Subgroups — #31819613

Order 17792.

1492 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984890	C₂	1	1	yes	maximal, characteristic
2	#22581435	C₂	1	4	yes	maximal, complemented
3	#22581433	C₂	1	6	yes	maximal, complemented
4	#22581429	C₂	1	4	yes	maximal, complemented
5	C₄×#14555786	C₂	1	4	yes	maximal, complemented
6	#22581421	C₂	1	6	yes	maximal, complemented
7	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₁₃₉	C₂	1	4	yes	maximal, complemented
9	#519×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#17984890	C₄	1	16	yes	complemented
11	C₂×#14555786	C₂²	1	4	yes	—
12	C₂×#14555785	C₂²	1	6	yes	—
13	C₂×Q₈×D₁₃₉	C₂²	1	4	yes	—
14	Q₈×Dic₁₃₉	C₂²	1	4	yes	—
15	Q₈×Dic₁₃₉	C₂²	1	4	yes	—
16	C₄×Dic₂₇₈	C₂²	1	12	yes	—
17	#17984807	C₂²	1	6	yes	—
18	#17984807	C₂²	1	6	yes	—
19	C₄×D₅₅₆	C₂²	1	12	yes	—
20	#17984804	C₂²	1	6	yes	—
21	#17984804	C₂²	1	6	yes	—
22	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	4	yes	—
23	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	12	yes	—
24	#17984800	C₂²	1	12	yes	—
25	#17984799	C₂²	1	12	yes	—
26	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	—
27	C₄⋊C₄×D₁₃₉	C₂²	1	12	yes	—
28	C₂×C₄×Dic₁₃₉	C₂²	1	3	yes	—
29	C₄²×D₁₃₉	C₂²	1	12	yes	—
30	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
31	C₄×Q₈×C₁₃₉	C₂²	1	4	yes	—
32	C₄×Q₈×C₁₃₉	C₂²	1	4	yes	—
33	#117×C₁₃₉	C₂²	1	6	yes	—
34	#14555786	C₂×C₄	1	32	yes	complemented
35	#14555785	C₂×C₄	1	48	yes	complemented
36	C₂×Dic₂₇₈	C₂³	1	6	yes	—
37	Q₈×D₁₃₉	C₂×C₄	1	32	yes	complemented
38	C₂×D₅₅₆	C₂³	1	6	yes	—
39	C₂×C₂²⋊D₁₃₉	C₂³	1	4	yes	—
40	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
42	C₂×C₄×D₁₃₉	C₂³	1	4	yes	—
43	C₂×C₄×D₁₃₉	C₂³	1	12	yes	—
44	C₂×C₄×D₁₃₉	C₂³	1	12	yes	—
45	C₄⋊Dic₁₃₉	C₂³	1	12	yes	—
46	Dic₁₃₉⋊C₄	C₂³	1	12	yes	—
47	Dic₁₃₉⋊C₄	C₂³	1	4	yes	—
48	(C₂×D₁₃₉)⋊C₄	C₂³	1	4	yes	—
49	(C₂×D₁₃₉)⋊C₄	C₂³	1	12	yes	—
50	C₄×Dic₁₃₉	C₂³	1	4	yes	—
51	C₄×Dic₁₃₉	C₂³	1	6	yes	—
52	C₄×Dic₁₃₉	C₂³	1	12	yes	—
53	C₄×Dic₁₃₉	C₂³	1	6	yes	—
54	C₂×Q₈×C₁₃₉	C₂×C₄	1	8	yes	complemented
55	C₂×Q₈×C₁₃₉	C₂³	1	4	yes	—
56	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
57	C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
58	C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
59	C₂²⋊C₄×C₁₃₉	C₂³	1	4	yes	—
60	C₄²×C₁₃₉	C₂³	1	12	yes	—
61	C₂²×D₁₃₉	C₂⁴	1	4	yes	—
62	Dic₂₇₈	C₂²×C₄	1	24	yes	—
63	D₅₅₆	C₂²×C₄	1	24	yes	—
64	C₂²⋊D₁₃₉	C₂²×C₄	1	16	yes	—
65	C₂×Dic₁₃₉	—	2	2	no	—
66	C₂×Dic₁₃₉	—	4	3	no	—
