Subgroups — #31815100

Order 17792.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984848	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581465	C₂	1	1	yes	maximal, characteristic
4	#22581026	C₂	1	4	yes	maximal, complemented
5	C₈×D₅₅₆	C₂	1	4	yes	maximal, complemented
6	#22580980	C₂	1	1	yes	maximal, characteristic, complemented
7	#22580921	C₂	1	1	yes	maximal, characteristic, complemented
8	#22580916	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14555785	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
12	#17984848	—	2	4	no	—
13	C₂×#14555769	C₂²	1	1	yes	characteristic
14	C₂×#14555768	C₂²	1	2	yes	complemented
15	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×D₁₃₉	—	2	4	no	—
17	C₂×C₈×D₁₃₉	C₂²	1	2	yes	—
18	C₂×C₈×D₁₃₉	C₂²	1	2	yes	—
19	C₂×C₈×D₁₃₉	C₂²	1	2	yes	complemented
20	#17984826	C₄	1	2	yes	—
21	#17984822	C₄	1	2	yes	—
22	#17984817	C₄	1	2	yes	—
23	#17984812	C₄	1	2	yes	—
24	C₄×D₅₅₆	C₂²	1	4	yes	—
25	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	4	yes	complemented
26	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	—
27	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
30	#117×C₁₃₉	C₂²	1	1	yes	characteristic
31	#97×C₁₃₉	C₂²	1	2	yes	complemented
32	C₄×C₈×C₁₃₉	C₂²	1	2	yes	complemented
33	C₂³×D₁₃₉	C₂×C₄	1	1	yes	characteristic
34	#14555785	—	2	2	no	—
35	#14555785	—	2	2	no	—
36	C₂×Dic₂₇₈	C₂×C₄	1	1	yes	characteristic
37	C₂×D₅₅₆	C₂×C₄	1	4	yes	—
38	C₂×D₅₅₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊D₁₃₉	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₃₉	—	2	2	no	—
42	C₂×C₄×D₁₃₉	—	2	2	no	—
43	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
45	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
46	#14555769	—	2	2	no	—
47	#14555768	—	2	4	no	—
48	C₂×C₁₃₉⋊C₈	—	2	2	no	—
49	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
50	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
51	C₈×D₁₃₉	—	2	4	no	—
52	C₈×D₁₃₉	—	2	4	no	—
53	C₈×D₁₃₉	—	2	4	no	—
54	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
55	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	4	yes	—
56	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	4	yes	—
57	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
58	M₁₆×C₁₃₉	—	2	2	no	—
59	C₂×C₈×C₁₃₉	D₄	1	4	yes	complemented
60	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
61	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₃₉	C₂×C₄	1	2	yes	—
65	C₄²×C₁₃₉	C₂³	1	2	yes	—
66	C₂²×D₁₃₉	—	2	2	no	—
67	C₂²×D₁₃₉	—	2	4	no	—
68	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
69	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
70	Dic₂₇₈	—	2	2	no	—
71	D₅₅₆	—	2	8	no	—
72	D₅₅₆	—	2	2	no	—
73	C₂²⋊D₁₃₉	—	2	4	no	—
74	C₂×Dic₁₃₉	—	2	2	no	—
75	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
76	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
77	C₄×D₁₃₉	—	2	4	no	—
78	C₄×D₁₃₉	—	2	2	no	—
79	C₄×D₁₃₉	—	2	2	no	—
80	C₄×D₁₃₉	—	2	2	no	—
81	C₄×D₁₃₉	—	2	2	no	—
82	C₁₃₉⋊C₈	—	2	2	no	—
83	C₁₃₉⋊C₈	—	2	2	no	—
84	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
86	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
88	C₂×C₄×C₁₃₉	#41	1	2	yes	—
89	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₃₉	—	2	2	no	—
91	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
92	C₈×C₁₃₉	C₂×D₄	1	2	yes	—
93	C₂²×C₁₃₉	—	2	2	no	—
94	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
95	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
96	C₂×D₁₃₉	—	2	8	no	—
97	C₂×D₁₃₉	—	2	2	no	—
98	C₂×D₁₃₉	—	2	2	no	—
99	C₂×D₁₃₉	—	2	2	no	—
100	C₂×D₁₃₉	—	2	4	no	—
101	C₂×D₁₃₉	#131	1	4	yes	complemented
102	C₄×C₁₃₉	—	2	4	no	—
103	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
104	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
106	Dic₁₃₉	—	2	2	no	—
107	Dic₁₃₉	—	2	2	no	—
108	C₂×C₁₃₉	—	2	2	no	—
109	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
110	C₂×C₁₃₉	C₂×#131	1	2	yes	—
111	D₁₃₉	—	2	4	no	—
112	D₁₃₉	—	4	2	no	—
113	C₁₃₉	#2929	1	1	yes	characteristic, complemented, Sylow
114	#2929	—	139	1	no	maximal, Sylow
115	C₂×#131	—	139	1	no	—
116	C₂³×C₈	—	139	1	no	—
