Subgroups — #31891114

Order 18176.

382 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	#22588456	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22586175	C₂	1	1	yes	maximal, characteristic
4	#22586174	C₂	1	2	yes	maximal
5	C₇₁×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₁	C₄	1	1	yes	characteristic
12	C₄²×Dic₇₁	C₂²	1	1	yes	characteristic
13	#17985753	—	2	2	no	—
14	#17985753	—	2	2	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	—	2	2	no	—
18	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₇₁	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₇₁	Q₈	1	1	yes	characteristic
26	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
28	C₄×C₇₁⋊C₈	—	2	2	no	—
29	C₈×Dic₇₁	—	2	2	no	—
30	C₈×Dic₇₁	—	2	2	no	—
31	C₈×Dic₇₁	—	2	2	no	—
32	C₈×Dic₇₁	C₈	1	4	yes	complemented
33	C₂²×C₈×C₇₁	—	2	2	no	—
34	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₇₁	—	2	2	no	—
38	C₄⋊C₈×C₇₁	—	2	2	no	—
39	C₄×C₈×C₇₁	—	2	2	no	—
40	C₂²×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₇₁⋊C₈	—	2	1	no	—
45	C₂×C₇₁⋊C₈	—	2	1	no	—
46	C₂×C₇₁⋊C₈	—	2	1	no	—
47	C₂×C₇₁⋊C₈	—	2	1	no	—
48	C₂×C₇₁⋊C₈	—	4	2	no	—
49	C₂×C₇₁⋊C₈	—	4	2	no	—
50	C₂×C₇₁⋊C₈	—	4	2	no	—
51	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
52	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
53	C₄×Dic₇₁	—	2	1	no	—
54	C₄×Dic₇₁	—	2	1	no	—
55	C₄×Dic₇₁	—	2	1	no	—
56	C₄×Dic₇₁	—	2	2	no	—
57	C₄×Dic₇₁	—	2	1	no	—
58	C₄×Dic₇₁	—	2	2	no	—
59	C₄×Dic₇₁	—	2	2	no	—
60	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
61	C₄×Dic₇₁	M₁₆	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	C₂×C₈×C₇₁	—	2	1	no	—
66	C₂×C₈×C₇₁	—	2	1	no	—
67	C₂×C₈×C₇₁	—	2	1	no	—
68	C₂×C₈×C₇₁	—	2	1	no	—
69	C₂×C₈×C₇₁	—	4	2	no	—
70	C₂×C₈×C₇₁	—	4	2	no	—
71	C₂×C₈×C₇₁	—	4	2	no	—
72	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
74	C₄²×C₇₁	#41	1	2	yes	—
75	C₄²×C₇₁	#41	1	2	yes	—
76	C₂³×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₇₁	—	2	1	no	—
78	C₂×C₄×C₇₁	—	2	1	no	—
79	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
80	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
81	C₂×C₄×C₇₁	#124	1	2	yes	—
82	C₂×C₄×C₇₁	#125	1	2	yes	—
83	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
88	C₂×C₄×C₇₁	#135	1	2	yes	—
89	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
90	C₂×Dic₇₁	—	2	1	no	—
91	C₂×Dic₇₁	—	2	1	no	—
92	C₂×Dic₇₁	—	2	1	no	—
93	C₂×Dic₇₁	—	2	1	no	—
94	C₂×Dic₇₁	—	2	1	no	—
95	C₂×Dic₇₁	—	2	1	no	—
96	C₂×Dic₇₁	—	2	1	no	—
97	C₂×Dic₇₁	—	2	1	no	—
98	C₂×Dic₇₁	C₄⋊C₈	1	2	yes	—
99	C₂×Dic₇₁	C₄⋊C₈	1	2	yes	—
100	C₂×Dic₇₁	#108	1	2	yes	—
101	C₂×Dic₇₁	#108	1	2	yes	—
