Subgroups — #3459×C₂₇₁ #46834045

Order 34688.

1160 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#543×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#539×C₂₇₁	C₂	1	6	yes	maximal, complemented
3	#538×C₂₇₁	C₂	1	6	yes	maximal, complemented
4	#534×C₂₇₁	C₂	1	3	yes	maximal, complemented
5	#529×C₂₇₁	C₂	1	6	yes	maximal, complemented
6	#525×C₂₇₁	C₂	1	6	yes	maximal, complemented
7	C₂×C₄×D₄×C₂₇₁	C₂	1	3	yes	maximal, complemented
8	C₂×#41×C₂₇₁	C₂²	1	2	yes	complemented
9	C₂²×D₄×C₂₇₁	C₂²	1	3	yes	—
10	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₇₁	C₂²	1	3	yes	complemented
12	#126×C₂₇₁	C₂²	1	12	yes	complemented
13	#125×C₂₇₁	C₂²	1	6	yes	complemented
14	#124×C₂₇₁	C₂²	1	6	yes	complemented
15	#123×C₂₇₁	C₂²	1	12	yes	complemented
16	#123×C₂₇₁	C₂²	1	12	yes	complemented
17	C₂²⋊Q₈×C₂₇₁	C₂²	1	6	yes	complemented
18	C₂²⋊Q₈×C₂₇₁	C₂²	1	6	yes	complemented
19	C₂²⋊Q₈×C₂₇₁	C₂²	1	6	yes	complemented
20	#121×C₂₇₁	C₂²	1	6	yes	—
21	#121×C₂₇₁	C₂²	1	6	yes	—
22	#121×C₂₇₁	C₂²	1	6	yes	complemented
23	#120×C₂₇₁	C₂²	1	12	yes	complemented
24	C₄×Q₈×C₂₇₁	C₂²	1	2	yes	complemented
25	C₄×D₄×C₂₇₁	C₂²	1	12	yes	complemented
26	C₄×D₄×C₂₇₁	C₂²	1	12	yes	complemented
27	C₄×D₄×C₂₇₁	C₂²	1	6	yes	complemented
28	#117×C₂₇₁	C₂²	1	6	yes	complemented
29	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	3	yes	complemented
30	C₂×C₂²⋊C₄×C₂₇₁	C₂²	1	3	yes	—
31	C₂×C₂²⋊C₄×C₂₇₁	C₂²	1	3	yes	complemented
32	C₂×C₄²×C₂₇₁	C₂²	1	3	yes	complemented
33	C₂⁴×C₂₇₁	C₂³	1	3	yes	—
34	C₂⁴×C₂₇₁	C₂³	1	1	yes	characteristic
35	#41×C₂₇₁	—	4	4	no	—
36	C₂×Q₈×C₂₇₁	C₂³	1	2	yes	—
37	C₂×Q₈×C₂₇₁	C₂³	1	2	yes	—
38	C₂×D₄×C₂₇₁	—	2	12	no	—
39	C₂×D₄×C₂₇₁	C₂³	1	12	yes	—
40	C₂×D₄×C₂₇₁	C₂³	1	6	yes	—
41	C₂×D₄×C₂₇₁	C₂³	1	6	yes	—
42	C₂²×C₄×C₂₇₁	—	4	2	no	—
43	C₂²×C₄×C₂₇₁	—	4	6	no	—
44	C₂²×C₄×C₂₇₁	C₂³	1	3	yes	—
45	C₂²×C₄×C₂₇₁	C₂³	1	3	yes	—
46	C₂²×C₄×C₂₇₁	C₂³	1	6	yes	—
47	C₂²×C₄×C₂₇₁	C₂³	1	3	yes	—
48	C₂²×C₄×C₂₇₁	C₂³	1	3	yes	—
49	C₂²×C₄×C₂₇₁	C₂³	1	3	yes	—
50	C₂²×C₄×C₂₇₁	C₂³	1	6	yes	—
51	C₄⋊C₄×C₂₇₁	C₂³	1	12	yes	—
52	C₄⋊C₄×C₂₇₁	C₂³	1	6	yes	—
53	C₄⋊C₄×C₂₇₁	C₂³	1	6	yes	—
54	C₄⋊C₄×C₂₇₁	C₂³	1	6	yes	—
55	C₄⋊C₄×C₂₇₁	C₂³	1	6	yes	—
56	C₂²⋊C₄×C₂₇₁	C₂³	1	12	yes	—
57	C₂²⋊C₄×C₂₇₁	C₂³	1	12	yes	—
58	C₂²⋊C₄×C₂₇₁	C₂³	1	12	yes	—
59	C₂²⋊C₄×C₂₇₁	C₂³	1	12	yes	—
60	C₄²×C₂₇₁	C₂³	1	6	yes	—
61	C₄²×C₂₇₁	C₂³	1	6	yes	—
