Subgroups — #31896608

Order 18176.

430 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598926	C₂	1	1	yes	maximal, characteristic
2	#22589284	C₂	1	2	yes	maximal
3	C₄×#14556062	C₂	1	1	yes	maximal, characteristic
4	#22586840	C₂	1	2	yes	maximal
5	C₈×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986556	C₂²	1	2	yes	—
7	#17986404	C₄	1	1	yes	characteristic, complemented
8	#17986379	C₄	1	1	yes	characteristic, complemented
9	#17986350	C₄	1	2	yes	complemented
10	#17986240	—	2	1	no	—
11	#17986240	—	2	1	no	—
12	C₄×Dic₂₈₄	—	2	1	no	—
13	C₄×Dic₂₈₄	—	2	1	no	—
14	C₂×#14556062	—	2	1	no	—
15	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
16	#17985762	—	2	1	no	—
17	#17985760	—	2	4	no	—
18	#17985737	—	2	4	no	—
19	#431×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	#322×C₇₁	—	2	1	no	—
23	C₈²×C₇₁	—	2	1	no	—
24	C₂×Dic₂₈₄	—	2	1	no	—
25	C₂×Dic₂₈₄	—	2	1	no	—
26	#14556128	C₂×C₄	1	2	yes	—
27	#14556128	C₂×C₄	1	2	yes	—
28	#14556120	—	2	2	no	—
29	#14556107	—	2	2	no	—
30	#14556107	—	2	2	no	—
31	C₄×Dic₁₄₂	—	2	2	no	—
32	C₄×Dic₁₄₂	—	2	2	no	—
33	#14556087	—	2	2	no	—
34	C₂×C₄⋊Dic₇₁	—	2	2	no	—
35	C₂×C₄×Dic₇₁	—	2	2	no	—
36	#14556062	—	2	4	no	—
37	#14556062	—	2	1	no	—
38	#14556062	—	2	1	no	—
39	#14556053	—	2	2	no	—
40	#14556053	—	2	2	no	—
41	C₇₁⋊#95	—	2	2	no	—
42	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
43	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₈×C₇₁	—	2	1	no	—
47	C₂²⋊C₈×C₇₁	D₄	1	2	yes	—
48	#97×C₇₁	—	2	1	no	—
49	C₄×C₈×C₇₁	—	2	1	no	—
50	C₄×C₈×C₇₁	—	2	1	no	—
51	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
55	C₂×Dic₁₄₂	—	2	2	no	—
56	C₂×Dic₁₄₂	—	2	2	no	—
57	C₂²×Dic₇₁	—	4	2	no	—
58	Dic₂₈₄	—	4	2	no	—
59	Dic₂₈₄	—	4	2	no	—
60	C₄⋊Dic₇₁	—	2	4	no	—
61	C₄⋊Dic₇₁	—	2	2	no	—
62	C₄⋊Dic₇₁	—	2	2	no	—
63	C₄⋊Dic₇₁	—	2	2	no	—
64	C₄⋊Dic₇₁	—	2	2	no	—
65	C₄⋊Dic₇₁	—	4	2	no	—
66	Dic₇₁⋊C₄	—	4	2	no	—
67	Dic₇₁⋊C₄	—	4	2	no	—
68	C₂²⋊Dic₇₁	—	4	2	no	—
69	C₄×Dic₇₁	—	4	2	no	—
70	C₄×Dic₇₁	—	4	4	no	—
71	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
74	M₁₆×C₇₁	—	2	2	no	—
75	M₁₆×C₇₁	Q₁₆	1	4	yes	complemented
76	C₂×C₈×C₇₁	—	2	2	no	—
77	C₂×C₈×C₇₁	—	2	2	no	—
78	C₂×C₈×C₇₁	—	2	2	no	—
79	C₂×C₈×C₇₁	—	4	1	no	—
80	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₇₁	#41	1	1	yes	characteristic
87	C₄²×C₇₁	#41	1	1	yes	characteristic
88	C₂³×C₇₁	#127	1	1	yes	characteristic
89	C₂×C₄×C₇₁	—	2	1	no	—
90	C₂×C₄×C₇₁	—	2	1	no	—
91	C₂×C₄×C₇₁	—	4	1	no	—
92	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
96	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
97	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
