Subgroups — #31808765

Order 17792.

780 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₁₃₉	C₂	1	2	yes	maximal
2	C₄⋊C₄×Dic₁₃₉	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₄×Dic₁₃₉	C₂	1	4	yes	maximal, complemented
5	C₄×C₂²⋊Dic₁₃₉	C₂	1	2	yes	maximal, complemented
6	#22580799	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃₉⋊#95	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₁₃₉	C₂²	1	4	yes	—
10	D₄×Dic₁₃₉	C₄	1	16	yes	complemented
11	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₁₃₉	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₃₉	C₂²	1	4	yes	—
14	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄×Dic₁₃₉	C₂²	1	2	yes	—
16	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
17	C₁₃₉⋊#95	C₂²	1	2	yes	—
18	C₁₃₉⋊#95	C₂²	1	4	yes	—
19	C₁₃₉⋊#95	C₂²	1	1	yes	characteristic
20	C₁₃₉⋊#95	C₂²	1	2	yes	—
21	C₂²×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₁₃₉	C₂²	1	2	yes	—
23	C₄×D₄×C₁₃₉	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₁₃₉	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
27	C₂²×Dic₁₃₉	—	4	4	no	—
28	C₂²×Dic₁₃₉	—	4	4	no	—
29	C₂²×Dic₁₃₉	C₂×C₄	1	16	yes	complemented
30	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
31	C₂²×Dic₁₃₉	C₂³	1	4	yes	—
32	C₂²×Dic₁₃₉	C₂³	1	2	yes	—
33	C₄⋊Dic₁₃₉	C₂×C₄	1	8	yes	complemented
34	C₂²⋊Dic₁₃₉	C₂×C₄	1	16	yes	complemented
35	C₄×Dic₁₃₉	—	4	4	no	—
36	C₄×Dic₁₃₉	—	4	2	no	—
37	C₄×Dic₁₃₉	C₂×C₄	1	8	yes	complemented
38	C₂⁴×C₁₃₉	C₂³	1	2	yes	—
39	C₂×D₄×C₁₃₉	C₂×C₄	1	8	yes	—
40	C₂×D₄×C₁₃₉	C₂×C₄	1	4	yes	—
41	C₂²×C₄×C₁₃₉	—	4	4	no	—
42	C₂²×C₄×C₁₃₉	C₂×C₄	1	8	yes	complemented
43	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₁₃₉	C₂×C₄	1	4	yes	complemented
48	C₂²⋊C₄×C₁₃₉	C₂×C₄	1	8	yes	complemented
49	C₄²×C₁₃₉	C₂×C₄	1	4	yes	complemented
50	C₂×Dic₁₃₉	—	2	2	no	—
51	C₂×Dic₁₃₉	—	2	4	no	—
52	C₂×Dic₁₃₉	—	2	4	no	—
53	C₂×Dic₁₃₉	—	2	2	no	—
54	C₂×Dic₁₃₉	—	2	2	no	—
55	C₂×Dic₁₃₉	—	2	2	no	—
56	C₂×Dic₁₃₉	—	4	16	no	—
57	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
58	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
59	C₂×Dic₁₃₉	C₂²×C₄	1	8	yes	—
60	C₂³×C₁₃₉	—	2	4	no	—
61	C₂³×C₁₃₉	—	2	2	no	—
62	C₂³×C₁₃₉	—	2	2	no	—
63	C₂³×C₁₃₉	C₂²×C₄	1	8	yes	—
64	C₂³×C₁₃₉	C₂²×C₄	1	4	yes	—
65	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
66	D₄×C₁₃₉	C₄²	1	16	yes	complemented
67	C₂×C₄×C₁₃₉	—	2	2	no	—
68	C₂×C₄×C₁₃₉	—	2	4	no	—
69	C₂×C₄×C₁₃₉	—	2	1	no	—
70	C₂×C₄×C₁₃₉	—	2	1	no	—
71	C₂×C₄×C₁₃₉	—	4	8	no	—
72	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