62	C₂³×C₂₇₁	—	2	6	no	—
63	C₂³×C₂₇₁	—	4	1	no	—
64	C₂³×C₂₇₁	—	4	3	no	—
65	C₂³×C₂₇₁	—	4	1	no	—
66	C₂³×C₂₇₁	—	4	3	no	—
67	C₂³×C₂₇₁	#41	1	4	yes	complemented
68	C₂³×C₂₇₁	C₂⁴	1	6	yes	—
69	C₂³×C₂₇₁	C₂⁴	1	3	yes	—
70	Q₈×C₂₇₁	—	4	2	no	—
71	Q₈×C₂₇₁	—	4	2	no	—
72	D₄×C₂₇₁	—	2	24	no	—
73	D₄×C₂₇₁	—	4	6	no	—
74	D₄×C₂₇₁	—	4	6	no	—
75	C₂×C₄×C₂₇₁	—	2	6	no	—
76	C₂×C₄×C₂₇₁	—	2	6	no	—
77	C₂×C₄×C₂₇₁	—	2	12	no	—
78	C₂×C₄×C₂₇₁	—	4	3	no	—
79	C₂×C₄×C₂₇₁	—	4	6	no	—
80	C₂×C₄×C₂₇₁	—	4	3	no	—
81	C₂×C₄×C₂₇₁	—	4	3	no	—
82	C₂×C₄×C₂₇₁	C₂⁴	1	6	yes	—
83	C₂×C₄×C₂₇₁	C₂⁴	1	6	yes	—
84	C₂×C₄×C₂₇₁	C₂⁴	1	3	yes	—
85	C₂×C₄×C₂₇₁	C₂⁴	1	6	yes	—
86	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
87	C₂²×C₂₇₁	—	2	3	no	—
88	C₂²×C₂₇₁	—	2	3	no	—
89	C₂²×C₂₇₁	—	2	6	no	—
90	C₂²×C₂₇₁	—	2	6	no	—
91	C₂²×C₂₇₁	—	2	6	no	—
92	C₂²×C₂₇₁	—	4	4	no	—
93	C₂²×C₂₇₁	—	4	12	no	—
94	C₂²×C₂₇₁	C₂×#41	1	6	yes	—
95	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
96	C₄×C₂₇₁	—	2	6	no	—
97	C₄×C₂₇₁	—	2	3	no	—
98	C₄×C₂₇₁	—	2	6	no	—
99	C₄×C₂₇₁	—	2	6	no	—
100	C₄×C₂₇₁	H(5,𝔽₂)	1	2	yes	—
101	C₂×C₂₇₁	—	2	6	no	—
102	C₂×C₂₇₁	—	4	6	no	—
103	C₂×C₂₇₁	C₂²×#41	1	1	yes	characteristic
104	C₂×C₂₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×C₂₇₁	#585	1	1	yes	characteristic
106	C₂₇₁	#3459	1	1	yes	characteristic, direct product factor, Sylow
107	#3459	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	#543	C₂×C₂₇₁	1	1	yes	characteristic, complemented
109	#539	C₂×C₂₇₁	1	6	yes	complemented
110	#538	C₂×C₂₇₁	1	6	yes	complemented
111	#534	C₂×C₂₇₁	1	3	yes	complemented
112	#529	C₂×C₂₇₁	1	6	yes	complemented
113	#525	C₂×C₂₇₁	1	6	yes	complemented
114	C₂×C₄×D₄	C₂×C₂₇₁	1	3	yes	complemented
115	C₂×#41	C₂²×C₂₇₁	1	2	yes	complemented
116	C₂²×D₄	C₂²×C₂₇₁	1	3	yes	—
117	C₂³×C₄	C₂²×C₂₇₁	1	1	yes	characteristic
118	C₂³×C₄	C₂²×C₂₇₁	1	3	yes	complemented
119	#126	C₂²×C₂₇₁	1	12	yes	complemented
120	#125	C₂²×C₂₇₁	1	6	yes	complemented
121	#124	C₂²×C₂₇₁	1	6	yes	complemented
122	#123	C₂²×C₂₇₁	1	12	yes	complemented
123	#123	C₂²×C₂₇₁	1	12	yes	complemented
124	C₂²⋊Q₈	C₂²×C₂₇₁	1	6	yes	complemented
125	C₂²⋊Q₈	C₂²×C₂₇₁	1	6	yes	complemented
126	C₂²⋊Q₈	C₂²×C₂₇₁	1	6	yes	complemented
127	#121	C₂²×C₂₇₁	1	6	yes	—
128	#121	C₂²×C₂₇₁	1	6	yes	—
129	#121	C₂²×C₂₇₁	1	6	yes	complemented
130	#120	C₂²×C₂₇₁	1	12	yes	complemented