102	C₈×C₇₁	—	2	2	no	—
103	C₈×C₇₁	—	2	2	no	—
104	C₈×C₇₁	—	8	2	no	—
105	C₇₁⋊C₈	—	2	2	no	—
106	C₇₁⋊C₈	—	2	2	no	—
107	C₇₁⋊C₈	—	8	2	no	—
108	C₂²×C₇₁	#382	1	1	yes	characteristic
109	C₂²×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₇₁	#423	1	1	yes	characteristic
111	C₂²×C₇₁	#427	1	1	yes	characteristic
112	C₂²×C₇₁	#427	1	1	yes	characteristic
113	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
114	C₂²×C₇₁	C₂×#108	1	1	yes	characteristic
115	C₄×C₇₁	—	2	2	no	—
116	C₄×C₇₁	#431	1	1	yes	characteristic
117	C₄×C₇₁	#431	1	1	yes	characteristic
118	C₄×C₇₁	#432	1	1	yes	characteristic
119	C₄×C₇₁	#432	1	1	yes	characteristic
120	C₄×C₇₁	#488	1	2	yes	—
121	C₄×C₇₁	#488	1	2	yes	—
122	Dic₇₁	—	2	2	no	—
123	Dic₇₁	—	2	2	no	—
124	Dic₇₁	—	2	2	no	—
125	Dic₇₁	—	2	2	no	—
126	#8785	—	71	1	no	maximal, Sylow
127	C₂×C₇₁	#1564	1	1	yes	characteristic
128	C₂×C₇₁	#1564	1	1	yes	characteristic
129	C₂×C₇₁	#1568	1	1	yes	characteristic
130	C₂×C₇₁	#1568	1	1	yes	characteristic
131	C₂×C₇₁	#1831	1	1	yes	characteristic
132	C₂×C₇₁	#1835	1	1	yes	characteristic
133	C₂×C₇₁	#1837	1	1	yes	characteristic
134	#1831	—	71	2	no	—
135	C₄²×C₈	—	71	1	no	—
136	#1277	—	71	2	no	—
137	#1277	—	71	1	no	—
138	#1277	D₇₁	1	1	yes	characteristic
139	C₇₁	#8785	1	1	yes	characteristic, complemented, Sylow
140	C₂×C₄⋊C₈	—	71	2	no	—
141	C₂×C₄⋊C₈	C₂×D₇₁	1	2	yes	—
142	C₂×C₄×C₈	—	71	1	no	—
143	C₂×C₄×C₈	—	71	1	no	—
144	C₂×C₄×C₈	—	71	1	no	—
145	C₂×C₄×C₈	—	71	1	no	—
146	C₂×C₄×C₈	—	71	1	no	—
147	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
148	C₄³	—	71	1	no	—
149	#336	—	142	2	no	—
150	#336	—	142	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	71	1	no	—
153	C₂²×C₈	—	71	1	no	—
154	C₂²×C₈	—	142	2	no	—
155	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
156	C₂²×C₈	C₄×D₇₁	1	1	yes	characteristic
157	C₂×C₄²	—	71	1	no	—
158	C₂×C₄²	—	71	1	no	—
159	C₂×C₄²	—	71	1	no	—
160	C₂×C₄²	—	71	1	no	—
161	C₂×C₄²	—	71	1	no	—
162	C₂×C₄²	—	71	1	no	—
163	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	142	2	no	—
167	C₄⋊C₈	—	142	2	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	—	71	4	no	—
170	C₄×C₈	—	142	2	no	—
171	C₄×C₈	—	142	2	no	—
172	C₄×C₈	—	142	2	no	—
173	C₄×C₈	—	142	2	no	—
174	C₂²×C₄	—	71	1	no	—
175	C₂²×C₄	—	71	1	no	—
176	C₂²×C₄	—	71	1	no	—
177	C₂²×C₄	—	71	1	no	—
178	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
179	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
180	C₂²×C₄	Q₈×D₇₁	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	284	2	no	—
189	C₂×C₈	—	284	2	no	—
190	C₂×C₈	—	284	2	no	—
191	C₂×C₈	—	71	2	no	—
192	C₂×C₈	—	71	2	no	—
193	C₂×C₈	—	142	1	no	—
