Subgroups — #31897584

Order 18176.

450 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22596703	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#22586224	C₂	1	2	yes	maximal
4	#22586224	C₂	1	2	yes	maximal
5	C₇₁×#1842	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556120	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
9	#17985897	—	2	1	no	—
10	#17985897	—	2	1	no	—
11	#17985897	C₄	1	2	yes	—
12	#17985897	C₄	1	2	yes	—
13	#17985869	C₂²	1	1	yes	characteristic
14	#17985869	C₂²	1	1	yes	characteristic
15	#17985865	—	2	1	no	—
16	#17985754	—	2	1	no	—
17	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
18	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
19	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
20	C₄³×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	#14556120	—	4	2	no	—
23	#14556120	—	4	2	no	—
24	C₂×C₄⋊Dic₇₁	—	2	1	no	—
25	C₂×C₄⋊Dic₇₁	—	2	1	no	—
26	C₂×C₄⋊Dic₇₁	—	2	1	no	—
27	C₂×C₄⋊Dic₇₁	—	2	1	no	—
28	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
30	C₂×Dic₇₁⋊C₄	—	2	2	no	—
31	C₂×Dic₇₁⋊C₄	—	2	2	no	—
32	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	Q₈	1	4	yes	—
34	C₄×C₇₁⋊C₈	—	4	2	no	—
35	C₇₁⋊#95	—	2	2	no	—
36	C₇₁⋊#95	—	2	2	no	—
37	C₂²×C₈×C₇₁	—	2	2	no	—
38	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₇₁	—	2	4	no	—
46	C₄⋊C₈×C₇₁	—	4	2	no	—
47	C₄⋊C₈×C₇₁	—	4	2	no	—
48	C₂²×Dic₇₁	—	2	2	no	—
49	C₂²×Dic₇₁	—	2	2	no	—
50	C₂×C₇₁⋊C₈	—	2	4	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	4	2	no	—
53	C₄⋊Dic₇₁	—	4	1	no	—
54	C₄⋊Dic₇₁	—	4	1	no	—
55	C₄⋊Dic₇₁	—	4	1	no	—
56	C₄⋊Dic₇₁	—	4	1	no	—
57	C₄⋊Dic₇₁	—	4	2	no	—
58	C₄⋊Dic₇₁	—	4	1	no	—
59	C₄⋊Dic₇₁	—	4	1	no	—
60	C₄⋊Dic₇₁	—	4	2	no	—
61	C₄⋊Dic₇₁	—	4	2	no	—
62	Dic₇₁⋊C₄	—	4	4	no	—
63	Dic₇₁⋊C₄	—	4	4	no	—
64	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₁	—	4	2	no	—
72	C₂×C₈×C₇₁	—	4	2	no	—
73	C₂×C₈×C₇₁	—	4	2	no	—
74	C₄²×C₇₁	—	2	2	no	—
75	C₄²×C₇₁	—	2	1	no	—
76	C₄²×C₇₁	—	2	1	no	—
77	C₄²×C₇₁	—	4	1	no	—
78	C₄²×C₇₁	—	4	1	no	—
79	C₄²×C₇₁	D₈	1	2	yes	—
80	C₄²×C₇₁	SD₁₆	1	2	yes	—
81	C₄²×C₇₁	SD₁₆	1	2	yes	—
82	C₄²×C₇₁	Q₁₆	1	2	yes	—
83	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₇₁	#41	1	2	yes	—
85	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₇₁	—	2	1	no	—
87	C₂×C₄×C₇₁	—	2	1	no	—
88	C₂×C₄×C₇₁	—	2	1	no	—
89	C₂×C₄×C₇₁	—	2	1	no	—
90	C₂×C₄×C₇₁	—	2	1	no	—
91	C₂×C₄×C₇₁	—	2	1	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	1	no	—
94	C₂×C₄×C₇₁	—	2	1	no	—
95	C₂×C₄×C₇₁	—	2	1	no	—
96	C₂×C₄×C₇₁	#102	1	2	yes	—
97	C₂×C₄×C₇₁	#102	1	2	yes	—
98	C₂×C₄×C₇₁	#103	1	2	yes	—
99	C₂×C₄×C₇₁	#103	1	2	yes	—
100	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#135	1	2	yes	—
109	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
111	C₂×Dic₇₁	—	4	2	no	—
112	C₂×Dic₇₁	—	4	2	no	—
