Subgroups — #31815147

Order 17792.

492 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984848	C₂	1	2	yes	maximal, complemented
2	#22581503	C₂	1	1	yes	maximal, characteristic
3	#22581030	C₂	1	2	yes	maximal, complemented
4	#22581023	C₂	1	4	yes	maximal, complemented
5	#22581018	C₂	1	2	yes	maximal, complemented
6	#22580919	C₂	1	2	yes	maximal, complemented
7	C₄×#14555769	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555785	C₂²	1	2	yes	—
10	#17984848	—	2	8	no	—
11	C₂×#14555769	C₂²	1	1	yes	characteristic
12	C₂×#14555769	C₂²	1	1	yes	characteristic
13	C₂×#14555768	C₂²	1	4	yes	—
14	C₂×C₈×D₁₃₉	C₂²	1	4	yes	—
15	#17984842	C₄	1	1	yes	characteristic
16	#17984839	C₄	1	1	yes	characteristic
17	#17984830	C₄	1	2	yes	—
18	#17984816	C₄	1	2	yes	—
19	#17984816	C₄	1	2	yes	—
20	C₄×D₅₅₆	C₂²	1	2	yes	—
21	C₄×D₅₅₆	C₂²	1	2	yes	—
22	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	—
23	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	—
24	#17984753	C₂²	1	2	yes	—
25	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
26	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₃₉	C₂²	1	1	yes	characteristic
31	#14555785	—	2	4	no	—
32	#14555785	—	2	4	no	—
33	C₂×Dic₂₇₈	C₂×C₄	1	2	yes	—
34	C₂×D₅₅₆	C₂×C₄	1	2	yes	—
35	C₂×D₅₅₆	C₂×C₄	1	2	yes	—
36	C₂×D₅₅₆	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊D₁₃₉	C₂×C₄	1	4	yes	—
38	C₂×C₄×D₁₃₉	C₂³	1	4	yes	—
39	#14555769	—	2	2	no	—
40	#14555769	D₄	1	4	yes	complemented
41	#14555768	—	2	8	no	—
42	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
43	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
44	C₈×D₁₃₉	—	2	8	no	—
45	C₄⋊Dic₁₃₉	C₂×C₄	1	2	yes	—
46	C₄⋊Dic₁₃₉	C₂×C₄	1	2	yes	—
47	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	4	yes	—
48	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	4	yes	—
49	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₁₃₉	—	2	4	no	—
53	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
54	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
55	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
56	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
58	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
59	C₂²×D₁₃₉	C₂²×C₄	1	4	yes	—
60	Dic₂₇₈	—	2	4	no	—
61	D₅₅₆	—	2	4	no	—
62	D₅₅₆	—	2	4	no	—
63	D₅₅₆	—	4	2	no	—
64	C₂²⋊D₁₃₉	—	2	8	no	—
65	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
66	C₄×D₁₃₉	—	2	4	no	—
67	C₄×D₁₃₉	—	2	4	no	—
68	C₁₃₉⋊C₈	—	2	2	no	—
69	C₁₃₉⋊C₈	C₂×D₄	1	4	yes	—
70	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₃₉	—	2	2	no	—
72	C₂×C₄×C₁₃₉	—	4	1	no	—
73	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃₉	#41	1	2	yes	—
80	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
81	C₈×C₁₃₉	—	2	2	no	—
82	C₈×C₁₃₉	—	2	2	no	—
83	C₂²×C₁₃₉	—	2	1	no	—
84	C₂²×C₁₃₉	—	2	1	no	—
85	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
86	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
87	C₂×D₁₃₉	—	2	4	no	—
88	C₂×D₁₃₉	—	2	4	no	—
89	C₂×D₁₃₉	—	2	4	no	—
90	C₄×C₁₃₉	—	2	1	no	—
91	C₄×C₁₃₉	—	2	1	no	—
92	C₄×C₁₃₉	—	2	1	no	—
93	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
94	C₄×C₁₃₉	#131	1	2	yes	complemented
95	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
96	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
97	Dic₁₃₉	—	2	4	no	—
98	C₂×C₁₃₉	—	2	2	no	—
99	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
100	C₂×C₁₃₉	C₂×#131	1	1	yes	characteristic
101	C₂×C₁₃₉	#568	1	1	yes	characteristic
102	D₁₃₉	—	4	4	no	—
103	C₁₃₉	#2935	1	1	yes	characteristic, complemented, Sylow
104	#2935	—	139	1	no	maximal, Sylow
105	C₂×#131	—	139	2	no	—
106	#532	—	139	1	no	—
107	#436	—	139	2	no	—
108	#435	—	139	4	no	—
109	C₈×D₄	—	139	2	no	—
110	C₂×C₄⋊C₈	D₁₃₉	1	1	yes	characteristic, complemented
111	#408	—	139	2	no	—
112	C₄×M₁₆	—	139	1	no	—
