Subgroups — C₂×#22581435 #31819600

Order 17792.

1640 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555786	C₂	1	1	yes	maximal, characteristic
2	#22581435	C₂	1	16	yes	maximal, direct product factor
3	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₅₅₆	C₂	1	3	yes	maximal, complemented
5	C₂×#17984804	C₂	1	3	yes	maximal, complemented
6	C₂×#17984799	C₂	1	3	yes	maximal, complemented
7	C₂×C₄⋊C₄×D₁₃₉	C₂	1	3	yes	maximal, complemented
8	C₂×C₄×Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555786	C₄	1	16	yes	complemented
10	C₂×#14555786	C₂²	1	8	yes	—
11	C₂²×D₅₅₆	C₂²	1	3	yes	—
12	C₂²×C₄×D₁₃₉	C₂²	1	6	yes	—
13	C₂²×C₄×D₁₃₉	C₂²	1	3	yes	—
14	Q₈×Dic₁₃₉	C₂²	1	8	yes	complemented
15	C₄×D₅₅₆	C₂²	1	24	yes	complemented
16	#17984804	C₂²	1	24	yes	complemented
17	#17984799	C₂²	1	24	yes	complemented
18	C₂×C₄⋊Dic₁₃₉	C₂²	1	3	yes	—
19	C₂×Dic₁₃₉⋊C₄	C₂²	1	3	yes	—
20	C₂×Dic₁₃₉⋊C₄	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×D₁₃₉	C₂²	1	24	yes	complemented
22	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	3	yes	—
23	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	3	yes	—
24	C₂×C₄×Dic₁₃₉	C₂²	1	3	yes	—
25	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₃₉	C₂²	1	8	yes	complemented
27	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	—
28	C₂×C₄²×C₁₃₉	C₂²	1	3	yes	—
29	C₂³×D₁₃₉	C₂³	1	3	yes	—
30	#14555786	C₂×C₄	1	64	yes	complemented
31	C₂×D₅₅₆	C₂×C₄	1	24	yes	complemented
32	C₂×D₅₅₆	C₂³	1	12	yes	—
33	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
34	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
35	C₂²×Dic₁₃₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×D₁₃₉	—	4	12	no	—
37	C₂×C₄×D₁₃₉	C₂×C₄	1	24	yes	complemented
38	C₂×C₄×D₁₃₉	C₂³	1	24	yes	—
39	C₂×C₄×D₁₃₉	C₂³	1	12	yes	—
40	C₄⋊Dic₁₃₉	C₂³	1	12	yes	—
41	Dic₁₃₉⋊C₄	C₂³	1	12	yes	—
42	Dic₁₃₉⋊C₄	C₂³	1	4	yes	—
43	(C₂×D₁₃₉)⋊C₄	C₂³	1	12	yes	—
44	(C₂×D₁₃₉)⋊C₄	C₂³	1	12	yes	—
45	C₄×Dic₁₃₉	C₂³	1	12	yes	—
46	C₂×Q₈×C₁₃₉	C₂×C₄	1	8	yes	complemented
47	C₂×Q₈×C₁₃₉	C₂³	1	4	yes	—
48	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
50	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
51	C₄⋊C₄×C₁₃₉	C₂³	1	12	yes	—
52	C₄²×C₁₃₉	C₂³	1	12	yes	—
53	C₂²×D₁₃₉	—	2	6	no	—
54	C₂²×D₁₃₉	—	2	6	no	—
55	C₂²×D₁₃₉	C₂²×C₄	1	12	yes	—
56	C₂²×D₁₃₉	C₂⁴	1	6	yes	—
57	D₅₅₆	C₂²×C₄	1	48	yes	complemented
58	C₂×Dic₁₃₉	—	2	6	no	—
59	C₂×Dic₁₃₉	—	2	2	no	—
60	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
61	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
62	C₂×Dic₁₃₉	C₂⁴	1	6	yes	—
63	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
64	C₄×D₁₃₉	—	4	24	no	—
65	C₄×D₁₃₉	C₂²×C₄	1	48	yes	complemented
