Subgroups — #31897459

Order 18176.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14556128	C₂	1	1	yes	maximal, characteristic
2	#22588492	C₂	1	1	yes	maximal, characteristic
3	#22586180	C₂	1	4	yes	maximal
4	C₄²×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₄₂	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	#17985905	—	2	1	no	—
11	#17985897	—	2	1	no	—
12	#17985897	C₄	1	2	yes	—
13	#17985869	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₇₁	—	2	1	no	—
15	C₄×C₄⋊Dic₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	—	2	1	no	—
17	C₂×C₄×C₈×C₇₁	—	2	1	no	—
18	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
19	C₄³×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₄₂	—	2	1	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	#14556128	—	2	2	no	—
23	#14556128	C₈	1	4	yes	complemented
24	C₄×Dic₁₄₂	—	4	2	no	—
25	C₄×Dic₁₄₂	—	4	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	1	no	—
27	C₂×C₄⋊Dic₇₁	—	2	1	no	—
28	C₂×C₄⋊Dic₇₁	—	2	1	no	—
29	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
31	C₂×Dic₇₁⋊C₄	—	2	2	no	—
32	C₂×C₄×Dic₇₁	—	2	2	no	—
33	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
34	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
35	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
36	C₇₁⋊#95	—	2	2	no	—
37	C₇₁⋊#95	—	2	2	no	—
38	C₂²×C₈×C₇₁	—	2	2	no	—
39	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₁	C₂×C₄	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	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₇₁	—	2	4	no	—
47	C₄×C₈×C₇₁	—	2	4	no	—
48	C₄×C₈×C₇₁	—	2	2	no	—
49	C₄×C₈×C₇₁	Q₈	1	4	yes	complemented
50	C₂×Dic₁₄₂	—	2	4	no	—
51	C₂×Dic₁₄₂	—	4	2	no	—
52	C₂×Dic₁₄₂	—	4	2	no	—
53	C₂²×Dic₇₁	—	2	2	no	—
54	C₂²×Dic₇₁	—	2	2	no	—
55	C₂×C₇₁⋊C₈	—	4	2	no	—
56	C₂×C₇₁⋊C₈	—	4	2	no	—
57	C₂×C₇₁⋊C₈	—	4	2	no	—
58	C₄⋊Dic₇₁	—	2	4	no	—
59	C₄⋊Dic₇₁	—	2	2	no	—
60	C₄⋊Dic₇₁	—	2	4	no	—
61	C₄⋊Dic₇₁	—	4	2	no	—
62	C₄⋊Dic₇₁	—	4	2	no	—
63	C₄⋊Dic₇₁	C₂×C₈	1	4	yes	—
64	Dic₇₁⋊C₄	—	4	4	no	—
65	C₄×Dic₇₁	—	4	2	no	—
66	C₄×Dic₇₁	—	4	2	no	—
67	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₁	C₂²×C₄	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	C₂×C₈×C₇₁	—	2	4	no	—
75	C₂×C₈×C₇₁	—	2	4	no	—
76	C₂×C₈×C₇₁	—	2	4	no	—
77	C₄²×C₇₁	—	2	2	no	—
78	C₄²×C₇₁	—	2	2	no	—
79	C₄²×C₇₁	—	2	2	no	—
80	C₄²×C₇₁	C₂×C₈	1	2	yes	—
81	C₄²×C₇₁	M₁₆	1	2	yes	—
82	C₄²×C₇₁	D₈	1	2	yes	—
83	C₄²×C₇₁	SD₁₆	1	2	yes	—
84	C₄²×C₇₁	SD₁₆	1	2	yes	—
85	C₄²×C₇₁	Q₁₆	1	2	yes	—
86	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
87	C₄²×C₇₁	#41	1	2	yes	—
88	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₇₁	#102	1	2	yes	—
98	C₂×C₄×C₇₁	#102	1	2	yes	—
99	C₂×C₄×C₇₁	#103	1	2	yes	—
100	C₂×C₄×C₇₁	#103	1	2	yes	—
101	C₂×C₄×C₇₁	#104	1	2	yes	—
102	C₂×C₄×C₇₁	#104	1	2	yes	—
103	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#131	1	2	yes	—
110	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×Dic₇₁	—	2	4	no	—
