Subgroups — #31815172

Order 17792.

500 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984862	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#22581421	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581045	C₂	1	2	yes	maximal, complemented
5	#22581044	C₂	1	2	yes	maximal, complemented
6	#22581038	C₂	1	2	yes	maximal, complemented
7	#22581033	C₂	1	2	yes	maximal, complemented
8	#22580989	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#416×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₂²	1	1	yes	characteristic
13	#17984862	C₄	1	8	yes	complemented
14	C₂×#14555775	C₂²	1	1	yes	characteristic
15	C₂×#14555773	C₂²	1	1	yes	characteristic
16	C₂×#14555772	C₂²	1	1	yes	characteristic
17	C₂×#14555770	C₂²	1	1	yes	characteristic
18	C₂²×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
20	#17984807	C₂²	1	2	yes	—
21	D₄×Dic₁₃₉	—	2	2	no	—
22	D₄×Dic₁₃₉	C₂²	1	2	yes	—
23	#17984804	C₂²	1	2	yes	—
24	(C₂²⋊D₁₃₉)⋊C₄	—	2	2	no	—
25	#17984801	—	2	1	no	—
26	#17984801	C₂²	1	1	yes	characteristic
27	#17984799	—	2	1	no	—
28	C₂×C₄×Dic₁₃₉	—	2	1	no	—
29	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
30	C₄²×D₁₃₉	—	2	1	no	—
31	#17984775	C₂²	1	2	yes	—
32	#17984773	C₂²	1	2	yes	—
33	#17984767	C₂²	1	2	yes	—
34	#17984764	C₂²	1	2	yes	—
35	C₈×Dic₁₃₉	C₂²	1	2	yes	—
36	C₈×Dic₁₃₉	C₂²	1	2	yes	—
37	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
39	#117×C₁₃₉	C₂²	1	1	yes	characteristic
40	#103×C₁₃₉	C₂²	1	2	yes	—
41	#102×C₁₃₉	C₂²	1	2	yes	—
42	#14555785	—	2	2	no	—
43	#14555785	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
45	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
47	C₂²×Dic₁₃₉	—	2	2	no	—
48	C₂²×Dic₁₃₉	D₄	1	2	yes	—
49	C₂×C₄×D₁₃₉	—	2	2	no	—
50	C₂×C₄×D₁₃₉	—	2	1	no	—
51	#14555775	C₂×C₄	1	4	yes	complemented
52	#14555773	C₂×C₄	1	4	yes	complemented
53	#14555772	C₂×C₄	1	4	yes	complemented
54	#14555770	C₂×C₄	1	4	yes	complemented
55	C₂×C₁₃₉⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
57	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₁₃₉	—	2	2	no	—
59	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
60	Dic₁₃₉⋊C₄	—	2	2	no	—
61	C₂²⋊Dic₁₃₉	—	2	2	no	—
62	C₂²⋊Dic₁₃₉	—	2	1	no	—
63	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
64	C₄×Dic₁₃₉	—	2	2	no	—
65	C₄×Dic₁₃₉	—	2	2	no	—
66	C₄×Dic₁₃₉	—	2	1	no	—
67	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
68	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
69	C₄×Dic₁₃₉	C₂³	1	2	yes	—
70	#41×C₁₃₉	—	2	2	no	—
71	#41×C₁₃₉	C₂×C₄	1	4	yes	complemented
72	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₁₃₉	—	2	1	no	—
74	C₂×D₄×C₁₃₉	C₂³	1	1	yes	characteristic
75	C₂²×C₄×C₁₃₉	—	2	1	no	—
76	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₃₉	—	2	2	no	—
78	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
79	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
81	C₄²×C₁₃₉	—	2	1	no	—
82	C₂²×D₁₃₉	—	2	1	no	—
83	Dic₂₇₈	—	2	1	no	—
84	Dic₂₇₈	C₂²×C₄	1	2	yes	—
85	D₅₅₆	—	2	1	no	—
86	D₅₅₆	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₁₃₉	—	2	4	no	—
88	C₂×Dic₁₃₉	—	2	1	no	—
