Subgroups — #31968314

Order 18176.

544 conjugacy classes of subgroups.

The subgroup lattice has 264 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	#22597434	C₂	1	2	yes	maximal
3	#22587213	C₂	1	4	yes	maximal, complemented
4	#22587192	C₂	1	4	yes	maximal, complemented
5	#22587103	C₂	1	1	yes	maximal, characteristic
6	#22586814	C₂	1	2	yes	maximal
7	C₇₁×#1452	C₂	1	1	yes	maximal, characteristic
8	#17986556	C₂²	1	2	yes	—
9	#17986404	C₄	1	1	yes	characteristic
10	#17986379	C₄	1	1	yes	characteristic
11	#17986350	C₄	1	2	yes	—
12	#17986240	C₄	1	2	yes	—
13	#17986240	C₄	1	2	yes	—
14	#17986129	C₂²	1	4	yes	—
15	#17986128	—	2	4	no	—
16	C₈×Dic₁₄₂	C₂²	1	4	yes	—
17	C₄×Dic₂₈₄	C₂²	1	2	yes	—
18	C₄×Dic₂₈₄	C₂²	1	2	yes	—
19	#17986057	—	2	4	no	—
20	#17986056	C₂²	1	2	yes	—
21	#17985752	C₂²	1	2	yes	—
22	#17985750	C₂²	1	2	yes	—
23	#17985731	C₂²	1	8	yes	—
24	#431×C₇₁	C₂²	1	1	yes	characteristic
25	#431×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
27	#322×C₇₁	C₂²	1	2	yes	—
28	#322×C₇₁	C₂²	1	2	yes	—
29	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
30	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
31	#14556128	C₂×C₄	1	2	yes	—
32	#14556128	C₂×C₄	1	2	yes	—
33	#14556120	—	2	2	no	—
34	#14556107	—	2	2	no	—
35	#14556107	C₂×C₄	1	4	yes	—
36	C₄×Dic₁₄₂	—	2	2	no	—
37	C₄×Dic₁₄₂	C₂³	1	4	yes	—
38	#14556087	—	2	2	no	—
39	#14556062	C₂×C₄	1	2	yes	—
40	#14556062	C₂×C₄	1	2	yes	—
41	C₄⋊(C₇₁⋊C₈)	C₂³	1	4	yes	—
42	Dic₇₁⋊C₈	—	2	4	no	—
43	Dic₇₁⋊C₈	—	2	4	no	—
44	#14556053	C₂×C₄	1	4	yes	—
45	#14556053	C₂×C₄	1	4	yes	—
46	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
47	#14556039	—	2	2	no	—
48	#14556038	—	2	4	no	—
49	C₈×Dic₇₁	—	2	4	no	—
50	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₇₁	—	2	1	no	—
54	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
55	C₂²⋊C₈×C₇₁	—	2	1	no	—
56	C₂²⋊C₈×C₇₁	D₄	1	2	yes	—
57	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₁	C₂³	1	2	yes	—
60	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₇₁	C₂³	1	2	yes	—
63	C₂×Dic₁₄₂	—	2	2	no	—
64	C₂×Dic₁₄₂	C₂²×C₄	1	4	yes	—
65	Dic₂₈₄	—	4	2	no	—
66	Dic₂₈₄	—	4	2	no	—
67	C₂×C₇₁⋊C₈	—	2	4	no	—
68	C₄⋊Dic₇₁	—	2	2	no	—
69	C₄⋊Dic₇₁	—	2	2	no	—
70	C₄⋊Dic₇₁	—	2	2	no	—
71	C₄⋊Dic₇₁	C₂²×C₄	1	4	yes	—
72	Dic₇₁⋊C₄	—	2	4	no	—
73	Dic₇₁⋊C₄	—	2	4	no	—
74	C₂²⋊Dic₇₁	—	2	4	no	—
75	C₄×Dic₇₁	—	2	4	no	—
76	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
79	C₂×C₈×C₇₁	—	2	2	no	—
80	C₂×C₈×C₇₁	—	4	1	no	—
81	C₂×C₈×C₇₁	—	4	1	no	—
82	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
87	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₁	#41	1	1	yes	characteristic
90	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
91	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	1	no	—
94	C₂×C₄×C₇₁	—	2	2	no	—
95	C₂×C₄×C₇₁	—	4	1	no	—
96	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₇₁	—	2	4	no	—
103	C₂×Dic₇₁	—	2	4	no	—
104	Dic₁₄₂	—	2	4	no	—
105	Dic₁₄₂	—	2	4	no	—
106	Dic₁₄₂	—	8	2	no	—
107	C₈×C₇₁	—	2	1	no	—
108	C₈×C₇₁	—	2	1	no	—
109	C₈×C₇₁	—	2	1	no	—
110	C₈×C₇₁	—	2	1	no	—
111	C₈×C₇₁	—	2	4	no	—
112	C₈×C₇₁	—	4	2	no	—
113	C₇₁⋊C₈	—	4	4	no	—
114	C₂²×C₇₁	—	2	1	no	—
115	C₂²×C₇₁	—	2	1	no	—
116	C₂²×C₇₁	—	2	1	no	—
117	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₇₁	—	2	1	no	—
119	C₄×C₇₁	—	2	1	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	#568	1	1	yes	characteristic
123	C₄×C₇₁	#568	1	1	yes	characteristic
124	C₄×C₇₁	#578	1	2	yes	—
125	Dic₇₁	—	4	4	no	—
126	Dic₇₁	—	4	4	no	—
127	#17991	—	71	1	no	maximal, Sylow
128	C₂×C₇₁	—	4	1	no	—
129	C₂×C₇₁	#2932	1	1	yes	characteristic
130	C₂×C₇₁	#2950	1	1	yes	characteristic
131	C₂×C₇₁	#2957	1	1	yes	characteristic
132	#3170	—	71	1	no	—
133	#2985	—	71	2	no	—
134	#1587	—	71	4	no	—
135	#1584	—	71	4	no	—
136	#1568	—	71	1	no	—
137	#1497	—	71	2	no	—
138	#1452	D₇₁	1	1	yes	characteristic
139	C₇₁	#17991	1	1	yes	characteristic, complemented, Sylow
140	#533	—	71	2	no	—
141	#500	—	71	1	no	—
142	#494	—	71	1	no	—
143	#487	—	71	2	no	—