117	#525	—	139	1	no	—
118	#435	—	139	4	no	—
119	C₈×D₄	—	139	4	no	—
120	#424	—	139	1	no	—
121	#408	—	139	1	no	—
122	#407	—	139	1	no	—
123	#405	D₁₃₉	1	1	yes	characteristic, complemented
124	C₂×#41	—	139	1	no	—
125	C₂³×C₄	—	139	1	no	—
126	#131	—	278	4	no	—
127	C₂×M₁₆	—	139	1	no	—
128	C₂×M₁₆	—	139	2	no	—
129	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
130	C₂²×C₈	—	278	4	no	—
131	C₂²×C₈	—	139	1	no	—
132	C₂²×C₈	—	139	2	no	—
133	C₂²×C₈	—	139	2	no	—
134	C₂²×C₈	—	139	2	no	—
135	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
136	#123	—	139	2	no	—
137	C₂²⋊Q₈	—	139	2	no	—
138	#121	—	139	2	no	—
139	#120	—	139	2	no	—
140	C₄×D₄	—	139	4	no	—
141	#117	C₂×D₁₃₉	1	1	yes	characteristic
142	C₄⋊C₈	—	139	4	no	—
143	C₂²⋊C₈	—	139	4	no	—
144	C₂²⋊C₈	—	139	4	no	—
145	#97	C₂×D₁₃₉	1	2	yes	complemented
146	C₄×C₈	C₂×D₁₃₉	1	2	yes	complemented
147	C₂⁴	—	139	1	no	—
148	#41	—	278	2	no	—
149	#41	—	278	2	no	—
150	C₂×Q₈	—	139	1	no	—
151	C₂×D₄	—	139	2	no	—
152	C₂×D₄	—	139	4	no	—
153	C₂×D₄	—	139	1	no	—
154	C₂²×C₄	—	278	2	no	—
155	C₂²×C₄	—	278	2	no	—
156	C₂²×C₄	—	139	1	no	—
157	C₂²×C₄	—	139	2	no	—
158	C₂²×C₄	—	139	2	no	—
159	C₂²×C₄	—	139	2	no	—
160	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	278	2	no	—
163	M₁₆	—	278	4	no	—
164	C₂×C₈	—	278	2	no	—
165	C₂×C₈	—	278	4	no	—
166	C₂×C₈	—	278	4	no	—
167	C₂×C₈	—	278	4	no	—
168	C₂×C₈	—	139	2	no	—
169	C₂×C₈	—	139	2	no	—
170	C₂×C₈	D₅₅₆	1	4	yes	complemented
171	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
172	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
173	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
174	C₄⋊C₄	—	139	4	no	—
175	C₄⋊C₄	C₄×D₁₃₉	1	2	yes	—
176	C₂²⋊C₄	—	139	4	no	—
177	C₂²⋊C₄	—	139	4	no	—
178	C₂²⋊C₄	C₄×D₁₃₉	1	2	yes	—
179	C₄²	C₂²×D₁₃₉	1	2	yes	—
180	C₂³	—	278	4	no	—
181	C₂³	—	278	2	no	—
182	C₂³	—	139	2	no	—
183	C₂³	—	139	2	no	—
184	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
185	Q₈	—	278	2	no	—
186	D₄	—	278	8	no	—
187	D₄	—	278	4	no	—
188	D₄	—	278	2	no	—
189	C₂×C₄	—	278	2	no	—
190	C₂×C₄	—	278	4	no	—
191	C₂×C₄	—	278	2	no	—
192	C₂×C₄	—	278	2	no	—
193	C₂×C₄	—	278	2	no	—
194	C₂×C₄	—	278	2	no	—
195	C₂×C₄	—	139	2	no	—
196	C₂×C₄	—	139	2	no	—
197	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
198	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
199	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
200	C₂×C₄	#14555785	1	2	yes	—
201	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	278	2	no	—
204	C₈	—	278	2	no	—
205	C₈	C₂×D₅₅₆	1	2	yes	—
206	C₈	C₂×D₅₅₆	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	278	8	no	—
209	C₂²	—	278	2	no	—
210	C₂²	—	278	2	no	—
211	C₂²	—	278	2	no	—
212	C₂²	—	278	4	no	—
213	C₂²	—	139	4	no	—
214	C₂²	C₄×D₅₅₆	1	2	yes	—
215	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
216	C₄	—	2	4	no	—
217	C₄	—	278	2	no	—
218	C₄	—	278	2	no	—
219	C₄	C₄×D₅₅₆	1	2	yes	—
220	C₄	C₂²×D₅₅₆	1	1	yes	characteristic
221	C₄	C₂×#14555785	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	278	4	no	—
224	C₂	—	556	2	no	—
225	C₂	C₂×C₄×D₅₅₆	1	1	yes	characteristic
226	C₂	#131×D₁₃₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#405×C₁₃₉	C₂
prime radical	#22581465	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#405	D₁₃₉
hypocenter	C₁₃₉	#2929
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
1. C₂×#17984848	C₂	semidirect
2. C₂²×C₈×D₁₃₉	C₂	semidirect
4. #22581026	C₂	semidirect
5. C₈×D₅₅₆	C₂	semidirect
6. #22580980	C₂	semidirect
7. #22580921	C₂	semidirect
8. #22580916	C₂	semidirect
9. #405×C₁₃₉	C₂	semidirect
14. C₂×#14555768	C₂²	semidirect
19. C₂×C₈×D₁₃₉	C₂²	semidirect
25. C₄⋊(C₁₃₉⋊C₈)	C₂²	semidirect
27. (C₂×D₁₃₉)⋊C₈	C₂²	semidirect
31. #97×C₁₃₉	C₂²	semidirect
32. C₄×C₈×C₁₃₉	C₂²	semidirect
59. C₂×C₈×C₁₃₉	D₄	semidirect
101. C₂×D₁₃₉	#131	semidirect
113. C₁₃₉	#2929	semidirect
123. #405	D₁₃₉	semidirect
145. #97	C₂×D₁₃₉	semidirect
146. C₄×C₈	C₂×D₁₃₉	semidirect
170. C₂×C₈	D₅₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984848	2
C₂²×C₈×D₁₃₉	2
#22581465	2
#22581026	2
C₈×D₅₅₆	2
#22580980	2
#22580921	2
#22580916	2
#405×C₁₃₉	2
#2929	139