Subgroups — C₂²×#137×C₂₇₁ #46834165

Order 34688.

1464 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₈×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#137×C₂₇₁	C₂	1	24	yes	maximal, direct product factor
4	C₂²×Q₁₆×C₂₇₁	C₂	1	2	yes	maximal, complemented
5	C₂²×SD₁₆×C₂₇₁	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₂₇₁	—	2	6	no	—
8	C₂×#41×C₂₇₁	C₂²	1	12	yes	—
9	C₂²×Q₈×C₂₇₁	—	2	6	no	—
10	C₂²×Q₈×C₂₇₁	—	2	1	no	—
11	C₂²×Q₈×C₂₇₁	C₂²	1	2	yes	—
12	C₂²×Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₂₇₁	C₂²	1	12	yes	complemented
14	C₂²×D₄×C₂₇₁	—	2	1	no	—
15	C₂²×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₇₁	—	2	1	no	—
17	C₂³×C₄×C₂₇₁	—	2	1	no	—
18	C₂³×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
19	#137×C₂₇₁	C₂²	1	64	yes	direct product factor
20	C₂×Q₁₆×C₂₇₁	C₂²	1	24	yes	complemented
21	C₂×SD₁₆×C₂₇₁	C₂²	1	24	yes	complemented
22	C₂×M₁₆×C₂₇₁	C₂²	1	12	yes	complemented
23	C₂²×C₈×C₂₇₁	C₂²	1	2	yes	—
24	C₂⁴×C₂₇₁	—	2	1	no	—
25	C₂⁴×C₂₇₁	D₄	1	1	yes	characteristic
26	#41×C₂₇₁	—	2	24	no	—
27	#41×C₂₇₁	C₂³	1	16	yes	—
28	C₂×Q₈×C₂₇₁	—	2	6	no	—
29	C₂×Q₈×C₂₇₁	—	2	12	no	—
30	C₂×Q₈×C₂₇₁	—	2	24	no	—
31	C₂×Q₈×C₂₇₁	—	2	3	no	—
32	C₂×Q₈×C₂₇₁	C₂³	1	12	yes	—
33	C₂×Q₈×C₂₇₁	C₂³	1	6	yes	—
34	C₂×Q₈×C₂₇₁	C₂³	1	16	yes	complemented
35	C₂×D₄×C₂₇₁	—	2	12	no	—
36	C₂×D₄×C₂₇₁	—	2	3	no	—
37	C₂×D₄×C₂₇₁	C₂³	1	6	yes	—
38	C₂²×C₄×C₂₇₁	—	2	12	no	—
39	C₂²×C₄×C₂₇₁	—	2	1	no	—
40	C₂²×C₄×C₂₇₁	—	2	2	no	—
41	C₂²×C₄×C₂₇₁	—	2	12	no	—
42	C₂²×C₄×C₂₇₁	D₄	1	6	yes	—
43	C₂²×C₄×C₂₇₁	D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₇₁	C₂³	1	6	yes	—
45	C₂²×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
46	Q₁₆×C₂₇₁	C₂³	1	32	yes	complemented
47	SD₁₆×C₂₇₁	C₂³	1	32	yes	complemented
48	M₁₆×C₂₇₁	C₂³	1	16	yes	complemented
49	C₂×C₈×C₂₇₁	C₂³	1	12	yes	—
50	C₂³×C₂₇₁	—	2	4	no	—
51	C₂³×C₂₇₁	—	2	6	no	—
52	C₂³×C₂₇₁	—	4	4	no	—
53	C₂³×C₂₇₁	C₂×D₄	1	6	yes	—
54	C₂³×C₂₇₁	C₂×D₄	1	1	yes	characteristic
55	Q₈×C₂₇₁	—	2	12	no	—
56	Q₈×C₂₇₁	—	2	16	no	—
57	Q₈×C₂₇₁	—	2	6	no	—
58	Q₈×C₂₇₁	C₂⁴	1	8	yes	—
59	Q₈×C₂₇₁	C₂⁴	1	4	yes	—
60	D₄×C₂₇₁	—	2	16	no	—
61	D₄×C₂₇₁	—	2	6	no	—
62	D₄×C₂₇₁	C₂⁴	1	4	yes	—
63	C₂×C₄×C₂₇₁	—	2	6	no	—
64	C₂×C₄×C₂₇₁	—	2	12	no	—
65	C₂×C₄×C₂₇₁	—	2	16	no	—
66	C₂×C₄×C₂₇₁	—	2	16	no	—
67	C₂×C₄×C₂₇₁	C₂×D₄	1	6	yes	—
68	C₂×C₄×C₂₇₁	C₂×D₄	1	3	yes	—
69	C₂×C₄×C₂₇₁	C₂×D₄	1	12	yes	—
70	C₂×C₄×C₂₇₁	C₂⁴	1	3	yes	—
71	C₂×C₄×C₂₇₁	C₂⁴	1	4	yes	—
72	C₈×C₂₇₁	C₂⁴	1	8	yes	—
