Subgroups — #31890708

Order 18176.

362 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22597274	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985752	C₂	1	1	yes	maximal, characteristic
3	#22587031	C₂	1	2	yes	maximal
4	#22586214	C₂	1	2	yes	maximal
5	C₇₁×#1848	C₂	1	1	yes	maximal, characteristic
6	#17986556	C₂²	1	1	yes	characteristic
7	#17986129	—	2	1	no	—
8	#17986129	C₄	1	2	yes	—
9	C₈×Dic₁₄₂	—	2	1	no	—
10	C₈×Dic₁₄₂	C₄	1	2	yes	—
11	#17986025	—	2	1	no	—
12	#17986023	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
14	#17985954	—	2	1	no	—
15	#17985752	—	2	1	no	—
16	#17985752	—	2	1	no	—
17	#17985747	—	2	2	no	—
18	#17985744	—	2	2	no	—
19	C₂×#108×C₇₁	—	2	1	no	—
20	C₂×#108×C₇₁	—	2	1	no	—
21	#423×C₇₁	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#11765849	—	2	1	no	—
24	C₂²×C₇₁⋊C₈	—	2	2	no	—
25	#14556128	D₄	1	1	yes	characteristic
26	#14556128	D₄	1	1	yes	characteristic
27	#14556120	—	2	1	no	—
28	#14556107	—	2	1	no	—
29	#14556107	Q₈	1	2	yes	—
30	C₄×Dic₁₄₂	—	2	1	no	—
31	C₄×Dic₁₄₂	C₂×C₄	1	2	yes	—
32	#14556087	—	2	1	no	—
33	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
34	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
36	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	2	yes	—
37	Dic₇₁⋊C₈	—	2	2	no	—
38	Dic₇₁⋊C₈	—	2	2	no	—
39	#14556038	—	2	2	no	—
40	C₈×Dic₇₁	—	2	2	no	—
41	C₂×M₁₆×C₇₁	—	2	2	no	—
42	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
45	#108×C₇₁	—	4	2	no	—
46	#108×C₇₁	—	4	2	no	—
47	C₄⋊C₈×C₇₁	—	2	2	no	—
48	C₄⋊C₈×C₇₁	—	2	2	no	—
49	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
53	C₂×Dic₁₄₂	—	2	1	no	—
54	C₂×Dic₁₄₂	C₄⋊C₄	1	2	yes	—
55	#11765849	—	4	1	no	—
56	#11765849	—	4	1	no	—
57	C₂×C₇₁⋊C₈	—	2	2	no	—
58	C₂×C₇₁⋊C₈	—	4	2	no	—
59	C₂×C₇₁⋊C₈	—	8	2	no	—
60	C₄⋊Dic₇₁	—	2	1	no	—
61	C₄⋊Dic₇₁	—	2	1	no	—
62	C₄⋊Dic₇₁	—	2	1	no	—
63	C₄⋊Dic₇₁	C₄⋊C₄	1	2	yes	—
64	Dic₇₁⋊C₄	—	2	2	no	—
65	Dic₇₁⋊C₄	—	2	2	no	—
66	C₂²⋊Dic₇₁	—	2	2	no	—
67	C₄×Dic₇₁	—	2	2	no	—
68	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
71	M₁₆×C₇₁	—	4	2	no	—
72	M₁₆×C₇₁	—	4	2	no	—
73	C₂×C₈×C₇₁	—	2	1	no	—
74	C₂×C₈×C₇₁	—	2	1	no	—
75	C₂×C₈×C₇₁	—	4	2	no	—
76	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₁	SD₁₆	1	2	yes	—
81	C₂×C₈×C₇₁	Q₁₆	1	2	yes	—
82	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇₁	#41	1	1	yes	characteristic
86	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇₁	—	2	1	no	—
88	C₂×C₄×C₇₁	—	2	1	no	—
89	C₂×C₄×C₇₁	—	4	1	no	—
90	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
95	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
99	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
100	C₂×Dic₇₁	—	2	2	no	—
101	C₂×Dic₇₁	—	2	2	no	—
102	Dic₁₄₂	—	4	1	no	—
103	Dic₁₄₂	—	4	1	no	—
104	Dic₁₄₂	—	4	2	no	—
105	C₈×C₇₁	—	2	1	no	—
106	C₈×C₇₁	—	2	1	no	—
107	C₈×C₇₁	—	8	2	no	—
108	C₈×C₇₁	#103	1	2	yes	—
109	C₈×C₇₁	#103	1	2	yes	—
110	C₇₁⋊C₈	—	4	2	no	—
111	C₇₁⋊C₈	—	8	2	no	—
112	C₂²×C₇₁	—	2	1	no	—
113	C₂²×C₇₁	—	2	1	no	—
114	C₂²×C₇₁	#380	1	1	yes	characteristic
115	C₂²×C₇₁	#475	1	1	yes	characteristic
116	C₂²×C₇₁	#477	1	1	yes	characteristic
117	C₄×C₇₁	—	2	1	no	—
118	C₄×C₇₁	—	2	1	no	—
119	C₄×C₇₁	—	2	1	no	—
120	C₄×C₇₁	C₂×#103	1	1	yes	characteristic
121	C₄×C₇₁	#417	1	1	yes	characteristic
122	C₄×C₇₁	#460	1	1	yes	characteristic
123	C₄×C₇₁	#462	1	1	yes	characteristic
124	C₄×C₇₁	#472	1	1	yes	characteristic
125	C₄×C₇₁	#473	1	1	yes	characteristic
126	Dic₇₁	—	4	2	no	—
127	Dic₇₁	—	4	2	no	—
128	#8685	—	71	1	no	maximal, Sylow
129	C₂×C₇₁	—	2	1	no	—
130	C₂×C₇₁	—	2	1	no	—
131	C₂×C₇₁	#1803	1	1	yes	characteristic
132	C₂×C₇₁	#1813	1	1	yes	characteristic
133	C₂×C₇₁	#1816	1	1	yes	characteristic
134	#2960	—	71	1	no	—
135	#1848	D₇₁	1	1	yes	characteristic
136	C₂×#335	—	71	1	no	—
137	#1550	—	71	2	no	—
138	#1292	—	71	2	no	—
139	C₇₁	#8685	1	1	yes	characteristic, complemented, Sylow
140	#533	—	71	1	no	—
141	#446	—	71	2	no	—
142	#446	—	142	1	no	—
143	C₈×Q₈	—	71	2	no	—
144	C₈×Q₈	—	142	1	no	—
145	C₂×#108	—	2	1	no	—
146	C₂×#108	—	2	1	no	—
147	#424	—	142	1	no	—
148	#423	—	71	1	no	—
149	#423	C₂×D₇₁	1	2	yes	—
150	C₂×C₄⋊C₈	—	71	2	no	—
151	#405	—	142	1	no	—
