Subgroups — C₂²×#127×D₇₁ #32904137

Order 18176.

24030 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×D₄×D₇₁	C₂	1	6	yes	maximal, complemented
2	C₂²×#14556125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14556124	C₂	1	6	yes	maximal, complemented
4	C₂×#127×D₇₁	C₂	1	48	yes	maximal, direct product factor
5	C₂²×C₄²×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#127×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₁	C₂²	1	4	yes	—
8	C₂³×D₂₈₄	C₂²	1	6	yes	complemented
9	C₂³×C₂²⋊D₇₁	C₂²	1	12	yes	complemented
10	C₂²×D₄×D₇₁	—	2	96	no	—
11	C₂²×D₄×D₇₁	C₂²	1	144	yes	complemented
12	C₂³×C₄×D₇₁	C₂²	1	3	yes	—
13	C₂×#14556125	C₂²	1	24	yes	complemented
14	C₂×#14556124	C₂²	1	144	yes	complemented
15	#127×D₇₁	C₂²	1	256	yes	direct product factor
16	C₂²×C₄×Dic₇₁	C₂²	1	3	yes	complemented
17	C₂×C₄²×D₇₁	C₂²	1	24	yes	complemented
18	C₂³×D₄×C₇₁	C₂²	1	6	yes	complemented
19	C₂×#127×C₇₁	C₂²	1	24	yes	complemented
20	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic, complemented
21	C₂⁴×D₇₁	—	2	96	no	—
22	C₂⁴×D₇₁	C₂³	1	48	yes	—
23	C₂⁴×D₇₁	C₂³	1	4	yes	—
24	C₂⁴×D₇₁	C₂³	1	1	yes	characteristic
25	C₂²×D₂₈₄	—	2	48	no	—
26	C₂²×D₂₈₄	C₂³	1	72	yes	complemented
27	C₂²×C₂²⋊D₇₁	—	2	96	no	—
28	C₂²×C₂²⋊D₇₁	C₂³	1	144	yes	complemented
29	C₂×D₄×D₇₁	—	2	1152	no	—
30	C₂×D₄×D₇₁	C₂³	1	384	yes	complemented
31	C₂³×Dic₇₁	C₂³	1	3	yes	—
32	C₂²×C₄×D₇₁	D₄	1	48	yes	complemented
33	C₂²×C₄×D₇₁	C₂³	1	36	yes	—
34	#14556125	C₂³	1	64	yes	complemented
35	#14556124	C₂³	1	384	yes	complemented
36	C₂×C₄×Dic₇₁	C₂³	1	36	yes	complemented
37	C₄²×D₇₁	C₂³	1	64	yes	complemented
38	C₂⁵×C₇₁	C₂³	1	4	yes	—
39	C₂²×D₄×C₇₁	—	2	48	no	—
40	C₂²×D₄×C₇₁	C₂³	1	72	yes	complemented
41	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
42	#127×C₇₁	C₂³	1	64	yes	complemented
43	C₂×C₄²×C₇₁	C₂³	1	12	yes	complemented
44	C₂³×D₇₁	—	2	48	no	—
45	C₂³×D₇₁	—	2	576	no	—
46	C₂³×D₇₁	—	4	256	no	—
47	C₂³×D₇₁	C₂×D₄	1	24	yes	—
48	C₂³×D₇₁	C₂⁴	1	64	yes	—
49	C₂³×D₇₁	C₂⁴	1	24	yes	—
50	C₂³×D₇₁	C₂⁴	1	6	yes	—
51	C₂×D₂₈₄	—	2	288	no	—
52	C₂×D₂₈₄	C₂⁴	1	96	yes	complemented
53	C₂×C₂²⋊D₇₁	—	2	576	no	—
54	C₂×C₂²⋊D₇₁	C₂⁴	1	192	yes	complemented
55	D₄×D₇₁	—	2	1536	no	—
56	C₂²×Dic₇₁	C₂×D₄	1	24	yes	complemented
57	C₂²×Dic₇₁	C₂⁴	1	18	yes	—
58	C₂×C₄×D₇₁	C₂×D₄	1	288	yes	complemented
