Subgroups — #31912665

Order 18176.

526 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17986081	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985801	C₂	1	1	yes	maximal, characteristic
3	C₂×#17985799	C₂	1	1	yes	maximal, characteristic
4	C₂×#17985774	C₂	1	1	yes	maximal, characteristic, complemented
5	#107×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586431	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×#14556090	—	2	1	no	—
10	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#17986081	—	2	2	no	—
12	#17986081	C₄	1	4	yes	complemented
13	C₂×#14556062	C₂²	1	1	yes	characteristic
14	C₂×#14556048	—	2	1	no	—
15	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×Dic₇₁	—	2	1	no	—
17	#17985801	C₄	1	4	yes	complemented
18	#17985799	C₄	1	4	yes	complemented
19	#17985774	—	2	2	no	—
20	#17985761	—	2	1	no	—
21	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
22	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#107×C₇₁	C₂²	1	1	yes	characteristic
24	C₂²×D₂₈₄	—	2	1	no	—
25	C₂²×C₄×D₇₁	—	4	1	no	—
26	C₂×D₅₆₈	—	2	1	no	—
27	C₂×D₅₆₈	—	2	1	no	—
28	C₂×D₅₆₈	—	2	1	no	—
29	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
30	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
31	C₂×D₅₆₈	C₂×C₄	1	2	yes	—
32	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
33	#14556090	—	2	4	no	—
34	#14556090	—	4	2	no	—
35	C₂×C₄⋊Dic₇₁	—	2	1	no	—
36	C₂×(C₂×D₇₁)⋊C₄	—	4	1	no	—
37	C₂×C₄×Dic₇₁	—	4	1	no	—
38	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
39	C₂×C₇₁⋊C₁₆	—	2	1	no	—
40	C₂×C₇₁⋊C₁₆	—	2	1	no	—
41	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	complemented
42	#14556062	—	2	1	no	—
43	#14556062	C₂×C₄	1	2	yes	complemented
44	#14556048	—	2	4	no	—
45	C₈×Dic₇₁	C₂×C₄	1	2	yes	—
46	C₈×Dic₇₁	D₄	1	2	yes	—
47	C₇₁⋊#95	—	4	1	no	—
48	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
51	C₂×C₁₆×C₇₁	—	2	1	no	—
52	C₂×C₁₆×C₇₁	—	2	1	no	—
53	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
54	#107×C₇₁	—	2	1	no	—
55	#107×C₇₁	—	2	1	no	—
56	#107×C₇₁	C₂×C₄	1	2	yes	—
57	#107×C₇₁	C₂×C₄	1	2	yes	complemented
58	C₂³×D₇₁	—	4	1	no	—
59	C₂×D₂₈₄	—	2	2	no	—
60	C₂×D₂₈₄	—	2	2	no	—
61	C₂×D₂₈₄	—	2	2	no	—
62	C₂×D₂₈₄	—	4	1	no	—
63	C₂×D₂₈₄	—	4	1	no	—
64	C₂×D₂₈₄	—	4	1	no	—
65	C₂²×Dic₇₁	—	4	1	no	—
66	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
67	C₂²×Dic₇₁	D₈	1	1	yes	characteristic
68	C₂×C₄×D₇₁	—	4	4	no	—
69	C₂×C₄×D₇₁	—	8	2	no	—
70	C₂×C₄×D₇₁	—	8	2	no	—
71	D₅₆₈	—	2	2	no	—
72	D₅₆₈	—	2	2	no	—
73	D₅₆₈	—	2	2	no	—
74	D₅₆₈	C₄²	1	4	yes	complemented
75	C₂×C₇₁⋊C₈	—	2	1	no	—
76	C₂×C₇₁⋊C₈	—	2	1	no	—
77	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	2	yes	—
78	C₄⋊Dic₇₁	—	2	2	no	—
79	C₄⋊Dic₇₁	—	4	1	no	—
80	(C₂×D₇₁)⋊C₄	—	4	4	no	—
81	C₄×Dic₇₁	—	8	2	no	—
