Subgroups — #31890414

Order 18176.

418 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22587103	C₂	1	1	yes	maximal, characteristic, complemented
2	#22587103	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586681	C₂	1	2	yes	maximal
4	#22586210	C₂	1	2	yes	maximal, complemented
5	C₂×#436×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#17986056	C₂²	1	2	yes	—
7	#17986056	C₂²	1	2	yes	—
8	#17985752	—	2	1	no	—
9	#17985752	—	2	1	no	—
10	#17985750	—	2	1	no	—
11	#17985750	—	2	1	no	—
12	#17985723	—	2	4	no	—
13	C₂²×M₁₆×C₇₁	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
15	#436×C₇₁	—	2	1	no	—
16	#436×C₇₁	—	2	1	no	—
17	#436×C₇₁	—	2	1	no	—
18	#436×C₇₁	—	2	1	no	—
19	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₈×C₇₁	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
23	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
25	C₂²⋊(C₇₁⋊C₈)	—	2	2	no	—
26	#14556039	—	2	2	no	—
27	#14556039	—	2	2	no	—
28	C₂²×D₄×C₇₁	Q₈	1	1	yes	characteristic
29	C₂³×C₄×C₇₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₇₁	—	2	2	no	—
31	C₂×M₁₆×C₇₁	—	2	2	no	—
32	C₂×M₁₆×C₇₁	—	4	4	no	—
33	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₇₁	—	2	1	no	—
37	C₄×D₄×C₇₁	—	2	1	no	—
38	C₄×D₄×C₇₁	—	2	1	no	—
39	C₄×D₄×C₇₁	—	2	1	no	—
40	C₂×C₄⋊C₄×C₇₁	Q₈	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
42	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₇₁	—	2	1	no	—
44	C₄⋊C₈×C₇₁	—	2	1	no	—
45	C₂²⋊C₈×C₇₁	—	2	2	no	—
46	C₂²⋊C₈×C₇₁	—	2	2	no	—
47	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	4	2	no	—
53	C₂⁴×C₇₁	C₄⋊C₄	1	2	yes	—
54	C₂×D₄×C₇₁	—	2	1	no	—
55	C₂×D₄×C₇₁	—	2	1	no	—
56	C₂×D₄×C₇₁	—	4	2	no	—
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	—	2	2	no	—
59	C₂²×C₄×C₇₁	—	4	2	no	—
60	C₂²×C₄×C₇₁	—	4	2	no	—
61	C₂²×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
65	M₁₆×C₇₁	—	4	4	no	—
66	M₁₆×C₇₁	—	4	4	no	—
67	C₂×C₈×C₇₁	—	2	2	no	—
68	C₂×C₈×C₇₁	—	2	2	no	—
69	C₂×C₈×C₇₁	—	4	1	no	—
70	C₂×C₈×C₇₁	—	4	2	no	—
71	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
75	C₄⋊C₄×C₇₁	—	2	1	no	—
76	C₄⋊C₄×C₇₁	—	2	1	no	—
77	C₂²⋊C₄×C₇₁	—	2	2	no	—
78	C₂²⋊C₄×C₇₁	—	2	2	no	—
79	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₁	#41	1	1	yes	characteristic
82	C₄²×C₇₁	#41	1	1	yes	characteristic
83	C₂³×C₇₁	—	2	2	no	—
84	C₂³×C₇₁	—	2	4	no	—
85	C₂³×C₇₁	—	4	2	no	—
86	C₂³×C₇₁	—	4	2	no	—
87	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
88	D₄×C₇₁	—	4	2	no	—
89	D₄×C₇₁	—	4	2	no	—
90	C₂×C₄×C₇₁	—	2	2	no	—
91	C₂×C₄×C₇₁	—	2	1	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	2	no	—
94	C₂×C₄×C₇₁	—	4	1	no	—
95	C₂×C₄×C₇₁	—	4	4	no	—
96	C₂×C₄×C₇₁	—	4	2	no	—
97	C₂×C₄×C₇₁	—	4	2	no	—
98	C₂×C₄×C₇₁	—	4	2	no	—
99	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
101	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
102	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
103	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
105	C₈×C₇₁	—	2	1	no	—
106	C₈×C₇₁	—	2	1	no	—
107	C₈×C₇₁	—	2	1	no	—
108	C₈×C₇₁	—	4	2	no	—
109	C₈×C₇₁	#100	1	2	yes	—
110	C₇₁⋊C₈	—	8	2	no	—
111	C₇₁⋊C₈	—	8	2	no	—
112	C₂²×C₇₁	—	2	1	no	—
113	C₂²×C₇₁	—	2	1	no	—
114	C₂²×C₇₁	—	4	2	no	—
115	C₂²×C₇₁	—	4	2	no	—
116	C₂²×C₇₁	—	4	2	no	—
117	C₂²×C₇₁	—	8	4	no	—
118	C₂²×C₇₁	#380	1	1	yes	characteristic
119	C₂²×C₇₁	#430	1	2	yes	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	2	1	no	—
123	C₄×C₇₁	—	2	1	no	—
124	C₄×C₇₁	—	2	1	no	—
125	C₄×C₇₁	—	4	2	no	—
126	C₄×C₇₁	C₂×#100	1	1	yes	characteristic
127	C₄×C₇₁	#413	1	1	yes	characteristic
128	#8624	—	71	1	no	maximal, Sylow
129	C₂×C₇₁	—	2	2	no	—
130	C₂×C₇₁	—	8	2	no	—
131	C₂×C₇₁	#1800	1	1	yes	characteristic
132	C₂×C₇₁	#1811	1	1	yes	characteristic
133	C₂×C₇₁	#1812	1	1	yes	characteristic
134	C₂×#436	D₇₁	1	1	yes	characteristic
