Subgroups — #31815322

Order 17792.

452 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984864	C₂	1	1	yes	maximal, characteristic
2	#22581719	C₂	1	2	yes	maximal, complemented
3	#22581076	C₂	1	4	yes	maximal, complemented
4	#22581071	C₂	1	4	yes	maximal, complemented
5	C₂×#17984778	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580971	C₂	1	2	yes	maximal, complemented
7	#405×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14555785	C₂²	1	2	yes	—
9	#17984864	C₄	1	2	yes	complemented
10	#17984864	C₄	1	2	yes	complemented
11	#17984864	C₂²	1	2	yes	—
12	#17984864	C₂²	1	2	yes	—
13	C₂×Dic₅₅₆	C₄	1	1	yes	characteristic, complemented
14	C₂×#14555774	C₄	1	2	yes	complemented
15	C₂×D₁₁₁₂	C₄	1	1	yes	characteristic, complemented
16	#17984850	—	2	4	no	—
17	#17984848	C₂²	1	8	yes	complemented
18	C₂×#14555769	C₂²	1	2	yes	—
19	M₁₆×D₁₃₉	—	2	4	no	—
20	#17984778	C₂²	1	2	yes	complemented
21	#17984778	C₂²	1	2	yes	complemented
22	#17984770	C₂²	1	8	yes	complemented
23	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
25	#117×C₁₃₉	C₂²	1	1	yes	characteristic
26	#97×C₁₃₉	C₂²	1	2	yes	complemented
27	C₄×C₈×C₁₃₉	C₂²	1	2	yes	complemented
28	#14555785	—	2	2	no	—
29	#14555785	—	2	2	no	—
30	#14555785	C₂×C₄	1	4	yes	—
31	#14555785	C₂³	1	4	yes	—
32	C₂×Dic₂₇₈	C₂×C₄	1	2	yes	—
33	C₂×D₅₅₆	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊D₁₃₉	—	2	2	no	—
35	C₂×C₄×D₁₃₉	—	2	2	no	—
36	Dic₅₅₆	C₂×C₄	1	2	yes	—
37	Dic₅₅₆	C₂×C₄	1	2	yes	—
38	#14555774	C₂×C₄	1	4	yes	—
39	#14555774	C₂×C₄	1	4	yes	—
40	D₁₁₁₂	C₂×C₄	1	2	yes	—
41	D₁₁₁₂	C₂×C₄	1	2	yes	—
42	#14555769	—	2	2	no	—
43	#14555769	C₂³	1	4	yes	—
44	#14555768	—	2	8	no	—
45	C₂×C₁₃₉⋊C₈	—	2	2	no	—
46	C₈×D₁₃₉	—	2	8	no	—
47	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
48	M₁₆×C₁₃₉	D₄	1	2	yes	—
49	M₁₆×C₁₃₉	D₄	1	2	yes	—
50	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
57	C₄⋊C₄×C₁₃₉	—	2	1	no	—
58	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
59	C₄²×C₁₃₉	C₂³	1	2	yes	—
60	C₂²×D₁₃₉	—	2	2	no	—
61	Dic₂₇₈	—	2	2	no	—
62	Dic₂₇₈	C₂²×C₄	1	4	yes	—
63	D₅₅₆	—	2	2	no	—
64	D₅₅₆	C₂²×C₄	1	4	yes	—
65	C₂²⋊D₁₃₉	—	2	4	no	—
66	C₂²⋊D₁₃₉	—	2	4	no	—
67	C₂×Dic₁₃₉	—	2	2	no	—
68	C₄×D₁₃₉	—	2	4	no	—
69	C₄×D₁₃₉	—	2	4	no	—
70	C₁₃₉⋊C₈	—	2	4	no	—
71	C₂³×C₁₃₉	#41	1	1	yes	characteristic
72	C₂×C₄×C₁₃₉	—	2	2	no	—
73	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
78	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
79	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
80	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
81	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
82	C₈×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
83	C₈×C₁₃₉	C₂×D₄	1	4	yes	—
84	C₂²×C₁₃₉	—	4	1	no	—
85	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
86	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
87	C₂²×C₁₃₉	C₂×#41	1	1	yes	characteristic
88	C₂×D₁₃₉	—	2	4	no	—
89	C₂×D₁₃₉	—	8	2	no	—
90	C₄×C₁₃₉	—	4	2	no	—
91	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
92	C₄×C₁₃₉	C₄×D₄	1	1	yes	characteristic
93	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
94	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
95	Dic₁₃₉	—	2	4	no	—
96	C₂×C₁₃₉	—	2	1	no	—
97	C₂×C₁₃₉	—	2	1	no	—
98	C₂×C₁₃₉	—	2	1	no	—
99	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
100	D₁₃₉	—	4	4	no	—
101	C₁₃₉	#2957	1	1	yes	characteristic, complemented, Sylow
102	#2957	—	139	1	no	maximal, Sylow
103	C₂×#135	—	139	1	no	—
104	#568	—	139	2	no	—
105	#444	—	139	4	no	—
106	#443	—	139	4	no	—
107	C₂×#108	—	139	1	no	—
108	#421	—	139	2	no	—
109	#405	D₁₃₉	1	1	yes	characteristic, complemented
110	C₂×#41	—	139	2	no	—
111	#135	—	139	2	no	—
112	#135	—	139	2	no	—
113	#135	—	139	2	no	—
114	#135	—	139	2	no	—
115	C₂×Q₁₆	—	139	1	no	—
116	C₂×SD₁₆	—	139	2	no	—
117	C₂×D₈	—	139	1	no	—
118	#131	—	278	4	no	—
119	#131	—	139	8	no	—
120	C₂×M₁₆	—	278	4	no	—
