Subgroups — #31969228

Order 18176.

586 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	#22598926	C₂	1	1	yes	maximal, characteristic
2	#22597283	C₂	1	2	yes	maximal
3	#22587236	C₂	1	2	yes	maximal, complemented
4	#22587189	C₂	1	4	yes	maximal, complemented
5	#22587146	C₂	1	2	yes	maximal, complemented
6	#22586742	C₂	1	2	yes	maximal
7	C₈×#11765849	C₂	1	1	yes	maximal, characteristic
8	C₂×#335×C₇₁	C₂	1	1	yes	maximal, characteristic
9	#17986556	C₂²	1	2	yes	—
10	#17986404	C₄	1	1	yes	characteristic
11	#17986379	C₄	1	1	yes	characteristic
12	#17986350	C₄	1	2	yes	—
13	#17986240	C₄	1	2	yes	—
14	#17986240	C₄	1	2	yes	—
15	#17986128	—	2	4	no	—
16	#17986126	C₂²	1	4	yes	—
17	#17986123	C₂²	1	4	yes	—
18	C₄×Dic₂₈₄	C₂²	1	2	yes	—
19	C₄×Dic₂₈₄	C₂²	1	2	yes	—
20	#17986052	C₂²	1	1	yes	characteristic
21	#17986025	—	2	2	no	—
22	#17985954	—	2	2	no	—
23	C₄×#11765849	C₂²	1	1	yes	characteristic
24	#17985729	C₂²	1	4	yes	—
25	#17985729	C₂²	1	4	yes	—
26	#17985721	C₂²	1	2	yes	—
27	C₈×C₇₁⋊C₈	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
30	#335×C₇₁	C₂²	1	1	yes	characteristic
31	#335×C₇₁	C₂²	1	2	yes	—
32	#335×C₇₁	C₂²	1	1	yes	characteristic
33	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
34	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
35	C₂×#11765849	D₄	1	1	yes	characteristic
36	C₂×#11765849	D₄	1	1	yes	characteristic
37	#14556128	C₂×C₄	1	2	yes	—
38	#14556128	C₂×C₄	1	2	yes	—
39	#14556120	—	2	2	no	—
40	#14556107	—	2	2	no	—
41	#14556107	C₂×C₄	1	4	yes	—
42	C₄×Dic₁₄₂	—	2	2	no	—
43	C₄×Dic₁₄₂	C₂³	1	4	yes	—
44	#14556087	—	2	2	no	—
45	#14556062	C₂×C₄	1	2	yes	—
46	#14556062	C₂×C₄	1	2	yes	—
47	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
48	Dic₇₁⋊C₈	—	2	4	no	—
49	Dic₇₁⋊C₈	—	2	4	no	—
50	#14556053	C₂×C₄	1	4	yes	—
51	#14556053	C₂×C₄	1	4	yes	—
52	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
53	#14556039	D₄	1	2	yes	—
54	#14556038	—	2	4	no	—
55	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
56	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
57	C₈×Dic₇₁	—	2	4	no	—
58	C₂²×C₈×C₇₁	—	2	2	no	—
59	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₇₁	—	2	2	no	—
63	C₄⋊C₈×C₇₁	C₂³	1	2	yes	—
64	C₄⋊C₈×C₇₁	C₂³	1	2	yes	—
65	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
66	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₄₂	—	2	2	no	—
70	C₂×Dic₁₄₂	C₂²×C₄	1	4	yes	—
71	Dic₂₈₄	—	2	4	no	—
72	Dic₂₈₄	—	4	2	no	—
73	#11765849	—	2	2	no	—
74	#11765849	Q₁₆	1	4	yes	complemented
75	C₂×C₇₁⋊C₈	—	2	2	no	—
76	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
77	C₂×C₇₁⋊C₈	C₂×D₄	1	2	yes	—
78	C₄⋊Dic₇₁	—	2	2	no	—
79	C₄⋊Dic₇₁	—	2	2	no	—
80	C₄⋊Dic₇₁	—	2	2	no	—
81	C₄⋊Dic₇₁	C₂²×C₄	1	4	yes	—
82	Dic₇₁⋊C₄	—	2	4	no	—
83	Dic₇₁⋊C₄	—	2	4	no	—
84	C₂²⋊Dic₇₁	—	2	4	no	—
85	C₄×Dic₇₁	—	2	4	no	—
86	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
89	C₂×C₈×C₇₁	—	2	2	no	—
90	C₂×C₈×C₇₁	—	2	2	no	—
91	C₂×C₈×C₇₁	—	2	2	no	—
92	C₂×C₈×C₇₁	—	4	1	no	—
93	C₂×C₈×C₇₁	—	4	4	no	—