73	C₂²×C₂₇₁	—	2	12	no	—
74	C₂²×C₂₇₁	—	2	4	no	—
75	C₂²×C₂₇₁	—	4	12	no	—
76	C₂²×C₂₇₁	#137	1	4	yes	direct product factor
77	C₂²×C₂₇₁	C₂²×D₄	1	4	yes	—
78	C₂²×C₂₇₁	C₂²×D₄	1	3	yes	—
79	C₄×C₂₇₁	—	2	4	no	—
80	C₄×C₂₇₁	—	2	8	no	—
81	C₄×C₂₇₁	C₂²×D₄	1	4	yes	—
82	C₄×C₂₇₁	C₂²×D₄	1	3	yes	—
83	C₄×C₂₇₁	C₂⁵	1	1	yes	characteristic
84	C₂×C₂₇₁	—	2	4	no	—
85	C₂×C₂₇₁	—	4	4	no	—
86	C₂×C₂₇₁	C₂×#137	1	6	yes	direct product factor
87	C₂×C₂₇₁	C₂³×D₄	1	1	yes	characteristic
88	C₂₇₁	C₂²×#137	1	1	yes	characteristic, direct product factor, Sylow
89	C₂²×#137	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×#41	C₂×C₂₇₁	1	1	yes	characteristic
91	C₂³×Q₈	C₂×C₂₇₁	1	1	yes	characteristic, complemented
92	C₂×#137	C₂×C₂₇₁	1	24	yes	direct product factor
93	C₂²×Q₁₆	C₂×C₂₇₁	1	2	yes	complemented
94	C₂²×SD₁₆	C₂×C₂₇₁	1	2	yes	complemented
95	C₂²×M₁₆	C₂×C₂₇₁	1	1	yes	characteristic, complemented
96	C₂×#41	—	2	6	no	—
97	C₂×#41	C₂²×C₂₇₁	1	12	yes	—
98	C₂²×Q₈	—	2	1	no	—
99	C₂²×Q₈	—	2	6	no	—
100	C₂²×Q₈	C₂²×C₂₇₁	1	2	yes	—
101	C₂²×Q₈	C₂²×C₂₇₁	1	1	yes	characteristic
102	C₂²×Q₈	C₂²×C₂₇₁	1	12	yes	complemented
103	C₂²×D₄	—	2	1	no	—
104	C₂²×D₄	C₂²×C₂₇₁	1	1	yes	characteristic
105	C₂³×C₄	—	2	1	no	—
106	C₂³×C₄	—	2	1	no	—
107	C₂³×C₄	C₂²×C₂₇₁	1	1	yes	characteristic
108	#137	C₂²×C₂₇₁	1	64	yes	direct product factor
109	C₂×Q₁₆	C₂²×C₂₇₁	1	24	yes	complemented
110	C₂×SD₁₆	C₂²×C₂₇₁	1	24	yes	complemented
111	C₂×M₁₆	C₂²×C₂₇₁	1	12	yes	complemented
112	C₂²×C₈	C₂²×C₂₇₁	1	2	yes	—
113	C₂⁴	—	2	1	no	—
114	C₂⁴	D₄×C₂₇₁	1	1	yes	characteristic
115	#41	—	2	24	no	—
116	#41	C₂³×C₂₇₁	1	16	yes	—
117	C₂×Q₈	—	2	6	no	—
118	C₂×Q₈	—	2	12	no	—
119	C₂×Q₈	—	2	24	no	—
120	C₂×Q₈	—	2	3	no	—
121	C₂×Q₈	C₂³×C₂₇₁	1	12	yes	—
122	C₂×Q₈	C₂³×C₂₇₁	1	6	yes	—
123	C₂×Q₈	C₂³×C₂₇₁	1	16	yes	complemented
124	C₂×D₄	—	2	12	no	—
125	C₂×D₄	—	2	3	no	—
126	C₂×D₄	C₂³×C₂₇₁	1	6	yes	—
127	C₂²×C₄	—	2	1	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	12	no	—
130	C₂²×C₄	—	2	12	no	—
131	C₂²×C₄	D₄×C₂₇₁	1	1	yes	characteristic
132	C₂²×C₄	D₄×C₂₇₁	1	6	yes	—
133	C₂²×C₄	C₂³×C₂₇₁	1	1	yes	characteristic
134	C₂²×C₄	C₂³×C₂₇₁	1	6	yes	—
135	Q₁₆	C₂³×C₂₇₁	1	32	yes	complemented
136	SD₁₆	C₂³×C₂₇₁	1	32	yes	complemented
137	M₁₆	C₂³×C₂₇₁	1	16	yes	complemented
138	C₂×C₈	C₂³×C₂₇₁	1	12	yes	—
139	C₂³	—	2	4	no	—
140	C₂³	—	2	6	no	—
141	C₂³	—	4	4	no	—
142	C₂³	C₂×D₄×C₂₇₁	1	6	yes	—
143	C₂³	C₂×D₄×C₂₇₁	1	1	yes	characteristic
144	Q₈	—	2	12	no	—
145	Q₈	—	2	16	no	—
