Subgroups — #2951×C₂₇₁ #46833537

Order 34688.

484 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₂₇₁	C₂	1	1	yes	maximal, characteristic
2	#520×C₂₇₁	C₂	1	2	yes	maximal, complemented
3	#440×C₂₇₁	C₂	1	4	yes	maximal, complemented
4	C₄×Q₁₆×C₂₇₁	C₂	1	2	yes	maximal, complemented
5	C₄×D₈×C₂₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×#107×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#417×C₂₇₁	C₂	1	2	yes	maximal, complemented
8	#405×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₂₇₁	C₂²	1	2	yes	—
10	#135×C₂₇₁	C₄	1	8	yes	complemented
11	C₂×Q₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₂×SD₁₆×C₂₇₁	C₂²	1	2	yes	—
13	C₂×D₈×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₂₇₁	C₂²	1	4	yes	—
17	C₄×D₄×C₂₇₁	—	2	4	no	—
18	C₄×D₄×C₂₇₁	C₂²	1	4	yes	—
19	#117×C₂₇₁	—	2	2	no	—
20	#117×C₂₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₇₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
23	#107×C₂₇₁	C₂²	1	2	yes	—
24	#107×C₂₇₁	C₂²	1	2	yes	—
25	#103×C₂₇₁	C₂²	1	4	yes	—
26	#102×C₂₇₁	C₂²	1	4	yes	—
27	#97×C₂₇₁	C₂²	1	2	yes	—
28	C₄×C₈×C₂₇₁	C₂²	1	2	yes	—
29	#41×C₂₇₁	—	2	4	no	—
30	#41×C₂₇₁	C₂×C₄	1	8	yes	complemented
31	C₂×Q₈×C₂₇₁	C₂³	1	2	yes	—
32	C₂×D₄×C₂₇₁	—	2	2	no	—
33	C₂×D₄×C₂₇₁	C₂³	1	2	yes	—
34	C₂²×C₄×C₂₇₁	—	2	2	no	—
35	C₂²×C₄×C₂₇₁	—	2	2	no	—
36	C₂²×C₄×C₂₇₁	—	2	4	no	—
37	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
38	Q₁₆×C₂₇₁	C₂×C₄	1	4	yes	complemented
39	SD₁₆×C₂₇₁	C₂×C₄	1	8	yes	complemented
40	D₈×C₂₇₁	C₂×C₄	1	4	yes	complemented
41	M₁₆×C₂₇₁	—	2	2	no	—
42	C₂×C₈×C₂₇₁	C₂×C₄	1	4	yes	complemented
43	C₂×C₈×C₂₇₁	C₂³	1	2	yes	—
44	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₂₇₁	—	2	2	no	—
47	C₄⋊C₄×C₂₇₁	—	2	4	no	—
48	C₄⋊C₄×C₂₇₁	D₄	1	1	yes	characteristic
49	C₄⋊C₄×C₂₇₁	D₄	1	1	yes	characteristic
50	C₄⋊C₄×C₂₇₁	C₂³	1	4	yes	—
51	C₂²⋊C₄×C₂₇₁	—	2	4	no	—
52	C₂²⋊C₄×C₂₇₁	D₄	1	2	yes	—
53	C₄²×C₂₇₁	—	2	4	no	—
54	C₄²×C₂₇₁	C₂³	1	2	yes	—
55	C₂³×C₂₇₁	—	2	2	no	—
56	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
57	Q₈×C₂₇₁	—	2	2	no	—
58	Q₈×C₂₇₁	C₂²×C₄	1	4	yes	—
59	D₄×C₂₇₁	—	2	8	no	—
60	D₄×C₂₇₁	—	2	2	no	—
61	D₄×C₂₇₁	C₂²×C₄	1	4	yes	—
62	C₂×C₄×C₂₇₁	—	2	2	no	—
63	C₂×C₄×C₂₇₁	—	2	8	no	—
64	C₂×C₄×C₂₇₁	—	2	4	no	—
65	C₂×C₄×C₂₇₁	—	8	4	no	—
66	C₂×C₄×C₂₇₁	—	8	2	no	—
67	C₂×C₄×C₂₇₁	C₂²×C₄	1	2	yes	—
68	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
69	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
70	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₂₇₁	#41	1	2	yes	—
72	C₂×C₄×C₂₇₁	C₂⁴	1	1	yes	characteristic
73	C₈×C₂₇₁	—	2	2	no	—
74	C₈×C₂₇₁	C₂²×C₄	1	2	yes	—