82	C₄×Dic₇₁	C₂²⋊C₄	1	2	yes	—
83	C₄×Dic₇₁	SD₁₆	1	2	yes	—
84	C₇₁⋊C₁₆	—	2	2	no	—
85	C₂²×C₄×C₇₁	—	4	1	no	—
86	C₂²×C₄×C₇₁	—	4	1	no	—
87	C₂²×C₄×C₇₁	C₂×D₄	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	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
93	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
94	C₄⋊C₄×C₇₁	—	2	2	no	—
95	C₄⋊C₄×C₇₁	—	2	2	no	—
96	C₄⋊C₄×C₇₁	—	4	1	no	—
97	C₄⋊C₄×C₇₁	—	4	1	no	—
98	C₁₆×C₇₁	—	2	2	no	—
99	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₇₁	—	4	2	no	—
101	C₂²×D₇₁	—	4	2	no	—
102	C₂²×D₇₁	—	4	2	no	—
103	C₂²×D₇₁	—	8	1	no	—
104	C₂²×D₇₁	—	8	1	no	—
105	C₂²×D₇₁	—	8	1	no	—
106	C₂²×D₇₁	—	8	1	no	—
107	C₂×C₄×C₇₁	—	4	2	no	—
108	C₂×C₄×C₇₁	—	4	2	no	—
109	C₂×C₄×C₇₁	—	8	1	no	—
110	C₂×C₄×C₇₁	—	8	1	no	—
111	C₂×C₄×C₇₁	—	8	1	no	—
112	C₂×C₄×C₇₁	—	8	1	no	—
113	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
118	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
119	C₂×Dic₇₁	—	2	1	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	4	2	no	—
124	C₂×Dic₇₁	—	8	1	no	—
125	C₂×Dic₇₁	—	8	1	no	—
126	C₂×Dic₇₁	#102	1	2	yes	—
127	C₂×Dic₇₁	#102	1	2	yes	—
128	D₂₈₄	—	2	4	no	—
129	D₂₈₄	—	4	2	no	—
130	D₂₈₄	—	4	2	no	—
131	D₂₈₄	—	4	2	no	—
132	C₄×D₇₁	—	8	8	no	—
133	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
134	C₈×C₇₁	#117	1	1	yes	characteristic
135	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
136	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
137	C₇₁⋊C₈	—	2	2	no	—
138	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
139	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
140	C₂²×C₇₁	C₄×D₈	1	1	yes	characteristic
141	C₂²×C₇₁	#509	1	1	yes	characteristic
142	C₂²×C₇₁	#509	1	1	yes	characteristic
143	C₂²×C₇₁	#509	1	1	yes	characteristic
144	C₂²×C₇₁	#510	1	1	yes	characteristic
145	C₂×D₇₁	—	4	4	no	—
146	C₂×D₇₁	—	8	2	no	—
147	C₂×D₇₁	—	8	2	no	—
148	C₂×D₇₁	—	8	2	no	—
149	C₂×D₇₁	—	8	2	no	—
150	C₂×D₇₁	—	8	2	no	—
151	C₂×D₇₁	—	8	2	no	—
152	C₄×C₇₁	—	8	2	no	—
153	C₄×C₇₁	—	8	2	no	—
154	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₇₁	#416	1	1	yes	characteristic
156	C₄×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
157	C₄×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
158	Dic₇₁	—	2	2	no	—
159	Dic₇₁	—	2	2	no	—
160	Dic₇₁	—	8	2	no	—
161	#12637	—	71	1	no	maximal, Sylow
162	C₂×C₇₁	C₄×#102	1	1	yes	characteristic
163	C₂×C₇₁	#2139	1	1	yes	characteristic
164	C₂×C₇₁	#2140	1	1	yes	characteristic
165	C₂×C₇₁	#2175	1	1	yes	characteristic
166	C₂×C₇₁	#2175	1	1	yes	characteristic
167	C₂×C₇₁	#2177	1	1	yes	characteristic
168	C₂×C₇₁	#2177	1	1	yes	characteristic
169	D₇₁	—	8	4	no	—
170	C₂×C₄×D₈	—	71	1	no	—
171	C₂×#357	—	71	1	no	—
172	C₂×#357	—	71	1	no	—
173	C₂×#346	—	71	1	no	—
174	C₄×#107	—	71	1	no	—
175	#1380	—	71	1	no	—
