Subgroups — #46845227

Order 34688.

456 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×D₂₇₁	C₂	1	1	yes	maximal, characteristic
2	#31774373	C₂	1	2	yes	maximal
3	#31774002	C₂	1	2	yes	maximal, complemented
4	#31774001	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₀₈₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₁₆×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#31773991	C₂	1	1	yes	maximal, characteristic, complemented
8	#31773934	C₂	1	1	yes	maximal, characteristic
9	#31773904	C₂	1	2	yes	maximal
10	#31773851	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₂₇₁	C₂²	1	2	yes	—
13	C₂×Dic₁₀₈₄	C₂²	1	1	yes	characteristic
14	C₂×#16864644	C₂²	1	2	yes	—
15	Q₁₆×D₂₇₁	C₄	1	8	yes	complemented
16	C₂×#16864637	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₇₁	C₂²	1	1	yes	characteristic
18	Q₈×Dic₂₇₁	C₂²	1	2	yes	—
19	C₄×Dic₅₄₂	—	2	2	no	—
20	C₄×Dic₅₄₂	C₂²	1	2	yes	—
21	#22576125	—	2	2	no	—
22	#22576125	C₂²	1	2	yes	—
23	#22576118	—	2	2	no	—
24	#22576118	C₂²	1	2	yes	—
25	#22576117	—	2	2	no	—
26	#22576117	C₂²	1	1	yes	characteristic
27	#22576094	C₂²	1	1	yes	characteristic
28	#22576091	C₂²	1	2	yes	—
29	#22576085	C₂²	1	2	yes	—
30	#22576084	C₂²	1	2	yes	—
31	#22576083	C₂²	1	2	yes	—
32	#22576071	C₂²	1	1	yes	characteristic
33	#22576070	C₂²	1	1	yes	characteristic
34	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
36	C₄×Q₈×C₂₇₁	C₂²	1	2	yes	—
37	#107×C₂₇₁	C₂²	1	1	yes	characteristic
38	#103×C₂₇₁	C₂²	1	2	yes	—
39	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
40	C₂×Dic₅₄₂	—	2	2	no	—
41	C₂×Dic₅₄₂	C₂³	1	2	yes	—
42	Q₈×D₂₇₁	—	2	4	no	—
43	Q₈×D₂₇₁	C₂×C₄	1	8	yes	—
44	C₂×C₄×D₂₇₁	—	2	2	no	—
45	C₂×C₄×D₂₇₁	C₂³	1	1	yes	characteristic
46	Dic₁₀₈₄	C₂×C₄	1	4	yes	—
47	#16864644	C₂×C₄	1	8	yes	—
48	#16864637	—	2	2	no	—
49	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
50	C₂×C₂₇₁⋊C₈	C₂³	1	1	yes	characteristic
51	C₈×D₂₇₁	C₂×C₄	1	4	yes	—
52	C₄⋊Dic₂₇₁	—	2	2	no	—
53	C₄⋊Dic₂₇₁	C₂³	1	2	yes	—
54	Dic₂₇₁⋊C₄	—	2	2	no	—
55	Dic₂₇₁⋊C₄	—	2	2	no	—
56	Dic₂₇₁⋊C₄	D₄	1	1	yes	characteristic
57	Dic₂₇₁⋊C₄	D₄	1	1	yes	characteristic
58	(C₂×D₂₇₁)⋊C₄	—	2	2	no	—
59	(C₂×D₂₇₁)⋊C₄	D₄	1	1	yes	characteristic
60	(C₂×D₂₇₁)⋊C₄	D₄	1	1	yes	characteristic
61	C₄×Dic₂₇₁	—	2	2	no	—
62	C₄×Dic₂₇₁	—	2	2	no	—
63	C₄×Dic₂₇₁	—	2	2	no	—
64	C₄×Dic₂₇₁	—	2	2	no	—
65	C₄×Dic₂₇₁	C₂³	1	1	yes	characteristic
66	C₂×Q₈×C₂₇₁	C₂³	1	2	yes	—
67	Q₁₆×C₂₇₁	C₂×C₄	1	4	yes	—
68	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₇₁	—	2	2	no	—
71	C₄⋊C₄×C₂₇₁	C₂³	1	2	yes	—
72	C₄²×C₂₇₁	—	2	2	no	—
73	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