75	C₈×C₂₇₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₂₇₁	—	2	1	no	—
77	C₂²×C₂₇₁	—	2	1	no	—
78	C₂²×C₂₇₁	—	2	4	no	—
79	C₂²×C₂₇₁	—	4	2	no	—
80	C₂²×C₂₇₁	—	4	2	no	—
81	C₂²×C₂₇₁	C₄×D₄	1	2	yes	—
82	C₂²×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
83	C₄×C₂₇₁	—	2	4	no	—
84	C₄×C₂₇₁	—	2	2	no	—
85	C₄×C₂₇₁	—	2	2	no	—
86	C₄×C₂₇₁	—	4	4	no	—
87	C₄×C₂₇₁	C₄×D₄	1	2	yes	—
88	C₄×C₂₇₁	C₂³×C₄	1	1	yes	characteristic
89	C₄×C₂₇₁	C₂×#41	1	1	yes	characteristic
90	C₂×C₂₇₁	—	2	2	no	—
91	C₂×C₂₇₁	—	4	4	no	—
92	C₂×C₂₇₁	C₂×C₄×D₄	1	1	yes	characteristic
93	C₂×C₂₇₁	#576	1	1	yes	characteristic
94	C₂×C₂₇₁	#578	1	1	yes	characteristic
95	C₂₇₁	#2951	1	1	yes	characteristic, direct product factor, Sylow
96	#2951	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂×#135	C₂×C₂₇₁	1	1	yes	characteristic
98	#520	C₂×C₂₇₁	1	2	yes	complemented
99	#440	C₂×C₂₇₁	1	4	yes	complemented
100	C₄×Q₁₆	C₂×C₂₇₁	1	2	yes	complemented
101	C₄×D₈	C₂×C₂₇₁	1	2	yes	complemented
102	C₂×#107	C₂×C₂₇₁	1	1	yes	characteristic, complemented
103	#417	C₂×C₂₇₁	1	2	yes	complemented
104	#405	C₂×C₂₇₁	1	1	yes	characteristic, complemented
105	C₂×#41	C₂²×C₂₇₁	1	2	yes	—
106	#135	C₄×C₂₇₁	1	8	yes	complemented
107	C₂×Q₁₆	C₂²×C₂₇₁	1	1	yes	characteristic
108	C₂×SD₁₆	C₂²×C₂₇₁	1	2	yes	—
109	C₂×D₈	C₂²×C₂₇₁	1	1	yes	characteristic
110	C₂×M₁₆	C₂²×C₂₇₁	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₂₇₁	1	1	yes	characteristic
112	C₄×Q₈	C₂²×C₂₇₁	1	4	yes	—
113	C₄×D₄	—	2	4	no	—
114	C₄×D₄	C₂²×C₂₇₁	1	4	yes	—
115	#117	—	2	2	no	—
116	#117	C₂²×C₂₇₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	2	no	—
118	C₂×C₄⋊C₄	C₂²×C₂₇₁	1	2	yes	—
119	#107	C₂²×C₂₇₁	1	2	yes	—
120	#107	C₂²×C₂₇₁	1	2	yes	—
121	#103	C₂²×C₂₇₁	1	4	yes	—
122	#102	C₂²×C₂₇₁	1	4	yes	—
123	#97	C₂²×C₂₇₁	1	2	yes	—
124	C₄×C₈	C₂²×C₂₇₁	1	2	yes	—
125	#41	—	2	4	no	—
126	#41	C₂×C₄×C₂₇₁	1	8	yes	complemented
127	C₂×Q₈	C₂³×C₂₇₁	1	2	yes	—
128	C₂×D₄	—	2	2	no	—
129	C₂×D₄	C₂³×C₂₇₁	1	2	yes	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂³×C₂₇₁	1	1	yes	characteristic
134	Q₁₆	C₂×C₄×C₂₇₁	1	4	yes	complemented
135	SD₁₆	C₂×C₄×C₂₇₁	1	8	yes	complemented
136	D₈	C₂×C₄×C₂₇₁	1	4	yes	complemented
137	M₁₆	—	2	2	no	—
138	C₂×C₈	C₂×C₄×C₂₇₁	1	4	yes	complemented
139	C₂×C₈	C₂³×C₂₇₁	1	2	yes	—
140	C₂×C₈	C₂³×C₂₇₁	1	1	yes	characteristic
141	C₂×C₈	C₂³×C₂₇₁	1	1	yes	characteristic
142	C₄⋊C₄	—	2	2	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	D₄×C₂₇₁	1	1	yes	characteristic
145	C₄⋊C₄	D₄×C₂₇₁	1	1	yes	characteristic
146	C₄⋊C₄	C₂³×C₂₇₁	1	4	yes	—
147	C₂²⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	D₄×C₂₇₁	1	2	yes	—
