Subgroups — C₂×#22581081 #31815329

Order 17792.

400 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	C₂×C₄×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	#22581081	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊(C₁₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984752	C₂	1	1	yes	maximal, characteristic
7	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₂₇₈	C₄	1	1	yes	characteristic
10	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₇₈	C₄	1	4	yes	—
13	C₄×Dic₂₇₈	C₂²	1	4	yes	—
14	C₂×C₄⋊Dic₁₃₉	C₄	1	1	yes	characteristic
15	C₂×Dic₁₃₉⋊C₄	C₄	1	1	yes	characteristic
16	C₂×Dic₁₃₉⋊C₄	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
19	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	4	yes	—
20	Dic₁₃₉⋊C₈	C₂²	1	4	yes	—
21	Dic₁₃₉⋊C₈	C₂²	1	4	yes	—
22	#17984752	C₂²	1	4	yes	—
23	C₈×Dic₁₃₉	C₂²	1	4	yes	—
24	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
27	#97×C₁₃₉	C₂²	1	4	yes	—
28	C₂×Dic₂₇₈	—	2	4	no	—
29	C₂×Dic₂₇₈	C₂×C₄	1	4	yes	—
30	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₃₉⋊C₈	—	2	2	no	—
35	C₂×C₁₃₉⋊C₈	—	2	2	no	—
36	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
37	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
38	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
39	Dic₁₃₉⋊C₄	C₂×C₄	1	4	yes	—
40	Dic₁₃₉⋊C₄	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
43	C₄×Dic₁₃₉	C₂³	1	2	yes	—
44	C₄×Dic₁₃₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₃₉	—	2	2	no	—
49	C₂×C₈×C₁₃₉	Q₈	1	4	yes	complemented
50	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
51	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
52	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
53	C₄²×C₁₃₉	C₂³	1	2	yes	—
54	Dic₂₇₈	—	2	8	no	—
55	C₂×Dic₁₃₉	—	2	2	no	—
56	C₂×Dic₁₃₉	—	2	2	no	—
57	C₂×Dic₁₃₉	—	2	2	no	—
58	C₂×Dic₁₃₉	M₁₆	1	4	yes	—
59	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
60	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
61	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
62	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
63	C₁₃₉⋊C₈	—	2	2	no	—
64	C₁₃₉⋊C₈	—	2	2	no	—
65	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₁₃₉	—	2	2	no	—
67	C₂×C₄×C₁₃₉	—	2	2	no	—
68	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
69	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
70	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
72	C₂×C₄×C₁₃₉	#41	1	2	yes	—
73	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
74	C₈×C₁₃₉	—	2	2	no	—
75	C₈×C₁₃₉	C₂×Q₈	1	4	yes	complemented
76	C₂²×C₁₃₉	C₄×Q₈	1	2	yes	—
77	C₂²×C₁₃₉	C₂×M₁₆	1	2	yes	—
78	C₂²×C₁₃₉	#131	1	2	yes	—
79	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
80	C₄×C₁₃₉	—	2	2	no	—
81	C₄×C₁₃₉	—	2	2	no	—
82	C₄×C₁₃₉	C₄×Q₈	1	2	yes	—
83	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
84	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
85	Dic₁₃₉	—	2	2	no	—
86	Dic₁₃₉	—	2	2	no	—
87	Dic₁₃₉	—	2	2	no	—
88	Dic₁₃₉	C₂×M₁₆	1	4	yes	—
89	C₂×C₁₃₉	#446	1	4	yes	complemented
90	C₂×C₁₃₉	C₂×C₄×Q₈	1	1	yes	characteristic
91	C₂×C₁₃₉	C₂²×M₁₆	1	1	yes	characteristic
92	C₂×C₁₃₉	C₂×#131	1	1	yes	characteristic
93	C₁₃₉	C₂×#446	1	1	yes	characteristic, complemented, Sylow
94	C₂×#446	—	139	1	no	maximal, Sylow
95	C₂×C₄×Q₈	—	139	1	no	—
96	#446	—	139	8	no	—
97	C₂×C₄⋊C₈	—	139	1	no	—
98	C₂×C₄⋊C₈	—	139	1	no	—
99	C₂×C₄⋊C₈	—	139	1	no	—
100	C₂×#97	—	139	1	no	—
101	C₂×#97	D₁₃₉	1	1	yes	characteristic