115	C₂×Dic₇₁	—	4	2	no	—
116	C₂×Dic₇₁	—	4	2	no	—
117	C₂×Dic₇₁	—	4	2	no	—
118	C₂×Dic₇₁	—	4	2	no	—
119	C₂×Dic₇₁	—	4	2	no	—
120	Dic₁₄₂	—	4	8	no	—
121	C₈×C₇₁	—	2	8	no	—
122	C₇₁⋊C₈	—	8	2	no	—
123	C₂²×C₇₁	#391	1	1	yes	characteristic
124	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂²×C₇₁	#408	1	1	yes	characteristic
126	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
127	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
128	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
129	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
130	C₄×C₇₁	—	2	2	no	—
131	C₄×C₇₁	—	2	2	no	—
132	C₄×C₇₁	—	2	2	no	—
133	C₄×C₇₁	D₄⋊C₈	1	4	yes	—
134	C₄×C₇₁	Q₈⋊C₈	1	4	yes	—
135	C₄×C₇₁	C₈×Q₈	1	2	yes	—
136	C₄×C₇₁	#446	1	2	yes	—
137	C₄×C₇₁	#486	1	1	yes	characteristic
138	C₄×C₇₁	#487	1	1	yes	characteristic
139	C₄×C₇₁	#498	1	1	yes	characteristic
140	C₄×C₇₁	#500	1	1	yes	characteristic
141	Dic₇₁	—	4	4	no	—
142	Dic₇₁	—	8	2	no	—
143	#9870	—	71	1	no	maximal, Sylow
144	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
145	C₂×C₇₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
146	C₂×C₇₁	#1492	1	1	yes	characteristic
147	C₂×C₇₁	#1493	1	1	yes	characteristic
148	C₂×C₇₁	#1984	1	1	yes	characteristic
149	C₂×C₇₁	#1990	1	1	yes	characteristic
150	C₂×C₇₁	#1994	1	1	yes	characteristic
151	C₄×C₄⋊Q₈	—	71	1	no	—
152	#1842	—	71	1	no	—
153	C₄²×C₈	D₇₁	1	1	yes	characteristic
154	#1279	—	71	4	no	—
155	C₇₁	#9870	1	1	yes	characteristic, complemented, Sylow
156	C₂×C₄⋊Q₈	—	71	1	no	—
157	C₂×C₄×Q₈	—	142	1	no	—
158	C₂×C₄⋊C₈	—	71	2	no	—
159	C₂×C₄⋊C₈	—	142	1	no	—
160	C₂×C₄⋊C₈	—	142	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	—	2	1	no	—
163	C₂×C₄×C₈	C₂×D₇₁	1	2	yes	—
164	#391	—	142	1	no	—
165	#389	—	71	2	no	—
166	#389	—	142	1	no	—
167	#384	—	71	1	no	—
168	C₄×C₄⋊C₄	—	71	2	no	—
169	C₄×C₄⋊C₄	—	142	1	no	—
170	C₄³	C₂×D₇₁	1	1	yes	characteristic
171	C₂²×Q₈	—	142	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	142	2	no	—
174	C₄⋊Q₈	—	71	4	no	—
175	C₄⋊Q₈	—	142	2	no	—
176	C₄×Q₈	—	284	2	no	—
177	C₄×Q₈	—	284	2	no	—
178	C₂×C₄⋊C₄	—	71	2	no	—
179	C₂×C₄⋊C₄	—	71	2	no	—
180	C₂×C₄⋊C₄	—	142	1	no	—
181	C₂×C₄⋊C₄	—	142	2	no	—
182	C₂×C₄⋊C₄	—	142	1	no	—
183	C₂×C₄⋊C₄	—	142	1	no	—
184	C₂×C₄²	—	142	2	no	—
185	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
187	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
188	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
192	C₄⋊C₈	—	284	2	no	—
193	C₄⋊C₈	—	284	2	no	—
194	C₄⋊C₈	—	142	4	no	—
195	C₄×C₈	—	2	4	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	Dic₁₄₂	1	4	yes	complemented
199	#95	—	142	2	no	—
200	#95	—	142	2	no	—
201	C₂×Q₈	—	284	2	no	—
202	C₂×Q₈	—	284	2	no	—
203	C₂×Q₈	—	142	4	no	—
204	C₂²×C₄	—	142	2	no	—
205	C₂²×C₄	—	142	2	no	—
206	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	284	2	no	—
217	C₂×C₈	—	284	2	no	—
218	C₂×C₈	—	284	2	no	—
219	C₄⋊C₄	—	284	4	no	—
