Subgroups — C₂×#22581430 #31819594

Order 17792.

1488 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	#22581430	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Dic₂₇₈	C₂	1	4	yes	maximal, complemented
4	C₂×#17984807	C₂	1	4	yes	maximal, complemented
5	C₂×#17984801	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984798	C₂	1	4	yes	maximal
7	C₂×#117×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₇₈	C₄	1	16	yes	complemented
9	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
10	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
11	C₂²×Dic₂₇₈	C₂²	1	8	yes	—
12	C₂³×Dic₁₃₉	C₂²	1	2	yes	—
13	C₄×Dic₂₇₈	C₂²	1	32	yes	complemented
14	#17984807	C₂²	1	32	yes	complemented
15	#17984801	C₂²	1	8	yes	—
16	#17984798	C₂²	1	32	yes	—
17	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
18	C₂×Dic₁₃₉⋊C₄	C₂²	1	8	yes	—
19	C₂×C₂²⋊Dic₁₃₉	C₂²	1	2	yes	—
20	C₂×C₄×Dic₁₃₉	C₂²	1	8	yes	—
21	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
22	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
23	#117×C₁₃₉	C₂²	1	8	yes	—
24	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	—
27	C₂×Dic₂₇₈	C₂×C₄	1	16	yes	—
28	C₂×Dic₂₇₈	C₂×C₄	1	16	yes	—
29	C₂×Dic₂₇₈	C₂×C₄	1	64	yes	complemented
30	C₂×Dic₂₇₈	C₂³	1	8	yes	—
31	C₂×Dic₂₇₈	C₂³	1	8	yes	—
32	C₂²×Dic₁₃₉	C₂×C₄	1	16	yes	—
33	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
34	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
35	C₂²×Dic₁₃₉	C₂³	1	8	yes	—
36	C₄⋊Dic₁₃₉	C₂³	1	8	yes	—
37	Dic₁₃₉⋊C₄	C₂³	1	32	yes	—
38	C₂²⋊Dic₁₃₉	C₂³	1	8	yes	—
39	C₄×Dic₁₃₉	C₂³	1	32	yes	—
40	C₄×Dic₁₃₉	C₂³	1	8	yes	—
41	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃₉	C₂×C₄	1	8	yes	—
43	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
44	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
47	C₄⋊C₄×C₁₃₉	C₂³	1	8	yes	—
48	C₂²⋊C₄×C₁₃₉	C₂³	1	8	yes	—
49	C₄²×C₁₃₉	C₂³	1	8	yes	—
50	Dic₂₇₈	C₂²×C₄	1	32	yes	—
51	Dic₂₇₈	C₂²×C₄	1	32	yes	—
52	C₂×Dic₁₃₉	—	2	8	no	—
53	C₂×Dic₁₃₉	C₂²×C₄	1	16	yes	—
54	C₂×Dic₁₃₉	C₂²×C₄	1	32	yes	—
55	C₂×Dic₁₃₉	C₂⁴	1	8	yes	—
56	C₂×Dic₁₃₉	C₂⁴	1	8	yes	—
57	C₂³×C₁₃₉	—	2	4	no	—
58	C₂³×C₁₃₉	C₂²×C₄	1	4	yes	—
59	C₂³×C₁₃₉	C₂⁴	1	2	yes	—
60	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
61	C₂×C₄×C₁₃₉	—	2	8	no	—
62	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
63	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
64	C₂×C₄×C₁₃₉	C₂²×C₄	1	16	yes	—
65	C₂×C₄×C₁₃₉	C₂⁴	1	8	yes	—
66	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
67	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
68	C₂²×C₁₃₉	—	2	8	no	—
69	C₂²×C₁₃₉	—	2	4	no	—
70	C₂²×C₁₃₉	C₂³×C₄	1	4	yes	—
71	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
72	C₂²×C₁₃₉	2₋¹⁺⁴	1	4	yes	—
73	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
74	C₄×C₁₃₉	—	2	8	no	—
75	C₄×C₁₃₉	C₂³×C₄	1	4	yes	—
76	C₄×C₁₃₉	C₂³×C₄	1	4	yes	—
77	Dic₁₃₉	—	2	8	no	—
78	Dic₁₃₉	C₂³×C₄	1	16	yes	—
79	C₂×C₁₃₉	—	2	4	no	—
80	C₂×C₁₃₉	#519	1	4	yes	complemented
81	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
82	C₂×C₁₃₉	C₂×2₋¹⁺⁴	1	2	yes	—
83	C₁₃₉	C₂×#519	1	1	yes	characteristic, complemented, Sylow
84	C₂×#519	—	139	1	no	maximal, Sylow
85	C₂³×Q₈	—	139	1	no	—
86	#519	—	139	16	no	—
87	C₂×C₄×Q₈	—	139	4	no	—
88	C₂×C₄×Q₈	—	139	4	no	—
