Subgroups — #31815219

Order 17792.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984873	C₂	1	1	yes	maximal, characteristic
2	#22581421	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581057	C₂	1	2	yes	maximal, complemented
5	#22581056	C₂	1	2	yes	maximal, complemented
6	#22581045	C₂	1	2	yes	maximal, complemented
7	#22581038	C₂	1	2	yes	maximal, complemented
8	#22580996	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580955	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#419×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₂²	1	1	yes	characteristic
13	#17984873	C₄	1	8	yes	complemented
14	C₂×#14555775	C₂²	1	2	yes	—
15	C₂×#14555772	C₂²	1	2	yes	—
16	C₂×#14555769	C₂²	1	1	yes	characteristic
17	Q₈×Dic₁₃₉	—	2	1	no	—
18	Q₈×Dic₁₃₉	—	2	1	no	—
19	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
20	Q₈×Dic₁₃₉	C₂²	1	2	yes	—
21	#17984807	C₂²	1	1	yes	characteristic
22	#17984807	C₂²	1	1	yes	characteristic
23	#17984804	C₂²	1	1	yes	characteristic
24	#17984804	C₂²	1	1	yes	characteristic
25	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
26	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
27	#17984799	—	2	1	no	—
28	C₂×C₄⋊Dic₁₃₉	—	2	1	no	—
29	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃₉	—	2	1	no	—
31	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	C₄²×D₁₃₉	—	2	1	no	—
33	#17984775	C₂²	1	2	yes	—
34	#17984773	C₂²	1	2	yes	—
35	#17984767	C₂²	1	2	yes	—
36	#17984764	C₂²	1	2	yes	—
37	#17984752	C₂²	1	2	yes	—
38	C₈×Dic₁₃₉	C₂²	1	2	yes	—
39	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
41	#117×C₁₃₉	C₂²	1	1	yes	characteristic
42	#103×C₁₃₉	C₂²	1	2	yes	—
43	#103×C₁₃₉	C₂²	1	2	yes	—
44	#14555785	—	2	2	no	—
45	#14555785	C₂×C₄	1	4	yes	—
46	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
47	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
48	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
49	C₂²×Dic₁₃₉	—	2	1	no	—
50	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
51	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
52	C₂×C₄×D₁₃₉	—	2	1	no	—
53	C₂×C₄×D₁₃₉	—	2	1	no	—
54	C₂×C₄×D₁₃₉	—	2	1	no	—
55	#14555775	C₂×C₄	1	8	yes	complemented
56	#14555772	C₂×C₄	1	8	yes	complemented
57	#14555769	C₂×C₄	1	4	yes	complemented
58	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
59	C₄⋊Dic₁₃₉	—	2	2	no	—
60	C₄⋊Dic₁₃₉	—	2	2	no	—
61	C₄⋊Dic₁₃₉	—	2	1	no	—
62	C₄⋊Dic₁₃₉	C₂³	1	2	yes	—
63	Dic₁₃₉⋊C₄	—	2	1	no	—
64	Dic₁₃₉⋊C₄	—	2	1	no	—
65	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
66	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
67	C₄×Dic₁₃₉	—	2	1	no	—
68	C₄×Dic₁₃₉	—	2	1	no	—
69	C₄×Dic₁₃₉	—	2	2	no	—
70	C₄×Dic₁₃₉	—	2	2	no	—
71	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
72	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
73	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
74	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₂×Q₈×C₁₃₉	—	2	1	no	—
76	C₂×Q₈×C₁₃₉	—	2	2	no	—
77	C₂×Q₈×C₁₃₉	C₂×C₄	1	4	yes	complemented
78	C₂×Q₈×C₁₃₉	C₂³	1	2	yes	—
79	C₂²×C₄×C₁₃₉	—	2	1	no	—
80	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
81	M₁₆×C₁₃₉	—	2	2	no	—
82	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
83	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
86	C₄²×C₁₃₉	—	2	1	no	—