74	C₂²×D₂₇₁	C₂×D₄	1	1	yes	characteristic
75	Dic₅₄₂	—	2	2	no	—
76	Dic₅₄₂	—	2	8	no	—
77	Dic₅₄₂	C₂²×C₄	1	4	yes	—
78	C₂×Dic₂₇₁	—	2	2	no	—
79	C₂×Dic₂₇₁	—	2	2	no	—
80	C₂×Dic₂₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×Dic₂₇₁	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₂₇₁	C₂×D₄	1	1	yes	characteristic
83	C₄×D₂₇₁	—	2	8	no	—
84	C₄×D₂₇₁	C₂²×C₄	1	2	yes	—
85	C₄×D₂₇₁	#41	1	2	yes	—
86	C₂₇₁⋊C₈	—	2	1	no	—
87	C₂₇₁⋊C₈	C₂²×C₄	1	2	yes	—
88	Q₈×C₂₇₁	—	2	2	no	—
89	Q₈×C₂₇₁	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₂₇₁	—	2	2	no	—
91	C₂×C₄×C₂₇₁	—	2	2	no	—
92	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
95	C₈×C₂₇₁	—	2	1	no	—
96	C₈×C₂₇₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×D₂₇₁	—	2	1	no	—
99	C₂×D₂₇₁	—	2	1	no	—
100	C₂×D₂₇₁	C₄×D₄	1	2	yes	—
101	C₄×C₂₇₁	—	2	4	no	—
102	C₄×C₂₇₁	—	2	1	no	—
103	C₄×C₂₇₁	—	2	1	no	—
104	C₄×C₂₇₁	—	4	2	no	—
105	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
107	Dic₂₇₁	—	2	4	no	—
108	Dic₂₇₁	—	2	1	no	—
109	Dic₂₇₁	—	2	1	no	—
110	Dic₂₇₁	—	4	2	no	—
111	Dic₂₇₁	C₄×D₄	1	2	yes	—
112	C₂×C₂₇₁	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₂₇₁	#578	1	1	yes	characteristic
114	C₂×C₂₇₁	#578	1	1	yes	characteristic
115	D₂₇₁	—	2	2	no	—
116	C₂₇₁	#2950	1	1	yes	characteristic, complemented, Sylow
117	#2950	—	271	1	no	maximal, Sylow
118	C₂²×Q₁₆	—	271	1	no	—
119	#519	—	271	2	no	—
120	#441	—	271	2	no	—
121	#441	—	271	2	no	—
122	C₄×Q₁₆	—	271	1	no	—
123	C₄×Q₁₆	—	271	1	no	—
124	C₄×Q₁₆	—	271	1	no	—
125	C₄×Q₁₆	D₂₇₁	1	1	yes	characteristic, complemented
126	#427	—	271	1	no	—
127	#419	—	271	2	no	—
128	#405	—	271	1	no	—
129	C₂²×Q₈	—	271	2	no	—
130	C₂×Q₁₆	—	271	2	no	—
131	C₂×Q₁₆	—	271	8	no	—
132	C₂×Q₁₆	—	271	1	no	—
133	C₂×Q₁₆	C₂×D₂₇₁	1	1	yes	characteristic
134	C₂×M₁₆	—	271	1	no	—
135	C₂²×C₈	—	271	1	no	—
136	C₄×Q₈	—	271	2	no	—
137	C₄×Q₈	—	271	2	no	—
138	C₄×Q₈	—	271	2	no	—
139	C₄×Q₈	—	542	2	no	—
140	C₄×Q₈	—	542	2	no	—
141	C₄×Q₈	C₂×D₂₇₁	1	2	yes	—
142	#117	—	271	1	no	—
143	#117	—	271	2	no	—
144	#117	—	542	2	no	—
145	#117	—	542	2	no	—
146	#107	—	271	1	no	—
147	#107	C₂×D₂₇₁	1	1	yes	characteristic
148	#106	—	271	2	no	—
149	#103	—	271	2	no	—
150	#103	—	271	2	no	—
151	#103	—	271	2	no	—
152	#103	C₂×D₂₇₁	1	2	yes	—
153	#97	—	271	1	no	—
154	#97	—	271	1	no	—
155	C₄×C₈	—	271	1	no	—
156	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
157	C₂×Q₈	—	271	8	no	—
158	C₂×Q₈	—	271	2	no	—
159	C₂×Q₈	—	542	2	no	—
160	C₂×Q₈	—	542	4	no	—
161	C₂×Q₈	C₂²×D₂₇₁	1	2	yes	—
162	C₂²×C₄	—	271	1	no	—
163	C₂²×C₄	—	542	2	no	—
164	Q₁₆	—	271	8	no	—
