Subgroups — #30680674

Order 16768.

1424 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554148	C₂	1	1	yes	maximal, characteristic
2	#22554078	C₂	1	6	yes	maximal, complemented
3	#22554077	C₂	1	8	yes	maximal, complemented
4	C₂×#16862254	C₂	1	6	yes	maximal, complemented
5	C₂×#16862253	C₂	1	1	yes	maximal, characteristic, complemented
6	#131×D₁₃₁	C₂	1	8	yes	maximal, complemented
7	#568×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#16862294	C₄	1	2	yes	—
9	#16862293	C₄	1	4	yes	—
10	#16862292	C₄	1	4	yes	—
11	#16862291	C₄	1	6	yes	—
12	C₂×#14523513	C₂²	1	6	yes	—
13	#41×D₁₃₁	C₂²	1	8	yes	—
14	#16862254	C₂²	1	12	yes	complemented
15	#16862254	C₂²	1	4	yes	complemented
16	#16862254	C₂²	1	12	yes	complemented
17	#16862254	C₂²	1	12	yes	complemented
18	#16862253	C₂²	1	4	yes	complemented
19	#16862253	C₂²	1	4	yes	complemented
20	#16862252	C₂²	1	24	yes	complemented
21	C₂×#14523497	C₂²	1	3	yes	complemented
22	C₂×#14523496	C₂²	1	12	yes	complemented
23	M₁₆×D₁₃₁	C₂²	1	24	yes	complemented
24	C₂²×C₁₃₁⋊C₈	C₂²	1	3	yes	complemented
25	C₂×C₈×D₁₃₁	C₂²	1	12	yes	complemented
26	C₂×#41×C₁₃₁	C₂²	1	1	yes	characteristic
27	#131×C₁₃₁	C₂²	1	8	yes	complemented
28	C₂×M₁₆×C₁₃₁	C₂²	1	6	yes	complemented
29	#14523514	C₂×C₄	1	8	yes	—
30	#14523513	C₂×C₄	1	12	yes	—
31	#14523513	C₂×C₄	1	12	yes	—
32	#14523513	C₂³	1	12	yes	—
33	#14523513	C₂³	1	4	yes	—
34	#14523512	C₂×C₄	1	24	yes	—
35	C₂×Dic₂₆₂	C₂×C₄	1	6	yes	—
36	Q₈×D₁₃₁	C₂×C₄	1	8	yes	—
37	C₂×D₅₂₄	C₂×C₄	1	6	yes	—
38	C₂×C₂²⋊D₁₃₁	C₂×C₄	1	12	yes	—
39	D₄×D₁₃₁	C₂×C₄	1	24	yes	—
40	C₂×C₄×D₁₃₁	C₂³	1	12	yes	—
41	#14523497	C₂³	1	12	yes	—
42	#14523496	C₂³	1	24	yes	—
43	#14523496	C₂³	1	8	yes	—
44	C₂×C₁₃₁⋊C₈	C₂³	1	6	yes	—
45	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₁₃₁⋊C₈	C₂³	1	6	yes	—
47	C₂×C₁₃₁⋊C₈	C₂³	1	3	yes	—
48	C₈×D₁₃₁	C₂³	1	8	yes	—
49	C₈×D₁₃₁	C₂³	1	24	yes	—
50	#41×C₁₃₁	C₂×C₄	1	4	yes	—
51	#41×C₁₃₁	C₂³	1	4	yes	—
52	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×D₄×C₁₃₁	C₂×C₄	1	3	yes	—
54	C₂²×C₄×C₁₃₁	C₂³	1	3	yes	—
55	M₁₆×C₁₃₁	C₂³	1	12	yes	—
56	M₁₆×C₁₃₁	C₂³	1	4	yes	—
57	C₂×C₈×C₁₃₁	C₂³	1	12	yes	—
58	C₂²×D₁₃₁	C₂²×C₄	1	12	yes	—
59	Dic₂₆₂	C₂²×C₄	1	12	yes	—
60	Dic₂₆₂	C₂²×C₄	1	4	yes	—
61	D₅₂₄	C₂²×C₄	1	12	yes	—
62	D₅₂₄	C₂²×C₄	1	4	yes	—
63	C₂²⋊D₁₃₁	C₂²×C₄	1	24	yes	—
64	C₂²⋊D₁₃₁	C₂²×C₄	1	8	yes	—