66	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
67	Q₈×C₁₃₉	C₂²×C₄	1	16	yes	complemented
68	C₂×C₄×C₁₃₉	—	2	6	no	—
69	C₂×C₄×C₁₃₉	—	2	2	no	—
70	C₂×C₄×C₁₃₉	C₂²×C₄	1	12	yes	—
71	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
72	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
73	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
74	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
75	C₂²×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
76	C₂²×C₁₃₉	2₋¹⁺⁴	1	2	yes	—
77	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
78	C₂×D₁₃₉	—	2	24	no	—
79	C₂×D₁₃₉	—	2	6	no	—
80	C₂×D₁₃₉	—	2	6	no	—
81	C₂×D₁₃₉	C₂³×C₄	1	12	yes	—
82	C₄×C₁₃₉	—	2	6	no	—
83	C₄×C₁₃₉	—	2	2	no	—
84	C₄×C₁₃₉	C₂³×C₄	1	12	yes	—
85	Dic₁₃₉	—	2	6	no	—
86	Dic₁₃₉	—	2	2	no	—
87	Dic₁₃₉	C₂³×C₄	1	4	yes	—
88	C₂×C₁₃₉	#520	1	4	yes	complemented
89	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
90	C₂×C₁₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
91	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
92	D₁₃₉	—	2	12	no	—
93	C₁₃₉	C₂×#520	1	1	yes	characteristic, complemented, Sylow
94	C₂×#520	—	139	1	no	maximal, Sylow
95	C₂²×#41	—	139	1	no	—
96	#520	—	139	16	no	—
97	C₂×C₄×Q₈	—	139	1	no	—
98	C₂×C₄×Q₈	D₁₃₉	1	1	yes	characteristic, complemented
99	C₂×C₄×D₄	—	139	3	no	—
100	C₂×C₄×D₄	—	139	3	no	—
101	C₂×#117	—	139	3	no	—
102	C₂²×C₄⋊C₄	—	139	3	no	—
103	C₂×#41	—	139	16	no	—
104	C₂×#41	—	139	8	no	—
105	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
106	C₂²×D₄	—	139	3	no	—
107	C₂³×C₄	—	139	6	no	—
108	C₂³×C₄	—	139	3	no	—
109	C₄×Q₈	—	139	8	no	—
110	C₄×Q₈	C₂×D₁₃₉	1	8	yes	complemented
111	C₄×D₄	—	139	24	no	—
112	C₄×D₄	—	139	24	no	—
113	#117	—	139	24	no	—
114	C₂×C₄⋊C₄	—	139	3	no	—
115	C₂×C₄⋊C₄	—	139	3	no	—
116	C₂×C₄⋊C₄	—	139	1	no	—
117	C₂×C₄⋊C₄	—	139	24	no	—
118	C₂×C₄⋊C₄	C₂×D₁₃₉	1	3	yes	—
119	C₂×C₂²⋊C₄	—	139	3	no	—
120	C₂×C₂²⋊C₄	—	139	3	no	—
121	C₂×C₄²	—	139	3	no	—
122	C₂×C₄²	C₂×D₁₃₉	1	3	yes	—
123	C₂⁴	—	139	3	no	—
124	#41	—	139	64	no	—
125	C₂×Q₈	C₄×D₁₃₉	1	8	yes	complemented
126	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
127	C₂×D₄	—	139	24	no	—
128	C₂×D₄	—	139	12	no	—
129	C₂²×C₄	—	556	12	no	—
130	C₂²×C₄	—	139	1	no	—
131	C₂²×C₄	—	139	3	no	—
132	C₂²×C₄	—	139	24	no	—
133	C₂²×C₄	—	139	1	no	—
134	C₂²×C₄	—	139	24	no	—
135	C₂²×C₄	—	139	12	no	—
136	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
137	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
138	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
139	C₄⋊C₄	—	139	12	no	—
140	C₄⋊C₄	—	139	12	no	—
141	C₄⋊C₄	—	139	4	no	—
142	C₄⋊C₄	C₂²×D₁₃₉	1	12	yes	—
143	C₂²⋊C₄	—	139	12	no	—
144	C₂²⋊C₄	—	139	12	no	—
145	C₄²	—	139	12	no	—
