Subgroups — #31885913

Order 18176.

448 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588193	C₂	1	1	yes	maximal, characteristic
2	#22588086	C₂	1	1	yes	maximal, characteristic
3	#22586166	C₂	1	4	yes	maximal
4	C₇₁×#2862	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₂×#14556039	—	2	1	no	—
7	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
8	#17985754	—	2	1	no	—
9	#17985754	C₂²	1	2	yes	—
10	#17985754	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
13	#391×C₇₁	—	2	1	no	—
14	#391×C₇₁	—	2	1	no	—
15	#391×C₇₁	C₄	1	2	yes	—
16	#391×C₇₁	C₂²	1	2	yes	—
17	#382×C₇₁	—	2	1	no	—
18	#382×C₇₁	—	2	1	no	—
19	C₄³×C₇₁	C₂²	1	1	yes	characteristic
20	C₂²×C₇₁⋊C₈	—	2	2	no	—
21	C₂²×C₇₁⋊C₈	—	2	2	no	—
22	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
23	#14556039	—	4	2	no	—
24	C₄×C₇₁⋊C₈	—	2	4	no	—
25	C₂²×Q₈×C₇₁	—	2	1	no	—
26	C₂²×Q₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₄⋊Q₈×C₇₁	—	4	2	no	—
28	C₄⋊Q₈×C₇₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
31	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
40	#95×C₇₁	—	2	2	no	—
41	#95×C₇₁	—	2	2	no	—
42	#95×C₇₁	—	2	2	no	—
43	#95×C₇₁	—	2	2	no	—
44	C₂×C₇₁⋊C₈	—	2	4	no	—
45	C₂×C₇₁⋊C₈	—	2	4	no	—
46	C₂×C₇₁⋊C₈	—	4	2	no	—
47	C₂×C₇₁⋊C₈	—	4	2	no	—
48	C₂×C₇₁⋊C₈	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×Q₈×C₇₁	—	2	4	no	—
51	C₂×Q₈×C₇₁	—	4	2	no	—
52	C₂×Q₈×C₇₁	—	4	2	no	—
53	C₂×Q₈×C₇₁	—	4	2	no	—
54	C₂×Q₈×C₇₁	—	4	2	no	—
55	C₂×Q₈×C₇₁	—	4	2	no	—
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	—	2	2	no	—
58	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
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×D₄	1	1	yes	characteristic
65	C₄⋊C₄×C₇₁	—	4	2	no	—
66	C₄⋊C₄×C₇₁	—	4	2	no	—
67	C₄⋊C₄×C₇₁	—	4	2	no	—
68	C₄⋊C₄×C₇₁	—	4	2	no	—
69	C₄²×C₇₁	—	2	2	no	—
70	C₄²×C₇₁	—	2	2	no	—
71	C₄²×C₇₁	—	2	2	no	—
72	C₄²×C₇₁	—	2	2	no	—
73	C₄²×C₇₁	—	2	2	no	—
74	C₄²×C₇₁	—	4	1	no	—
75	C₄²×C₇₁	SD₁₆	1	2	yes	—
76	C₄²×C₇₁	Q₁₆	1	2	yes	—
77	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈×C₇₁	—	8	4	no	—
79	Q₈×C₇₁	—	8	4	no	—
80	C₂×C₄×C₇₁	—	2	1	no	—
81	C₂×C₄×C₇₁	—	2	1	no	—
82	C₂×C₄×C₇₁	—	2	1	no	—
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	—	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	—	4	2	no	—
94	C₂×C₄×C₇₁	—	4	2	no	—
95	C₂×C₄×C₇₁	—	4	2	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	#100	1	2	yes	—
100	C₂×C₄×C₇₁	#103	1	2	yes	—
101	C₂×C₄×C₇₁	#103	1	2	yes	—
102	C₂×C₄×C₇₁	#104	1	2	yes	—
103	C₂×C₄×C₇₁	#104	1	2	yes	—
104	C₂×C₄×C₇₁	#131	1	2	yes	—
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	C₇₁⋊C₈	—	4	4	no	—
109	C₇₁⋊C₈	—	4	4	no	—
110	C₂²×C₇₁	#408	1	1	yes	characteristic
111	C₂²×C₇₁	#410	1	1	yes	characteristic
112	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
113	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
114	C₂²×C₇₁	#419	1	1	yes	characteristic
115	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
116	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
117	C₄×C₇₁	—	2	2	no	—
118	C₄×C₇₁	—	2	2	no	—
119	C₄×C₇₁	—	2	2	no	—
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	2	2	no	—
123	C₄×C₇₁	—	4	1	no	—
124	C₄×C₇₁	—	8	2	no	—
125	C₄×C₇₁	—	8	2	no	—
126	C₄×C₇₁	#329	1	4	yes	—
127	#7682	—	71	1	no	maximal, Sylow
128	C₂×C₇₁	#1469	1	1	yes	characteristic
129	C₂×C₇₁	#1485	1	1	yes	characteristic
130	C₂×C₇₁	#1493	1	1	yes	characteristic
131	C₂×C₇₁	#1503	1	1	yes	characteristic
132	C₂×C₇₁	#1506	1	1	yes	characteristic