113	C₂×#41	—	139	2	no	—
114	#131	—	278	8	no	—
115	C₂×M₁₆	—	139	1	no	—
116	C₂×M₁₆	—	139	1	no	—
117	C₂×M₁₆	—	139	4	no	—
118	C₂²×C₈	—	139	4	no	—
119	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
120	C₄⋊Q₈	—	139	1	no	—
121	#127	—	139	1	no	—
122	#124	—	139	2	no	—
123	#121	—	139	2	no	—
124	#121	—	139	2	no	—
125	C₄×D₄	—	139	2	no	—
126	C₄×D₄	—	139	2	no	—
127	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
128	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
129	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
130	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
131	C₂²⋊C₈	—	139	4	no	—
132	C₂²⋊C₈	—	139	4	no	—
133	#97	—	139	2	no	—
134	C₄×C₈	—	139	2	no	—
135	#41	—	278	4	no	—
136	#41	—	278	4	no	—
137	C₂×Q₈	—	139	2	no	—
138	C₂×D₄	—	139	4	no	—
139	C₂×D₄	—	139	2	no	—
140	C₂×D₄	—	139	2	no	—
141	C₂×D₄	—	139	2	no	—
142	C₂²×C₄	—	139	4	no	—
143	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
144	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
145	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
146	M₁₆	—	278	8	no	—
147	M₁₆	—	278	2	no	—
148	M₁₆	—	139	4	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	278	8	no	—
151	C₂×C₈	—	139	2	no	—
152	C₂×C₈	—	139	2	no	—
153	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
154	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
155	C₄⋊C₄	—	139	2	no	—
156	C₄⋊C₄	—	139	2	no	—
157	C₂²⋊C₄	—	139	4	no	—
158	C₂²⋊C₄	—	139	4	no	—
159	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
160	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
161	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
162	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
163	C₂³	—	139	4	no	—
164	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
165	Q₈	—	278	4	no	—
166	D₄	—	278	8	no	—
167	D₄	—	278	4	no	—
168	D₄	—	278	4	no	—
169	D₄	—	556	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	278	4	no	—
173	C₂×C₄	—	278	4	no	—
174	C₂×C₄	—	139	4	no	—
175	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
176	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
177	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
178	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
179	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
180	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
181	C₂×C₄	#14555786	1	2	yes	—
182	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	2	no	—
185	C₈	—	278	2	no	—
186	C₈	—	139	4	no	—
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	278	4	no	—
190	C₂²	—	278	4	no	—
191	C₂²	—	278	4	no	—
192	C₂²	#17984804	1	2	yes	—
193	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
194	C₄	—	2	1	no	—
195	C₄	—	2	1	no	—
196	C₄	—	2	1	no	—
197	C₄	—	278	4	no	—
198	C₄	#17984804	1	2	yes	—
199	C₄	#17984848	1	2	yes	complemented
200	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
201	C₄	C₂×#14555786	1	1	yes	characteristic
202	C₂	—	2	2	no	—
203	C₂	—	556	4	no	—
204	C₂	C₂×#17984804	1	1	yes	characteristic
205	C₂	C₂×#17984848	1	1	yes	characteristic
206	C₂	#22581719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂³×C₄
Fitting	C₂×C₄⋊C₈×C₁₃₉	C₂
prime radical	#22581503	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#2935
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#17984848	C₂	semidirect
3. #22581030	C₂	semidirect
4. #22581023	C₂	semidirect
5. #22581018	C₂	semidirect
6. #22580919	C₂	semidirect
7. C₄×#14555769	C₂	semidirect
8. C₂×C₄⋊C₈×C₁₃₉	C₂	semidirect
40. #14555769	D₄	semidirect
94. C₄×C₁₃₉	#131	semidirect
103. C₁₃₉	#2935	semidirect
110. C₂×C₄⋊C₈	D₁₃₉	semidirect
199. C₄	#17984848	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984848	2
#22581503	2
#22581030	2
#22581023	2
#22581018	2
#22580919	2
C₄×#14555769	2
C₂×C₄⋊C₈×C₁₃₉	2
#2935	139