67	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
68	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
69	C₂×Dic₁₃₉	C₂⁴	1	4	yes	—
70	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
71	C₄×D₁₃₉	—	2	8	no	—
72	C₄×D₁₃₉	—	4	12	no	—
73	C₄×D₁₃₉	C₂²×C₄	1	48	yes	—
74	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
75	Q₈×C₁₃₉	C₂²×C₄	1	16	yes	—
76	C₂×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
77	C₂×C₄×C₁₃₉	C₂⁴	1	4	yes	—
78	C₂×C₄×C₁₃₉	C₂⁴	1	4	yes	—
79	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
80	C₂²×C₁₃₉	—	2	1	no	—
81	C₂²×C₁₃₉	—	2	1	no	—
82	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
83	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
84	C₂×D₁₃₉	—	2	4	no	—
85	C₂×D₁₃₉	—	2	4	no	—
86	C₂×D₁₃₉	C₂³×C₄	1	8	yes	—
87	C₄×C₁₃₉	—	2	4	no	—
88	C₄×C₁₃₉	—	2	4	no	—
89	C₄×C₁₃₉	C₂³×C₄	1	12	yes	—
90	Dic₁₃₉	—	2	1	no	—
91	Dic₁₃₉	—	2	6	no	—
92	Dic₁₃₉	C₂³×C₄	1	8	yes	—
93	Dic₁₃₉	2₋¹⁺⁴	1	2	yes	complemented
94	C₂×C₁₃₉	—	2	2	no	—
95	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
96	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
97	C₂×C₁₃₉	#585	1	1	yes	characteristic
98	D₁₃₉	—	2	8	no	—
99	C₁₃₉	C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
100	C₄×2₋¹⁺⁴	—	139	1	no	maximal, Sylow
101	C₂×2₋¹⁺⁴	—	139	1	no	—
102	#520	—	139	4	no	—
103	#520	—	139	6	no	—
104	#519	—	139	4	no	—
105	#519	D₁₃₉	1	1	yes	characteristic, complemented
106	C₄×#41	—	139	4	no	—
107	C₄×#41	—	139	6	no	—
108	C₂×C₄×Q₈	—	139	1	no	—
109	C₂×C₄×Q₈	—	139	4	no	—
110	2₋¹⁺⁴	—	139	16	no	—
111	C₂×#41	—	139	4	no	—
112	C₂×#41	—	139	6	no	—
113	C₂²×Q₈	—	139	4	no	—
114	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
115	C₄×Q₈	—	139	4	no	—
116	C₄×Q₈	—	139	4	no	—
117	C₄×Q₈	—	139	6	no	—
118	C₄×Q₈	—	139	6	no	—
119	C₄×Q₈	—	139	12	no	—
120	C₄×Q₈	C₂×D₁₃₉	1	4	yes	—
121	C₄×Q₈	C₂×D₁₃₉	1	4	yes	—
122	C₄×D₄	—	139	6	no	—
123	C₄×D₄	—	139	6	no	—
124	C₄×D₄	—	139	4	no	—
125	C₄×D₄	—	139	12	no	—
126	C₄×D₄	—	139	12	no	—
127	#117	—	139	12	no	—
128	#117	—	139	12	no	—
129	#117	C₂×D₁₃₉	1	6	yes	—
130	C₂×C₄⋊C₄	—	139	3	no	—
131	C₂×C₄⋊C₄	—	139	12	no	—
132	C₂×C₄²	—	139	3	no	—
133	C₂×C₄²	—	139	12	no	—
134	#41	—	139	32	no	—
135	#41	—	139	48	no	—
136	C₂×Q₈	—	139	32	no	—
137	C₂×Q₈	—	139	6	no	—
138	C₂×Q₈	C₄×D₁₃₉	1	8	yes	complemented
139	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
140	C₂×D₄	—	139	4	no	—
141	C₂×D₄	—	139	6	no	—
142	C₂²×C₄	—	139	1	no	—
143	C₂²×C₄	—	139	4	no	—
144	C₂²×C₄	—	139	3	no	—
145	C₂²×C₄	—	139	12	no	—