146	Q₈	—	2	6	no	—
147	Q₈	C₂⁴×C₂₇₁	1	8	yes	—
148	Q₈	C₂⁴×C₂₇₁	1	4	yes	—
149	D₄	—	2	16	no	—
150	D₄	—	2	6	no	—
151	D₄	C₂⁴×C₂₇₁	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	2	16	no	—
155	C₂×C₄	—	2	16	no	—
156	C₂×C₄	C₂×D₄×C₂₇₁	1	6	yes	—
157	C₂×C₄	C₂×D₄×C₂₇₁	1	3	yes	—
158	C₂×C₄	C₂×D₄×C₂₇₁	1	12	yes	—
159	C₂×C₄	C₂⁴×C₂₇₁	1	3	yes	—
160	C₂×C₄	C₂⁴×C₂₇₁	1	4	yes	—
161	C₈	C₂⁴×C₂₇₁	1	8	yes	—
162	C₂²	—	2	12	no	—
163	C₂²	—	2	4	no	—
164	C₂²	—	4	12	no	—
165	C₂²	#137×C₂₇₁	1	4	yes	direct product factor
166	C₂²	C₂²×D₄×C₂₇₁	1	4	yes	—
167	C₂²	C₂²×D₄×C₂₇₁	1	3	yes	—
168	C₄	—	2	4	no	—
169	C₄	—	2	8	no	—
170	C₄	C₂²×D₄×C₂₇₁	1	4	yes	—
171	C₄	C₂²×D₄×C₂₇₁	1	3	yes	—
172	C₄	C₂⁵×C₂₇₁	1	1	yes	characteristic
173	C₂	—	2	4	no	—
174	C₂	—	4	4	no	—
175	C₂	C₂×#137×C₂₇₁	1	6	yes	direct product factor
176	C₂	C₂³×D₄×C₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₇₁	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₂₇₁
prime radical	C₂²×#41×C₂₇₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₇₁ ◁ C₂³×C₄×C₂₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₈×C₂₇₁	C₂	semidirect
3. C₂×#137×C₂₇₁	C₂	direct (N × H)
4. C₂²×Q₁₆×C₂₇₁	C₂	semidirect
5. C₂²×SD₁₆×C₂₇₁	C₂	semidirect
6. C₂²×M₁₆×C₂₇₁	C₂	semidirect
13. C₂²×Q₈×C₂₇₁	C₂²	semidirect
19. #137×C₂₇₁	C₂²	direct (N × H)
20. C₂×Q₁₆×C₂₇₁	C₂²	semidirect
21. C₂×SD₁₆×C₂₇₁	C₂²	semidirect
22. C₂×M₁₆×C₂₇₁	C₂²	semidirect
34. C₂×Q₈×C₂₇₁	C₂³	semidirect
46. Q₁₆×C₂₇₁	C₂³	semidirect
47. SD₁₆×C₂₇₁	C₂³	semidirect
48. M₁₆×C₂₇₁	C₂³	semidirect
76. C₂²×C₂₇₁	#137	direct (N × H)
86. C₂×C₂₇₁	C₂×#137	direct (N × H)
88. C₂₇₁	C₂²×#137	direct (N × H)
89. C₂²×#137	C₂₇₁	direct (N × H)
91. C₂³×Q₈	C₂×C₂₇₁	semidirect
92. C₂×#137	C₂×C₂₇₁	direct (N × H)
93. C₂²×Q₁₆	C₂×C₂₇₁	semidirect
94. C₂²×SD₁₆	C₂×C₂₇₁	semidirect
95. C₂²×M₁₆	C₂×C₂₇₁	semidirect
102. C₂²×Q₈	C₂²×C₂₇₁	semidirect
108. #137	C₂²×C₂₇₁	direct (N × H)
109. C₂×Q₁₆	C₂²×C₂₇₁	semidirect
110. C₂×SD₁₆	C₂²×C₂₇₁	semidirect
111. C₂×M₁₆	C₂²×C₂₇₁	semidirect
123. C₂×Q₈	C₂³×C₂₇₁	semidirect
135. Q₁₆	C₂³×C₂₇₁	semidirect
136. SD₁₆	C₂³×C₂₇₁	semidirect
137. M₁₆	C₂³×C₂₇₁	semidirect
165. C₂²	#137×C₂₇₁	direct (N × H)
175. C₂	C₂×#137×C₂₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41×C₂₇₁	2
C₂³×Q₈×C₂₇₁	2
C₂×#137×C₂₇₁	2
C₂²×Q₁₆×C₂₇₁	2
C₂²×SD₁₆×C₂₇₁	2
C₂²×M₁₆×C₂₇₁	2
C₂²×#137	271

Subgroups — C22×#137×C271 #46834165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#137×C₂₇₁ #46834165