Subgroups — C₂²×(C₂²⋊D₂₇₁)⋊C₄ #46849532

Order 34688.

2528 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊D₂₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₄×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×(C₂²⋊D₂₇₁)⋊C₄	C₂	1	24	yes	maximal, direct product factor
5	C₂²×Dic₂₇₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×(C₂×D₂₇₁)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×Dic₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊C₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×D₂₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₂₇₁	C₄	1	16	yes	complemented
11	C₂²×C₂²⋊D₂₇₁	C₂²	1	12	yes	—
12	C₂³×Dic₂₇₁	—	2	8	no	—
13	C₂³×Dic₂₇₁	C₂²	1	1	yes	characteristic
14	C₂³×Dic₂₇₁	C₂²	1	1	yes	characteristic
15	C₂³×Dic₂₇₁	C₂²	1	1	yes	characteristic
16	C₂³×Dic₂₇₁	C₂²	1	12	yes	complemented
17	C₂²×C₄×D₂₇₁	—	2	8	no	—
18	C₂²×C₄×D₂₇₁	C₂²	1	12	yes	complemented
19	(C₂²⋊D₂₇₁)⋊C₄	C₂²	1	64	yes	direct product factor
20	C₂×Dic₂₇₁⋊C₄	C₂²	1	12	yes	complemented
21	C₂×(C₂×D₂₇₁)⋊C₄	C₂²	1	12	yes	complemented
22	C₂×C₄×Dic₂₇₁	C₂²	1	12	yes	complemented
23	C₂⁵×C₂₇₁	C₂²	1	1	yes	characteristic
24	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
25	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊C₄×C₂₇₁	C₂²	1	12	yes	complemented
27	C₂³×D₂₇₁	—	2	4	no	—
28	C₂³×D₂₇₁	—	2	4	no	—
29	C₂³×D₂₇₁	C₂×C₄	1	8	yes	complemented
30	C₂³×D₂₇₁	C₂³	1	6	yes	—
31	C₂×C₂²⋊D₂₇₁	C₂×C₄	1	96	yes	complemented
32	C₂×C₂²⋊D₂₇₁	C₂³	1	16	yes	—
33	C₂²×Dic₂₇₁	—	2	48	no	—
34	C₂²×Dic₂₇₁	—	2	4	no	—
35	C₂²×Dic₂₇₁	C₂×C₄	1	8	yes	complemented
36	C₂²×Dic₂₇₁	D₄	1	8	yes	complemented
37	C₂²×Dic₂₇₁	C₂³	1	6	yes	—
38	C₂²×Dic₂₇₁	C₂³	1	6	yes	—
39	C₂²×Dic₂₇₁	C₂³	1	6	yes	—
40	C₂²×Dic₂₇₁	C₂³	1	16	yes	complemented
41	C₂×C₄×D₂₇₁	—	2	48	no	—
42	C₂×C₄×D₂₇₁	C₂³	1	16	yes	complemented
43	Dic₂₇₁⋊C₄	C₂³	1	16	yes	complemented
44	(C₂×D₂₇₁)⋊C₄	C₂³	1	16	yes	complemented
45	C₄×Dic₂₇₁	C₂³	1	16	yes	complemented
46	C₂⁴×C₂₇₁	—	2	4	no	—
47	C₂⁴×C₂₇₁	—	2	4	no	—
48	C₂⁴×C₂₇₁	C₂×C₄	1	8	yes	complemented
49	C₂⁴×C₂₇₁	C₂³	1	6	yes	—
50	C₂⁴×C₂₇₁	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₇₁	—	2	4	no	—
52	C₂²×C₄×C₂₇₁	—	2	4	no	—
53	C₂²×C₄×C₂₇₁	C₂³	1	6	yes	—
54	C₂²×C₄×C₂₇₁	C₂³	1	6	yes	—
55	C₂²⋊C₄×C₂₇₁	C₂³	1	16	yes	complemented
