Subgroups — #31819809

Order 17792.

996 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22581584	C₂	1	4	yes	maximal, complemented
2	#22581581	C₂	1	8	yes	maximal, complemented
3	#22581579	C₂	1	4	yes	maximal
4	#22581510	C₂	1	4	yes	maximal, complemented
5	#22581506	C₂	1	8	yes	maximal, complemented
6	C₂×#17984801	C₂	1	2	yes	maximal
7	C₂×#117×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	#17984842	C₂²	1	8	yes	complemented
9	#17984841	C₂²	1	16	yes	complemented
10	#17984834	C₂²	1	8	yes	complemented
11	#17984833	C₂²	1	16	yes	complemented
12	#17984821	C₂²	1	16	yes	—
13	#17984819	C₂²	1	32	yes	—
14	C₄×Dic₂₇₈	C₂²	1	16	yes	complemented
15	#17984801	C₂²	1	4	yes	—
16	#17984801	C₂²	1	4	yes	—
17	#17984801	C₂²	1	4	yes	—
18	#17984801	C₂²	1	4	yes	—
19	C₂×C₄⋊Dic₁₃₉	C₂²	1	4	yes	—
20	C₂×C₂²⋊Dic₁₃₉	C₂²	1	4	yes	—
21	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
22	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
23	#117×C₁₃₉	C₂²	1	4	yes	—
24	#117×C₁₃₉	C₂²	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₃₉	C₂²	1	2	yes	—
28	C₂×Dic₂₇₈	C₂³	1	16	yes	—
29	C₂²×Dic₁₃₉	C₂³	1	8	yes	—
30	C₄⋊Dic₁₃₉	C₂³	1	8	yes	—
31	C₄⋊Dic₁₃₉	C₂³	1	8	yes	—
32	C₄⋊Dic₁₃₉	C₂³	1	16	yes	—
33	Dic₁₃₉⋊C₄	C₂³	1	32	yes	—
34	C₂²⋊Dic₁₃₉	C₂³	1	8	yes	—
35	C₂²⋊Dic₁₃₉	C₂³	1	8	yes	—
36	C₄×Dic₁₃₉	C₂³	1	8	yes	—
37	C₄×Dic₁₃₉	C₂³	1	8	yes	—
38	C₂⁴×C₁₃₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₁₃₉	Q₈	1	8	yes	complemented
40	C₂²×C₄×C₁₃₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₃₉	C₂³	1	8	yes	—
46	C₂²⋊C₄×C₁₃₉	C₂³	1	8	yes	—
47	C₄²×C₁₃₉	C₂³	1	4	yes	—
48	C₄²×C₁₃₉	C₂³	1	4	yes	—
49	Dic₂₇₈	—	4	16	no	—
50	C₂×Dic₁₃₉	—	4	8	no	—
51	C₂×Dic₁₃₉	C₂⁴	1	16	yes	—
52	C₂³×C₁₃₉	—	4	2	no	—
53	C₂³×C₁₃₉	C₂×Q₈	1	4	yes	—
54	C₂³×C₁₃₉	C₂⁴	1	2	yes	—
55	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	—	4	4	no	—
57	C₂×C₄×C₁₃₉	C₂×Q₈	1	8	yes	—
58	C₂×C₄×C₁₃₉	C₂×Q₈	1	8	yes	—
59	C₂×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
60	C₂×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
61	C₂×C₄×C₁₃₉	C₂⁴	1	8	yes	—
62	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
63	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
64	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
65	C₂²×C₁₃₉	—	2	2	no	—
66	C₂²×C₁₃₉	—	2	4	no	—
67	C₂²×C₁₃₉	—	4	4	no	—
68	C₂²×C₁₃₉	C₂²×Q₈	1	4	yes	—
69	C₂²×C₁₃₉	C₂²×Q₈	1	2	yes	—
70	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
71	C₄×C₁₃₉	—	2	8	no	—
72	C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
73	C₄×C₁₃₉	C₂²×Q₈	1	4	yes	—
74	C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
75	Dic₁₃₉	—	2	16	no	—
76	C₂×C₁₃₉	—	2	2	no	—
77	C₂×C₁₃₉	—	2	4	no	—
78	C₂×C₁₃₉	C₂³×Q₈	1	1	yes	characteristic
79	C₂×C₁₃₉	#585	1	2	yes	—
80	C₁₃₉	#3458	1	1	yes	characteristic, complemented, Sylow
81	#3458	—	139	1	no	maximal, Sylow
82	#544	—	139	8	no	—
83	#544	—	139	4	no	—
84	#543	—	139	4	no	—
85	#533	—	139	8	no	—
86	#533	—	139	4	no	—
87	C₂×#117	—	139	2	no	—
88	C₂×#117	D₁₃₉	1	1	yes	characteristic
89	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
