Subgroups — #31897577

Order 18176.

538 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22596738	C₂	1	1	yes	maximal, characteristic
2	#22588492	C₂	1	1	yes	maximal, characteristic
3	#22586224	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556128	—	2	1	no	—
6	C₂×#14556128	—	2	1	no	—
7	C₂×#14556128	C₄	1	2	yes	—
8	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#17985869	—	2	1	no	—
12	#17985869	—	2	1	no	—
13	#17985869	C₄	1	2	yes	—
14	#17985869	C₂²	1	2	yes	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	—	2	1	no	—
17	C₄³×C₇₁	C₂²	1	1	yes	characteristic
18	#336×C₇₁	—	2	1	no	—
19	C₂²×Dic₁₄₂	—	2	2	no	—
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	#14556128	—	4	4	no	—
22	#14556128	—	4	2	no	—
23	#14556128	—	4	2	no	—
24	#14556128	—	4	2	no	—
25	#14556128	—	4	2	no	—
26	C₂×C₄⋊Dic₇₁	—	2	2	no	—
27	C₂×C₄⋊Dic₇₁	—	2	2	no	—
28	C₂×C₄⋊Dic₇₁	—	2	2	no	—
29	C₂×C₄⋊Dic₇₁	—	2	2	no	—
30	C₂×C₄⋊Dic₇₁	—	2	2	no	—
31	C₂×C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
32	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
33	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
34	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
35	C₂²×C₈×C₇₁	—	2	2	no	—
36	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	D₄	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	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₁	—	2	2	no	—
44	C₄⋊C₈×C₇₁	Q₈	1	4	yes	complemented
45	C₄×C₈×C₇₁	—	4	2	no	—
46	C₂×Dic₁₄₂	—	4	4	no	—
47	C₂×Dic₁₄₂	—	4	4	no	—
48	C₂×Dic₁₄₂	—	4	4	no	—
49	C₂²×Dic₇₁	—	2	4	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	4	2	no	—
53	C₄⋊Dic₇₁	—	4	4	no	—
54	C₄⋊Dic₇₁	—	4	4	no	—
55	C₄⋊Dic₇₁	—	4	4	no	—
56	C₄⋊Dic₇₁	—	4	2	no	—
57	C₄⋊Dic₇₁	—	4	2	no	—
58	C₄⋊Dic₇₁	—	4	2	no	—
59	C₄⋊Dic₇₁	—	4	2	no	—
60	C₄⋊Dic₇₁	—	4	4	no	—
61	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₁	—	2	4	no	—
69	C₂×C₈×C₇₁	—	4	2	no	—
70	C₂×C₈×C₇₁	—	4	2	no	—
71	C₄²×C₇₁	—	2	2	no	—
72	C₄²×C₇₁	—	2	2	no	—
73	C₄²×C₇₁	—	2	2	no	—
74	C₄²×C₇₁	D₈	1	2	yes	—
75	C₄²×C₇₁	D₈	1	2	yes	—
76	C₄²×C₇₁	SD₁₆	1	2	yes	—
77	C₄²×C₇₁	SD₁₆	1	2	yes	—
78	C₄²×C₇₁	SD₁₆	1	2	yes	—
79	C₄²×C₇₁	Q₁₆	1	2	yes	—
80	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
81	C₄²×C₇₁	#41	1	2	yes	—
82	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₁	—	2	1	no	—
84	C₂×C₄×C₇₁	—	2	1	no	—
85	C₂×C₄×C₇₁	—	2	1	no	—
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	#100	1	2	yes	—
90	C₂×C₄×C₇₁	#101	1	2	yes	—
91	C₂×C₄×C₇₁	#102	1	2	yes	—
92	C₂×C₄×C₇₁	#102	1	2	yes	—
93	C₂×C₄×C₇₁	#102	1	2	yes	—
94	C₂×C₄×C₇₁	#102	1	2	yes	—
95	C₂×C₄×C₇₁	#103	1	2	yes	—
96	C₂×C₄×C₇₁	#103	1	2	yes	—
97	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
98	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
99	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#137	1	2	yes	—