152	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
153	#335	—	142	1	no	—
154	#335	—	142	1	no	—
155	#333	—	142	2	no	—
156	#332	—	142	2	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	142	1	no	—
159	C₂²×C₈	—	142	2	no	—
160	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
162	C₄⋊Q₈	—	71	1	no	—
163	C₄⋊Q₈	—	71	1	no	—
164	#125	—	142	1	no	—
165	C₂²⋊Q₈	—	71	2	no	—
166	C₂²⋊Q₈	—	142	1	no	—
167	C₄×Q₈	—	71	2	no	—
168	C₄×Q₈	—	142	1	no	—
169	#117	—	142	1	no	—
170	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
171	#108	—	4	2	no	—
172	#108	—	4	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	71	2	no	—
176	C₄⋊C₈	—	142	2	no	—
177	C₄⋊C₈	—	142	2	no	—
178	C₄⋊C₈	—	142	2	no	—
179	C₄⋊C₈	—	142	2	no	—
180	C₄⋊C₈	—	142	1	no	—
181	#97	—	142	2	no	—
182	C₄×C₈	—	142	2	no	—
183	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
184	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
185	C₄×C₈	D₂₈₄	1	1	yes	characteristic
186	C₄×C₈	D₂₈₄	1	1	yes	characteristic
187	C₂×Q₈	—	71	2	no	—
188	C₂×Q₈	—	142	1	no	—
189	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
190	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
191	C₂²×C₄	#11765866	1	1	yes	characteristic
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	M₁₆	—	284	1	no	—
195	M₁₆	—	284	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	4	2	no	—
199	C₂×C₈	—	284	2	no	—
200	C₂×C₈	—	568	2	no	—
201	C₂×C₈	—	142	2	no	—
202	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
204	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
205	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
206	C₂×C₈	#11765853	1	2	yes	—
207	C₂×C₈	#11765855	1	2	yes	—
208	C₄⋊C₄	—	71	2	no	—
209	C₄⋊C₄	—	142	2	no	—
210	C₄⋊C₄	—	142	1	no	—
211	C₄⋊C₄	—	142	1	no	—
212	C₄⋊C₄	—	142	1	no	—
213	C₄⋊C₄	—	142	2	no	—
214	C₂²⋊C₄	—	142	2	no	—
215	C₄²	—	142	2	no	—
216	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
217	C₄²	D₄×D₇₁	1	1	yes	characteristic
218	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
219	C₄²	#11765864	1	1	yes	characteristic
220	C₂³	#14556104	1	1	yes	characteristic
221	Q₈	—	284	2	no	—
222	Q₈	—	284	1	no	—
223	Q₈	—	284	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	142	2	no	—
228	C₂×C₄	—	142	2	no	—
229	C₂×C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
230	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
231	C₂×C₄	#14556086	1	1	yes	characteristic
232	C₂×C₄	#14556099	1	1	yes	characteristic
233	C₂×C₄	#14556103	1	1	yes	characteristic
234	C₂×C₄	#14556108	1	1	yes	characteristic
235	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
236	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
237	C₂×C₄	#14556154	1	1	yes	characteristic
238	C₂×C₄	#14556159	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	8	2	no	—
242	C₈	—	284	2	no	—
243	C₈	—	568	2	no	—
244	C₈	#14556051	1	2	yes	—
245	C₈	#14556051	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#17985856	1	1	yes	characteristic
249	C₂²	#17986281	1	1	yes	characteristic
250	C₂²	#17986291	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	284	2	no	—
255	C₄	—	284	2	no	—
256	C₄	C₂×#14556051	1	1	yes	characteristic
257	C₄	#17985998	1	1	yes	characteristic
258	C₄	#17986196	1	1	yes	characteristic
259	C₄	#17986206	1	1	yes	characteristic
260	C₄	#17986270	1	1	yes	characteristic
261	C₄	#17986275	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	#22588342	1	1	yes	characteristic
265	C₂	#22588387	1	1	yes	characteristic
266	C₂	#22588401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985856
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1848	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂³×C₇₁	C₂²⋊Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1848	D₇₁
hypocenter	C₇₁	#8685
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1848

Complements

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

N	H ≅ G/N	type
139. C₇₁	#8685	semidirect

Maximal subgroups

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

subgroup	index
#22597274	2
C₂×#17985752	2
#22587031	2
#22586214	2
C₇₁×#1848	2
#8685	71