194	C₂×C₈	—	142	1	no	—
195	C₂×C₈	—	142	1	no	—
196	C₂×C₈	—	142	1	no	—
197	C₂×C₈	C₈×D₇₁	1	2	yes	—
198	C₂×C₈	#11765848	1	2	yes	—
199	C₄²	—	71	2	no	—
200	C₄²	—	71	2	no	—
201	C₄²	—	142	2	no	—
202	C₄²	—	142	2	no	—
203	C₄²	—	142	1	no	—
204	C₄²	—	142	1	no	—
205	C₄²	—	142	1	no	—
206	C₄²	—	142	2	no	—
207	C₄²	—	142	1	no	—
208	C₄²	#11765864	1	2	yes	—
209	C₄²	#11765865	1	2	yes	—
210	C₂³	C₄⋊C₄×D₇₁	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	71	2	no	—
214	C₂×C₄	—	71	2	no	—
215	C₂×C₄	—	71	2	no	—
216	C₂×C₄	—	71	2	no	—
217	C₂×C₄	—	142	1	no	—
218	C₂×C₄	—	142	1	no	—
219	C₂×C₄	—	142	1	no	—
220	C₂×C₄	—	142	1	no	—
221	C₂×C₄	—	142	1	no	—
222	C₂×C₄	—	142	1	no	—
223	C₂×C₄	—	142	1	no	—
224	C₂×C₄	—	142	1	no	—
225	C₂×C₄	#14556084	1	1	yes	characteristic
226	C₂×C₄	#14556085	1	1	yes	characteristic
227	C₂×C₄	#14556114	1	2	yes	—
228	C₂×C₄	#14556118	1	2	yes	—
229	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
230	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
231	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
232	C₂×C₄	C₂×#11765848	1	1	yes	characteristic
233	C₂×C₄	#14556147	1	1	yes	characteristic
234	C₂×C₄	#14556149	1	2	yes	—
235	C₂×C₄	#14556151	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	8	2	no	—
239	C₈	—	568	2	no	—
240	C₈	—	142	2	no	—
241	C₈	—	142	2	no	—
242	C₂²	#17985863	1	1	yes	characteristic
243	C₂²	C₄⋊C₈×D₇₁	1	1	yes	characteristic
244	C₂²	#17986021	1	1	yes	characteristic
245	C₂²	#17986034	1	1	yes	characteristic
246	C₂²	#17986036	1	1	yes	characteristic
247	C₂²	#108×D₇₁	1	1	yes	characteristic
248	C₂²	#108×D₇₁	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	142	2	no	—
251	C₄	—	142	2	no	—
252	C₄	—	142	2	no	—
253	C₄	—	142	2	no	—
254	C₄	#17986049	1	1	yes	characteristic
255	C₄	#17986050	1	1	yes	characteristic
256	C₄	#17986053	1	1	yes	characteristic
257	C₄	#17986054	1	1	yes	characteristic
258	C₄	#17986352	1	2	yes	—
259	C₄	#17986355	1	2	yes	—
260	C₂	#22587085	1	1	yes	characteristic
261	C₂	#22587087	1	1	yes	characteristic
262	C₂	#22587100	1	1	yes	characteristic
263	C₂	#22587102	1	1	yes	characteristic
264	C₂	#22588455	1	1	yes	characteristic
265	C₂	#22588469	1	1	yes	characteristic
266	C₂	#22588476	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1277	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1277	D₇₁
hypocenter	C₇₁	#8785
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
32. C₈×Dic₇₁	C₈	semidirect
139. C₇₁	#8785	semidirect

Maximal subgroups

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

subgroup	index
#22588456	2
C₄×C₈×Dic₇₁	2
#22586175	2
#22586174	2
C₇₁×#1277	2
#8785	71