56	C₂²×D₂₇₁	—	2	32	no	—
57	C₂²×D₂₇₁	—	2	12	no	—
58	C₂²×D₂₇₁	—	2	12	no	—
59	C₂²×D₂₇₁	C₂²×C₄	1	24	yes	complemented
60	C₂²×D₂₇₁	C₂⁴	1	4	yes	—
61	C₂²⋊D₂₇₁	C₂²×C₄	1	64	yes	complemented
62	C₂×Dic₂₇₁	—	2	12	no	—
63	C₂×Dic₂₇₁	—	2	32	no	—
64	C₂×Dic₂₇₁	C₂²×C₄	1	24	yes	complemented
65	C₂×Dic₂₇₁	C₂×D₄	1	24	yes	complemented
66	C₂×Dic₂₇₁	C₂⁴	1	4	yes	—
67	C₂×Dic₂₇₁	C₂⁴	1	4	yes	—
68	C₂×Dic₂₇₁	C₂⁴	1	4	yes	—
69	C₄×D₂₇₁	—	2	32	no	—
70	C₂³×C₂₇₁	—	2	32	no	—
71	C₂³×C₂₇₁	—	2	12	no	—
72	C₂³×C₂₇₁	—	2	12	no	—
73	C₂³×C₂₇₁	C₂²×C₄	1	4	yes	—
74	C₂³×C₂₇₁	C₂²×C₄	1	24	yes	complemented
75	C₂³×C₂₇₁	C₂×D₄	1	4	yes	—
76	C₂³×C₂₇₁	#41	1	4	yes	—
77	C₂³×C₂₇₁	C₂⁴	1	4	yes	—
78	C₂³×C₂₇₁	C₂⁴	1	3	yes	—
79	C₂×C₄×C₂₇₁	—	2	12	no	—
80	C₂×C₄×C₂₇₁	—	2	12	no	—
81	C₂×C₄×C₂₇₁	C₂⁴	1	4	yes	—
82	C₂×C₄×C₂₇₁	C₂⁴	1	4	yes	—
83	C₂²×C₂₇₁	—	2	48	no	—
84	C₂²×C₂₇₁	—	2	4	no	—
85	C₂²×C₂₇₁	—	2	4	no	—
86	C₂²×C₂₇₁	C₄×D₄	1	16	yes	complemented
87	C₂²×C₂₇₁	C₂³×C₄	1	6	yes	—
88	C₂²×C₂₇₁	C₂³×C₄	1	8	yes	complemented
89	C₂²×C₂₇₁	C₂²×D₄	1	6	yes	—
90	C₂²×C₂₇₁	C₂×#41	1	6	yes	—
91	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
92	C₂×D₂₇₁	—	2	48	no	—
93	C₂×D₂₇₁	—	2	4	no	—
94	C₂×D₂₇₁	—	2	4	no	—
95	C₂×D₂₇₁	C₂³×C₄	1	8	yes	complemented
96	C₄×C₂₇₁	—	2	4	no	—
97	C₄×C₂₇₁	—	2	4	no	—
98	Dic₂₇₁	—	2	4	no	—
99	Dic₂₇₁	C₂³×C₄	1	8	yes	complemented
100	Dic₂₇₁	C₂²×D₄	1	8	yes	complemented
101	C₂×C₂₇₁	—	2	8	no	—
102	C₂×C₂₇₁	C₂×C₄×D₄	1	12	yes	complemented
103	C₂×C₂₇₁	C₂⁴×C₄	1	1	yes	characteristic
104	C₂×C₂₇₁	C₂³×D₄	1	1	yes	characteristic
105	C₂×C₂₇₁	C₂²×#41	1	1	yes	characteristic
106	D₂₇₁	—	2	8	no	—
107	C₂₇₁	C₂²×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₄×D₄	—	271	1	no	maximal, Sylow
109	C₂³×D₄	—	271	1	no	—
110	C₂⁴×C₄	—	271	1	no	—
111	C₂⁴×C₄	—	271	1	no	—
112	C₂×C₄×D₄	—	271	24	no	—
113	C₂²×C₄⋊C₄	—	271	1	no	—
114	C₂²×C₂²⋊C₄	—	271	1	no	—
115	C₂²×C₂²⋊C₄	D₂₇₁	1	1	yes	characteristic, complemented
116	C₂²×C₄²	—	271	1	no	—
117	C₂⁵	—	271	1	no	—
118	C₂⁵	C₂×D₂₇₁	1	1	yes	characteristic
119	C₂²×D₄	—	271	16	no	—
120	C₂²×D₄	—	271	12	no	—
121	C₂³×C₄	—	271	1	no	—
122	C₂³×C₄	—	271	1	no	—
123	C₂³×C₄	—	271	12	no	—
124	C₂³×C₄	—	271	1	no	—
125	C₂³×C₄	—	271	12	no	—
126	C₂³×C₄	—	542	8	no	—
127	C₂³×C₄	—	542	8	no	—
128	C₂³×C₄	C₂×D₂₇₁	1	1	yes	characteristic