144	#467	—	71	2	no	—
145	#467	—	71	2	no	—
146	#446	—	71	4	no	—
147	#446	—	142	4	no	—
148	C₈×Q₈	—	71	4	no	—
149	C₄×Q₁₆	—	71	2	no	—
150	C₄×Q₁₆	—	71	2	no	—
151	#433	—	142	4	no	—
152	#432	—	71	2	no	—
153	#431	C₂×D₇₁	1	1	yes	characteristic
154	#431	C₂×D₇₁	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
156	#335	—	71	2	no	—
157	#334	—	71	2	no	—
158	Q₈⋊C₈	—	71	8	no	—
159	#322	C₂×D₇₁	1	2	yes	—
160	#322	C₂×D₇₁	1	2	yes	—
161	C₂×Q₁₆	—	71	2	no	—
162	C₂×Q₁₆	—	71	2	no	—
163	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
164	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
165	C₄⋊Q₈	—	71	2	no	—
166	C₄⋊Q₈	—	71	2	no	—
167	#125	—	142	2	no	—
168	C₂²⋊Q₈	—	71	4	no	—
169	C₂²⋊Q₈	—	142	2	no	—
170	C₄×Q₈	—	71	4	no	—
171	C₄×Q₈	—	142	2	no	—
172	#117	—	142	2	no	—
173	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
174	#107	—	71	2	no	—
175	#107	—	71	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	71	4	no	—
178	C₄⋊C₈	—	142	4	no	—
179	C₄⋊C₈	—	142	4	no	—
180	C₄⋊C₈	D₂₈₄	1	2	yes	—
181	#103	—	71	4	no	—
182	#103	—	71	4	no	—
183	C₂²⋊C₈	—	2	1	no	—
184	C₂²⋊C₈	—	142	2	no	—
185	C₂²⋊C₈	D₂₈₄	1	2	yes	—
186	#97	—	142	4	no	—
187	#97	—	142	2	no	—
188	C₄×C₈	—	142	4	no	—
189	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
190	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₇₁	1	2	yes	—
192	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
193	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₇₁	1	2	yes	—
195	C₂×Q₈	—	71	4	no	—
196	C₂×Q₈	—	142	2	no	—
197	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
199	C₂²×C₄	#11765865	1	1	yes	characteristic
200	Q₁₆	—	284	2	no	—
201	Q₁₆	—	284	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	142	4	no	—
206	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
207	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
208	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
209	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
210	C₂×C₈	C₂×D₂₈₄	1	4	yes	—
211	C₄⋊C₄	—	71	4	no	—
212	C₄⋊C₄	—	142	4	no	—
213	C₄⋊C₄	—	142	4	no	—
214	C₄⋊C₄	—	142	2	no	—
215	C₄⋊C₄	—	142	2	no	—
216	C₄⋊C₄	—	142	2	no	—
217	C₂²⋊C₄	—	142	4	no	—
218	C₄²	—	142	4	no	—
219	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
220	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
221	C₄²	#11765865	1	1	yes	characteristic
222	C₄²	C₂³×D₇₁	1	1	yes	characteristic
223	C₂³	C₄×D₂₈₄	1	1	yes	characteristic
224	Q₈	—	568	2	no	—
225	Q₈	—	142	4	no	—
226	Q₈	—	142	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	142	4	no	—
232	C₂×C₄	—	142	4	no	—
233	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
234	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
235	C₂×C₄	C₄×D₂₈₄	1	1	yes	characteristic
236	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
237	C₂×C₄	C₂²×D₂₈₄	1	1	yes	characteristic
238	C₂×C₄	C₂×#11765865	1	1	yes	characteristic
239	C₈	—	2	4	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	—	284	4	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	C₂×C₄×D₂₈₄	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	284	4	no	—
255	C₄	—	284	4	no	—
256	C₄	#17986765	1	1	yes	characteristic
257	C₄	#17986765	1	1	yes	characteristic
258	C₄	#17986816	1	2	yes	—
259	C₂	—	4	1	no	—
260	C₂	#22597059	1	1	yes	characteristic
261	C₂	#22597209	1	1	yes	characteristic
262	C₂	#22597258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₇₁×#1452	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	C₂³×C₇₁	C₄×D₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	#1452	D₇₁
hypocenter	C₇₁	#17991
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1452

Complements

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

N	H ≅ G/N	type
3. #22587213	C₂	semidirect
4. #22587192	C₂	semidirect
139. C₇₁	#17991	semidirect

Maximal subgroups

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

subgroup	index
#22598926	2
#22597434	2
#22587213	2
#22587192	2
#22587103	2
#22586814	2
C₇₁×#1452	2
#17991	71