121	C₂×M₁₆	—	139	2	no	—
122	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
123	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
124	#117	C₂×D₁₃₉	1	1	yes	characteristic
125	#108	—	139	2	no	—
126	#108	—	139	2	no	—
127	#104	—	139	8	no	—
128	#97	C₂×D₁₃₉	1	2	yes	complemented
129	C₄×C₈	C₂×D₁₃₉	1	2	yes	complemented
130	#41	—	278	2	no	—
131	#41	—	278	2	no	—
132	#41	—	139	4	no	—
133	#41	—	139	4	no	—
134	C₂×Q₈	—	139	2	no	—
135	C₂×D₄	—	278	2	no	—
136	C₂×D₄	—	139	2	no	—
137	C₂²×C₄	—	278	2	no	—
138	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
139	Q₁₆	—	139	2	no	—
140	Q₁₆	—	139	2	no	—
141	SD₁₆	—	139	4	no	—
142	SD₁₆	—	139	4	no	—
143	D₈	—	139	2	no	—
144	D₈	—	139	2	no	—
145	M₁₆	—	278	2	no	—
146	M₁₆	—	278	8	no	—
147	M₁₆	—	139	4	no	—
148	M₁₆	D₅₅₆	1	2	yes	—
149	M₁₆	D₅₅₆	1	2	yes	—
150	C₂×C₈	—	278	2	no	—
151	C₂×C₈	—	278	8	no	—
152	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
153	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
154	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
155	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
156	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
157	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
158	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
159	C₄⋊C₄	—	2	1	no	—
160	C₂²⋊C₄	—	2	1	no	—
161	C₄²	C₂²×D₁₃₉	1	2	yes	—
162	C₂³	—	278	2	no	—
163	C₂³	#14555785	1	1	yes	characteristic
164	Q₈	—	278	2	no	—
165	Q₈	—	139	4	no	—
166	D₄	—	278	4	no	—
167	D₄	—	278	4	no	—
168	D₄	—	278	2	no	—
169	D₄	—	139	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	278	2	no	—
172	C₂×C₄	—	278	4	no	—
173	C₂×C₄	—	278	4	no	—
174	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
175	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
176	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
177	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
178	C₂×C₄	#14555785	1	1	yes	characteristic
179	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
180	C₈	—	278	4	no	—
181	C₈	C₂×C₄×D₁₃₉	1	1	yes	characteristic
182	C₈	C₂×C₄×D₁₃₉	1	1	yes	characteristic
183	C₈	C₂×C₄×D₁₃₉	1	1	yes	characteristic
184	C₈	C₂×C₄×D₁₃₉	1	1	yes	characteristic
185	C₈	C₂×D₅₅₆	1	4	yes	—
186	C₂²	—	4	1	no	—
187	C₂²	—	278	4	no	—
188	C₂²	—	1112	2	no	—
189	C₂²	C₄×D₅₅₆	1	1	yes	characteristic
190	C₂²	C₄×D₅₅₆	1	1	yes	characteristic
191	C₂²	C₂×#14555785	1	1	yes	characteristic
192	C₄	—	4	2	no	—
193	C₄	—	278	4	no	—
194	C₄	C₄×D₅₅₆	1	1	yes	characteristic
195	C₄	C₄×D₅₅₆	1	1	yes	characteristic
196	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
197	C₄	C₂²×D₅₅₆	1	1	yes	characteristic
198	C₂	—	2	1	no	—
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	—	556	4	no	—
202	C₂	C₂×C₄×D₅₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₅₅₆
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#405×C₁₃₉	C₂
socle	C₂×C₁₃₉	C₂×C₄×D₄
prime radical	C₂×#17984864	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#405	D₁₃₉
hypocenter	C₁₃₉	#2957
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405

Complements

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

N	H ≅ G/N	type
2. #22581719	C₂	semidirect
3. #22581076	C₂	semidirect
4. #22581071	C₂	semidirect
5. C₂×#17984778	C₂	semidirect
6. #22580971	C₂	semidirect
7. #405×C₁₃₉	C₂	semidirect
9. #17984864	C₄	semidirect
10. #17984864	C₄	semidirect
13. C₂×Dic₅₅₆	C₄	semidirect
14. C₂×#14555774	C₄	semidirect
15. C₂×D₁₁₁₂	C₄	semidirect
17. #17984848	C₂²	semidirect
20. #17984778	C₂²	semidirect
21. #17984778	C₂²	semidirect
22. #17984770	C₂²	semidirect
26. #97×C₁₃₉	C₂²	semidirect
27. C₄×C₈×C₁₃₉	C₂²	semidirect
101. C₁₃₉	#2957	semidirect
109. #405	D₁₃₉	semidirect
128. #97	C₂×D₁₃₉	semidirect
129. C₄×C₈	C₂×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984864	2
#22581719	2
#22581076	2
#22581071	2
C₂×#17984778	2
#22580971	2
#405×C₁₃₉	2
#2957	139