Subgroups — C₂×#22580920 #31814844

Order 17792.

792 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555769	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22580920	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊(C₁₃₉⋊C₈)	C₂	1	4	yes	maximal, complemented
5	C₂²×#41×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	—	2	4	no	—
7	C₂×#14555769	C₂²	1	4	yes	complemented
8	C₂²×C₁₃₉⋊C₈	—	2	4	no	—
9	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
10	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₁₃₉⋊C₈	C₂²	1	4	yes	complemented
12	C₂²⋊(C₁₃₉⋊C₈)	C₂²	1	16	yes	complemented
13	C₂×#41×C₁₃₉	—	2	4	no	—
14	C₂×#41×C₁₃₉	—	2	4	no	—
15	C₂×#41×C₁₃₉	C₄	1	4	yes	—
16	C₂×#41×C₁₃₉	C₂²	1	4	yes	—
17	C₂²×Q₈×C₁₃₉	C₄	1	1	yes	characteristic
18	C₂²×D₄×C₁₃₉	C₄	1	2	yes	—
19	C₂²×D₄×C₁₃₉	C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃₉	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
22	#14555769	—	2	8	no	—
23	C₂×C₁₃₉⋊C₈	—	2	4	no	—
24	C₂×C₁₃₉⋊C₈	—	2	8	no	—
25	C₂×C₁₃₉⋊C₈	—	2	4	no	—
26	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
27	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
28	C₂⁴×C₁₃₉	C₂×C₄	1	2	yes	—
29	C₂⁴×C₁₃₉	C₂×C₄	1	1	yes	characteristic
30	#41×C₁₃₉	—	2	16	no	—
31	#41×C₁₃₉	—	2	8	no	—
32	#41×C₁₃₉	—	2	8	no	—
33	C₂×Q₈×C₁₃₉	—	2	4	no	—
34	C₂×Q₈×C₁₃₉	C₂×C₄	1	4	yes	—
35	C₂×D₄×C₁₃₉	—	2	8	no	—
36	C₂×D₄×C₁₃₉	—	2	4	no	—
37	C₂×D₄×C₁₃₉	C₂×C₄	1	8	yes	—
38	C₂×D₄×C₁₃₉	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₁₃₉	—	2	4	no	—
40	C₂²×C₄×C₁₃₉	—	2	4	no	—
41	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
42	C₂²×C₄×C₁₃₉	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
45	C₂²×C₄×C₁₃₉	D₄	1	4	yes	—
46	C₂²×C₄×C₁₃₉	D₄	1	4	yes	—
47	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
48	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
49	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
50	C₁₃₉⋊C₈	—	2	4	no	—
51	C₁₃₉⋊C₈	—	2	4	no	—
52	C₂³×C₁₃₉	—	2	4	no	—
53	C₂³×C₁₃₉	—	2	4	no	—
54	C₂³×C₁₃₉	—	2	8	no	—
55	C₂³×C₁₃₉	C₂²⋊C₄	1	4	yes	—
56	C₂³×C₁₃₉	C₂²×C₄	1	4	yes	—
57	C₂³×C₁₃₉	C₂²×C₄	1	2	yes	—
58	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
59	Q₈×C₁₃₉	—	2	8	no	—
60	D₄×C₁₃₉	—	2	16	no	—
61	D₄×C₁₃₉	—	2	8	no	—
62	C₂×C₄×C₁₃₉	—	2	4	no	—
63	C₂×C₄×C₁₃₉	—	2	8	no	—
64	C₂×C₄×C₁₃₉	—	2	4	no	—
65	C₂×C₄×C₁₃₉	—	2	4	no	—
66	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
67	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	8	yes	—
68	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
69	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
71	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
73	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
74	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
75	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
76	C₂²×C₁₃₉	—	2	4	no	—
77	C₂²×C₁₃₉	—	2	8	no	—
78	C₂²×C₁₃₉	—	2	4	no	—
79	C₂²×C₁₃₉	—	2	8	no	—
80	C₂²×C₁₃₉	—	4	8	no	—
81	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	4	yes	—
82	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂²×C₁₃₉	#131	1	4	yes	—
84	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
85	C₄×C₁₃₉	—	2	4	no	—
86	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	2	yes	—
87	C₄×C₁₃₉	C₂×C₂²⋊C₄	1	4	yes	—
88	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
89	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₁₃₉	—	2	4	no	—
91	C₂×C₁₃₉	—	4	4	no	—
92	C₂×C₁₃₉	#408	1	4	yes	complemented
93	C₂×C₁₃₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₁₃₉	C₂×#131	1	2	yes	—
95	C₁₃₉	C₂×#408	1	1	yes	characteristic, complemented, Sylow
96	C₂×#408	—	139	1	no	maximal, Sylow
97	C₂²×#41	D₁₃₉	1	1	yes	characteristic
98	C₂²×M₁₆	—	139	1	no	—
99	C₂³×C₈	—	139	1	no	—
100	#408	—	139	8	no	—