94	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₁	#41	1	1	yes	characteristic
101	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
102	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	—	2	1	no	—
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	4	1	no	—
106	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₇₁	#135	1	2	yes	—
111	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
114	C₂×Dic₇₁	—	2	4	no	—
115	C₂×Dic₇₁	—	2	4	no	—
116	Dic₁₄₂	—	2	4	no	—
117	Dic₁₄₂	—	2	4	no	—
118	Dic₁₄₂	—	8	2	no	—
119	C₈×C₇₁	—	2	1	no	—
120	C₈×C₇₁	—	2	1	no	—
121	C₈×C₇₁	—	2	1	no	—
122	C₈×C₇₁	—	4	2	no	—
123	C₈×C₇₁	—	4	2	no	—
124	C₈×C₇₁	#131	1	2	yes	—
125	C₇₁⋊C₈	—	2	2	no	—
126	C₇₁⋊C₈	—	4	2	no	—
127	C₇₁⋊C₈	C₂×Q₁₆	1	4	yes	—
128	C₂²×C₇₁	—	2	1	no	—
129	C₂²×C₇₁	—	2	1	no	—
130	C₂²×C₇₁	C₄×Q₁₆	1	2	yes	—
131	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₇₁	—	2	1	no	—
133	C₄×C₇₁	—	2	1	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	C₄×Q₁₆	1	2	yes	—
136	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
137	C₄×C₇₁	#568	1	1	yes	characteristic
138	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
139	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
140	Dic₇₁	—	4	4	no	—
141	Dic₇₁	—	4	4	no	—
142	#18068	—	71	1	no	maximal, Sylow
143	C₂×C₇₁	—	2	2	no	—
144	C₂×C₇₁	#2935	1	1	yes	characteristic
145	C₂×C₇₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₇₁	#2957	1	1	yes	characteristic
147	#3170	—	71	1	no	—
148	#2962	—	71	2	no	—
149	#1591	—	71	2	no	—
150	#1584	—	71	4	no	—
151	C₈×Q₁₆	—	71	2	no	—
152	C₂×#335	D₇₁	1	1	yes	characteristic
153	#1481	—	71	2	no	—
154	C₈×M₁₆	—	71	1	no	—
155	C₇₁	#18068	1	1	yes	characteristic, complemented, Sylow
156	#533	—	71	2	no	—
157	#500	—	71	1	no	—
158	#494	—	71	1	no	—
159	#487	—	71	2	no	—
160	#467	—	71	2	no	—
161	#467	—	71	2	no	—
162	#446	—	71	4	no	—
163	#446	—	142	4	no	—
164	C₈×Q₈	—	71	4	no	—
165	C₄×Q₁₆	—	71	2	no	—
166	C₄×Q₁₆	—	71	2	no	—
167	#431	—	71	1	no	—
168	#424	—	142	2	no	—
169	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
170	#405	—	142	2	no	—
171	C₄×M₁₆	—	71	1	no	—
172	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
173	#335	C₂×D₇₁	1	1	yes	characteristic
174	#335	C₂×D₇₁	1	2	yes	—
175	#335	C₂×D₇₁	1	1	yes	characteristic
176	Q₈⋊C₈	—	71	4	no	—
177	Q₈⋊C₈	—	71	4	no	—
178	#322	—	71	2	no	—
179	C₈²	—	71	2	no	—
180	C₂×Q₁₆	—	71	2	no	—
181	C₂×Q₁₆	—	71	2	no	—
182	C₂×M₁₆	—	71	1	no	—
183	C₂×M₁₆	—	71	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
186	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
187	C₄⋊Q₈	—	71	2	no	—
188	C₄⋊Q₈	—	71	2	no	—
189	#125	—	142	2	no	—
190	C₂²⋊Q₈	—	71	4	no	—
191	C₂²⋊Q₈	—	142	2	no	—
192	C₄×Q₈	—	71	4	no	—
193	C₄×Q₈	—	142	2	no	—
194	#117	—	142	2	no	—
195	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
196	#107	—	71	2	no	—
197	#107	—	71	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	142	4	no	—
200	C₄⋊C₈	—	142	4	no	—
201	C₄⋊C₈	—	142	1	no	—
202	C₄⋊C₈	C₂²×D₇₁	1	2	yes	—