89	C₂×Dic₁₃₉	—	2	4	no	—
90	C₂×Dic₁₃₉	—	2	2	no	—
91	C₂×Dic₁₃₉	—	4	4	no	—
92	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
93	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
94	C₄×D₁₃₉	—	2	4	no	—
95	C₄×D₁₃₉	—	4	4	no	—
96	C₁₃₉⋊C₈	C₂²×C₄	1	2	yes	—
97	C₁₃₉⋊C₈	C₂²×C₄	1	2	yes	—
98	C₂³×C₁₃₉	—	2	1	no	—
99	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₁₃₉	—	2	1	no	—
101	Q₈×C₁₃₉	C₂²×C₄	1	2	yes	—
102	D₄×C₁₃₉	—	2	4	no	—
103	D₄×C₁₃₉	—	2	1	no	—
104	D₄×C₁₃₉	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₁₃₉	—	2	1	no	—
106	C₂×C₄×C₁₃₉	—	2	4	no	—
107	C₂×C₄×C₁₃₉	—	2	2	no	—
108	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃₉	#41	1	2	yes	—
111	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
112	C₈×C₁₃₉	—	2	2	no	—
113	C₂²×C₁₃₉	—	2	2	no	—
114	C₂²×C₁₃₉	—	2	2	no	—
115	C₂²×C₁₃₉	—	4	2	no	—
116	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
117	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₃₉	—	2	2	no	—
119	C₂×D₁₃₉	—	4	2	no	—
120	C₄×C₁₃₉	—	2	2	no	—
121	C₄×C₁₃₉	—	4	2	no	—
122	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
123	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
125	Dic₁₃₉	—	2	2	no	—
126	Dic₁₃₉	—	2	2	no	—
127	Dic₁₃₉	—	4	2	no	—
128	Dic₁₃₉	#135	1	4	yes	complemented
129	C₂×C₁₃₉	—	2	2	no	—
130	C₂×C₁₃₉	—	4	2	no	—
131	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
132	C₂×C₁₃₉	C₂×#135	1	2	yes	—
133	D₁₃₉	—	4	2	no	—
134	C₁₃₉	C₄×#135	1	1	yes	characteristic, complemented, Sylow
135	C₄×#135	—	139	1	no	maximal, Sylow
136	C₂×#135	—	139	1	no	—
137	C₄×#41	—	139	1	no	—
138	C₄×#41	—	139	1	no	—
139	C₄×Q₁₆	—	139	2	no	—
140	C₄×SD₁₆	—	139	2	no	—
141	C₄×SD₁₆	—	139	2	no	—
142	C₄×D₈	—	139	2	no	—
143	#427	—	139	1	no	—
144	#416	—	139	1	no	—
145	#416	D₁₃₉	1	1	yes	characteristic, complemented
146	C₂×C₄×C₈	—	139	1	no	—
147	C₂×#41	—	139	1	no	—
148	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
149	#135	—	139	8	no	—
150	C₂×Q₁₆	—	139	1	no	—
151	C₂×SD₁₆	—	139	1	no	—
152	C₂×SD₁₆	—	139	1	no	—
153	C₂×D₈	—	139	1	no	—
154	C₂²×C₈	—	139	1	no	—
155	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
156	C₄×Q₈	—	139	2	no	—
157	C₄×Q₈	—	139	2	no	—
158	C₄×D₄	—	278	2	no	—
159	C₄×D₄	—	278	2	no	—
160	C₄×D₄	—	139	2	no	—
161	C₄×D₄	—	139	2	no	—
162	#117	—	278	1	no	—
163	#117	—	278	1	no	—
164	#117	—	139	1	no	—
165	#117	C₂×D₁₃₉	1	1	yes	characteristic
166	C₂×C₄²	—	278	1	no	—
167	C₂×C₄²	—	278	1	no	—
168	C₂×C₄²	—	139	1	no	—
169	#107	—	139	2	no	—
170	#106	—	139	2	no	—
171	#103	—	139	2	no	—
172	#103	C₂×D₁₃₉	1	2	yes	—
173	#102	—	139	2	no	—
174	#102	C₂×D₁₃₉	1	2	yes	—
175	C₄×C₈	—	139	2	no	—
176	C₄×C₈	—	139	2	no	—
177	#41	—	2	2	no	—
178	#41	—	278	2	no	—
179	#41	—	139	4	no	—
180	#41	C₄×D₁₃₉	1	4	yes	complemented
181	C₂×Q₈	—	139	1	no	—
182	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	278	1	no	—
185	C₂×D₄	—	139	1	no	—
186	C₂×D₄	C₂²×D₁₃₉	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	278	2	no	—
189	C₂²×C₄	—	278	1	no	—
190	C₂²×C₄	—	278	2	no	—
191	C₂²×C₄	—	139	2	no	—
192	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
193	Q₁₆	—	139	4	no	—