108	C₂×Dic₇₁	—	4	4	no	—
109	C₂×Dic₇₁	—	4	4	no	—
110	C₂×Dic₇₁	—	4	4	no	—
111	Dic₁₄₂	—	8	8	no	—
112	C₈×C₇₁	—	4	4	no	—
113	C₇₁⋊C₈	—	8	2	no	—
114	C₂²×C₇₁	#391	1	1	yes	characteristic
115	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
116	C₂²×C₇₁	C₂×#101	1	1	yes	characteristic
117	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
118	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
120	C₂²×C₇₁	#419	1	1	yes	characteristic
121	C₄×C₇₁	—	2	2	no	—
122	C₄×C₇₁	—	2	2	no	—
123	C₄×C₇₁	—	2	2	no	—
124	C₄×C₇₁	#331	1	4	yes	—
125	C₄×C₇₁	#332	1	4	yes	—
126	C₄×C₇₁	#474	1	2	yes	—
127	C₄×C₇₁	#475	1	2	yes	—
128	C₄×C₇₁	#486	1	1	yes	characteristic
129	C₄×C₇₁	#487	1	1	yes	characteristic
130	C₄×C₇₁	#498	1	1	yes	characteristic
131	C₄×C₇₁	#500	1	1	yes	characteristic
132	Dic₇₁	—	8	4	no	—
133	#9898	—	71	1	no	maximal, Sylow
134	C₂×C₇₁	C₂×#331	1	1	yes	characteristic
135	C₂×C₇₁	C₂×#332	1	1	yes	characteristic
136	C₂×C₇₁	#1546	1	1	yes	characteristic
137	C₂×C₇₁	#1548	1	1	yes	characteristic
138	C₂×C₇₁	#1988	1	1	yes	characteristic
139	C₂×C₇₁	#1990	1	1	yes	characteristic
140	C₂×C₇₁	#1997	1	1	yes	characteristic
141	#2868	—	71	1	no	—
142	#1842	—	71	1	no	—
143	C₄×C₄⋊C₈	D₇₁	1	1	yes	characteristic
144	#1296	—	71	4	no	—
145	C₇₁	#9898	1	1	yes	characteristic, complemented, Sylow
146	C₂×C₄⋊Q₈	—	71	2	no	—
147	C₂×C₄⋊Q₈	—	142	1	no	—
148	C₂×C₄⋊Q₈	—	142	1	no	—
149	C₂×C₄⋊C₈	—	71	2	no	—
150	C₂×C₄⋊C₈	—	142	1	no	—
151	C₂×C₄⋊C₈	—	142	1	no	—
152	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	#384	—	71	2	no	—
155	#384	—	71	2	no	—
156	#384	—	142	1	no	—
157	#384	—	142	1	no	—
158	C₄³	C₂×D₇₁	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	C₂²×Q₈	—	142	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	142	2	no	—
163	C₄⋊Q₈	—	284	4	no	—
164	C₄⋊Q₈	—	284	2	no	—
165	C₄⋊Q₈	—	284	2	no	—
166	C₄⋊Q₈	—	284	2	no	—
167	C₄⋊Q₈	—	284	2	no	—
168	C₂×C₄⋊C₄	—	71	4	no	—
169	C₂×C₄⋊C₄	—	142	2	no	—
170	C₂×C₄⋊C₄	—	142	2	no	—
171	C₂×C₄⋊C₄	—	142	2	no	—
172	C₂×C₄⋊C₄	—	142	2	no	—
173	C₂×C₄⋊C₄	—	142	2	no	—
174	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₇₁	1	1	yes	characteristic
176	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
177	C₂×C₄²	D₂₈₄	1	1	yes	characteristic
178	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
180	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	—	284	2	no	—
183	C₄⋊C₈	—	284	2	no	—
184	C₄⋊C₈	—	142	4	no	—
185	C₄⋊C₈	Dic₁₄₂	1	4	yes	complemented
186	C₄×C₈	—	4	2	no	—
187	C₂×Q₈	—	284	4	no	—
188	C₂×Q₈	—	284	4	no	—
189	C₂×Q₈	—	284	4	no	—
190	C₂²×C₄	—	142	4	no	—
191	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
192	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
193	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