87	C₂²×D₁₃₉	—	2	1	no	—
88	Dic₂₇₈	—	2	1	no	—
89	Dic₂₇₈	C₂²×C₄	1	2	yes	—
90	D₅₅₆	—	2	1	no	—
91	D₅₅₆	C₂²×C₄	1	2	yes	—
92	C₂²⋊D₁₃₉	—	2	4	no	—
93	C₂×Dic₁₃₉	—	2	1	no	—
94	C₂×Dic₁₃₉	—	2	2	no	—
95	C₂×Dic₁₃₉	—	2	2	no	—
96	C₂×Dic₁₃₉	—	4	1	no	—
97	C₂×Dic₁₃₉	—	4	2	no	—
98	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
101	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
102	C₄×D₁₃₉	—	2	4	no	—
103	C₄×D₁₃₉	—	4	2	no	—
104	C₄×D₁₃₉	—	4	2	no	—
105	C₁₃₉⋊C₈	C₂²×C₄	1	4	yes	—
106	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₁₃₉	—	2	4	no	—
108	Q₈×C₁₃₉	—	2	2	no	—
109	Q₈×C₁₃₉	C₂²×C₄	1	4	yes	—
110	C₂×C₄×C₁₃₉	—	2	4	no	—
111	C₂×C₄×C₁₃₉	—	2	2	no	—
112	C₂×C₄×C₁₃₉	—	2	1	no	—
113	C₂×C₄×C₁₃₉	—	2	1	no	—
114	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
115	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₃₉	#41	1	2	yes	—
117	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
118	C₈×C₁₃₉	—	2	2	no	—
119	C₂²×C₁₃₉	—	2	1	no	—
120	C₂²×C₁₃₉	—	2	1	no	—
121	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
122	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₁₃₉	—	2	2	no	—
124	C₂×D₁₃₉	—	4	1	no	—
125	C₂×D₁₃₉	—	4	1	no	—
126	C₄×C₁₃₉	—	2	4	no	—
127	C₄×C₁₃₉	—	4	1	no	—
128	C₄×C₁₃₉	—	4	1	no	—
129	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
130	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
132	Dic₁₃₉	—	2	2	no	—
133	Dic₁₃₉	—	2	1	no	—
134	Dic₁₃₉	—	2	1	no	—
135	Dic₁₃₉	—	2	1	no	—
136	Dic₁₃₉	—	4	2	no	—
137	Dic₁₃₉	#137	1	2	yes	complemented
138	C₂×C₁₃₉	—	2	2	no	—
139	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₁₃₉	C₂×#137	1	1	yes	characteristic
141	C₂×C₁₃₉	#575	1	1	yes	characteristic
142	D₁₃₉	—	4	2	no	—
143	C₁₃₉	C₄×#137	1	1	yes	characteristic, complemented, Sylow
144	C₄×#137	—	139	1	no	maximal, Sylow
145	C₂×#137	—	139	1	no	—
146	C₄×#41	—	139	1	no	—
147	C₂×C₄×Q₈	—	139	1	no	—
148	#441	—	139	2	no	—
149	#440	—	139	2	no	—
150	C₄×Q₁₆	—	139	2	no	—
151	C₄×SD₁₆	—	139	2	no	—
152	#428	—	139	1	no	—
153	#419	D₁₃₉	1	1	yes	characteristic, complemented
154	#417	—	139	1	no	—
155	C₄×M₁₆	—	139	1	no	—
156	C₂×#41	—	139	1	no	—
157	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
158	#137	—	139	8	no	—
159	C₂×Q₁₆	—	139	2	no	—
160	C₂×SD₁₆	—	139	2	no	—
161	C₂×M₁₆	—	139	1	no	—
162	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
163	C₄×Q₈	—	278	1	no	—
164	C₄×Q₈	—	278	1	no	—
165	C₄×Q₈	—	139	2	no	—
166	C₄×Q₈	—	139	2	no	—
167	C₄×Q₈	—	139	1	no	—
168	C₄×Q₈	—	139	1	no	—
169	C₄×D₄	—	278	1	no	—
170	C₄×D₄	—	278	1	no	—
171	C₄×D₄	—	139	1	no	—
172	C₄×D₄	—	139	1	no	—
173	#117	—	278	1	no	—
174	#117	C₂×D₁₃₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	278	1	no	—
176	C₂×C₄⋊C₄	—	139	1	no	—
177	C₂×C₄²	—	278	1	no	—
178	C₂×C₄²	—	278	1	no	—
179	C₂×C₄²	—	139	1	no	—
180	#107	—	139	2	no	—
181	#106	—	139	2	no	—
182	#103	—	139	2	no	—
183	#103	C₂×D₁₃₉	1	2	yes	—
184	#103	C₂×D₁₃₉	1	2	yes	—
185	#102	—	139	2	no	—
186	#97	—	139	2	no	—
187	C₄×C₈	—	139	2	no	—
188	#41	—	278	2	no	—
189	#41	—	139	4	no	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	—	139	1	no	—
193	C₂×Q₈	C₄×D₁₃₉	1	4	yes	complemented
194	C₂×Q₈	C₂²×D₁₃₉	1	2	yes	—
195	C₂×D₄	—	278	1	no	—
196	C₂×D₄	—	139	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	278	1	no	—
199	C₂²×C₄	—	278	1	no	—