176	#1380	D₇₁	1	1	yes	characteristic, complemented
177	C₇₁	#12637	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₈	—	71	1	no	—
179	C₂×C₄×D₄	—	142	1	no	—
180	C₂²×C₁₆	—	71	1	no	—
181	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
182	C₄×D₈	—	71	4	no	—
183	C₄×D₈	—	142	2	no	—
184	C₂×#107	—	71	1	no	—
185	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
186	C₂×#107	C₂×D₇₁	1	1	yes	characteristic
187	C₂×#102	—	142	1	no	—
188	C₂×C₄×C₈	—	71	1	no	—
189	C₄×C₄⋊C₄	—	142	1	no	—
190	#357	—	71	4	no	—
191	#357	—	71	4	no	—
192	#346	—	142	2	no	—
193	#340	—	142	1	no	—
194	C₂²×D₄	—	142	1	no	—
195	C₂³×C₄	—	284	1	no	—
196	C₂×D₈	—	71	2	no	—
197	C₂×D₈	—	71	2	no	—
198	C₂×D₈	—	71	2	no	—
199	C₂×D₈	—	142	1	no	—
200	C₂×D₈	—	142	1	no	—
201	C₂×D₈	—	142	1	no	—
202	C₂²×C₈	—	71	1	no	—
203	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
204	C₄×D₄	—	284	2	no	—
205	C₄×D₄	—	142	4	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	142	1	no	—
209	C₂×C₂²⋊C₄	—	284	1	no	—
210	C₂×C₄²	—	284	1	no	—
211	C₂×C₄²	—	71	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	71	2	no	—
215	C₂×C₁₆	—	142	1	no	—
216	C₂×C₁₆	—	142	1	no	—
217	C₂×C₁₆	C₄×D₇₁	1	2	yes	complemented
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	71	2	no	—
221	#107	—	142	1	no	—
222	#107	C₄×D₇₁	1	2	yes	—
223	#107	C₄×D₇₁	1	2	yes	complemented
224	#102	—	142	4	no	—
225	C₄×C₈	—	71	2	no	—
226	C₄×C₈	—	71	2	no	—
227	#95	—	284	1	no	—
228	C₂⁴	—	284	1	no	—
229	C₂×D₄	—	284	1	no	—
230	C₂×D₄	—	284	1	no	—
231	C₂×D₄	—	284	1	no	—
232	C₂×D₄	—	142	2	no	—
233	C₂×D₄	—	142	2	no	—
234	C₂×D₄	—	142	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	284	4	no	—
238	C₂²×C₄	—	284	1	no	—
239	C₂²×C₄	—	568	2	no	—
240	C₂²×C₄	—	568	2	no	—
241	C₂²×C₄	—	71	1	no	—
242	C₂²×C₄	—	71	1	no	—
243	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
244	D₈	—	71	4	no	—
245	D₈	—	142	2	no	—
246	D₈	—	142	2	no	—
247	D₈	—	142	2	no	—
248	C₂×C₈	—	71	2	no	—
249	C₂×C₈	—	142	1	no	—
250	C₂×C₈	—	142	1	no	—
251	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
252	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
254	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
255	C₂×C₈	#11765864	1	1	yes	characteristic
256	C₂×C₈	#11765866	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	284	1	no	—
262	C₄⋊C₄	—	142	2	no	—
263	C₂²⋊C₄	—	284	4	no	—
264	C₄²	—	568	2	no	—
265	C₄²	—	71	2	no	—
266	C₄²	—	71	2	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	142	2	no	—
269	C₂³	—	284	2	no	—
270	C₂³	—	284	2	no	—
271	C₂³	—	284	2	no	—
272	C₂³	—	568	1	no	—
273	C₂³	—	568	1	no	—
274	C₂³	—	568	1	no	—
275	C₂³	—	568	1	no	—
276	C₂³	D₈×D₇₁	1	1	yes	characteristic
277	D₄	—	284	2	no	—
278	D₄	—	284	2	no	—
279	D₄	—	284	2	no	—
280	D₄	—	142	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	284	2	no	—