73	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
76	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
77	C₂²×C₁₃₉	—	2	4	no	—
78	C₂²×C₁₃₉	—	2	8	no	—
79	C₂²×C₁₃₉	—	2	8	no	—
80	C₂²×C₁₃₉	—	2	4	no	—
81	C₂²×C₁₃₉	C₂×C₄²	1	8	yes	—
82	C₂²×C₁₃₉	C₂³×C₄	1	2	yes	—
83	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
85	C₂²×C₁₃₉	H(5,𝔽₂)	1	1	yes	characteristic
86	C₂²×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
87	C₄×C₁₃₉	—	2	2	no	—
88	C₄×C₁₃₉	—	2	4	no	—
89	C₄×C₁₃₉	—	2	2	no	—
90	C₄×C₁₃₉	C₂×C₄²	1	4	yes	—
91	Dic₁₃₉	—	2	4	no	—
92	Dic₁₃₉	—	2	8	no	—
93	Dic₁₃₉	—	2	4	no	—
94	C₂×C₁₃₉	—	2	8	no	—
95	C₂×C₁₃₉	C₂²×C₄²	1	1	yes	characteristic
96	C₂×C₁₃₉	#518	1	2	yes	—
97	C₂×C₁₃₉	#518	1	1	yes	characteristic
98	C₂×C₁₃₉	#520	1	2	yes	—
99	C₂×C₁₃₉	#520	1	1	yes	characteristic
100	C₁₃₉	#2275	1	1	yes	characteristic, complemented, Sylow
101	#2275	—	139	1	no	maximal, Sylow
102	C₂×C₄×D₄	—	139	2	no	—
103	C₂×C₄×D₄	D₁₃₉	1	1	yes	characteristic
104	C₄×C₄⋊C₄	—	139	2	no	—
105	C₄×C₄⋊C₄	—	139	1	no	—
106	C₄×C₂²⋊C₄	—	139	2	no	—
107	C₄×C₂²⋊C₄	—	139	4	no	—
108	#376	—	139	1	no	—
109	C₂×#95	—	139	2	no	—
110	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
111	C₂³×C₄	—	139	4	no	—
112	C₂³×C₄	C₂×D₁₃₉	1	2	yes	—
113	C₄×D₄	—	139	16	no	—
114	C₄×D₄	Dic₁₃₉	1	8	yes	complemented
115	C₂×C₄⋊C₄	—	139	2	no	—
116	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
117	C₂×C₂²⋊C₄	—	139	4	no	—
118	C₂×C₂²⋊C₄	C₂×D₁₃₉	1	2	yes	—
119	C₂×C₄²	—	139	4	no	—
120	C₂×C₄²	—	139	2	no	—
121	C₂×C₄²	—	139	2	no	—
122	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
123	#95	—	139	1	no	—
124	#95	—	139	4	no	—
125	#95	—	139	2	no	—
126	#95	—	139	2	no	—
127	#95	—	139	1	no	—
128	C₂⁴	C₂²×D₁₃₉	1	2	yes	—
129	C₂×D₄	C₄×D₁₃₉	1	8	yes	—
130	C₂×D₄	C₂×Dic₁₃₉	1	4	yes	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	—	556	4	no	—
133	C₂²×C₄	—	556	4	no	—
134	C₂²×C₄	—	139	2	no	—
135	C₂²×C₄	—	139	2	no	—
136	C₂²×C₄	—	139	4	no	—
137	C₂²×C₄	—	139	16	no	—
138	C₂²×C₄	C₂×Dic₁₃₉	1	8	yes	complemented
139	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
140	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
141	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
142	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
143	C₄⋊C₄	—	139	8	no	—
144	C₄⋊C₄	C₂×Dic₁₃₉	1	4	yes	complemented
145	C₂²⋊C₄	—	139	16	no	—
146	C₂²⋊C₄	C₂×Dic₁₃₉	1	8	yes	complemented
147	C₄²	—	556	4	no	—
148	C₄²	—	556	2	no	—
149	C₄²	—	139	8	no	—
150	C₄²	C₂×Dic₁₃₉	1	4	yes	complemented
151	C₂³	—	2	4	no	—
152	C₂³	—	2	2	no	—
153	C₂³	—	2	2	no	—
