Subgroups — #31897544

Order 18176.

484 conjugacy classes of subgroups.

The subgroup lattice has 276 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	C₄×#14556039	C₂	1	1	yes	maximal, characteristic
3	#22586221	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×#14556039	C₂²	1	2	yes	—
8	C₂×#14556039	C₂²	1	2	yes	—
9	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
10	#17985754	—	2	1	no	—
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	—	2	1	no	—
16	#391×C₇₁	—	2	1	no	—
17	#382×C₇₁	C₄	1	2	yes	—
18	#382×C₇₁	C₂²	1	2	yes	—
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	—	2	4	no	—
24	#14556039	—	2	2	no	—
25	#14556039	—	2	4	no	—
26	#14556039	Q₈	1	4	yes	complemented
27	C₄×C₇₁⋊C₈	—	2	4	no	—
28	C₂²×Q₈×C₇₁	—	2	1	no	—
29	C₂²×Q₈×C₇₁	—	2	1	no	—
30	C₄⋊Q₈×C₇₁	—	4	2	no	—
31	C₄⋊Q₈×C₇₁	—	4	2	no	—
32	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
34	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
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	#95×C₇₁	—	2	2	no	—
44	#95×C₇₁	—	2	2	no	—
45	#95×C₇₁	—	2	2	no	—
46	#95×C₇₁	—	2	2	no	—
47	C₂×C₇₁⋊C₈	—	2	4	no	—
48	C₂×C₇₁⋊C₈	—	2	4	no	—
49	C₂×C₇₁⋊C₈	—	2	4	no	—
50	C₂×C₇₁⋊C₈	—	2	4	no	—
51	C₂×C₇₁⋊C₈	—	4	2	no	—
52	C₂×C₇₁⋊C₈	—	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	C₂×Q₈×C₇₁	—	4	2	no	—
57	C₂×Q₈×C₇₁	—	4	2	no	—
58	C₂×Q₈×C₇₁	—	4	2	no	—
59	C₂²×C₄×C₇₁	—	2	2	no	—
60	C₂²×C₄×C₇₁	—	2	2	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	C₄⋊C₄×C₇₁	—	4	2	no	—
69	C₄⋊C₄×C₇₁	—	4	2	no	—
70	C₄⋊C₄×C₇₁	—	4	2	no	—
71	C₄⋊C₄×C₇₁	—	4	2	no	—
72	C₄²×C₇₁	—	2	2	no	—
73	C₄²×C₇₁	—	2	2	no	—
74	C₄²×C₇₁	—	2	2	no	—
75	C₄²×C₇₁	—	2	2	no	—
76	C₄²×C₇₁	—	2	2	no	—
77	C₄²×C₇₁	—	4	1	no	—
78	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
79	C₄²×C₇₁	#41	1	2	yes	—
80	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	Q₈×C₇₁	—	8	4	no	—
82	Q₈×C₇₁	—	8	4	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	—	2	1	no	—
94	C₂×C₄×C₇₁	—	2	1	no	—
95	C₂×C₄×C₇₁	—	2	1	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	—	4	2	no	—
100	C₂×C₄×C₇₁	—	4	2	no	—
101	C₂×C₄×C₇₁	—	4	2	no	—
102	C₂×C₄×C₇₁	#100	1	2	yes	—
103	C₂×C₄×C₇₁	#101	1	2	yes	—
104	C₂×C₄×C₇₁	#104	1	2	yes	—
105	C₂×C₄×C₇₁	#104	1	2	yes	—
106	C₂×C₄×C₇₁	#104	1	2	yes	—
107	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
111	C₇₁⋊C₈	—	2	8	no	—
112	C₇₁⋊C₈	—	4	4	no	—
113	C₂²×C₇₁	#391	1	1	yes	characteristic
114	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
115	C₂²×C₇₁	C₂×#101	1	1	yes	characteristic
116	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
117	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
118	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
119	C₂²×C₇₁	#421	1	1	yes	characteristic
120	C₄×C₇₁	—	2	1	no	—
121	C₄×C₇₁	—	2	1	no	—
122	C₄×C₇₁	—	2	2	no	—
123	C₄×C₇₁	—	2	2	no	—
124	C₄×C₇₁	—	2	2	no	—
125	C₄×C₇₁	—	2	2	no	—
126	C₄×C₇₁	—	4	1	no	—
127	C₄×C₇₁	—	8	2	no	—
128	C₄×C₇₁	—	8	2	no	—
129	C₄×C₇₁	#329	1	4	yes	—
130	#9890	—	71	1	no	maximal, Sylow
131	C₂×C₇₁	C₂×#329	1	1	yes	characteristic
132	C₂×C₇₁	#1526	1	1	yes	characteristic
133	C₂×C₇₁	#1527	1	1	yes	characteristic
134	C₂×C₇₁	#1531	1	1	yes	characteristic
135	C₂×C₇₁	#1986	1	1	yes	characteristic
136	C₂×C₇₁	#1994	1	1	yes	characteristic
137	C₂×C₇₁	#1997	1	1	yes	characteristic
138	#2862	D₇₁	1	1	yes	characteristic
139	#1767	—	71	1	no	—
140	C₄×#97	—	71	1	no	—
141	#1295	—	71	4	no	—