99	C₂×C₄×C₇₁	#135	1	2	yes	—
100	C₂×Dic₇₁	—	4	4	no	—
101	C₂×Dic₇₁	—	4	2	no	—
102	C₂×Dic₇₁	—	4	2	no	—
103	C₂×Dic₇₁	—	8	2	no	—
104	C₂×Dic₇₁	—	8	2	no	—
105	Dic₁₄₂	—	4	2	no	—
106	Dic₁₄₂	—	4	2	no	—
107	Dic₁₄₂	—	8	2	no	—
108	C₈×C₇₁	—	2	1	no	—
109	C₈×C₇₁	—	2	1	no	—
110	C₈×C₇₁	—	2	1	no	—
111	C₈×C₇₁	—	2	4	no	—
112	C₈×C₇₁	—	4	2	no	—
113	C₈×C₇₁	#104	1	2	yes	—
114	C₂²×C₇₁	—	2	1	no	—
115	C₂²×C₇₁	—	2	1	no	—
116	C₂²×C₇₁	#390	1	1	yes	characteristic
117	C₂²×C₇₁	#494	1	1	yes	characteristic
118	C₂²×C₇₁	#495	1	1	yes	characteristic
119	C₄×C₇₁	—	2	1	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
123	C₄×C₇₁	#421	1	1	yes	characteristic
124	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
125	C₄×C₇₁	#467	1	2	yes	—
126	Dic₇₁	—	8	4	no	—
127	Dic₇₁	—	8	2	no	—
128	Dic₇₁	—	8	2	no	—
129	#9718	—	71	1	no	maximal, Sylow
130	C₂×C₇₁	—	2	1	no	—
131	C₂×C₇₁	—	2	1	no	—
132	C₂×C₇₁	#1969	1	1	yes	characteristic
133	C₂×C₇₁	#1976	1	1	yes	characteristic
134	C₂×C₇₁	#1980	1	1	yes	characteristic
135	#3170	—	71	1	no	—
136	#1986	—	71	2	no	—
137	C₄×#107	—	71	1	no	—
138	#1503	—	71	2	no	—
139	C₈×M₁₆	D₇₁	1	1	yes	characteristic
140	C₇₁	#9718	1	1	yes	characteristic, complemented, Sylow
141	#533	—	71	2	no	—
142	#500	—	71	1	no	—
143	#494	—	71	1	no	—
144	#487	—	71	2	no	—
145	#467	—	142	1	no	—
146	#467	—	142	1	no	—
147	C₄×Q₁₆	—	142	1	no	—
148	C₄×Q₁₆	—	142	1	no	—
149	#431	C₂×D₇₁	1	1	yes	characteristic
150	C₂×#107	—	142	1	no	—
151	C₄×M₁₆	C₂×D₇₁	1	1	yes	characteristic
152	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
153	C₄×C₄⋊C₄	—	71	2	no	—
154	#340	—	142	1	no	—
155	#339	—	142	4	no	—
156	#328	—	142	4	no	—
157	#322	—	2	1	no	—
158	C₈²	—	2	1	no	—
159	C₂×Q₁₆	—	142	1	no	—
160	C₂×Q₁₆	—	142	1	no	—
161	C₂×M₁₆	D₂₈₄	1	2	yes	—
162	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
163	C₂²×C₈	D₂₈₄	1	1	yes	characteristic
164	C₄⋊Q₈	—	71	2	no	—
165	C₄⋊Q₈	—	71	2	no	—
166	#125	—	142	2	no	—
167	C₂²⋊Q₈	—	142	2	no	—
168	C₂²⋊Q₈	—	142	2	no	—
169	C₄×Q₈	—	142	2	no	—
170	C₄×Q₈	—	142	2	no	—
171	#117	—	142	2	no	—
172	C₂×C₄⋊C₄	—	142	2	no	—
173	C₂×C₄²	—	142	2	no	—
174	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
175	#107	—	142	4	no	—
176	#107	—	142	1	no	—
177	#107	—	142	1	no	—
178	C₄⋊C₈	—	2	1	no	—
179	#103	—	142	2	no	—
180	#103	—	142	2	no	—
181	C₂²⋊C₈	D₂₈₄	1	2	yes	—
182	#97	—	2	1	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	—	2	1	no	—
185	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
186	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
188	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
189	#95	—	142	2	no	—
190	C₂×Q₈	—	142	2	no	—
