208	C₂²×C₄	#11765852	1	2	yes	—
209	C₂²×C₄	#11765853	1	2	yes	—
210	C₂²×C₄	#11765855	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	284	4	no	—
216	M₁₆	—	284	2	no	—
217	M₁₆	—	142	4	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	284	4	no	—
221	C₂×C₈	—	284	2	no	—
222	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
223	C₂×C₈	C₄×Dic₇₁	1	2	yes	—
224	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
225	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
226	C₂×C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
227	C₂×C₈	C₂²⋊Dic₇₁	1	2	yes	—
228	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
229	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
230	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
231	C₂×C₈	Dic₇₁⋊C₄	1	2	yes	—
232	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
233	C₂×C₈	C₄⋊Dic₇₁	1	2	yes	—
234	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
236	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
237	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
238	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
239	C₂×C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
240	C₁₆	—	142	4	no	—
241	C₂³	—	4	2	no	—
242	C₂³	#14556051	1	2	yes	—
243	C₂³	#14556052	1	2	yes	—
244	C₂³	#14556061	1	1	yes	characteristic
245	C₂³	#14556061	1	1	yes	characteristic
246	C₂³	C₂×#11765855	1	1	yes	characteristic
247	C₂×C₄	C₇₁⋊#95	1	2	yes	—
248	C₂×C₄	C₇₁⋊#95	1	2	yes	—
249	C₂×C₄	#14556048	1	2	yes	—
250	C₂×C₄	#14556048	1	2	yes	—
251	C₂×C₄	#14556049	1	2	yes	—
252	C₂×C₄	#14556049	1	2	yes	—
253	C₂×C₄	#14556051	1	2	yes	—
254	C₂×C₄	#14556052	1	2	yes	—
255	C₂×C₄	#14556059	1	2	yes	—
256	C₂×C₄	#14556059	1	2	yes	—
257	C₂×C₄	#14556061	1	1	yes	characteristic
258	C₂×C₄	#14556061	1	1	yes	characteristic
259	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
261	C₂×C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
263	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
264	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
265	C₈	—	4	4	no	—
266	C₈	—	284	4	no	—
267	C₈	C₇₁⋊#95	1	2	yes	—
268	C₈	C₇₁⋊#95	1	2	yes	—
269	C₈	C₂×C₄×Dic₇₁	1	1	yes	characteristic
270	C₈	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
271	C₈	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
272	C₈	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	#17985761	1	2	yes	—
279	C₂²	#17985761	1	2	yes	—
280	C₂²	C₂×#14556051	1	1	yes	characteristic
281	C₂²	C₂×#14556052	1	1	yes	characteristic
282	C₂²	C₂×#14556061	1	1	yes	characteristic
283	C₄	#17985761	1	2	yes	—
284	C₄	#17985761	1	2	yes	—
285	C₄	C₂×C₇₁⋊#95	1	1	yes	characteristic
286	C₄	C₂×#14556048	1	1	yes	characteristic
287	C₄	C₂×#14556049	1	1	yes	characteristic
288	C₄	C₂×#14556059	1	1	yes	characteristic
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	2	no	—
292	C₂	—	2	2	no	—
293	C₂	C₂×#17985761	1	1	yes	characteristic
294	C₂	#22590134	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11765850
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂²×#108×C₇₁	C₂
socle	C₂²×C₇₁	C₂×#107
prime radical	C₂⁴×C₇₁	Q₁₆
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂²×#108	D₇₁
hypocenter	C₇₁	#12559
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. #22586433	C₂	semidirect
152. C₇₁	#12559	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556068	2
C₂²×#14556064	2
#22586433	2
C₂²×#108×C₇₁	2
#12559	71