142	C₇₁	#9890	1	1	yes	characteristic, complemented, Sylow
143	C₂×C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
144	C₂×C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
145	C₂×C₄⋊C₈	—	142	1	no	—
146	C₂×#97	—	71	2	no	—
147	C₂×#97	—	71	2	no	—
148	C₂×#97	—	142	1	no	—
149	C₂×C₄×C₈	—	142	1	no	—
150	#391	—	2	1	no	—
151	#391	—	2	1	no	—
152	#391	—	2	1	no	—
153	#391	—	2	1	no	—
154	#382	Dic₇₁	1	2	yes	—
155	#382	C₂×D₇₁	1	2	yes	—
156	C₄³	C₂×D₇₁	1	1	yes	characteristic
157	#336	—	142	1	no	—
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×Q₈	—	2	1	no	—
160	C₂²×C₈	—	142	2	no	—
161	C₂²×C₈	—	142	2	no	—
162	C₄⋊Q₈	—	4	2	no	—
163	C₄⋊Q₈	—	4	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
167	C₂×C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
168	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
169	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
170	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
171	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
172	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
174	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
175	C₄⋊C₈	—	142	4	no	—
176	#97	—	71	4	no	—
177	#97	—	142	4	no	—
178	#97	—	142	4	no	—
179	#97	—	142	2	no	—
180	C₄×C₈	—	142	4	no	—
181	#95	—	2	2	no	—
182	#95	—	2	2	no	—
183	#95	—	2	2	no	—
184	#95	—	2	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
194	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
195	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
196	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
197	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
200	C₂×C₈	—	284	2	no	—
201	C₂×C₈	—	284	2	no	—
202	C₂×C₈	—	142	4	no	—
203	C₂×C₈	—	142	4	no	—
204	C₂×C₈	—	142	4	no	—
205	C₂×C₈	—	142	4	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	4	1	no	—
216	C₄²	Q₈×D₇₁	1	2	yes	—
217	C₄²	#11765866	1	2	yes	—
218	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
219	Q₈	—	8	4	no	—
220	Q₈	—	8	4	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	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	#14556045	1	2	yes	—
241	C₂×C₄	#14556047	1	2	yes	—
242	C₂×C₄	#14556055	1	2	yes	—
243	C₂×C₄	#14556055	1	2	yes	—
244	C₂×C₄	#14556055	1	2	yes	—
245	C₂×C₄	Q₈×Dic₇₁	1	2	yes	—
246	C₂×C₄	#14556104	1	2	yes	—
247	C₂×C₄	#14556117	1	1	yes	characteristic
248	C₂×C₄	#14556126	1	1	yes	characteristic
249	C₈	—	284	4	no	—
250	C₈	—	142	8	no	—
251	C₂²	#17985906	1	1	yes	characteristic
252	C₂²	C₂×#14556045	1	1	yes	characteristic
253	C₂²	C₂×#14556047	1	1	yes	characteristic
254	C₂²	C₂×#14556055	1	1	yes	characteristic
255	C₂²	C₂×#14556055	1	1	yes	characteristic
256	C₂²	C₂×#14556055	1	1	yes	characteristic
257	C₂²	#17986016	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	8	2	no	—
266	C₄	—	8	2	no	—
267	C₄	#17985739	1	4	yes	—
268	C₂	C₂×#17985739	1	1	yes	characteristic
269	C₂	#22586935	1	1	yes	characteristic
270	C₂	#22586938	1	1	yes	characteristic
271	C₂	#22586955	1	1	yes	characteristic
272	C₂	#22589285	1	1	yes	characteristic
273	C₂	#22589318	1	1	yes	characteristic
274	C₂	#22589330	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	#9890
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
26. #14556039	Q₈	semidirect
142. C₇₁	#9890	semidirect

Maximal subgroups

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

subgroup	index
#22588193	2
C₄×#14556039	2
#22586221	2
C₇₁×#2862	2
#9890	71