220	C₄⋊C₄	—	284	2	no	—
221	C₄⋊C₄	—	284	2	no	—
222	C₄⋊C₄	—	71	4	no	—
223	C₄⋊C₄	—	142	4	no	—
224	C₄⋊C₄	—	142	4	no	—
225	C₄⋊C₄	—	142	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	284	2	no	—
230	C₄²	—	284	2	no	—
231	C₄²	C₈×D₇₁	1	2	yes	—
232	C₄²	#11765848	1	2	yes	—
233	C₄²	D₅₆₈	1	2	yes	—
234	C₄²	#11765854	1	2	yes	—
235	C₄²	#11765854	1	2	yes	—
236	C₄²	Dic₂₈₄	1	2	yes	—
237	C₄²	C₂×Dic₁₄₂	1	2	yes	—
238	C₄²	#11765865	1	2	yes	—
239	C₂³	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
240	Q₈	—	284	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	284	2	no	—
248	C₂×C₄	—	284	2	no	—
249	C₂×C₄	—	284	2	no	—
250	C₂×C₄	—	284	2	no	—
251	C₂×C₄	—	284	2	no	—
252	C₂×C₄	—	142	4	no	—
253	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
254	C₂×C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
255	C₂×C₄	#14556050	1	2	yes	—
256	C₂×C₄	#14556050	1	2	yes	—
257	C₂×C₄	#14556053	1	2	yes	—
258	C₂×C₄	#14556053	1	2	yes	—
259	C₂×C₄	#14556056	1	2	yes	—
260	C₂×C₄	#14556056	1	2	yes	—
261	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
262	C₂×C₄	#14556107	1	2	yes	—
263	C₂×C₄	#14556116	1	1	yes	characteristic
264	C₂×C₄	#14556128	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
266	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
267	C₂×C₄	#14556134	1	2	yes	—
268	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
269	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
270	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
271	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	568	2	no	—
274	C₂²	#17985905	1	1	yes	characteristic
275	C₂²	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
276	C₂²	#17985965	1	1	yes	characteristic
277	C₂²	C₂×#14556050	1	1	yes	characteristic
278	C₂²	C₂×#14556053	1	1	yes	characteristic
279	C₂²	C₂×#14556056	1	1	yes	characteristic
280	C₂²	C₂×#14556056	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	284	4	no	—
285	C₄	—	568	2	no	—
286	C₄	#17985728	1	4	yes	—
287	C₄	#17985731	1	4	yes	—
288	C₄	C₈×Dic₁₄₂	1	2	yes	—
289	C₄	#17986129	1	2	yes	—
290	C₄	#17986346	1	1	yes	characteristic
291	C₄	#17986350	1	1	yes	characteristic
292	C₄	#17986397	1	1	yes	characteristic
293	C₄	#17986404	1	1	yes	characteristic
294	C₂	C₂×#17985728	1	1	yes	characteristic
295	C₂	C₂×#17985731	1	1	yes	characteristic
296	C₂	#22586792	1	1	yes	characteristic
297	C₂	#22586796	1	1	yes	characteristic
298	C₂	#22589276	1	1	yes	characteristic
299	C₂	#22589301	1	1	yes	characteristic
300	C₂	#22589317	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄²×C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄²×C₈	D₇₁
hypocenter	C₇₁	#9870
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
23. #14556128	C₈	semidirect
49. C₄×C₈×C₇₁	Q₈	semidirect
155. C₇₁	#9870	semidirect
198. C₄×C₈	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14556128	2
#22588492	2
#22586180	2
C₄²×C₈×C₇₁	2
#9870	71