129	C₂³×C₄	C₂×D₂₇₁	1	1	yes	characteristic
130	C₄×D₄	—	271	64	no	—
131	C₂×C₄⋊C₄	—	271	12	no	—
132	C₂×C₂²⋊C₄	—	271	12	no	—
133	C₂×C₂²⋊C₄	C₂×D₂₇₁	1	12	yes	complemented
134	C₂×C₄²	—	271	12	no	—
135	C₂⁴	—	2	4	no	—
136	C₂⁴	—	2	4	no	—
137	C₂⁴	—	271	8	no	—
138	C₂⁴	—	271	6	no	—
139	C₂⁴	—	542	4	no	—
140	C₂⁴	—	542	4	no	—
141	C₂⁴	C₄×D₂₇₁	1	8	yes	complemented
142	C₂⁴	C₂²×D₂₇₁	1	6	yes	—
143	C₂⁴	C₂²×D₂₇₁	1	1	yes	characteristic
144	C₂×D₄	—	271	96	no	—
145	C₂×D₄	—	271	16	no	—
146	C₂²×C₄	—	2	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	271	8	no	—
149	C₂²×C₄	—	271	8	no	—
150	C₂²×C₄	—	271	6	no	—
151	C₂²×C₄	—	271	6	no	—
152	C₂²×C₄	—	271	6	no	—
153	C₂²×C₄	—	271	16	no	—
154	C₂²×C₄	—	271	16	no	—
155	C₂²×C₄	—	542	48	no	—
156	C₂²×C₄	—	542	48	no	—
157	C₂²×C₄	—	542	4	no	—
158	C₂²×C₄	C₂²×D₂₇₁	1	6	yes	—
159	C₂²×C₄	C₂²×D₂₇₁	1	6	yes	—
160	C₄⋊C₄	—	271	16	no	—
161	C₂²⋊C₄	—	271	16	no	—
162	C₂²⋊C₄	C₂²×D₂₇₁	1	16	yes	complemented
163	C₄²	—	271	16	no	—
164	C₂³	—	2	12	no	—
165	C₂³	—	2	12	no	—
166	C₂³	—	2	32	no	—
167	C₂³	—	271	24	no	—
168	C₂³	—	271	4	no	—
169	C₂³	—	542	12	no	—
170	C₂³	—	542	12	no	—
171	C₂³	—	542	32	no	—
172	C₂³	C₂×C₄×D₂₇₁	1	4	yes	—
173	C₂³	C₂×C₄×D₂₇₁	1	24	yes	complemented
174	C₂³	D₄×D₂₇₁	1	4	yes	—
175	C₂³	#16864653	1	4	yes	—
176	C₂³	C₂³×D₂₇₁	1	4	yes	—
177	C₂³	C₂³×D₂₇₁	1	3	yes	—
178	D₄	—	271	64	no	—
179	C₂×C₄	—	2	12	no	—
180	C₂×C₄	—	2	12	no	—
181	C₂×C₄	—	271	24	no	—
182	C₂×C₄	—	271	24	no	—
183	C₂×C₄	—	271	4	no	—
184	C₂×C₄	—	271	4	no	—
185	C₂×C₄	—	271	4	no	—
186	C₂×C₄	—	542	32	no	—
187	C₂×C₄	—	542	32	no	—
188	C₂×C₄	—	542	12	no	—
189	C₂×C₄	C₂³×D₂₇₁	1	4	yes	—
190	C₂×C₄	C₂³×D₂₇₁	1	4	yes	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	48	no	—
193	C₂²	—	2	4	no	—
194	C₂²	—	271	8	no	—
195	C₂²	—	542	4	no	—
196	C₂²	—	542	48	no	—
197	C₂²	—	542	4	no	—
198	C₂²	(C₂²⋊D₂₇₁)⋊C₄	1	16	yes	direct product factor
199	C₂²	C₂²×C₄×D₂₇₁	1	6	yes	—
200	C₂²	C₂²×C₄×D₂₇₁	1	8	yes	complemented
201	C₂²	C₂×D₄×D₂₇₁	1	6	yes	—
202	C₂²	C₂×#16864653	1	6	yes	—
203	C₂²	C₂⁴×D₂₇₁	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	2	4	no	—
206	C₄	—	271	8	no	—
207	C₄	—	271	8	no	—
208	C₄	—	542	4	no	—
209	C₂	—	2	8	no	—
210	C₂	—	542	8	no	—
211	C₂	C₂×(C₂²⋊D₂₇₁)⋊C₄	1	12	yes	direct product factor