113	C₂×Dic₇₁	—	4	2	no	—
114	C₂×Dic₇₁	—	4	2	no	—
115	C₂×Dic₇₁	—	4	2	no	—
116	C₂×Dic₇₁	—	4	2	no	—
117	C₈×C₇₁	—	8	2	no	—
118	C₇₁⋊C₈	—	4	4	no	—
119	C₂²×C₇₁	#391	1	1	yes	characteristic
120	C₂²×C₇₁	#413	1	1	yes	characteristic
121	C₂²×C₇₁	#413	1	1	yes	characteristic
122	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
123	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
124	C₂²×C₇₁	#416	1	1	yes	characteristic
125	C₂²×C₇₁	#417	1	1	yes	characteristic
126	C₄×C₇₁	—	2	2	no	—
127	C₄×C₇₁	—	2	2	no	—
128	C₄×C₇₁	—	2	2	no	—
129	C₄×C₇₁	—	2	2	no	—
130	C₄×C₇₁	—	2	2	no	—
131	C₄×C₇₁	#478	1	2	yes	—
132	C₄×C₇₁	#479	1	2	yes	—
133	C₄×C₇₁	#486	1	1	yes	characteristic
134	C₄×C₇₁	#487	1	1	yes	characteristic
135	C₄×C₇₁	#498	1	1	yes	characteristic
136	C₄×C₇₁	#500	1	1	yes	characteristic
137	Dic₇₁	—	8	2	no	—
138	Dic₇₁	—	8	2	no	—
139	#9900	—	71	1	no	maximal, Sylow
140	C₂×C₇₁	#1540	1	1	yes	characteristic
141	C₂×C₇₁	#1542	1	1	yes	characteristic
142	C₂×C₇₁	#1547	1	1	yes	characteristic
143	C₂×C₇₁	#1550	1	1	yes	characteristic
144	C₂×C₇₁	#1989	1	1	yes	characteristic
145	C₂×C₇₁	#1990	1	1	yes	characteristic
146	C₂×C₇₁	#1998	1	1	yes	characteristic
147	#2859	—	71	1	no	—
148	#1842	D₇₁	1	1	yes	characteristic
149	C₄×C₄⋊C₈	—	71	1	no	—
150	#1296	—	71	2	no	—
151	#1296	—	71	2	no	—
152	C₇₁	#9900	1	1	yes	characteristic, complemented, Sylow
153	C₂×#125	—	142	1	no	—
154	C₂×C₄⋊C₈	—	2	1	no	—
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	—	71	2	no	—
157	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
158	C₂×C₄×C₈	—	142	1	no	—
159	#389	—	71	2	no	—
160	#389	—	71	2	no	—
161	#389	—	142	1	no	—
162	#389	—	142	1	no	—
163	#384	—	71	1	no	—
164	#384	—	71	1	no	—
165	#382	—	142	1	no	—
166	C₄³	C₂×D₇₁	1	1	yes	characteristic
167	#336	—	142	1	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	142	2	no	—
170	#125	—	284	2	no	—
171	#125	—	284	2	no	—
172	C₂×C₄⋊C₄	—	71	2	no	—
173	C₂×C₄⋊C₄	—	71	2	no	—
174	C₂×C₄⋊C₄	—	142	2	no	—
175	C₂×C₄⋊C₄	—	142	2	no	—
176	C₂×C₄⋊C₄	—	142	1	no	—
177	C₂×C₄⋊C₄	—	142	1	no	—
178	C₂×C₄⋊C₄	—	142	1	no	—
179	C₂×C₄⋊C₄	—	142	1	no	—
180	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
182	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
183	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
184	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
186	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
187	C₄⋊C₈	—	2	4	no	—
188	C₄⋊C₈	—	4	2	no	—
189	C₄⋊C₈	—	4	2	no	—
190	C₄⋊C₈	—	71	4	no	—
191	C₄⋊C₈	—	142	2	no	—
192	C₄×C₈	—	284	2	no	—
193	#95	—	142	2	no	—
194	#95	—	142	2	no	—
195	C₂²×C₄	—	142	2	no	—
196	C₂²×C₄	—	142	2	no	—
197	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
199	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
200	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	284	2	no	—