65	C₂×Dic₁₃₁	C₂²×C₄	1	12	yes	—
66	C₄×D₁₃₁	C₂²×C₄	1	24	yes	—
67	C₄×D₁₃₁	C₂⁴	1	8	yes	—
68	C₁₃₁⋊C₈	C₂⁴	1	1	yes	characteristic
69	C₁₃₁⋊C₈	C₂⁴	1	6	yes	—
70	C₁₃₁⋊C₈	C₂⁴	1	1	yes	characteristic
71	C₂³×C₁₃₁	C₂²×C₄	1	3	yes	—
72	Q₈×C₁₃₁	C₂²×C₄	1	4	yes	—
73	D₄×C₁₃₁	C₂²×C₄	1	12	yes	—
74	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
75	C₂×C₄×C₁₃₁	C₂²×C₄	1	6	yes	—
76	C₂×C₄×C₁₃₁	C₂⁴	1	6	yes	—
77	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
78	C₈×C₁₃₁	C₂⁴	1	8	yes	—
79	C₂²×C₁₃₁	—	4	3	no	—
80	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
81	C₂²×C₁₃₁	C₂³×C₄	1	6	yes	—
82	C₂×D₁₃₁	—	4	12	no	—
83	C₂×D₁₃₁	C₂³×C₄	1	8	yes	—
84	C₄×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₁₃₁	C₂³×C₄	1	6	yes	—
86	C₄×C₁₃₁	C₂⁵	1	1	yes	characteristic
87	Dic₁₃₁	C₂³×C₄	1	8	yes	—
88	C₂×C₁₃₁	—	2	6	no	—
89	C₂×C₁₃₁	—	2	1	no	—
90	C₂×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
91	D₁₃₁	—	2	8	no	—
92	C₁₃₁	#3573	1	1	yes	characteristic, complemented, Sylow
93	#3573	—	131	1	no	maximal, Sylow
94	#585	—	131	1	no	—
95	#568	—	131	8	no	—
96	#568	—	131	6	no	—
97	#568	D₁₃₁	1	1	yes	characteristic, complemented
98	C₂×#131	—	131	1	no	—
99	C₂×#131	—	131	8	no	—
100	C₂×#131	—	131	6	no	—
101	2₋¹⁺⁴	—	131	4	no	—
102	2₋¹⁺⁴	—	131	2	no	—
103	H(5,𝔽₂)	—	131	4	no	—
104	H(5,𝔽₂)	—	131	6	no	—
105	C₂×#41	—	131	8	no	—
106	C₂×#41	—	131	6	no	—
107	C₂×#41	C₂×D₁₃₁	1	1	yes	characteristic
108	#131	—	131	4	no	—
109	#131	—	131	4	no	—
110	#131	—	131	12	no	—
111	#131	—	131	4	no	—
112	#131	—	131	12	no	—
113	#131	—	131	12	no	—
114	#131	—	131	24	no	—
115	#131	C₂×D₁₃₁	1	8	yes	complemented
116	C₂×M₁₆	—	131	3	no	—
117	C₂×M₁₆	—	131	24	no	—
118	C₂×M₁₆	—	131	12	no	—
119	C₂×M₁₆	C₂×D₁₃₁	1	6	yes	complemented
120	C₂²×C₈	—	131	3	no	—
121	C₂²×C₈	—	131	12	no	—
122	#41	—	131	8	no	—
123	#41	—	131	24	no	—
124	#41	—	131	12	no	—
125	#41	—	131	12	no	—
126	#41	—	131	4	no	—
127	#41	—	131	12	no	—
128	#41	C₄×D₁₃₁	1	4	yes	—
129	#41	C₂²×D₁₃₁	1	4	yes	—
130	C₂×Q₈	—	131	8	no	—
131	C₂×Q₈	—	131	6	no	—
132	C₂×Q₈	C₄×D₁₃₁	1	1	yes	characteristic
133	C₂×D₄	—	131	12	no	—
134	C₂×D₄	—	131	24	no	—
135	C₂×D₄	—	131	6	no	—
136	C₂×D₄	C₄×D₁₃₁	1	3	yes	—
137	C₂²×C₄	—	131	12	no	—
138	C₂²×C₄	C₂²×D₁₃₁	1	3	yes	—
139	M₁₆	—	131	12	no	—
140	M₁₆	—	131	24	no	—
141	M₁₆	—	131	8	no	—
142	M₁₆	C₂²×D₁₃₁	1	12	yes	—
143	M₁₆	C₂²×D₁₃₁	1	4	yes	—