59	C₂×C₄×D₇₁	C₂⁴	1	48	yes	—
60	C₄×Dic₇₁	C₂⁴	1	48	yes	complemented
61	C₂⁴×C₇₁	—	2	48	no	—
62	C₂⁴×C₇₁	C₂⁴	1	24	yes	—
63	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
64	C₂×D₄×C₇₁	—	2	288	no	—
65	C₂×D₄×C₇₁	C₂⁴	1	96	yes	complemented
66	C₂²×C₄×C₇₁	C₂×D₄	1	24	yes	complemented
67	C₂²×C₄×C₇₁	C₂⁴	1	18	yes	—
68	C₄²×C₇₁	C₂⁴	1	16	yes	complemented
69	C₂³×C₇₁	—	2	144	no	—
70	C₂³×C₇₁	—	4	64	no	—
71	C₂³×C₇₁	C₂²×D₄	1	12	yes	—
72	C₂³×C₇₁	C₂⁵	1	16	yes	—
73	C₂³×C₇₁	C₂⁵	1	3	yes	—
74	C₂²×D₇₁	—	2	144	no	—
75	C₂²×D₇₁	—	2	384	no	—
76	C₂²×D₇₁	—	4	64	no	—
77	C₂²×D₇₁	—	4	768	no	—
78	C₂²×D₇₁	#127	1	64	yes	direct product factor
79	C₂²×D₇₁	C₂²×D₄	1	72	yes	—
80	C₂²×D₇₁	C₂⁵	1	16	yes	—
81	C₂²×D₇₁	C₂⁵	1	4	yes	—
82	D₄×C₇₁	—	2	192	no	—
83	C₂×C₄×C₇₁	C₂²×D₄	1	72	yes	complemented
84	C₂×C₄×C₇₁	C₂⁵	1	12	yes	—
85	C₂²⋊D₇₁	—	2	384	no	—
86	C₂×Dic₇₁	C₂²×D₄	1	72	yes	complemented
87	C₂×Dic₇₁	C₂⁵	1	12	yes	—
88	D₂₈₄	—	2	192	no	—
89	C₄×D₇₁	C₂²×D₄	1	192	yes	complemented
90	C₂²×C₇₁	—	2	48	no	—
91	C₂²×C₇₁	—	4	96	no	—
92	C₂²×C₇₁	C₂×#127	1	16	yes	complemented
93	C₂²×C₇₁	C₂³×D₄	1	18	yes	—
94	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
95	C₂×D₇₁	—	2	48	no	—
96	C₂×D₇₁	—	4	96	no	—
97	C₂×D₇₁	—	4	256	no	—
98	C₂×D₇₁	C₂×#127	1	96	yes	direct product factor
99	C₂×D₇₁	C₂³×D₄	1	24	yes	—
100	C₄×C₇₁	C₂³×D₄	1	24	yes	complemented
101	Dic₇₁	C₂³×D₄	1	24	yes	complemented
102	C₂³×#127	—	71	1	no	maximal, Sylow
103	C₂×C₇₁	—	4	16	no	—
104	C₂×C₇₁	C₂²×#127	1	12	yes	complemented
105	C₂×C₇₁	C₂⁴×D₄	1	3	yes	—
106	D₇₁	—	4	16	no	—
107	D₇₁	C₂²×#127	1	16	yes	direct product factor
108	C₂⁴×D₄	—	71	6	no	—
109	C₂²×#127	—	71	1	no	—
110	C₂²×#127	—	71	6	no	—
111	C₂²×#127	—	71	48	no	—
112	C₂²×#127	D₇₁	1	1	yes	characteristic, direct product factor
113	C₂³×C₄²	—	71	1	no	—
114	C₇₁	C₂³×#127	1	1	yes	characteristic, complemented, Sylow
115	C₂⁶	—	71	4	no	—
116	C₂³×D₄	—	71	12	no	—
117	C₂³×D₄	—	71	144	no	—
118	C₂³×D₄	—	71	6	no	—
119	C₂³×D₄	—	142	96	no	—
120	C₂³×D₄	C₂×D₇₁	1	6	yes	complemented
121	C₂⁴×C₄	—	71	3	no	—
122	C₂×#127	—	71	144	no	—
123	C₂×#127	—	71	256	no	—
124	C₂×#127	—	71	24	no	—
125	C₂×#127	C₂×D₇₁	1	24	yes	direct product factor
126	C₂²×C₄²	—	71	3	no	—
127	C₂²×C₄²	—	71	24	no	—
128	C₂²×C₄²	C₂×D₇₁	1	1	yes	characteristic, complemented
129	C₂⁵	—	71	1	no	—
130	C₂⁵	—	71	48	no	—
