Subgroups — #31815069

Order 17792.

392 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14555785	C₂	1	1	yes	maximal, characteristic
2	#22581013	C₂	1	4	yes	maximal, complemented
3	#22581009	C₂	1	2	yes	maximal, complemented
4	#22581005	C₂	1	2	yes	maximal, complemented
5	#22580977	C₂	1	1	yes	maximal, characteristic, complemented
6	#22580919	C₂	1	2	yes	maximal
7	#22580908	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14555785	C₂²	1	1	yes	characteristic
10	C₂×#14555769	C₂²	1	2	yes	—
11	C₄×Dic₂₇₈	C₄	1	1	yes	characteristic
12	C₄×Dic₂₇₈	C₄	1	1	yes	characteristic
13	C₄×D₅₅₆	C₄	1	1	yes	characteristic
14	C₄×D₅₅₆	C₄	1	1	yes	characteristic
15	C₄×C₂²⋊D₁₃₉	C₄	1	2	yes	—
16	C₄×C₂²⋊D₁₃₉	C₄	1	2	yes	—
17	#17984801	C₂²	1	1	yes	characteristic
18	#17984799	C₂²	1	2	yes	—
19	C₄²×D₁₃₉	C₂²	1	2	yes	—
20	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
21	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
22	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	—
23	(C₂×D₁₃₉)⋊C₈	C₂²	1	4	yes	—
24	#17984752	C₂²	1	4	yes	—
25	C₈×Dic₁₃₉	C₂²	1	4	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	1	yes	characteristic
30	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
31	#14555785	—	2	2	no	—
32	#14555785	—	2	2	no	—
33	C₂×Dic₂₇₈	C₂×C₄	1	1	yes	characteristic
34	C₂×D₅₅₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₂²⋊D₁₃₉	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
39	#14555769	—	2	4	no	—
40	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
41	C₄⋊Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
43	Dic₁₃₉⋊C₄	C₂×C₄	1	2	yes	—
44	Dic₁₃₉⋊C₄	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₁₃₉	C₂×C₄	1	2	yes	—
46	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	2	yes	—
47	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	2	yes	—
48	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
49	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
50	C₄×Dic₁₃₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃₉	—	2	4	no	—
55	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
57	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
60	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
61	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
62	Dic₂₇₈	—	2	2	no	—
63	D₅₅₆	—	2	2	no	—
64	C₂²⋊D₁₃₉	—	2	4	no	—
65	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
66	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
67	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
68	C₄×D₁₃₉	—	2	4	no	—
69	C₄×D₁₃₉	—	2	2	no	—
70	C₄×D₁₃₉	—	2	2	no	—
71	C₄×D₁₃₉	—	4	2	no	—
72	C₁₃₉⋊C₈	—	2	4	no	—
73	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₁₃₉	—	2	2	no	—
75	C₂×C₄×C₁₃₉	—	4	1	no	—
76	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃₉	#41	1	2	yes	—
82	C₂×C₄×C₁₃₉	#41	1	2	yes	—
83	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
84	C₈×C₁₃₉	—	2	2	no	—
85	C₈×C₁₃₉	—	2	2	no	—
86	C₂²×C₁₃₉	—	2	1	no	—
87	C₂²×C₁₃₉	—	2	1	no	—
88	C₂²×C₁₃₉	#117	1	2	yes	—
89	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×D₁₃₉	—	2	2	no	—
91	C₂×D₁₃₉	—	2	2	no	—
92	C₂×D₁₃₉	—	2	2	no	—
93	C₄×C₁₃₉	—	2	1	no	—
94	C₄×C₁₃₉	—	2	1	no	—
95	C₄×C₁₃₉	—	2	1	no	—
96	C₄×C₁₃₉	#117	1	2	yes	—
97	C₄×C₁₃₉	#131	1	2	yes	—
98	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
99	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
100	Dic₁₃₉	—	2	2	no	—
101	Dic₁₃₉	—	2	2	no	—
102	Dic₁₃₉	—	2	2	no	—
103	C₂×C₁₃₉	—	2	2	no	—
104	C₂×C₁₃₉	C₂×#117	1	1	yes	characteristic
105	C₂×C₁₃₉	C₂×#131	1	1	yes	characteristic
106	C₂×C₁₃₉	#568	1	1	yes	characteristic
107	D₁₃₉	—	4	2	no	—
108	C₁₃₉	#2925	1	1	yes	characteristic, complemented, Sylow