191	C₂×Q₈	—	142	2	no	—
192	C₂²×C₄	—	284	2	no	—
193	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
196	Q₁₆	—	284	2	no	—
197	Q₁₆	—	284	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	Dic₂₈₄	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
207	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
208	C₄⋊C₄	—	284	2	no	—
209	C₄⋊C₄	—	284	2	no	—
210	C₄⋊C₄	—	284	2	no	—
211	C₄⋊C₄	—	142	4	no	—
212	C₄⋊C₄	—	142	2	no	—
213	C₄⋊C₄	—	142	2	no	—
214	C₄⋊C₄	—	142	2	no	—
215	C₄⋊C₄	—	142	2	no	—
216	C₂²⋊C₄	—	284	2	no	—
217	C₄²	—	284	2	no	—
218	C₄²	—	284	4	no	—
219	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
221	C₄²	#11765865	1	1	yes	characteristic
222	C₄²	#11765865	1	1	yes	characteristic
223	C₂³	#14556125	1	1	yes	characteristic
224	Q₈	—	284	2	no	—
225	Q₈	—	284	2	no	—
226	Q₈	—	568	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	284	4	no	—
231	C₂×C₄	—	284	2	no	—
232	C₂×C₄	—	284	2	no	—
233	C₂×C₄	—	568	2	no	—
234	C₂×C₄	—	568	2	no	—
235	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
237	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
238	C₂×C₄	#14556102	1	1	yes	characteristic
239	C₂×C₄	#14556102	1	1	yes	characteristic
240	C₂×C₄	#14556116	1	1	yes	characteristic
241	C₂×C₄	C₂×Dic₂₈₄	1	2	yes	—
242	C₂×C₄	#14556150	1	2	yes	—
243	C₈	—	2	4	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	4	2	no	—
248	C₈	#14556054	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	#17985901	1	1	yes	characteristic
252	C₂²	#17986379	1	1	yes	characteristic
253	C₂²	#17986383	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	568	4	no	—
258	C₄	—	568	2	no	—
259	C₄	—	568	2	no	—
260	C₄	C₂×#14556054	1	1	yes	characteristic
261	C₄	#17986015	1	1	yes	characteristic
262	C₄	C₄×Dic₂₈₄	1	2	yes	—
263	C₄	#17986240	1	2	yes	—
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	#22589201	1	1	yes	characteristic
267	C₂	#22589238	1	1	yes	characteristic
268	C₂	#22589259	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556102
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₈×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#390
prime radical	C₂³×C₇₁	#127
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₈×M₁₆	D₇₁
hypocenter	C₇₁	#9718
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
7. #17986404	C₄	semidirect
8. #17986379	C₄	semidirect
9. #17986350	C₄	semidirect
75. M₁₆×C₇₁	Q₁₆	semidirect
140. C₇₁	#9718	semidirect
199. M₁₆	Dic₂₈₄	semidirect

Maximal subgroups

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

subgroup	index
#22598926	2
#22589284	2
C₄×#14556062	2
#22586840	2
C₈×M₁₆×C₇₁	2
#9718	71