133	C₂×C₇₁	C₂×#329	1	1	yes	characteristic
134	C₂×C₇₁	#1548	1	1	yes	characteristic
135	#2862	D₇₁	1	1	yes	characteristic
136	#1767	—	71	1	no	—
137	#1744	—	71	1	no	—
138	#1272	—	71	4	no	—
139	C₇₁	#7682	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
141	C₂×C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
142	C₂×C₄⋊C₈	—	142	1	no	—
143	C₂×#97	—	142	1	no	—
144	C₂×C₄×C₈	—	142	1	no	—
145	#391	—	2	1	no	—
146	#391	—	2	1	no	—
147	#391	Dic₇₁	1	2	yes	—
148	#391	C₂×D₇₁	1	2	yes	—
149	#382	—	2	1	no	—
150	#382	—	2	1	no	—
151	C₄³	C₂×D₇₁	1	1	yes	characteristic
152	#336	—	71	2	no	—
153	#336	—	71	2	no	—
154	#336	—	142	1	no	—
155	C₂²×Q₈	—	2	1	no	—
156	C₂²×Q₈	C₂×Dic₇₁	1	2	yes	—
157	C₂²×C₈	—	142	2	no	—
158	C₂²×C₈	—	142	2	no	—
159	C₄⋊Q₈	—	4	2	no	—
160	C₄⋊Q₈	—	4	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
165	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
166	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
167	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
168	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
172	C₄⋊C₈	—	142	4	no	—
173	#97	—	284	2	no	—
174	C₄×C₈	—	142	4	no	—
175	#95	—	2	2	no	—
176	#95	—	2	2	no	—
177	#95	—	2	2	no	—
178	#95	—	2	2	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	4	2	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
188	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
191	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
194	C₂×C₈	—	284	2	no	—
195	C₂×C₈	—	284	2	no	—
196	C₂×C₈	—	284	2	no	—
197	C₂×C₈	—	284	2	no	—
198	C₂×C₈	—	142	4	no	—
199	C₂×C₈	—	142	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	4	1	no	—
210	C₄²	#11765852	1	2	yes	—
211	C₄²	#11765855	1	2	yes	—
212	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
213	Q₈	—	8	4	no	—
214	Q₈	—	8	4	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	#14556045	1	2	yes	—
235	C₂×C₄	#14556052	1	2	yes	—
236	C₂×C₄	#14556052	1	2	yes	—
237	C₂×C₄	#14556055	1	2	yes	—
238	C₂×C₄	#14556055	1	2	yes	—
239	C₂×C₄	#14556135	1	2	yes	—
240	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
241	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
242	C₂×C₄	#14556159	1	2	yes	—
243	C₈	—	284	4	no	—
244	C₈	—	284	4	no	—
245	C₂²	#17985966	1	1	yes	characteristic
246	C₂²	#17985973	1	1	yes	characteristic
247	C₂²	C₂×#14556045	1	1	yes	characteristic
248	C₂²	C₂×#14556052	1	1	yes	characteristic
249	C₂²	#17986008	1	1	yes	characteristic
250	C₂²	C₂×#14556055	1	1	yes	characteristic
251	C₂²	C₂×#14556055	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	8	2	no	—
260	C₄	—	8	2	no	—
261	C₄	#17985739	1	4	yes	—
262	C₂	#22586692	1	1	yes	characteristic
263	C₂	#22586761	1	1	yes	characteristic
264	C₂	#22586795	1	1	yes	characteristic
265	C₂	#22586839	1	1	yes	characteristic
266	C₂	#22586852	1	1	yes	characteristic
267	C₂	C₂×#17985739	1	1	yes	characteristic
268	C₂	#22587021	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₁
commutator	C₂²×C₄×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#2862	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#2862	D₇₁
hypocenter	C₇₁	#7682
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2862

Complements

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

N	H ≅ G/N	type
139. C₇₁	#7682	semidirect

Maximal subgroups

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

subgroup	index
#22588193	2
#22588086	2
#22586166	2
C₇₁×#2862	2
#7682	71