149	C₄²	—	2	4	no	—
150	C₄²	C₂³×C₂₇₁	1	2	yes	—
151	C₂³	—	2	2	no	—
152	C₂³	C₂×D₄×C₂₇₁	1	1	yes	characteristic
153	Q₈	—	2	2	no	—
154	Q₈	C₂²×C₄×C₂₇₁	1	4	yes	—
155	D₄	—	2	8	no	—
156	D₄	—	2	2	no	—
157	D₄	C₂²×C₄×C₂₇₁	1	4	yes	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	8	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	C₂²×C₄×C₂₇₁	1	2	yes	—
164	C₂×C₄	C₂×D₄×C₂₇₁	1	2	yes	—
165	C₂×C₄	C₂×D₄×C₂₇₁	1	2	yes	—
166	C₂×C₄	C₂×D₄×C₂₇₁	1	1	yes	characteristic
167	C₂×C₄	#41×C₂₇₁	1	2	yes	—
168	C₂×C₄	C₂⁴×C₂₇₁	1	1	yes	characteristic
169	C₈	—	2	2	no	—
170	C₈	C₂²×C₄×C₂₇₁	1	2	yes	—
171	C₈	C₂²×C₄×C₂₇₁	1	2	yes	—
172	C₂²	—	2	1	no	—
173	C₂²	—	2	1	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	4	2	no	—
176	C₂²	—	4	2	no	—
177	C₂²	C₄×D₄×C₂₇₁	1	2	yes	—
178	C₂²	C₂²×D₄×C₂₇₁	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	2	2	no	—
181	C₄	—	2	2	no	—
182	C₄	—	4	4	no	—
183	C₄	C₄×D₄×C₂₇₁	1	2	yes	—
184	C₄	C₂³×C₄×C₂₇₁	1	1	yes	characteristic
185	C₄	C₂×#41×C₂₇₁	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	4	4	no	—
188	C₂	C₂×C₄×D₄×C₂₇₁	1	1	yes	characteristic
189	C₂	#576×C₂₇₁	1	1	yes	characteristic
190	C₂	#578×C₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₇₁	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₇₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₇₁
prime radical	C₂×#135×C₂₇₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇₁ ◁ #117×C₂₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #520×C₂₇₁	C₂	semidirect
3. #440×C₂₇₁	C₂	semidirect
4. C₄×Q₁₆×C₂₇₁	C₂	semidirect
5. C₄×D₈×C₂₇₁	C₂	semidirect
6. C₂×#107×C₂₇₁	C₂	semidirect
7. #417×C₂₇₁	C₂	semidirect
8. #405×C₂₇₁	C₂	semidirect
10. #135×C₂₇₁	C₄	semidirect
30. #41×C₂₇₁	C₂×C₄	semidirect
38. Q₁₆×C₂₇₁	C₂×C₄	semidirect
39. SD₁₆×C₂₇₁	C₂×C₄	semidirect
40. D₈×C₂₇₁	C₂×C₄	semidirect
42. C₂×C₈×C₂₇₁	C₂×C₄	semidirect
95. C₂₇₁	#2951	direct (N × H)
96. #2951	C₂₇₁	direct (N × H)
98. #520	C₂×C₂₇₁	semidirect
99. #440	C₂×C₂₇₁	semidirect
100. C₄×Q₁₆	C₂×C₂₇₁	semidirect
101. C₄×D₈	C₂×C₂₇₁	semidirect
102. C₂×#107	C₂×C₂₇₁	semidirect
103. #417	C₂×C₂₇₁	semidirect
104. #405	C₂×C₂₇₁	semidirect
106. #135	C₄×C₂₇₁	semidirect
126. #41	C₂×C₄×C₂₇₁	semidirect
134. Q₁₆	C₂×C₄×C₂₇₁	semidirect
135. SD₁₆	C₂×C₄×C₂₇₁	semidirect
136. D₈	C₂×C₄×C₂₇₁	semidirect
138. C₂×C₈	C₂×C₄×C₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#135×C₂₇₁	2
#520×C₂₇₁	2
#440×C₂₇₁	2
C₄×Q₁₆×C₂₇₁	2
C₄×D₈×C₂₇₁	2
C₂×#107×C₂₇₁	2
#417×C₂₇₁	2
#405×C₂₇₁	2
#2951	271

Subgroups — #2951×C271 #46833537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2951×C₂₇₁ #46833537