144	C₂×C₈	—	131	1	no	—
145	C₂×C₈	—	131	6	no	—
146	C₂×C₈	—	131	3	no	—
147	C₂×C₈	—	131	6	no	—
148	C₂×C₈	—	131	8	no	—
149	C₂×C₈	—	131	24	no	—
150	C₂×C₈	C₂²×D₁₃₁	1	12	yes	—
151	C₂³	—	131	12	no	—
152	C₂³	C₂×C₄×D₁₃₁	1	3	yes	—
153	Q₈	—	131	12	no	—
154	Q₈	—	131	4	no	—
155	Q₈	C₂×C₄×D₁₃₁	1	4	yes	—
156	D₄	—	131	24	no	—
157	D₄	—	131	8	no	—
158	D₄	—	131	12	no	—
159	D₄	—	131	4	no	—
160	D₄	C₂×C₄×D₁₃₁	1	12	yes	—
161	C₂×C₄	—	131	12	no	—
162	C₂×C₄	—	131	24	no	—
163	C₂×C₄	—	131	8	no	—
164	C₂×C₄	C₂×C₄×D₁₃₁	1	6	yes	—
165	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
166	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
167	C₂×C₄	C₂³×D₁₃₁	1	6	yes	—
168	C₈	—	131	1	no	—
169	C₈	—	131	1	no	—
170	C₈	—	131	6	no	—
171	C₈	C₂³×D₁₃₁	1	8	yes	—
172	C₂²	—	4	3	no	—
173	C₂²	—	524	12	no	—
174	C₂²	—	131	8	no	—
175	C₂²	C₂²×C₄×D₁₃₁	1	6	yes	—
176	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
177	C₄	—	131	8	no	—
178	C₄	C₂²×C₄×D₁₃₁	1	6	yes	—
179	C₄	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
180	C₄	C₂⁴×D₁₃₁	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	6	no	—
183	C₂	—	262	8	no	—
184	C₂	C₂³×C₄×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₁₃₁
commutator, socle	C₂×C₁₃₁	C₂⁴×C₄
Fitting	#568×C₁₃₁	C₂
prime radical	#22554148	C₂
cosocle	C₄×C₁₃₁	C₂⁵
hypercenter	#568	D₁₃₁
hypocenter	C₁₃₁	#3573
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₄ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #22554078	C₂	semidirect
3. #22554077	C₂	semidirect
4. C₂×#16862254	C₂	semidirect
5. C₂×#16862253	C₂	semidirect
6. #131×D₁₃₁	C₂	semidirect
7. #568×C₁₃₁	C₂	semidirect
14. #16862254	C₂²	semidirect
15. #16862254	C₂²	semidirect
16. #16862254	C₂²	semidirect
17. #16862254	C₂²	semidirect
18. #16862253	C₂²	semidirect
19. #16862253	C₂²	semidirect
20. #16862252	C₂²	semidirect
21. C₂×#14523497	C₂²	semidirect
22. C₂×#14523496	C₂²	semidirect
23. M₁₆×D₁₃₁	C₂²	semidirect
24. C₂²×C₁₃₁⋊C₈	C₂²	semidirect
25. C₂×C₈×D₁₃₁	C₂²	semidirect
27. #131×C₁₃₁	C₂²	semidirect
28. C₂×M₁₆×C₁₃₁	C₂²	semidirect
92. C₁₃₁	#3573	semidirect
97. #568	D₁₃₁	semidirect
115. #131	C₂×D₁₃₁	semidirect
119. C₂×M₁₆	C₂×D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22554148	2
#22554078	2
#22554077	2
C₂×#16862254	2
C₂×#16862253	2
#131×D₁₃₁	2
#568×C₁₃₁	2
#3573	131