288	C₂×C₄	—	568	8	no	—
289	C₂×C₄	—	568	1	no	—
290	C₂×C₄	—	568	1	no	—
291	C₂×C₄	—	71	2	no	—
292	C₂×C₄	—	71	2	no	—
293	C₂×C₄	—	142	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	—	142	1	no	—
296	C₂×C₄	—	142	1	no	—
297	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
298	C₂×C₄	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	#14556090	1	1	yes	characteristic
300	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
301	C₂×C₄	#14556149	1	1	yes	characteristic
302	C₂×C₄	#14556151	1	1	yes	characteristic
303	C₈	—	142	2	no	—
304	C₈	C₄²×D₇₁	1	1	yes	characteristic
305	C₈	#14556086	1	1	yes	characteristic
306	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
307	C₈	#14556090	1	1	yes	characteristic
308	C₂²	—	284	4	no	—
309	C₂²	—	568	2	no	—
310	C₂²	—	568	2	no	—
311	C₂²	—	568	2	no	—
312	C₂²	—	568	2	no	—
313	C₂²	—	568	2	no	—
314	C₂²	—	568	2	no	—
315	C₂²	#102×D₇₁	1	1	yes	characteristic
316	C₂²	#17986079	1	1	yes	characteristic
317	C₂²	#17986081	1	1	yes	characteristic
318	C₂²	#17986433	1	1	yes	characteristic
319	C₂²	#17986436	1	1	yes	characteristic
320	C₂²	#17986436	1	1	yes	characteristic
321	C₂²	#17986441	1	1	yes	characteristic
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	568	2	no	—
325	C₄	—	142	2	no	—
326	C₄	—	142	2	no	—
327	C₄	#17985846	1	1	yes	characteristic
328	C₄	#17985994	1	1	yes	characteristic
329	C₄	#17986084	1	1	yes	characteristic
330	C₄	#17986086	1	1	yes	characteristic
331	C₂	—	568	4	no	—
332	C₂	#22588154	1	1	yes	characteristic
333	C₂	#22590051	1	1	yes	characteristic
334	C₂	#22590054	1	1	yes	characteristic
335	C₂	#22590203	1	1	yes	characteristic
336	C₂	#22590207	1	1	yes	characteristic
337	C₂	#22590213	1	1	yes	characteristic
338	C₂	#22590215	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₁
commutator	C₈×C₇₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1380	C₂
socle	C₂³×C₇₁	C₂×D₈
prime radical	C₂²×D₅₆₈	C₂²
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1380	D₇₁
hypocenter	C₇₁	#12637
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
4. C₂×#17985774	C₂	semidirect
5. #107×Dic₇₁	C₂	semidirect
6. #22586431	C₂	semidirect
7. C₇₁×#1380	C₂	semidirect
12. #17986081	C₄	semidirect
17. #17985801	C₄	semidirect
18. #17985799	C₄	semidirect
41. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
43. #14556062	C₂×C₄	semidirect
53. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
57. #107×C₇₁	C₂×C₄	semidirect
74. D₅₆₈	C₄²	semidirect
176. #1380	D₇₁	semidirect
177. C₇₁	#12637	semidirect
217. C₂×C₁₆	C₄×D₇₁	semidirect
223. #107	C₄×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17986081	2
C₂×#17985801	2
C₂×#17985799	2
C₂×#17985774	2
#107×Dic₇₁	2
#22586431	2
C₇₁×#1380	2
#12637	71