Subgroups — C₂×#444×C₁₃₁ #30663137

Order 16768.

464 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#135×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#131×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#444×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#104×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#41×C₁₃₁	C₂²	1	2	yes	—
8	#135×C₁₃₁	C₄	1	4	yes	complemented
9	#135×C₁₃₁	C₂²	1	4	yes	—
10	C₂×Q₁₆×C₁₃₁	C₄	1	1	yes	characteristic, complemented
11	C₂×SD₁₆×C₁₃₁	C₄	1	2	yes	complemented
12	C₂×D₈×C₁₃₁	C₄	1	1	yes	characteristic, complemented
13	#131×C₁₃₁	—	2	4	no	—
14	#131×C₁₃₁	C₂²	1	8	yes	complemented
15	C₂×M₁₆×C₁₃₁	—	2	2	no	—
16	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	—
17	C₂²×C₈×C₁₃₁	—	2	2	no	—
18	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
21	#108×C₁₃₁	C₂²	1	4	yes	complemented
22	#104×C₁₃₁	C₂²	1	8	yes	complemented
23	#97×C₁₃₁	C₂²	1	4	yes	complemented
24	#41×C₁₃₁	—	2	2	no	—
25	#41×C₁₃₁	—	2	2	no	—
26	#41×C₁₃₁	C₂×C₄	1	4	yes	—
27	#41×C₁₃₁	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×D₄×C₁₃₁	—	2	2	no	—
30	C₂×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₁₃₁	—	2	2	no	—
32	C₂²×C₄×C₁₃₁	—	2	1	no	—
33	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
34	Q₁₆×C₁₃₁	C₂×C₄	1	4	yes	complemented
35	SD₁₆×C₁₃₁	C₂×C₄	1	8	yes	complemented
36	D₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
37	M₁₆×C₁₃₁	—	2	2	no	—
38	M₁₆×C₁₃₁	—	2	8	no	—
39	M₁₆×C₁₃₁	C₂³	1	4	yes	—
40	C₂×C₈×C₁₃₁	—	2	2	no	—
41	C₂×C₈×C₁₃₁	—	2	8	no	—
42	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
46	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
47	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
48	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
49	C₄²×C₁₃₁	—	2	1	no	—
50	C₄²×C₁₃₁	C₂³	1	2	yes	—
51	C₂³×C₁₃₁	—	2	2	no	—
52	C₂³×C₁₃₁	#41	1	1	yes	characteristic
53	Q₈×C₁₃₁	—	2	2	no	—
54	Q₈×C₁₃₁	C₂²×C₄	1	4	yes	—
55	D₄×C₁₃₁	—	2	4	no	—
56	D₄×C₁₃₁	—	2	4	no	—
57	D₄×C₁₃₁	—	2	2	no	—
58	D₄×C₁₃₁	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₁₃₁	—	2	2	no	—
60	C₂×C₄×C₁₃₁	—	2	4	no	—
61	C₂×C₄×C₁₃₁	—	2	4	no	—
62	C₂×C₄×C₁₃₁	—	2	2	no	—
63	C₂×C₄×C₁₃₁	—	4	2	no	—
64	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
69	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
70	C₈×C₁₃₁	—	2	4	no	—
71	C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
72	C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
73	C₈×C₁₃₁	C₂×D₄	1	4	yes	—
74	C₂²×C₁₃₁	—	2	2	no	—
75	C₂²×C₁₃₁	—	2	4	no	—
76	C₂²×C₁₃₁	—	4	4	no	—
77	C₂²×C₁₃₁	C₄×D₄	1	1	yes	characteristic
78	C₂²×C₁₃₁	C₄×D₄	1	1	yes	characteristic
79	C₂²×C₁₃₁	C₂×#41	1	1	yes	characteristic
80	C₄×C₁₃₁	—	2	4	no	—
81	C₄×C₁₃₁	—	4	2	no	—
82	C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
83	C₄×C₁₃₁	C₄×D₄	1	1	yes	characteristic
84	C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₁₃₁	—	2	1	no	—
87	C₂×C₁₃₁	—	2	1	no	—
88	C₂×C₁₃₁	—	4	4	no	—
89	C₂×C₁₃₁	#444	1	2	yes	direct product factor
90	C₂×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
91	C₁₃₁	C₂×#444	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×#444	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂×#135	C₂×C₁₃₁	1	1	yes	characteristic
94	C₂×#131	C₂×C₁₃₁	1	2	yes	complemented
95	#444	C₂×C₁₃₁	1	8	yes	direct product factor
96	C₂×#108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
97	C₂×#104	C₂×C₁₃₁	1	2	yes	complemented
98	C₂×#97	C₂×C₁₃₁	1	1	yes	characteristic, complemented
99	C₂×#41	C₂²×C₁₃₁	1	2	yes	—
100	#135	C₄×C₁₃₁	1	4	yes	complemented
101	#135	C₂²×C₁₃₁	1	4	yes	—
102	C₂×Q₁₆	C₄×C₁₃₁	1	1	yes	characteristic, complemented
103	C₂×SD₁₆	C₄×C₁₃₁	1	2	yes	complemented
104	C₂×D₈	C₄×C₁₃₁	1	1	yes	characteristic, complemented
105	#131	—	2	4	no	—
106	#131	C₂²×C₁₃₁	1	8	yes	complemented
107	C₂×M₁₆	—	2	2	no	—
108	C₂×M₁₆	C₂²×C₁₃₁	1	2	yes	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
111	C₂²×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