146	C₄²	C₂²×D₁₃₉	1	12	yes	—
147	C₂³	—	278	6	no	—
148	C₂³	—	278	6	no	—
149	C₂³	—	139	12	no	—
150	C₂³	—	139	6	no	—
151	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
152	Q₈	C₂×C₄×D₁₃₉	1	16	yes	complemented
153	D₄	—	139	48	no	—
154	C₂×C₄	—	2	6	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	278	6	no	—
157	C₂×C₄	—	278	2	no	—
158	C₂×C₄	—	556	24	no	—
159	C₂×C₄	—	139	2	no	—
160	C₂×C₄	—	139	6	no	—
161	C₂×C₄	—	139	4	no	—
162	C₂×C₄	—	139	2	no	—
163	C₂×C₄	—	139	48	no	—
164	C₂×C₄	C₂×C₄×D₁₃₉	1	12	yes	—
165	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
166	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
167	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
168	C₂²	—	278	6	no	—
169	C₂²	—	278	6	no	—
170	C₂²	—	278	24	no	—
171	C₂²	—	139	12	no	—
172	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
173	C₂²	#17984888	1	2	yes	—
174	C₂²	#17984890	1	2	yes	—
175	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
176	C₄	—	2	6	no	—
177	C₄	—	2	2	no	—
178	C₄	—	278	6	no	—
179	C₄	—	278	2	no	—
180	C₄	—	139	4	no	—
181	C₄	C₂²×C₄×D₁₃₉	1	12	yes	—
182	C₂	—	278	12	no	—
183	C₂	#22581435	1	4	yes	direct product factor
184	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
185	C₂	C₂×#17984888	1	1	yes	characteristic
186	C₂	C₂×#17984890	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×C₄×Q₈×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂²×#14555786	C₂
cosocle	C₂²×C₁₃₉	C₂⁵
hypercenter	C₂×C₄×Q₈	D₁₃₉
hypocenter	C₁₃₉	C₂×#520
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #22581435	C₂	direct (N × H)
3. C₂×Q₈×Dic₁₃₉	C₂	semidirect
4. C₂×C₄×D₅₅₆	C₂	semidirect
5. C₂×#17984804	C₂	semidirect
6. C₂×#17984799	C₂	semidirect
7. C₂×C₄⋊C₄×D₁₃₉	C₂	semidirect
8. C₂×C₄×Q₈×C₁₃₉	C₂	semidirect
9. C₂×#14555786	C₄	semidirect
14. Q₈×Dic₁₃₉	C₂²	semidirect
15. C₄×D₅₅₆	C₂²	semidirect
16. #17984804	C₂²	semidirect
17. #17984799	C₂²	semidirect
21. C₄⋊C₄×D₁₃₉	C₂²	semidirect
26. C₄×Q₈×C₁₃₉	C₂²	semidirect
30. #14555786	C₂×C₄	semidirect
31. C₂×D₅₅₆	C₂×C₄	semidirect
37. C₂×C₄×D₁₃₉	C₂×C₄	semidirect
46. C₂×Q₈×C₁₃₉	C₂×C₄	semidirect
57. D₅₅₆	C₂²×C₄	semidirect
65. C₄×D₁₃₉	C₂²×C₄	semidirect
67. Q₈×C₁₃₉	C₂²×C₄	semidirect
88. C₂×C₁₃₉	#520	semidirect
93. C₁₃₉	C₂×#520	semidirect
98. C₂×C₄×Q₈	D₁₃₉	semidirect
110. C₄×Q₈	C₂×D₁₃₉	semidirect
125. C₂×Q₈	C₄×D₁₃₉	semidirect
152. Q₈	C₂×C₄×D₁₃₉	semidirect
183. C₂	#22581435	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14555786	2
#22581435	2
C₂×Q₈×Dic₁₃₉	2
C₂×C₄×D₅₅₆	2
C₂×#17984804	2
C₂×#17984799	2
C₂×C₄⋊C₄×D₁₃₉	2
C₂×C₄×Q₈×C₁₃₉	2
C₂×#520	139

Subgroups — C2×#22581435 #31819600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581435 #31819600