154	C₂³	C₂×C₄×D₁₃₉	1	8	yes	—
155	C₂³	C₂²×Dic₁₃₉	1	4	yes	—
156	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
157	D₄	C₄×Dic₁₃₉	1	16	yes	complemented
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	4	8	no	—
163	C₂×C₄	—	278	2	no	—
164	C₂×C₄	—	278	4	no	—
165	C₂×C₄	—	278	2	no	—
166	C₂×C₄	—	278	4	no	—
167	C₂×C₄	—	278	2	no	—
168	C₂×C₄	—	278	2	no	—
169	C₂×C₄	—	556	16	no	—
170	C₂×C₄	—	139	4	no	—
171	C₂×C₄	—	139	4	no	—
172	C₂×C₄	—	139	8	no	—
173	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
174	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
175	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
176	C₂×C₄	C₂²×Dic₁₃₉	1	4	yes	—
177	C₂×C₄	C₂²×Dic₁₃₉	1	2	yes	—
178	C₂²	—	2	8	no	—
179	C₂²	—	2	4	no	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	4	no	—
182	C₂²	C₂×C₄×Dic₁₃₉	1	8	yes	—
183	C₂²	C₂²×C₄×D₁₃₉	1	2	yes	—
184	C₂²	C₂³×Dic₁₃₉	1	1	yes	characteristic
185	C₂²	#17984887	1	2	yes	—
186	C₂²	#17984888	1	1	yes	characteristic
187	C₂²	#17984889	1	1	yes	characteristic
188	C₄	—	2	2	no	—
189	C₄	—	2	4	no	—
190	C₄	—	2	2	no	—
191	C₄	—	278	4	no	—
192	C₄	—	278	8	no	—
193	C₄	—	278	4	no	—
194	C₄	C₂×C₄×Dic₁₃₉	1	4	yes	—
195	C₂	—	2	8	no	—
196	C₂	C₂²×C₄×Dic₁₃₉	1	1	yes	characteristic
197	C₂	#22581425	1	2	yes	—
198	C₂	#22581428	1	1	yes	characteristic
199	C₂	#22581432	1	2	yes	—
200	C₂	#22581436	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator	C₂×C₁₃₉	C₂²×C₄²
Fitting	C₂×C₄×D₄×C₁₃₉	C₂
socle, cosocle	C₂³×C₁₃₉	C₂⁴
prime radical	C₂²×D₄×C₁₃₉	C₂²
hypercenter	C₂×C₄×D₄	D₁₃₉
hypocenter	C₁₃₉	#2275
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×Dic₁₃₉	C₂	semidirect
3. C₄×C₄⋊Dic₁₃₉	C₂	semidirect
4. C₂²⋊C₄×Dic₁₃₉	C₂	semidirect
5. C₄×C₂²⋊Dic₁₃₉	C₂	semidirect
6. #22580799	C₂	semidirect
7. C₂×C₁₃₉⋊#95	C₂	semidirect
10. D₄×Dic₁₃₉	C₄	semidirect
23. C₄×D₄×C₁₃₉	C₄	semidirect
29. C₂²×Dic₁₃₉	C₂×C₄	semidirect
33. C₄⋊Dic₁₃₉	C₂×C₄	semidirect
34. C₂²⋊Dic₁₃₉	C₂×C₄	semidirect
37. C₄×Dic₁₃₉	C₂×C₄	semidirect
42. C₂²×C₄×C₁₃₉	C₂×C₄	semidirect
47. C₄⋊C₄×C₁₃₉	C₂×C₄	semidirect
48. C₂²⋊C₄×C₁₃₉	C₂×C₄	semidirect
49. C₄²×C₁₃₉	C₂×C₄	semidirect
66. D₄×C₁₃₉	C₄²	semidirect
100. C₁₃₉	#2275	semidirect
114. C₄×D₄	Dic₁₃₉	semidirect
138. C₂²×C₄	C₂×Dic₁₃₉	semidirect
144. C₄⋊C₄	C₂×Dic₁₃₉	semidirect
146. C₂²⋊C₄	C₂×Dic₁₃₉	semidirect
150. C₄²	C₂×Dic₁₃₉	semidirect
157. D₄	C₄×Dic₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Dic₁₃₉	2
C₄⋊C₄×Dic₁₃₉	2
C₄×C₄⋊Dic₁₃₉	2
C₂²⋊C₄×Dic₁₃₉	2
C₄×C₂²⋊Dic₁₃₉	2
#22580799	2
C₂×C₁₃₉⋊#95	2
C₂×C₄×D₄×C₁₃₉	2
#2275	139