208	C₂×C₈	—	284	2	no	—
209	C₂×C₈	—	142	4	no	—
210	C₄⋊C₄	—	284	2	no	—
211	C₄⋊C₄	—	284	2	no	—
212	C₄⋊C₄	—	284	1	no	—
213	C₄⋊C₄	—	284	4	no	—
214	C₄⋊C₄	—	284	4	no	—
215	C₄⋊C₄	—	284	1	no	—
216	C₄⋊C₄	—	284	1	no	—
217	C₄⋊C₄	—	284	1	no	—
218	C₄⋊C₄	—	284	1	no	—
219	C₄⋊C₄	—	284	2	no	—
220	C₄⋊C₄	—	284	1	no	—
221	C₄²	—	2	1	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₄²	#11765850	1	2	yes	—
227	C₄²	#11765852	1	2	yes	—
228	C₄²	#11765853	1	2	yes	—
229	C₄²	#11765855	1	2	yes	—
230	C₄²	Q₈×D₇₁	1	2	yes	—
231	C₄²	#11765864	1	2	yes	—
232	C₂³	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	284	2	no	—
244	C₂×C₄	—	284	2	no	—
245	C₂×C₄	—	284	2	no	—
246	C₂×C₄	—	284	2	no	—
247	C₂×C₄	—	284	2	no	—
248	C₂×C₄	—	284	2	no	—
249	C₂×C₄	#14556048	1	2	yes	—
250	C₂×C₄	#14556048	1	2	yes	—
251	C₂×C₄	#14556051	1	2	yes	—
252	C₂×C₄	#14556051	1	2	yes	—
253	C₂×C₄	#14556093	1	2	yes	—
254	C₂×C₄	#14556108	1	2	yes	—
255	C₂×C₄	#14556115	1	1	yes	characteristic
256	C₂×C₄	#14556126	1	1	yes	characteristic
257	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
258	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
259	C₂×C₄	C₂×#11765853	1	1	yes	characteristic
260	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
261	C₂×C₄	#14556150	1	2	yes	—
262	C₂×C₄	#14556156	1	1	yes	characteristic
263	C₂×C₄	#14556161	1	1	yes	characteristic
264	C₈	—	8	2	no	—
265	C₈	—	284	4	no	—
266	C₂²	#17985904	1	1	yes	characteristic
267	C₂²	#17985981	1	1	yes	characteristic
268	C₂²	#17985981	1	1	yes	characteristic
269	C₂²	C₂×#14556048	1	1	yes	characteristic
270	C₂²	C₂×#14556051	1	1	yes	characteristic
271	C₂²	#17985996	1	1	yes	characteristic
272	C₂²	#17986001	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	568	2	no	—
279	C₄	—	568	2	no	—
280	C₄	#17986300	1	2	yes	—
281	C₄	#17986307	1	2	yes	—
282	C₄	#17986345	1	1	yes	characteristic
283	C₄	#17986349	1	1	yes	characteristic
284	C₄	#17986395	1	1	yes	characteristic
285	C₄	#17986402	1	1	yes	characteristic
286	C₂	#22586990	1	1	yes	characteristic
287	C₂	#22586998	1	1	yes	characteristic
288	C₂	#22587018	1	1	yes	characteristic
289	C₂	#22587031	1	1	yes	characteristic
290	C₂	#22589295	1	1	yes	characteristic
291	C₂	#22589300	1	1	yes	characteristic
292	C₂	#22589332	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₁)⋊C₄
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1842	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1842	D₇₁
hypocenter	C₇₁	#9900
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1842

Complements

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

N	H ≅ G/N	type
152. C₇₁	#9900	semidirect

Maximal subgroups

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

subgroup	index
#22596703	2
C₄×C₄⋊(C₇₁⋊C₈)	2
#22586224	2
C₇₁×#1842	2
#9900	71