131	C₂⁵	—	71	4	no	—
132	C₂⁵	—	142	96	no	—
133	C₂⁵	C₂²×D₇₁	1	4	yes	—
134	C₂²×D₄	—	2	48	no	—
135	C₂²×D₄	—	71	144	no	—
136	C₂²×D₄	—	71	384	no	—
137	C₂²×D₄	—	71	72	no	—
138	C₂²×D₄	—	142	1152	no	—
139	C₂²×D₄	—	142	96	no	—
140	C₂²×D₄	—	142	48	no	—
141	C₂²×D₄	C₂²×D₇₁	1	72	yes	complemented
142	C₂³×C₄	—	71	3	no	—
143	C₂³×C₄	—	71	48	no	—
144	C₂³×C₄	—	71	36	no	—
145	C₂³×C₄	C₂²×D₇₁	1	3	yes	—
146	#127	—	71	384	no	—
147	#127	—	71	64	no	—
148	#127	C₂²×D₇₁	1	64	yes	direct product factor
149	C₂×C₄²	—	71	36	no	—
150	C₂×C₄²	—	71	64	no	—
151	C₂×C₄²	C₂²×D₇₁	1	12	yes	complemented
152	C₂⁴	—	2	48	no	—
153	C₂⁴	—	284	256	no	—
154	C₂⁴	—	71	24	no	—
155	C₂⁴	—	71	6	no	—
156	C₂⁴	—	71	64	no	—
157	C₂⁴	—	71	24	no	—
158	C₂⁴	—	142	576	no	—
159	C₂⁴	—	142	48	no	—
160	C₂⁴	C₂³×D₇₁	1	24	yes	—
161	C₂⁴	C₂³×D₇₁	1	1	yes	characteristic
162	C₂×D₄	—	2	288	no	—
163	C₂×D₄	—	71	192	no	—
164	C₂×D₄	—	71	96	no	—
165	C₂×D₄	—	142	1536	no	—
166	C₂×D₄	—	142	576	no	—
167	C₂×D₄	—	142	288	no	—
168	C₂×D₄	C₂³×D₇₁	1	96	yes	complemented
169	C₂²×C₄	—	71	24	no	—
170	C₂²×C₄	—	71	18	no	—
171	C₂²×C₄	—	71	288	no	—
172	C₂²×C₄	—	71	48	no	—
173	C₂²×C₄	D₄×D₇₁	1	24	yes	complemented
174	C₂²×C₄	C₂³×D₇₁	1	18	yes	—
175	C₄²	—	71	48	no	—
176	C₄²	C₂³×D₇₁	1	16	yes	complemented
177	C₂³	—	2	144	no	—
178	C₂³	—	4	64	no	—
179	C₂³	—	284	768	no	—
180	C₂³	—	284	64	no	—
181	C₂³	—	71	72	no	—
182	C₂³	—	71	4	no	—
183	C₂³	—	71	64	no	—
184	C₂³	—	71	16	no	—
185	C₂³	—	142	384	no	—
186	C₂³	—	142	144	no	—
187	C₂³	C₂×D₄×D₇₁	1	12	yes	—
188	C₂³	C₂⁴×D₇₁	1	16	yes	—
189	C₂³	C₂⁴×D₇₁	1	3	yes	—
190	D₄	—	2	192	no	—
191	D₄	—	142	384	no	—
192	D₄	—	142	192	no	—
193	C₂×C₄	—	71	72	no	—
194	C₂×C₄	—	71	12	no	—
195	C₂×C₄	—	71	192	no	—
196	C₂×C₄	C₂×D₄×D₇₁	1	72	yes	complemented
197	C₂×C₄	C₂⁴×D₇₁	1	12	yes	—
198	C₂²	—	2	48	no	—
199	C₂²	—	4	96	no	—
200	C₂²	—	284	256	no	—
201	C₂²	—	284	96	no	—
202	C₂²	—	71	24	no	—
203	C₂²	—	71	96	no	—
204	C₂²	—	142	48	no	—
205	C₂²	#127×D₇₁	1	16	yes	direct product factor
206	C₂²	C₂²×D₄×D₇₁	1	18	yes	—
207	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
208	C₄	—	71	24	no	—
209	C₄	C₂²×D₄×D₇₁	1	24	yes	complemented
210	C₂	—	4	16	no	—
211	C₂	—	284	16	no	—
212	C₂	—	71	16	no	—
213	C₂	C₂×#127×D₇₁	1	12	yes	direct product factor
214	C₂	C₂³×D₄×D₇₁	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂²×#127×C₇₁	C₂