165	Q₁₆	—	271	4	no	—
166	Q₁₆	C₄×D₂₇₁	1	4	yes	—
167	M₁₆	—	542	2	no	—
168	C₂×C₈	—	271	1	no	—
169	C₂×C₈	—	271	4	no	—
170	C₂×C₈	—	271	1	no	—
171	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
172	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	271	1	no	—
175	C₄⋊C₄	—	271	1	no	—
176	C₄⋊C₄	—	271	2	no	—
177	C₄⋊C₄	—	542	2	no	—
178	C₄⋊C₄	—	542	2	no	—
179	C₄⋊C₄	—	542	2	no	—
180	C₄⋊C₄	C₂²×D₂₇₁	1	2	yes	—
181	C₂²⋊C₄	—	271	1	no	—
182	C₂²⋊C₄	—	271	1	no	—
183	C₂²⋊C₄	—	542	2	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	271	1	no	—
186	C₄²	—	542	2	no	—
187	C₄²	—	542	2	no	—
188	C₄²	—	542	2	no	—
189	C₄²	—	542	2	no	—
190	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
191	C₂³	—	271	1	no	—
192	Q₈	—	2	2	no	—
193	Q₈	—	271	4	no	—
194	Q₈	—	542	8	no	—
195	Q₈	—	542	2	no	—
196	Q₈	C₂×C₄×D₂₇₁	1	4	yes	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	271	1	no	—
200	C₂×C₄	—	271	2	no	—
201	C₂×C₄	—	271	2	no	—
202	C₂×C₄	—	271	1	no	—
203	C₂×C₄	—	271	1	no	—
204	C₂×C₄	—	542	8	no	—
205	C₂×C₄	—	542	2	no	—
206	C₂×C₄	—	542	2	no	—
207	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
208	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
209	C₂×C₄	C₂³×D₂₇₁	1	1	yes	characteristic
210	C₈	—	2	1	no	—
211	C₈	—	271	2	no	—
212	C₈	—	542	1	no	—
213	C₈	C₂×C₄×D₂₇₁	1	2	yes	—
214	C₂²	—	271	2	no	—
215	C₂²	—	542	1	no	—
216	C₂²	—	542	1	no	—
217	C₂²	C₂×D₄×D₂₇₁	1	1	yes	characteristic
218	C₄	—	2	4	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	2	no	—
222	C₄	—	271	2	no	—
223	C₄	—	542	4	no	—
224	C₄	—	542	1	no	—
225	C₄	—	542	1	no	—
226	C₄	—	1084	2	no	—
227	C₄	C₂²×C₄×D₂₇₁	1	1	yes	characteristic
228	C₄	#41×D₂₇₁	1	1	yes	characteristic
229	C₂	—	542	2	no	—
230	C₂	C₄×D₄×D₂₇₁	1	1	yes	characteristic
231	C₂	#31774710	1	1	yes	characteristic
232	C₂	#31774711	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₇₁
commutator	C₄×C₂₇₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₇₁
Fitting	C₄×Q₁₆×C₂₇₁	C₂
socle	C₂²×C₂₇₁	C₂²×D₄
prime radical	C₂²×D₂₇₁	C₂×D₄
cosocle	C₂×C₄×C₂₇₁	C₂⁴
hypercenter	C₄×Q₁₆	D₂₇₁
hypocenter	C₂₇₁	#2950
LCS	G ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #31774002	C₂	semidirect
4. #31774001	C₂	semidirect
5. C₄×Dic₁₀₈₄	C₂	semidirect
6. Q₁₆×Dic₂₇₁	C₂	semidirect
7. #31773991	C₂	semidirect
11. C₄×Q₁₆×C₂₇₁	C₂	semidirect
15. Q₁₆×D₂₇₁	C₄	semidirect
116. C₂₇₁	#2950	semidirect
125. C₄×Q₁₆	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×D₂₇₁	2
#31774373	2
#31774002	2
#31774001	2
C₄×Dic₁₀₈₄	2
Q₁₆×Dic₂₇₁	2
#31773991	2
#31773934	2
#31773904	2
#31773851	2
C₄×Q₁₆×C₂₇₁	2
#2950	271