101	C₂×C₂²⋊C₈	—	139	4	no	—
102	C₂×#41	—	2	4	no	—
103	C₂×#41	—	2	4	no	—
104	C₂×#41	Dic₁₃₉	1	4	yes	—
105	C₂×#41	C₂×D₁₃₉	1	4	yes	—
106	C₂²×Q₈	Dic₁₃₉	1	1	yes	characteristic
107	C₂²×D₄	Dic₁₃₉	1	2	yes	—
108	C₂²×D₄	Dic₁₃₉	1	1	yes	characteristic
109	C₂³×C₄	C₂×D₁₃₉	1	2	yes	—
110	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
111	C₂×M₁₆	—	278	4	no	—
112	C₂×M₁₆	—	139	4	no	—
113	C₂²×C₈	—	278	4	no	—
114	C₂²×C₈	—	139	4	no	—
115	C₂²×C₈	—	139	2	no	—
116	C₂²×C₈	—	139	2	no	—
117	C₂²⋊C₈	—	139	16	no	—
118	C₂⁴	C₂×Dic₁₃₉	1	2	yes	—
119	C₂⁴	C₂×Dic₁₃₉	1	1	yes	characteristic
120	#41	—	2	16	no	—
121	#41	—	2	8	no	—
122	#41	—	2	8	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	C₂×Dic₁₃₉	1	4	yes	—
125	C₂×D₄	—	2	8	no	—
126	C₂×D₄	—	2	4	no	—
127	C₂×D₄	C₂×Dic₁₃₉	1	8	yes	—
128	C₂×D₄	C₂×Dic₁₃₉	1	4	yes	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	C₂×Dic₁₃₉	1	2	yes	—
132	C₂²×C₄	C₂×Dic₁₃₉	1	4	yes	—
133	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
134	C₂²×C₄	C₂×Dic₁₃₉	1	2	yes	—
135	C₂²×C₄	C₂²⋊D₁₃₉	1	4	yes	—
136	C₂²×C₄	C₂²⋊D₁₃₉	1	4	yes	—
137	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
138	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
139	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
140	M₁₆	—	278	8	no	—
141	C₂×C₈	—	278	4	no	—
142	C₂×C₈	—	278	8	no	—
143	C₂×C₈	—	278	4	no	—
144	C₂×C₈	—	139	4	no	—
145	C₂×C₈	—	139	4	no	—
146	C₂³	—	2	4	no	—
147	C₂³	—	2	4	no	—
148	C₂³	—	2	8	no	—
149	C₂³	C₂²⋊Dic₁₃₉	1	4	yes	—
150	C₂³	C₂²×Dic₁₃₉	1	4	yes	—
151	C₂³	C₂²×Dic₁₃₉	1	2	yes	—
152	C₂³	C₂²×Dic₁₃₉	1	1	yes	characteristic
153	Q₈	—	2	8	no	—
154	D₄	—	2	16	no	—
155	D₄	—	2	8	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	C₂²⋊Dic₁₃₉	1	4	yes	—
161	C₂×C₄	C₂²⋊Dic₁₃₉	1	8	yes	—
162	C₂×C₄	C₂²×Dic₁₃₉	1	4	yes	—
163	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
164	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
165	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
166	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
167	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	4	yes	—
168	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	4	yes	—
169	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
170	C₈	—	278	4	no	—
171	C₈	—	278	4	no	—
172	C₂²	—	2	8	no	—
173	C₂²	—	2	4	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	4	8	no	—
177	C₂²	C₂×C₂²⋊Dic₁₃₉	1	4	yes	—
178	C₂²	C₂×C₂²⋊Dic₁₃₉	1	2	yes	—
179	C₂²	#17984849	1	4	yes	—
180	C₂²	C₂³×Dic₁₃₉	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	C₂×C₂²⋊Dic₁₃₉	1	2	yes	—
183	C₄	C₂×C₂²⋊Dic₁₃₉	1	4	yes	—
184	C₄	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
185	C₄	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
186	C₂	—	2	4	no	—
187	C₂	—	4	4	no	—
188	C₂	#22580920	1	4	yes	direct product factor
189	C₂	C₂²×C₂²⋊Dic₁₃₉	1	1	yes	characteristic
190	C₂	C₂×#17984849	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃₉
commutator	C₂²×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting, prime radical	C₂²×#41×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂²×C₄
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂²×#41	D₁₃₉
hypocenter	C₁₃₉	C₂×#408
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555769	C₂	semidirect
2. C₂³×C₁₃₉⋊C₈	C₂	semidirect
3. #22580920	C₂	direct (N × H)
4. C₂×C₂²⋊(C₁₃₉⋊C₈)	C₂	semidirect
7. C₂×#14555769	C₂²	semidirect
11. C₂²×C₁₃₉⋊C₈	C₂²	semidirect
12. C₂²⋊(C₁₃₉⋊C₈)	C₂²	semidirect
92. C₂×C₁₃₉	#408	semidirect
95. C₁₃₉	C₂×#408	semidirect
188. C₂	#22580920	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×#14555769	2
C₂³×C₁₃₉⋊C₈	2
#22580920	2
C₂×C₂²⋊(C₁₃₉⋊C₈)	2
C₂²×#41×C₁₃₉	2
C₂×#408	139

Subgroups — C2×#22580920 #31814844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22580920 #31814844