131	C₄×Q₈	C₂²×C₂₇₁	1	2	yes	complemented
132	C₄×D₄	C₂²×C₂₇₁	1	12	yes	complemented
133	C₄×D₄	C₂²×C₂₇₁	1	12	yes	complemented
134	C₄×D₄	C₂²×C₂₇₁	1	6	yes	complemented
135	#117	C₂²×C₂₇₁	1	6	yes	complemented
136	C₂×C₄⋊C₄	C₂²×C₂₇₁	1	3	yes	complemented
137	C₂×C₂²⋊C₄	C₂²×C₂₇₁	1	3	yes	—
138	C₂×C₂²⋊C₄	C₂²×C₂₇₁	1	3	yes	complemented
139	C₂×C₄²	C₂²×C₂₇₁	1	3	yes	complemented
140	C₂⁴	C₂³×C₂₇₁	1	3	yes	—
141	C₂⁴	C₂³×C₂₇₁	1	1	yes	characteristic
142	#41	—	4	4	no	—
143	C₂×Q₈	C₂³×C₂₇₁	1	2	yes	—
144	C₂×Q₈	C₂³×C₂₇₁	1	2	yes	—
145	C₂×D₄	—	2	12	no	—
146	C₂×D₄	C₂³×C₂₇₁	1	12	yes	—
147	C₂×D₄	C₂³×C₂₇₁	1	6	yes	—
148	C₂×D₄	C₂³×C₂₇₁	1	6	yes	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	4	6	no	—
151	C₂²×C₄	C₂³×C₂₇₁	1	3	yes	—
152	C₂²×C₄	C₂³×C₂₇₁	1	3	yes	—
153	C₂²×C₄	C₂³×C₂₇₁	1	6	yes	—
154	C₂²×C₄	C₂³×C₂₇₁	1	3	yes	—
155	C₂²×C₄	C₂³×C₂₇₁	1	3	yes	—
156	C₂²×C₄	C₂³×C₂₇₁	1	3	yes	—
157	C₂²×C₄	C₂³×C₂₇₁	1	6	yes	—
158	C₄⋊C₄	C₂³×C₂₇₁	1	12	yes	—
159	C₄⋊C₄	C₂³×C₂₇₁	1	6	yes	—
160	C₄⋊C₄	C₂³×C₂₇₁	1	6	yes	—
161	C₄⋊C₄	C₂³×C₂₇₁	1	6	yes	—
162	C₄⋊C₄	C₂³×C₂₇₁	1	6	yes	—
163	C₂²⋊C₄	C₂³×C₂₇₁	1	12	yes	—
164	C₂²⋊C₄	C₂³×C₂₇₁	1	12	yes	—
165	C₂²⋊C₄	C₂³×C₂₇₁	1	12	yes	—
166	C₂²⋊C₄	C₂³×C₂₇₁	1	12	yes	—
167	C₄²	C₂³×C₂₇₁	1	6	yes	—
168	C₄²	C₂³×C₂₇₁	1	6	yes	—
169	C₂³	—	2	6	no	—
170	C₂³	—	4	1	no	—
171	C₂³	—	4	3	no	—
172	C₂³	—	4	1	no	—
173	C₂³	—	4	3	no	—
174	C₂³	#41×C₂₇₁	1	4	yes	complemented
175	C₂³	C₂⁴×C₂₇₁	1	6	yes	—
176	C₂³	C₂⁴×C₂₇₁	1	3	yes	—
177	Q₈	—	4	2	no	—
178	Q₈	—	4	2	no	—
179	D₄	—	2	24	no	—
180	D₄	—	4	6	no	—
181	D₄	—	4	6	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	6	no	—
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	4	3	no	—
186	C₂×C₄	—	4	6	no	—
187	C₂×C₄	—	4	3	no	—
188	C₂×C₄	—	4	3	no	—
189	C₂×C₄	C₂⁴×C₂₇₁	1	6	yes	—
190	C₂×C₄	C₂⁴×C₂₇₁	1	6	yes	—
191	C₂×C₄	C₂⁴×C₂₇₁	1	3	yes	—
192	C₂×C₄	C₂⁴×C₂₇₁	1	6	yes	—
193	C₂×C₄	C₂⁴×C₂₇₁	1	1	yes	characteristic
194	C₂²	—	2	3	no	—
195	C₂²	—	2	3	no	—
196	C₂²	—	2	6	no	—
197	C₂²	—	2	6	no	—
198	C₂²	—	2	6	no	—
199	C₂²	—	4	4	no	—
200	C₂²	—	4	12	no	—
201	C₂²	C₂×#41×C₂₇₁	1	6	yes	—
202	C₂²	C₂⁵×C₂₇₁	1	1	yes	characteristic
203	C₄	—	2	6	no	—
204	C₄	—	2	3	no	—
205	C₄	—	2	6	no	—
206	C₄	—	2	6	no	—
207	C₄	H(5,𝔽₂)×C₂₇₁	1	2	yes	—
208	C₂	—	2	6	no	—
209	C₂	—	4	6	no	—
210	C₂	C₂²×#41×C₂₇₁	1	1	yes	characteristic