135	#1568	—	71	1	no	—
136	#1568	—	71	1	no	—
137	#1466	—	71	2	no	—
138	#1290	—	71	2	no	—
139	C₇₁	#8624	1	1	yes	characteristic, complemented, Sylow
140	C₂²×M₁₆	Dic₇₁	1	2	yes	—
141	C₂×C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
142	#436	—	2	1	no	—
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	#436	—	2	1	no	—
146	#432	—	71	2	no	—
147	#432	—	71	2	no	—
148	C₂×C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
149	C₂×C₂²⋊C₈	Dic₇₁	1	2	yes	—
150	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
151	#335	—	142	1	no	—
152	#335	—	142	1	no	—
153	#334	—	142	1	no	—
154	#334	—	142	1	no	—
155	#323	—	142	4	no	—
156	C₂²×D₄	Dic₁₄₂	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₇₁	1	2	yes	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	4	4	no	—
161	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
162	C₂²×C₈	C₂×Dic₇₁	1	2	yes	—
163	C₂²×C₈	C₂×Dic₇₁	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	C₄×D₄	—	2	1	no	—
168	C₂×C₄⋊C₄	Dic₁₄₂	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	D₂₈₄	1	2	yes	—
170	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	2	1	no	—
173	C₄⋊C₈	—	142	2	no	—
174	C₄⋊C₈	—	142	2	no	—
175	C₂²⋊C₈	—	2	2	no	—
176	C₂²⋊C₈	—	2	2	no	—
177	C₂²⋊C₈	—	142	2	no	—
178	C₂²⋊C₈	—	142	2	no	—
179	#97	—	142	2	no	—
180	#97	—	142	2	no	—
181	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
184	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
185	C₂⁴	C₄⋊Dic₇₁	1	2	yes	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	C₄⋊Dic₇₁	1	2	yes	—
194	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₂₈₄	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
197	M₁₆	—	4	4	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	284	2	no	—
204	C₂×C₈	—	284	2	no	—
205	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
206	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
207	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
208	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
214	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
215	C₄²	#11765865	1	1	yes	characteristic
216	C₄²	#11765865	1	1	yes	characteristic
217	C₂³	—	2	2	no	—
218	C₂³	—	2	4	no	—
219	C₂³	—	4	2	no	—
220	C₂³	—	4	2	no	—
221	C₂³	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
222	D₄	—	4	2	no	—
223	D₄	—	4	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
234	C₂×C₄	#14556087	1	1	yes	characteristic
235	C₂×C₄	#14556102	1	1	yes	characteristic
236	C₂×C₄	#14556102	1	1	yes	characteristic
237	C₂×C₄	#14556107	1	1	yes	characteristic
238	C₂×C₄	#14556107	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	2	1	no	—
241	C₈	—	2	1	no	—
242	C₈	—	4	2	no	—
243	C₈	—	568	2	no	—
244	C₈	—	568	2	no	—
245	C₈	#14556045	1	2	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	8	4	no	—
252	C₂²	#17985859	1	1	yes	characteristic
253	C₂²	#17986048	1	2	yes	—
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	4	2	no	—
260	C₄	C₂×#14556045	1	1	yes	characteristic
261	C₄	#17985983	1	1	yes	characteristic
262	C₂	—	2	2	no	—
263	C₂	—	8	2	no	—
264	C₂	#22588332	1	1	yes	characteristic
265	C₂	#22588382	1	1	yes	characteristic
266	C₂	#22588386	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985859
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₂×#436×C₇₁	C₂
socle	C₂²×C₇₁	#380
prime radical	C₂²×D₄×C₇₁	Q₈
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₂×#436	D₇₁
hypocenter	C₇₁	#8624
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. #22587103	C₂	semidirect
2. #22587103	C₂	semidirect
4. #22586210	C₂	semidirect
139. C₇₁	#8624	semidirect

Maximal subgroups

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

subgroup	index
#22587103	2
#22586681	2
#22586210	2
C₂×#436×C₇₁	2
#8624	71