203	C₄⋊C₈	C₂²×D₇₁	1	2	yes	—
204	#103	—	71	4	no	—
205	#103	—	71	4	no	—
206	C₂²⋊C₈	—	142	1	no	—
207	#97	—	71	2	no	—
208	#97	—	142	4	no	—
209	C₄×C₈	—	71	2	no	—
210	C₄×C₈	—	71	2	no	—
211	C₄×C₈	—	142	4	no	—
212	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
213	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
214	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
215	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
216	C₂×Q₈	—	71	4	no	—
217	C₂×Q₈	—	142	2	no	—
218	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
220	C₂²×C₄	#11765866	1	1	yes	characteristic
221	Q₁₆	—	284	2	no	—
222	Q₁₆	—	142	4	no	—
223	M₁₆	—	71	4	no	—
224	M₁₆	—	142	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	71	2	no	—
231	C₂×C₈	—	71	2	no	—
232	C₂×C₈	—	142	2	no	—
233	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
234	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
237	C₄⋊C₄	—	71	4	no	—
238	C₄⋊C₄	—	142	4	no	—
239	C₄⋊C₄	—	142	4	no	—
240	C₄⋊C₄	—	142	2	no	—
241	C₄⋊C₄	—	142	2	no	—
242	C₄⋊C₄	—	142	2	no	—
243	C₂²⋊C₄	—	142	4	no	—
244	C₄²	—	142	4	no	—
245	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
246	C₄²	D₄×D₇₁	1	1	yes	characteristic
247	C₄²	#11765866	1	1	yes	characteristic
248	C₄²	C₂³×D₇₁	1	1	yes	characteristic
249	C₂³	#14556090	1	1	yes	characteristic
250	Q₈	—	568	2	no	—
251	Q₈	—	142	4	no	—
252	Q₈	—	142	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	142	4	no	—
257	C₂×C₄	—	142	4	no	—
258	C₂×C₄	#14556090	1	1	yes	characteristic
259	C₂×C₄	#14556090	1	1	yes	characteristic
260	C₂×C₄	#14556090	1	1	yes	characteristic
261	C₂×C₄	Q₁₆×D₇₁	1	2	yes	—
262	C₂×C₄	#14556147	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
265	C₂×C₄	C₂×#11765866	1	1	yes	characteristic
266	C₈	—	2	1	no	—
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	4	2	no	—
270	C₈	—	4	2	no	—
271	C₈	—	284	2	no	—
272	C₈	—	71	4	no	—
273	C₈	—	142	2	no	—
274	C₈	#14556134	1	2	yes	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	#17986093	1	2	yes	—
278	C₂²	C₂×#14556090	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	—	284	4	no	—
283	C₄	—	284	4	no	—
284	C₄	#17986093	1	2	yes	—
285	C₄	C₂×#14556134	1	1	yes	characteristic
286	C₄	#17986765	1	1	yes	characteristic
287	C₄	C₂×Q₁₆×D₇₁	1	1	yes	characteristic
288	C₄	C₂×#14556147	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	#22597083	1	1	yes	characteristic
291	C₂	C₂×#17986093	1	1	yes	characteristic
292	C₂	#22597256	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₂×#335×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	C₂×#335	D₇₁
hypocenter	C₇₁	#18068
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. #22587236	C₂	semidirect
4. #22587189	C₂	semidirect
5. #22587146	C₂	semidirect
74. #11765849	Q₁₆	semidirect
155. C₇₁	#18068	semidirect

Maximal subgroups

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

subgroup	index
#22598926	2
#22597283	2
#22587236	2
#22587189	2
#22587146	2
#22586742	2
C₈×#11765849	2
C₂×#335×C₇₁	2
#18068	71