194	C₂²×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	4	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	284	2	no	—
202	C₂×C₈	—	284	2	no	—
203	C₂×C₈	—	284	2	no	—
204	C₄⋊C₄	—	284	4	no	—
205	C₄⋊C₄	—	284	4	no	—
206	C₄⋊C₄	—	284	4	no	—
207	C₄⋊C₄	—	284	4	no	—
208	C₄⋊C₄	—	284	2	no	—
209	C₄⋊C₄	—	284	2	no	—
210	C₄⋊C₄	—	284	2	no	—
211	C₄⋊C₄	—	284	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	#11765850	1	2	yes	—
216	C₄²	D₅₆₈	1	2	yes	—
217	C₄²	#11765852	1	2	yes	—
218	C₄²	#11765854	1	2	yes	—
219	C₄²	#11765854	1	2	yes	—
220	C₄²	Dic₂₈₄	1	2	yes	—
221	C₄²	C₂×Dic₁₄₂	1	2	yes	—
222	C₄²	#11765865	1	2	yes	—
223	C₂³	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
224	Q₈	—	568	8	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	284	4	no	—
232	C₂×C₄	—	284	4	no	—
233	C₂×C₄	—	284	4	no	—
234	C₂×C₄	#14556044	1	2	yes	—
235	C₂×C₄	#14556046	1	2	yes	—
236	C₂×C₄	#14556048	1	2	yes	—
237	C₂×C₄	#14556048	1	2	yes	—
238	C₂×C₄	#14556050	1	2	yes	—
239	C₂×C₄	#14556050	1	2	yes	—
240	C₂×C₄	#14556053	1	2	yes	—
241	C₂×C₄	#14556053	1	2	yes	—
242	C₂×C₄	C₄×Dic₁₄₂	1	2	yes	—
243	C₂×C₄	#14556107	1	2	yes	—
244	C₂×C₄	#14556116	1	1	yes	characteristic
245	C₂×C₄	#14556128	1	1	yes	characteristic
246	C₂×C₄	C₂×#11765850	1	1	yes	characteristic
247	C₂×C₄	C₂×D₅₆₈	1	1	yes	characteristic
248	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
249	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
250	C₂×C₄	C₂×#11765854	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
252	C₂×C₄	#14556158	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	568	2	no	—
255	C₂²	#17985905	1	1	yes	characteristic
256	C₂²	C₂×#14556044	1	1	yes	characteristic
257	C₂²	C₂×#14556046	1	1	yes	characteristic
258	C₂²	C₂×#14556048	1	1	yes	characteristic
259	C₂²	C₂×#14556050	1	1	yes	characteristic
260	C₂²	C₂×#14556053	1	1	yes	characteristic
261	C₂²	#17986007	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	568	4	no	—
266	C₄	#17985742	1	4	yes	—
267	C₄	#17985746	1	4	yes	—
268	C₄	#17986279	1	2	yes	—
269	C₄	#17986284	1	2	yes	—
270	C₄	#17986346	1	1	yes	characteristic
271	C₄	#17986350	1	1	yes	characteristic
272	C₄	#17986397	1	1	yes	characteristic
273	C₄	#17986404	1	1	yes	characteristic
274	C₂	C₂×#17985742	1	1	yes	characteristic
275	C₂	C₂×#17985746	1	1	yes	characteristic
276	C₂	#22587011	1	1	yes	characteristic
277	C₂	#22587020	1	1	yes	characteristic
278	C₂	#22589292	1	1	yes	characteristic
279	C₂	#22589301	1	1	yes	characteristic
280	C₂	#22589329	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	C₄×C₄⋊C₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#9898
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
44. C₄⋊C₈×C₇₁	Q₈	semidirect
145. C₇₁	#9898	semidirect
185. 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
#22596738	2
#22588492	2
#22586224	2
C₄×C₄⋊C₈×C₇₁	2
#9898	71