socle	C₂⁴×C₇₁	C₂⁴
hypercenter	C₂²×#127	D₇₁
hypocenter	C₇₁	C₂³×#127
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₇₁	C₂	semidirect
2. C₂²×#14556125	C₂	semidirect
3. C₂²×#14556124	C₂	semidirect
4. C₂×#127×D₇₁	C₂	direct (N × H)
5. C₂²×C₄²×D₇₁	C₂	semidirect
6. C₂²×#127×C₇₁	C₂	semidirect
8. C₂³×D₂₈₄	C₂²	semidirect
9. C₂³×C₂²⋊D₇₁	C₂²	semidirect
11. C₂²×D₄×D₇₁	C₂²	semidirect
13. C₂×#14556125	C₂²	semidirect
14. C₂×#14556124	C₂²	semidirect
15. #127×D₇₁	C₂²	direct (N × H)
16. C₂²×C₄×Dic₇₁	C₂²	semidirect
17. C₂×C₄²×D₇₁	C₂²	semidirect
18. C₂³×D₄×C₇₁	C₂²	semidirect
19. C₂×#127×C₇₁	C₂²	semidirect
20. C₂²×C₄²×C₇₁	C₂²	semidirect
26. C₂²×D₂₈₄	C₂³	semidirect
28. C₂²×C₂²⋊D₇₁	C₂³	semidirect
30. C₂×D₄×D₇₁	C₂³	semidirect
32. C₂²×C₄×D₇₁	D₄	semidirect
34. #14556125	C₂³	semidirect
35. #14556124	C₂³	semidirect
36. C₂×C₄×Dic₇₁	C₂³	semidirect
37. C₄²×D₇₁	C₂³	semidirect
40. C₂²×D₄×C₇₁	C₂³	semidirect
42. #127×C₇₁	C₂³	semidirect
43. C₂×C₄²×C₇₁	C₂³	semidirect
52. C₂×D₂₈₄	C₂⁴	semidirect
54. C₂×C₂²⋊D₇₁	C₂⁴	semidirect
56. C₂²×Dic₇₁	C₂×D₄	semidirect
58. C₂×C₄×D₇₁	C₂×D₄	semidirect
60. C₄×Dic₇₁	C₂⁴	semidirect
65. C₂×D₄×C₇₁	C₂⁴	semidirect
66. C₂²×C₄×C₇₁	C₂×D₄	semidirect
68. C₄²×C₇₁	C₂⁴	semidirect
78. C₂²×D₇₁	#127	direct (N × H)
83. C₂×C₄×C₇₁	C₂²×D₄	semidirect
86. C₂×Dic₇₁	C₂²×D₄	semidirect
89. C₄×D₇₁	C₂²×D₄	semidirect
92. C₂²×C₇₁	C₂×#127	semidirect
98. C₂×D₇₁	C₂×#127	direct (N × H)
100. C₄×C₇₁	C₂³×D₄	semidirect
101. Dic₇₁	C₂³×D₄	semidirect
104. C₂×C₇₁	C₂²×#127	semidirect
107. D₇₁	C₂²×#127	direct (N × H)
112. C₂²×#127	D₇₁	direct (N × H)
114. C₇₁	C₂³×#127	semidirect
120. C₂³×D₄	C₂×D₇₁	semidirect
125. C₂×#127	C₂×D₇₁	direct (N × H)
128. C₂²×C₄²	C₂×D₇₁	semidirect
141. C₂²×D₄	C₂²×D₇₁	semidirect
148. #127	C₂²×D₇₁	direct (N × H)
151. C₂×C₄²	C₂²×D₇₁	semidirect
168. C₂×D₄	C₂³×D₇₁	semidirect
173. C₂²×C₄	D₄×D₇₁	semidirect
176. C₄²	C₂³×D₇₁	semidirect
196. C₂×C₄	C₂×D₄×D₇₁	semidirect
205. C₂²	#127×D₇₁	direct (N × H)
209. C₄	C₂²×D₄×D₇₁	semidirect
213. C₂	C₂×#127×D₇₁	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×D₄×D₇₁	2
C₂²×#14556125	2
C₂²×#14556124	2
C₂×#127×D₇₁	2
C₂²×C₄²×D₇₁	2
C₂²×#127×C₇₁	2
C₂³×#127	71

Subgroups — C22×#127×D71 #32904137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#127×D₇₁ #32904137