212	C₂	C₂³×C₄×D₂₇₁	1	1	yes	characteristic
213	C₂	C₂²×D₄×D₂₇₁	1	1	yes	characteristic
214	C₂	C₂²×#16864653	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₂₇₁
commutator	C₂×C₂₇₁	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₂₇₁
Fitting	C₂²×C₂²⋊C₄×C₂₇₁	C₂
socle	C₂⁴×C₂₇₁	C₂³
prime radical	C₂³×C₂²⋊D₂₇₁	C₂
cosocle	C₂²×C₂₇₁	C₂⁵
hypercenter	C₂²×C₂²⋊C₄	D₂₇₁
hypocenter	C₂₇₁	C₂²×C₄×D₄
LCS	G ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₂₇₁	C₂	semidirect
3. C₂³×C₄×D₂₇₁	C₂	semidirect
4. C₂×(C₂²⋊D₂₇₁)⋊C₄	C₂	direct (N × H)
5. C₂²×Dic₂₇₁⋊C₄	C₂	semidirect
6. C₂²×(C₂×D₂₇₁)⋊C₄	C₂	semidirect
7. C₂²×C₄×Dic₂₇₁	C₂	semidirect
8. C₂²×C₂²⋊C₄×C₂₇₁	C₂	semidirect
10. C₂²×C₂²⋊D₂₇₁	C₄	semidirect
16. C₂³×Dic₂₇₁	C₂²	semidirect
18. C₂²×C₄×D₂₇₁	C₂²	semidirect
19. (C₂²⋊D₂₇₁)⋊C₄	C₂²	direct (N × H)
20. C₂×Dic₂₇₁⋊C₄	C₂²	semidirect
21. C₂×(C₂×D₂₇₁)⋊C₄	C₂²	semidirect
22. C₂×C₄×Dic₂₇₁	C₂²	semidirect
26. C₂×C₂²⋊C₄×C₂₇₁	C₂²	semidirect
29. C₂³×D₂₇₁	C₂×C₄	semidirect
31. C₂×C₂²⋊D₂₇₁	C₂×C₄	semidirect
35. C₂²×Dic₂₇₁	C₂×C₄	semidirect
36. C₂²×Dic₂₇₁	D₄	semidirect
40. C₂²×Dic₂₇₁	C₂³	semidirect
42. C₂×C₄×D₂₇₁	C₂³	semidirect
43. Dic₂₇₁⋊C₄	C₂³	semidirect
44. (C₂×D₂₇₁)⋊C₄	C₂³	semidirect
45. C₄×Dic₂₇₁	C₂³	semidirect
48. C₂⁴×C₂₇₁	C₂×C₄	semidirect
55. C₂²⋊C₄×C₂₇₁	C₂³	semidirect
59. C₂²×D₂₇₁	C₂²×C₄	semidirect
61. C₂²⋊D₂₇₁	C₂²×C₄	semidirect
64. C₂×Dic₂₇₁	C₂²×C₄	semidirect
65. C₂×Dic₂₇₁	C₂×D₄	semidirect
74. C₂³×C₂₇₁	C₂²×C₄	semidirect
86. C₂²×C₂₇₁	C₄×D₄	semidirect
88. C₂²×C₂₇₁	C₂³×C₄	semidirect
95. C₂×D₂₇₁	C₂³×C₄	semidirect
99. Dic₂₇₁	C₂³×C₄	semidirect
100. Dic₂₇₁	C₂²×D₄	semidirect
102. C₂×C₂₇₁	C₂×C₄×D₄	semidirect
107. C₂₇₁	C₂²×C₄×D₄	semidirect
115. C₂²×C₂²⋊C₄	D₂₇₁	semidirect
133. C₂×C₂²⋊C₄	C₂×D₂₇₁	semidirect
141. C₂⁴	C₄×D₂₇₁	semidirect
162. C₂²⋊C₄	C₂²×D₂₇₁	semidirect
173. C₂³	C₂×C₄×D₂₇₁	semidirect
198. C₂²	(C₂²⋊D₂₇₁)⋊C₄	direct (N × H)
200. C₂²	C₂²×C₄×D₂₇₁	semidirect
211. C₂	C₂×(C₂²⋊D₂₇₁)⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₂²⋊D₂₇₁	2
C₂⁴×Dic₂₇₁	2
C₂³×C₄×D₂₇₁	2
C₂×(C₂²⋊D₂₇₁)⋊C₄	2
C₂²×Dic₂₇₁⋊C₄	2
C₂²×(C₂×D₂₇₁)⋊C₄	2
C₂²×C₄×Dic₂₇₁	2
C₂²×C₂²⋊C₄×C₂₇₁	2
C₂²×C₄×D₄	271

Subgroups — C22×(C22⋊D271)⋊C4 #46849532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×(C₂²⋊D₂₇₁)⋊C₄ #46849532