90	C₄⋊Q₈	—	139	16	no	—
91	C₄⋊Q₈	—	139	8	no	—
92	#125	—	139	16	no	—
93	#125	—	139	8	no	—
94	C₂²⋊Q₈	—	139	16	no	—
95	C₂²⋊Q₈	—	139	32	no	—
96	C₄×Q₈	—	139	16	no	—
97	#117	—	139	4	no	—
98	#117	—	139	4	no	—
99	#117	—	139	4	no	—
100	#117	—	139	4	no	—
101	#117	C₂×D₁₃₉	1	4	yes	—
102	#117	C₂×D₁₃₉	1	4	yes	—
103	C₂×C₄⋊C₄	—	139	4	no	—
104	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
105	C₂×C₂²⋊C₄	—	139	4	no	—
106	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
107	C₂×C₄²	—	139	4	no	—
108	C₂×C₄²	C₂×D₁₃₉	1	2	yes	—
109	C₂⁴	C₂²×D₁₃₉	1	1	yes	characteristic
110	C₂×Q₈	—	139	16	no	—
111	C₂²×C₄	—	139	8	no	—
112	C₂²×C₄	Dic₂₇₈	1	8	yes	complemented
113	C₂²×C₄	C₂²×D₁₃₉	1	4	yes	—
114	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
115	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
117	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
118	C₄⋊C₄	—	139	8	no	—
119	C₄⋊C₄	—	139	8	no	—
120	C₄⋊C₄	—	139	32	no	—
121	C₄⋊C₄	—	139	16	no	—
122	C₄⋊C₄	C₂²×D₁₃₉	1	8	yes	—
123	C₂²⋊C₄	—	139	8	no	—
124	C₂²⋊C₄	—	139	8	no	—
125	C₂²⋊C₄	C₂²×D₁₃₉	1	8	yes	—
126	C₄²	—	139	8	no	—
127	C₄²	—	139	8	no	—
128	C₄²	C₂²×D₁₃₉	1	4	yes	—
129	C₄²	C₂²×D₁₃₉	1	4	yes	—
130	C₂³	—	4	2	no	—
131	C₂³	C₂×Dic₂₇₈	1	4	yes	—
132	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
133	C₂³	C₂³×D₁₃₉	1	2	yes	—
134	Q₈	—	556	16	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	556	8	no	—
137	C₂×C₄	—	139	16	no	—
138	C₂×C₄	C₂×Dic₂₇₈	1	8	yes	—
139	C₂×C₄	C₂×Dic₂₇₈	1	4	yes	—
140	C₂×C₄	C₂×Dic₂₇₈	1	8	yes	—
141	C₂×C₄	C₂×Dic₂₇₈	1	4	yes	—
142	C₂×C₄	C₂³×D₁₃₉	1	8	yes	—
143	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
144	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
145	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
146	C₂²	—	2	2	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	4	4	no	—
149	C₂²	C₂²×Dic₂₇₈	1	2	yes	—
150	C₂²	C₂²×Dic₂₇₈	1	4	yes	—
151	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
152	C₄	—	2	8	no	—
153	C₄	—	278	16	no	—
154	C₄	C₂²×Dic₂₇₈	1	4	yes	—
155	C₄	C₂²×Dic₂₇₈	1	2	yes	—
156	C₄	C₂²×Dic₂₇₈	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	2	4	no	—
159	C₂	C₂³×Dic₂₇₈	1	1	yes	characteristic
160	C₂	#22581789	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₃₉
commutator, socle, cosocle	C₂²×C₁₃₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₃₉
Fitting	C₂×#117×C₁₃₉	C₂
prime radical	C₂⁴×C₁₃₉	C₂³
hypercenter	C₂×#117	D₁₃₉
hypocenter	C₁₃₉	#3458
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
1. #22581584	C₂	semidirect
2. #22581581	C₂	semidirect
4. #22581510	C₂	semidirect
5. #22581506	C₂	semidirect
8. #17984842	C₂²	semidirect
9. #17984841	C₂²	semidirect
10. #17984834	C₂²	semidirect
11. #17984833	C₂²	semidirect
14. C₄×Dic₂₇₈	C₂²	semidirect
39. C₂²×C₄×C₁₃₉	Q₈	semidirect
80. C₁₃₉	#3458	semidirect
112. C₂²×C₄	Dic₂₇₈	semidirect

Maximal subgroups

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

subgroup	index
#22581584	2
#22581581	2
#22581579	2
#22581510	2
#22581506	2
C₂×#17984801	2
C₂×#117×C₁₃₉	2
#3458	139