146	C₂²×C₄	—	139	12	no	—
147	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
148	C₄⋊C₄	—	139	12	no	—
149	C₄⋊C₄	—	139	12	no	—
150	C₄⋊C₄	—	139	4	no	—
151	C₄⋊C₄	C₂²×D₁₃₉	1	6	yes	—
152	C₄⋊C₄	C₂²×D₁₃₉	1	6	yes	—
153	C₂²⋊C₄	—	139	4	no	—
154	C₂²⋊C₄	—	139	12	no	—
155	C₂²⋊C₄	C₂²×D₁₃₉	1	4	yes	—
156	C₄²	—	139	6	no	—
157	C₄²	—	139	4	no	—
158	C₄²	—	139	6	no	—
159	C₄²	—	139	12	no	—
160	C₄²	C₂²×D₁₃₉	1	12	yes	—
161	C₂³	—	139	4	no	—
162	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
163	Q₈	—	139	24	no	—
164	Q₈	C₂×C₄×D₁₃₉	1	16	yes	—
165	D₄	—	139	16	no	—
166	D₄	—	139	24	no	—
167	C₂×C₄	—	278	8	no	—
168	C₂×C₄	—	278	2	no	—
169	C₂×C₄	—	556	12	no	—
170	C₂×C₄	—	556	3	no	—
171	C₂×C₄	—	139	1	no	—
172	C₂×C₄	—	139	6	no	—
173	C₂×C₄	—	139	1	no	—
174	C₂×C₄	—	139	4	no	—
175	C₂×C₄	—	139	48	no	—
176	C₂×C₄	C₂×C₄×D₁₃₉	1	12	yes	—
177	C₂×C₄	C₂³×D₁₃₉	1	4	yes	—
178	C₂×C₄	C₂³×D₁₃₉	1	4	yes	—
179	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
180	C₂²	—	2	1	no	—
181	C₂²	—	2	1	no	—
182	C₂²	—	278	4	no	—
183	C₂²	—	278	4	no	—
184	C₂²	—	139	8	no	—
185	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
186	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
187	C₄	—	2	4	no	—
188	C₄	—	2	4	no	—
189	C₄	—	278	1	no	—
190	C₄	—	278	6	no	—
191	C₄	—	139	8	no	—
192	C₄	—	139	2	no	—
193	C₄	C₂²×C₄×D₁₃₉	1	12	yes	—
194	C₂	—	2	2	no	—
195	C₂	—	278	8	no	—
196	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
197	C₂	2₋¹⁺⁴×D₁₃₉	1	1	yes	characteristic
198	C₂	#22581790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Fitting	#519×C₁₃₉	C₂
socle, cosocle	C₂²×C₁₃₉	C₂⁵
prime radical	C₂×#17984890	C₂
hypercenter	#519	D₁₃₉
hypocenter	C₁₃₉	C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #519

Complements

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

N	H ≅ G/N	type
2. #22581435	C₂	semidirect
3. #22581433	C₂	semidirect
4. #22581429	C₂	semidirect
5. C₄×#14555786	C₂	semidirect
6. #22581421	C₂	semidirect
7. C₂×Q₈×Dic₁₃₉	C₂	semidirect
8. C₄×Q₈×D₁₃₉	C₂	semidirect
9. #519×C₁₃₉	C₂	semidirect
10. #17984890	C₄	semidirect
34. #14555786	C₂×C₄	semidirect
35. #14555785	C₂×C₄	semidirect
37. Q₈×D₁₃₉	C₂×C₄	semidirect
54. C₂×Q₈×C₁₃₉	C₂×C₄	semidirect
93. Dic₁₃₉	2₋¹⁺⁴	semidirect
99. C₁₃₉	C₄×2₋¹⁺⁴	semidirect
105. #519	D₁₃₉	semidirect
138. C₂×Q₈	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984890	2
#22581435	2
#22581433	2
#22581429	2
C₄×#14555786	2
#22581421	2
C₂×Q₈×Dic₁₃₉	2
C₄×Q₈×D₁₃₉	2
#519×C₁₃₉	2
C₄×2₋¹⁺⁴	139