109	#2925	—	139	1	no	maximal, Sylow
110	C₄×#41	—	139	1	no	—
111	#433	—	139	4	no	—
112	#432	—	139	2	no	—
113	#431	—	139	2	no	—
114	#423	—	139	1	no	—
115	C₂×C₄⋊C₈	D₁₃₉	1	1	yes	characteristic, complemented
116	#408	—	139	2	no	—
117	#405	—	139	2	no	—
118	C₂×#41	—	139	1	no	—
119	C₂×M₁₆	—	139	2	no	—
120	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
121	C₄×Q₈	—	139	1	no	—
122	C₄×Q₈	—	139	1	no	—
123	C₄×D₄	—	139	2	no	—
124	C₄×D₄	—	139	2	no	—
125	C₄×D₄	—	139	1	no	—
126	C₄×D₄	—	139	1	no	—
127	#117	—	139	1	no	—
128	#117	—	139	2	no	—
129	C₂×C₄²	—	139	2	no	—
130	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
131	C₄⋊C₈	—	139	2	no	—
132	C₄⋊C₈	—	139	2	no	—
133	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₁₃₉	1	1	yes	characteristic
135	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
136	C₂²⋊C₈	—	139	4	no	—
137	C₂²⋊C₈	—	139	4	no	—
138	#97	—	139	4	no	—
139	C₄×C₈	—	139	4	no	—
140	#41	—	278	2	no	—
141	#41	—	278	2	no	—
142	C₂×Q₈	—	139	1	no	—
143	C₂×D₄	—	139	2	no	—
144	C₂×D₄	—	139	1	no	—
145	C₂²×C₄	—	139	2	no	—
146	C₂²×C₄	—	139	2	no	—
147	C₂²×C₄	—	139	2	no	—
148	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
149	C₂²×C₄	C₄×D₁₃₉	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
151	M₁₆	—	278	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	139	4	no	—
154	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
155	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
156	C₄⋊C₄	—	139	2	no	—
157	C₄⋊C₄	—	139	2	no	—
158	C₄⋊C₄	—	139	1	no	—
159	C₄⋊C₄	—	139	1	no	—
160	C₂²⋊C₄	—	139	2	no	—
161	C₂²⋊C₄	—	139	2	no	—
162	C₂²⋊C₄	—	139	2	no	—
163	C₄²	—	139	2	no	—
164	C₄²	—	139	2	no	—
165	C₄²	—	139	2	no	—
166	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
167	C₄²	C₄×D₁₃₉	1	1	yes	characteristic
168	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
169	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
170	C₂³	—	139	2	no	—
171	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
172	Q₈	—	278	2	no	—
173	D₄	—	278	4	no	—
174	D₄	—	278	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	4	1	no	—
177	C₂×C₄	—	278	4	no	—
178	C₂×C₄	—	278	2	no	—
179	C₂×C₄	—	278	2	no	—
180	C₂×C₄	—	556	2	no	—
181	C₂×C₄	—	139	2	no	—
182	C₂×C₄	—	139	2	no	—
183	C₂×C₄	—	139	2	no	—
184	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
189	C₂×C₄	#14555784	1	2	yes	—
190	C₂×C₄	#14555786	1	2	yes	—
191	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	278	4	no	—
195	C₂²	—	2	1	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	278	2	no	—
198	C₂²	—	278	2	no	—
199	C₂²	—	278	2	no	—
200	C₂²	#17984800	1	2	yes	—
201	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
202	C₄	—	2	1	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	278	2	no	—
206	C₄	—	278	2	no	—
207	C₄	—	278	2	no	—
208	C₄	#17984800	1	2	yes	—
209	C₄	#17984848	1	2	yes	—
210	C₄	C₂×#14555784	1	1	yes	characteristic
211	C₄	C₂×#14555786	1	1	yes	characteristic
212	C₂	—	2	2	no	—
213	C₂	—	556	2	no	—
214	C₂	C₂×#17984800	1	1	yes	characteristic
215	C₂	C₂×#17984848	1	1	yes	characteristic
216	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	C₂×#14555785	C₂²
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₁₃₉
hypocenter	C₁₃₉	#2925
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
2. #22581013	C₂	semidirect
3. #22581009	C₂	semidirect
4. #22581005	C₂	semidirect
5. #22580977	C₂	semidirect
7. #22580908	C₂	semidirect
8. C₂×C₄⋊C₈×C₁₃₉	C₂	semidirect
108. C₁₃₉	#2925	semidirect
115. C₂×C₄⋊C₈	D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14555785	2
#22581013	2
#22581009	2
#22581005	2
#22580977	2
#22580919	2
#22580908	2
C₂×C₄⋊C₈×C₁₃₉	2
#2925	139