211	C₂	C₂×H(5,𝔽₂)×C₂₇₁	1	1	yes	characteristic
212	C₂	#585×C₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₇₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₇₁
socle	C₂²×C₂₇₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #543×C₂₇₁	C₂	semidirect
2. #539×C₂₇₁	C₂	semidirect
3. #538×C₂₇₁	C₂	semidirect
4. #534×C₂₇₁	C₂	semidirect
5. #529×C₂₇₁	C₂	semidirect
6. #525×C₂₇₁	C₂	semidirect
7. C₂×C₄×D₄×C₂₇₁	C₂	semidirect
8. C₂×#41×C₂₇₁	C₂²	semidirect
11. C₂³×C₄×C₂₇₁	C₂²	semidirect
12. #126×C₂₇₁	C₂²	semidirect
13. #125×C₂₇₁	C₂²	semidirect
14. #124×C₂₇₁	C₂²	semidirect
15. #123×C₂₇₁	C₂²	semidirect
16. #123×C₂₇₁	C₂²	semidirect
17. C₂²⋊Q₈×C₂₇₁	C₂²	semidirect
18. C₂²⋊Q₈×C₂₇₁	C₂²	semidirect
19. C₂²⋊Q₈×C₂₇₁	C₂²	semidirect
22. #121×C₂₇₁	C₂²	semidirect
23. #120×C₂₇₁	C₂²	semidirect
24. C₄×Q₈×C₂₇₁	C₂²	semidirect
25. C₄×D₄×C₂₇₁	C₂²	semidirect
26. C₄×D₄×C₂₇₁	C₂²	semidirect
27. C₄×D₄×C₂₇₁	C₂²	semidirect
28. #117×C₂₇₁	C₂²	semidirect
29. C₂×C₄⋊C₄×C₂₇₁	C₂²	semidirect
31. C₂×C₂²⋊C₄×C₂₇₁	C₂²	semidirect
32. C₂×C₄²×C₂₇₁	C₂²	semidirect
67. C₂³×C₂₇₁	#41	semidirect
106. C₂₇₁	#3459	direct (N × H)
107. #3459	C₂₇₁	direct (N × H)
108. #543	C₂×C₂₇₁	semidirect
109. #539	C₂×C₂₇₁	semidirect
110. #538	C₂×C₂₇₁	semidirect
111. #534	C₂×C₂₇₁	semidirect
112. #529	C₂×C₂₇₁	semidirect
113. #525	C₂×C₂₇₁	semidirect
114. C₂×C₄×D₄	C₂×C₂₇₁	semidirect
115. C₂×#41	C₂²×C₂₇₁	semidirect
118. C₂³×C₄	C₂²×C₂₇₁	semidirect
119. #126	C₂²×C₂₇₁	semidirect
120. #125	C₂²×C₂₇₁	semidirect
121. #124	C₂²×C₂₇₁	semidirect
122. #123	C₂²×C₂₇₁	semidirect
123. #123	C₂²×C₂₇₁	semidirect
124. C₂²⋊Q₈	C₂²×C₂₇₁	semidirect
125. C₂²⋊Q₈	C₂²×C₂₇₁	semidirect
126. C₂²⋊Q₈	C₂²×C₂₇₁	semidirect
129. #121	C₂²×C₂₇₁	semidirect
130. #120	C₂²×C₂₇₁	semidirect
131. C₄×Q₈	C₂²×C₂₇₁	semidirect
132. C₄×D₄	C₂²×C₂₇₁	semidirect
133. C₄×D₄	C₂²×C₂₇₁	semidirect
134. C₄×D₄	C₂²×C₂₇₁	semidirect
135. #117	C₂²×C₂₇₁	semidirect
136. C₂×C₄⋊C₄	C₂²×C₂₇₁	semidirect
138. C₂×C₂²⋊C₄	C₂²×C₂₇₁	semidirect
139. C₂×C₄²	C₂²×C₂₇₁	semidirect
174. C₂³	#41×C₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#543×C₂₇₁	2
#539×C₂₇₁	2
#538×C₂₇₁	2
#534×C₂₇₁	2
#529×C₂₇₁	2
#525×C₂₇₁	2
C₂×C₄×D₄×C₂₇₁	2
#3459	271

Subgroups — #3459×C271 #46834045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3459×C₂₇₁ #46834045