89	C₂×#117	—	139	1	no	—
90	C₂×#117	—	139	4	no	—
91	C₂×#117	D₁₃₉	1	1	yes	characteristic
92	C₂²×Q₈	—	139	16	no	—
93	C₂²×Q₈	—	139	2	no	—
94	C₂²×Q₈	—	139	2	no	—
95	C₂²×Q₈	—	139	8	no	—
96	C₂³×C₄	—	139	2	no	—
97	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
98	C₄×Q₈	—	139	32	no	—
99	C₄×Q₈	—	139	32	no	—
100	#117	—	139	8	no	—
101	#117	—	139	32	no	—
102	#117	C₂×D₁₃₉	1	8	yes	—
103	C₂×C₄⋊C₄	—	139	8	no	—
104	C₂×C₄⋊C₄	—	139	2	no	—
105	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
106	C₂×C₂²⋊C₄	—	139	2	no	—
107	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
108	C₂×C₄²	—	139	2	no	—
109	C₂×C₄²	—	139	8	no	—
110	C₂×C₄²	C₂×D₁₃₉	1	2	yes	—
111	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
112	C₂×Q₈	—	139	16	no	—
113	C₂×Q₈	—	139	16	no	—
114	C₂×Q₈	—	139	64	no	—
115	C₂×Q₈	—	139	8	no	—
116	C₂×Q₈	—	139	8	no	—
117	C₂²×C₄	—	139	4	no	—
118	C₂²×C₄	—	139	16	no	—
119	C₂²×C₄	—	139	8	no	—
120	C₂²×C₄	—	139	4	no	—
121	C₂²×C₄	C₄×D₁₃₉	1	8	yes	—
122	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
123	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
124	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
126	C₄⋊C₄	—	139	32	no	—
127	C₄⋊C₄	—	139	8	no	—
128	C₄⋊C₄	C₂²×D₁₃₉	1	8	yes	—
129	C₂²⋊C₄	—	139	8	no	—
130	C₂²⋊C₄	C₂²×D₁₃₉	1	8	yes	—
131	C₄²	—	139	8	no	—
132	C₄²	—	139	32	no	—
133	C₄²	C₂²×D₁₃₉	1	8	yes	—
134	C₂³	—	2	4	no	—
135	C₂³	C₂×C₄×D₁₃₉	1	4	yes	—
136	C₂³	C₂³×D₁₃₉	1	2	yes	—
137	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
138	Q₈	—	139	32	no	—
139	Q₈	—	139	32	no	—
140	C₂×C₄	—	2	8	no	—
141	C₂×C₄	—	278	8	no	—
142	C₂×C₄	—	139	8	no	—
143	C₂×C₄	—	139	32	no	—
144	C₂×C₄	—	139	16	no	—
145	C₂×C₄	—	139	8	no	—
146	C₂×C₄	C₂×C₄×D₁₃₉	1	16	yes	—
147	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
148	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
149	C₂×C₄	C₂³×D₁₃₉	1	8	yes	—
150	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
151	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	8	no	—
154	C₂²	C₂²×C₄×D₁₃₉	1	4	yes	—
155	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
156	C₂²	#17984889	1	4	yes	—
157	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
158	C₄	—	2	8	no	—
159	C₄	—	278	8	no	—
160	C₄	—	139	16	no	—
161	C₄	C₂²×C₄×D₁₃₉	1	4	yes	—
162	C₄	C₂²×C₄×D₁₃₉	1	4	yes	—
163	C₂	—	2	4	no	—
164	C₂	#22581430	1	4	yes	direct product factor
165	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
166	C₂	C₂×#17984889	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×#117×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂⁴×C₁₃₉	C₂³
cosocle	C₂²×C₁₃₉	C₂⁵
hypercenter	C₂×#117	D₁₃₉
hypocenter	C₁₃₉	C₂×#519
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #22581430	C₂	direct (N × H)
3. C₂×C₄×Dic₂₇₈	C₂	semidirect
4. C₂×#17984807	C₂	semidirect
8. C₂²×Dic₂₇₈	C₄	semidirect
13. C₄×Dic₂₇₈	C₂²	semidirect
14. #17984807	C₂²	semidirect
29. C₂×Dic₂₇₈	C₂×C₄	semidirect
80. C₂×C₁₃₉	#519	semidirect
83. C₁₃₉	C₂×#519	semidirect
164. C₂	#22581430	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₂₇₈	2
#22581430	2
C₂×C₄×Dic₂₇₈	2
C₂×#17984807	2
C₂×#17984801	2
C₂×#17984798	2
C₂×#117×C₁₃₉	2
C₂×#519	139

Subgroups — C2×#22581430 #31819594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581430 #31819594