194	SD₁₆	—	139	4	no	—
195	SD₁₆	—	139	4	no	—
196	D₈	—	139	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	139	4	no	—
199	C₂×C₈	—	139	1	no	—
200	C₂×C₈	—	139	1	no	—
201	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
202	C₄⋊C₄	—	278	2	no	—
203	C₄⋊C₄	—	278	2	no	—
204	C₄⋊C₄	—	139	2	no	—
205	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	278	2	no	—
208	C₂²⋊C₄	—	278	1	no	—
209	C₂²⋊C₄	—	278	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	278	2	no	—
212	C₄²	—	278	2	no	—
213	C₄²	—	278	1	no	—
214	C₄²	—	139	1	no	—
215	C₄²	—	139	1	no	—
216	C₄²	—	139	2	no	—
217	C₂³	—	2	1	no	—
218	C₂³	—	278	1	no	—
219	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
220	Q₈	—	2	1	no	—
221	Q₈	—	278	1	no	—
222	Q₈	—	139	2	no	—
223	Q₈	C₂×C₄×D₁₃₉	1	2	yes	—
224	D₄	—	2	4	no	—
225	D₄	—	2	1	no	—
226	D₄	—	278	4	no	—
227	D₄	—	278	1	no	—
228	D₄	—	139	2	no	—
229	D₄	C₂×C₄×D₁₃₉	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	278	1	no	—
234	C₂×C₄	—	278	4	no	—
235	C₂×C₄	—	278	4	no	—
236	C₂×C₄	—	278	2	no	—
237	C₂×C₄	—	556	4	no	—
238	C₂×C₄	—	556	4	no	—
239	C₂×C₄	—	139	2	no	—
240	C₂×C₄	—	139	2	no	—
241	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
242	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
243	C₂×C₄	#14555784	1	2	yes	—
244	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	139	2	no	—
247	C₈	—	139	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	278	2	no	—
252	C₂²	—	556	2	no	—
253	C₂²	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
254	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	2	no	—
257	C₄	—	278	2	no	—
258	C₄	—	278	2	no	—
259	C₄	—	556	2	no	—
260	C₄	—	139	4	no	—
261	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
262	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
263	C₄	C₂×#14555784	1	1	yes	characteristic
264	C₂	—	2	2	no	—
265	C₂	—	4	2	no	—
266	C₂	—	556	2	no	—
267	C₂	C₂×(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
268	C₂	#135×D₁₃₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#416×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984862	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#416	D₁₃₉
hypocenter	C₁₃₉	C₄×#135
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
2. #41×Dic₁₃₉	C₂	semidirect
3. #22581421	C₂	semidirect
4. #22581045	C₂	semidirect
5. #22581044	C₂	semidirect
6. #22581038	C₂	semidirect
7. #22581033	C₂	semidirect
8. #22580989	C₂	semidirect
9. #22580950	C₂	semidirect
10. C₂×C₈×Dic₁₃₉	C₂	semidirect
11. #416×C₁₃₉	C₂	semidirect
13. #17984862	C₄	semidirect
43. #14555785	C₂×C₄	semidirect
51. #14555775	C₂×C₄	semidirect
52. #14555773	C₂×C₄	semidirect
53. #14555772	C₂×C₄	semidirect
54. #14555770	C₂×C₄	semidirect
55. C₂×C₁₃₉⋊C₈	C₂×C₄	semidirect
71. #41×C₁₃₉	C₂×C₄	semidirect
128. Dic₁₃₉	#135	semidirect
134. C₁₃₉	C₄×#135	semidirect
145. #416	D₁₃₉	semidirect
180. #41	C₄×D₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17984862	2
#41×Dic₁₃₉	2
#22581421	2
#22581045	2
#22581044	2
#22581038	2
#22581033	2
#22580989	2
#22580950	2
C₂×C₈×Dic₁₃₉	2
#416×C₁₃₉	2
C₄×#135	139