102	C₂×C₄×C₈	—	139	1	no	—
103	C₂²×Q₈	—	139	1	no	—
104	C₂²×C₈	—	139	1	no	—
105	C₂²×C₈	—	139	1	no	—
106	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
107	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
108	C₄×Q₈	—	139	4	no	—
109	C₄×Q₈	—	139	4	no	—
110	C₂×C₄⋊C₄	—	139	1	no	—
111	C₂×C₄⋊C₄	—	139	1	no	—
112	C₂×C₄⋊C₄	—	139	1	no	—
113	C₂×C₄²	—	139	1	no	—
114	C₂×C₄²	—	139	1	no	—
115	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
116	C₄⋊C₈	—	139	4	no	—
117	C₄⋊C₈	—	139	4	no	—
118	C₄⋊C₈	—	139	4	no	—
119	#97	—	139	4	no	—
120	#97	C₂×D₁₃₉	1	4	yes	—
121	C₄×C₈	—	139	4	no	—
122	C₂×Q₈	—	278	4	no	—
123	C₂×Q₈	—	139	4	no	—
124	C₂²×C₄	—	139	1	no	—
125	C₂²×C₄	—	139	1	no	—
126	C₂²×C₄	—	139	1	no	—
127	C₂²×C₄	—	139	1	no	—
128	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
129	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
130	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	278	2	no	—
133	C₂×C₈	—	278	2	no	—
134	C₂×C₈	—	139	2	no	—
135	C₂×C₈	—	139	2	no	—
136	C₂×C₈	Dic₂₇₈	1	4	yes	complemented
137	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
138	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
139	C₄⋊C₄	—	139	4	no	—
140	C₄⋊C₄	—	139	4	no	—
141	C₄⋊C₄	—	139	4	no	—
142	C₄²	—	139	2	no	—
143	C₄²	—	139	2	no	—
144	C₄²	—	139	2	no	—
145	C₄²	—	139	2	no	—
146	C₄²	C₄×D₁₃₉	1	2	yes	—
147	C₄²	C₂²×D₁₃₉	1	2	yes	—
148	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
149	Q₈	—	278	8	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	278	2	no	—
153	C₂×C₄	—	278	2	no	—
154	C₂×C₄	—	278	2	no	—
155	C₂×C₄	—	139	2	no	—
156	C₂×C₄	—	139	2	no	—
157	C₂×C₄	—	139	2	no	—
158	C₂×C₄	—	139	4	no	—
159	C₂×C₄	—	139	2	no	—
160	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
161	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
162	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
163	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
164	C₂×C₄	#14555785	1	2	yes	—
165	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
166	C₈	—	2	2	no	—
167	C₈	—	278	2	no	—
168	C₈	—	278	2	no	—
169	C₈	C₂×Dic₂₇₈	1	4	yes	complemented
170	C₂²	C₄×Dic₂₇₈	1	2	yes	—
171	C₂²	M₁₆×D₁₃₉	1	2	yes	—
172	C₂²	#17984850	1	2	yes	—
173	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
174	C₄	—	2	2	no	—
175	C₄	—	2	2	no	—
176	C₄	—	278	2	no	—
177	C₄	—	278	2	no	—
178	C₄	—	278	2	no	—
179	C₄	—	139	4	no	—
180	C₄	C₄×Dic₂₇₈	1	2	yes	—
181	C₄	C₂²×Dic₂₇₈	1	1	yes	characteristic
182	C₄	C₂×#14555785	1	1	yes	characteristic
183	C₂	#22581081	1	4	yes	direct product factor
184	C₂	C₂×C₄×Dic₂₇₈	1	1	yes	characteristic
185	C₂	C₂×M₁₆×D₁₃₉	1	1	yes	characteristic
186	C₂	C₂×#17984850	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×#97×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂²×C₄
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×#97	D₁₃₉
hypocenter	C₁₃₉	C₂×#446
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
2. #22581081	C₂	direct (N × H)
49. C₂×C₈×C₁₃₉	Q₈	semidirect
75. C₈×C₁₃₉	C₂×Q₈	semidirect
89. C₂×C₁₃₉	#446	semidirect
93. C₁₃₉	C₂×#446	semidirect
136. C₂×C₈	Dic₂₇₈	semidirect
169. C₈	C₂×Dic₂₇₈	semidirect
183. C₂	#22581081	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₂₇₈	2
#22581081	2
C₂×C₄⋊(C₁₃₉⋊C₈)	2
C₂×Dic₁₃₉⋊C₈	2
C₂×#17984752	2
C₂×C₈×Dic₁₃₉	2
C₂×#97×C₁₃₉	2
C₂×#446	139

Subgroups — C2×#22581081 #31815329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581081 #31815329