200	C₂²×C₄	—	278	1	no	—
201	C₂²×C₄	—	278	1	no	—
202	C₂²×C₄	—	139	1	no	—
203	C₂²×C₄	—	139	1	no	—
204	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
205	Q₁₆	—	139	8	no	—
206	SD₁₆	—	139	8	no	—
207	M₁₆	—	2	2	no	—
208	M₁₆	—	139	4	no	—
209	C₂×C₈	—	139	2	no	—
210	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
211	C₄⋊C₄	—	278	2	no	—
212	C₄⋊C₄	—	278	2	no	—
213	C₄⋊C₄	—	278	1	no	—
214	C₄⋊C₄	—	278	1	no	—
215	C₄⋊C₄	—	278	1	no	—
216	C₄⋊C₄	—	139	2	no	—
217	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	278	1	no	—
221	C₂²⋊C₄	—	278	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	278	1	no	—
224	C₄²	—	278	1	no	—
225	C₄²	—	278	2	no	—
226	C₄²	—	278	2	no	—
227	C₄²	—	139	1	no	—
228	C₄²	—	139	1	no	—
229	C₄²	—	139	1	no	—
230	C₄²	—	139	1	no	—
231	C₂³	—	278	1	no	—
232	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	278	1	no	—
236	Q₈	—	139	2	no	—
237	Q₈	C₂×C₄×D₁₃₉	1	4	yes	—
238	D₄	—	278	4	no	—
239	D₄	—	278	1	no	—
240	D₄	—	139	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	278	1	no	—
246	C₂×C₄	—	278	2	no	—
247	C₂×C₄	—	278	4	no	—
248	C₂×C₄	—	278	2	no	—
249	C₂×C₄	—	556	2	no	—
250	C₂×C₄	—	556	2	no	—
251	C₂×C₄	—	556	1	no	—
252	C₂×C₄	—	556	2	no	—
253	C₂×C₄	—	139	1	no	—
254	C₂×C₄	—	139	1	no	—
255	C₂×C₄	—	139	1	no	—
256	C₂×C₄	—	139	1	no	—
257	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
258	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
259	C₂×C₄	#14555784	1	2	yes	—
260	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
261	C₈	—	2	2	no	—
262	C₈	—	139	4	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	278	2	no	—
266	C₂²	—	556	1	no	—
267	C₂²	—	556	1	no	—
268	C₂²	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
269	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
270	C₄	—	2	4	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	278	2	no	—
274	C₄	—	278	1	no	—
275	C₄	—	278	1	no	—
276	C₄	—	278	1	no	—
277	C₄	—	556	2	no	—
278	C₄	—	139	2	no	—
279	C₄	(C₂²⋊D₁₃₉)⋊C₄	1	2	yes	—
280	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
281	C₄	C₂×#14555784	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	556	2	no	—
284	C₂	C₂×(C₂²⋊D₁₃₉)⋊C₄	1	1	yes	characteristic
285	C₂	#137×D₁₃₉	1	1	yes	characteristic
286	C₂	#22581752	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator	C₄×C₁₃₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#419×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#14555785	C₂²
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#419	D₁₃₉
hypocenter	C₁₃₉	C₄×#137
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈×Dic₁₃₉	C₂	semidirect
4. #22581057	C₂	semidirect
5. #22581056	C₂	semidirect
6. #22581045	C₂	semidirect
7. #22581038	C₂	semidirect
8. #22580996	C₂	semidirect
10. M₁₆×Dic₁₃₉	C₂	semidirect
11. #419×C₁₃₉	C₂	semidirect
13. #17984873	C₄	semidirect
55. #14555775	C₂×C₄	semidirect
56. #14555772	C₂×C₄	semidirect
57. #14555769	C₂×C₄	semidirect
77. C₂×Q₈×C₁₃₉	C₂×C₄	semidirect
137. Dic₁₃₉	#137	semidirect
143. C₁₃₉	C₄×#137	semidirect
153. #419	D₁₃₉	semidirect
193. C₂×Q₈	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×#17984873	2
#22581421	2
C₂×Q₈×Dic₁₃₉	2
#22581057	2
#22581056	2
#22581045	2
#22581038	2
#22580996	2
#22580955	2
M₁₆×Dic₁₃₉	2
#419×C₁₃₉	2
C₄×#137	139