112	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
113	#108	C₂²×C₁₃₁	1	4	yes	complemented
114	#104	C₂²×C₁₃₁	1	8	yes	complemented
115	#97	C₂²×C₁₃₁	1	4	yes	complemented
116	#41	—	2	2	no	—
117	#41	—	2	2	no	—
118	#41	C₂×C₄×C₁₃₁	1	4	yes	—
119	#41	C₂³×C₁₃₁	1	4	yes	—
120	C₂×Q₈	C₂×C₄×C₁₃₁	1	2	yes	—
121	C₂×D₄	—	2	2	no	—
122	C₂×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
123	C₂²×C₄	—	2	2	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
126	Q₁₆	C₂×C₄×C₁₃₁	1	4	yes	complemented
127	SD₁₆	C₂×C₄×C₁₃₁	1	8	yes	complemented
128	D₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
129	M₁₆	—	2	2	no	—
130	M₁₆	—	2	8	no	—
131	M₁₆	C₂³×C₁₃₁	1	4	yes	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	8	no	—
134	C₂×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
136	C₂×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
137	C₂×C₈	D₄×C₁₃₁	1	2	yes	—
138	C₂×C₈	D₄×C₁₃₁	1	2	yes	—
139	C₂×C₈	C₂³×C₁₃₁	1	2	yes	—
140	C₂×C₈	C₂³×C₁₃₁	1	2	yes	—
141	C₄²	—	2	1	no	—
142	C₄²	C₂³×C₁₃₁	1	2	yes	—
143	C₂³	—	2	2	no	—
144	C₂³	#41×C₁₃₁	1	1	yes	characteristic
145	Q₈	—	2	2	no	—
146	Q₈	C₂²×C₄×C₁₃₁	1	4	yes	—
147	D₄	—	2	4	no	—
148	D₄	—	2	4	no	—
149	D₄	—	2	2	no	—
150	D₄	C₂²×C₄×C₁₃₁	1	4	yes	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	4	2	no	—
156	C₂×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
157	C₂×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
158	C₂×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
159	C₂×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
160	C₂×C₄	#41×C₁₃₁	1	1	yes	characteristic
161	C₂×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
162	C₈	—	2	4	no	—
163	C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
164	C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
165	C₈	C₂×D₄×C₁₃₁	1	4	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	—	2	4	no	—
168	C₂²	—	4	4	no	—
169	C₂²	C₄×D₄×C₁₃₁	1	1	yes	characteristic
170	C₂²	C₄×D₄×C₁₃₁	1	1	yes	characteristic
171	C₂²	C₂×#41×C₁₃₁	1	1	yes	characteristic
172	C₄	—	2	4	no	—
173	C₄	—	4	2	no	—
174	C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
175	C₄	C₄×D₄×C₁₃₁	1	1	yes	characteristic
176	C₄	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
177	C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	—	4	4	no	—
181	C₂	#444×C₁₃₁	1	2	yes	direct product factor
182	C₂	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂×D₄
commutator	C₄	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₄×D₄
prime radical	C₂×#135×C₁₃₁	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂²×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#131×C₁₃₁	C₂	semidirect
3. #444×C₁₃₁	C₂	direct (N × H)
4. C₂×#108×C₁₃₁	C₂	semidirect
5. C₂×#104×C₁₃₁	C₂	semidirect
6. C₂×#97×C₁₃₁	C₂	semidirect
8. #135×C₁₃₁	C₄	semidirect
10. C₂×Q₁₆×C₁₃₁	C₄	semidirect
11. C₂×SD₁₆×C₁₃₁	C₄	semidirect
12. C₂×D₈×C₁₃₁	C₄	semidirect
14. #131×C₁₃₁	C₂²	semidirect
21. #108×C₁₃₁	C₂²	semidirect
22. #104×C₁₃₁	C₂²	semidirect
23. #97×C₁₃₁	C₂²	semidirect
34. Q₁₆×C₁₃₁	C₂×C₄	semidirect
35. SD₁₆×C₁₃₁	C₂×C₄	semidirect
36. D₈×C₁₃₁	C₂×C₄	semidirect
89. C₂×C₁₃₁	#444	direct (N × H)
91. C₁₃₁	C₂×#444	direct (N × H)
92. C₂×#444	C₁₃₁	direct (N × H)
94. C₂×#131	C₂×C₁₃₁	semidirect
95. #444	C₂×C₁₃₁	direct (N × H)
96. C₂×#108	C₂×C₁₃₁	semidirect
97. C₂×#104	C₂×C₁₃₁	semidirect
98. C₂×#97	C₂×C₁₃₁	semidirect
100. #135	C₄×C₁₃₁	semidirect
102. C₂×Q₁₆	C₄×C₁₃₁	semidirect
103. C₂×SD₁₆	C₄×C₁₃₁	semidirect
104. C₂×D₈	C₄×C₁₃₁	semidirect
106. #131	C₂²×C₁₃₁	semidirect
113. #108	C₂²×C₁₃₁	semidirect
114. #104	C₂²×C₁₃₁	semidirect
115. #97	C₂²×C₁₃₁	semidirect
126. Q₁₆	C₂×C₄×C₁₃₁	semidirect
127. SD₁₆	C₂×C₄×C₁₃₁	semidirect
128. D₈	C₂×C₄×C₁₃₁	semidirect
181. C₂	#444×C₁₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#135×C₁₃₁	2
C₂×#131×C₁₃₁	2
#444×C₁₃₁	2
C₂×#108×C₁₃₁	2
C₂×#104×C₁₃₁	2
C₂×#97×C₁₃₁	2
C₂×#444	131

Subgroups — C2×#444×C131 #30663137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#444×C₁₃₁ #30663137