Subgroups — #31815211

Order 17792.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984870	C₂	1	1	yes	maximal, characteristic
2	#22581421	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17984804	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581064	C₂	1	2	yes	maximal, complemented
5	#22581052	C₂	1	2	yes	maximal, complemented
6	#22581040	C₂	1	2	yes	maximal, complemented
7	#22581035	C₂	1	2	yes	maximal, complemented
8	#22580957	C₂	1	1	yes	maximal, characteristic, complemented
9	#22580952	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
11	#428×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14555785	C₂²	1	1	yes	characteristic
13	C₂²×D₅₅₆	C₂²	1	1	yes	characteristic
14	C₂²×C₄×D₁₃₉	—	2	1	no	—
15	#17984870	C₄	1	8	yes	complemented
16	C₂×#14555774	C₂²	1	2	yes	—
17	C₂×D₁₁₁₂	C₂²	1	2	yes	—
18	C₂×#14555769	C₂²	1	1	yes	characteristic
19	#17984807	C₂²	1	1	yes	characteristic, complemented
20	#17984807	C₂²	1	1	yes	characteristic, complemented
21	#17984804	—	2	1	no	—
22	#17984804	—	2	1	no	—
23	#17984804	C₂²	1	2	yes	—
24	#17984804	C₂²	1	2	yes	—
25	#17984804	C₂²	1	1	yes	characteristic, complemented
26	#17984804	C₂²	1	1	yes	characteristic, complemented
27	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
28	(C₂²⋊D₁₃₉)⋊C₄	—	2	1	no	—
29	#17984799	—	2	1	no	—
30	C₂×(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
32	C₄²×D₁₃₉	—	2	1	no	—
33	#17984765	C₂²	1	2	yes	complemented
34	#17984762	C₂²	1	2	yes	—
35	#17984762	C₂²	1	2	yes	—
36	#17984762	C₂²	1	2	yes	complemented
37	#17984752	C₂²	1	2	yes	complemented
38	C₈×Dic₁₃₉	C₂²	1	2	yes	complemented
39	C₂×M₁₆×C₁₃₉	C₂²	1	1	yes	characteristic
40	#117×C₁₃₉	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	1	yes	characteristic
42	#107×C₁₃₉	C₂²	1	2	yes	complemented
43	#106×C₁₃₉	C₂²	1	2	yes	complemented
44	C₂³×D₁₃₉	—	2	1	no	—
45	#14555785	—	2	2	no	—
46	#14555785	C₂×C₄	1	4	yes	complemented
47	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
48	C₂×D₅₅₆	—	2	1	no	—
49	C₂×D₅₅₆	—	2	2	no	—
50	C₂×D₅₅₆	C₂×C₄	1	4	yes	complemented
51	C₂×D₅₅₆	C₂³	1	2	yes	—
52	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
53	C₂×C₂²⋊D₁₃₉	—	2	1	no	—
54	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
55	C₂²×Dic₁₃₉	D₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃₉	—	2	1	no	—
57	C₂×C₄×D₁₃₉	—	2	1	no	—
58	C₂×C₄×D₁₃₉	—	2	2	no	—
59	C₂×C₄×D₁₃₉	—	2	2	no	—
60	C₂×C₄×D₁₃₉	—	2	1	no	—
61	C₂×C₄×D₁₃₉	—	4	4	no	—
62	#14555774	C₂×C₄	1	8	yes	complemented
63	D₁₁₁₂	C₂×C₄	1	8	yes	complemented
64	#14555769	—	2	2	no	—
65	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
66	Dic₁₃₉⋊C₄	—	2	1	no	—
67	Dic₁₃₉⋊C₄	—	2	1	no	—
68	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
69	(C₂×D₁₃₉)⋊C₄	—	2	2	no	—
70	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
71	(C₂×D₁₃₉)⋊C₄	—	2	1	no	—
72	C₄×Dic₁₃₉	—	2	1	no	—
73	C₄×Dic₁₃₉	—	2	1	no	—
74	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
75	C₄×Dic₁₃₉	D₄	1	1	yes	characteristic
76	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
77	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
78	C₂²×C₄×C₁₃₉	—	2	1	no	—
79	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
80	M₁₆×C₁₃₉	C₂×C₄	1	4	yes	complemented
81	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
82	C₄⋊C₄×C₁₃₉	—	2	1	no	—
83	C₄⋊C₄×C₁₃₉	C₂³	1	2	yes	—
84	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₁₃₉	—	2	1	no	—
87	C₄²×C₁₃₉	—	2	1	no	—
88	C₂²×D₁₃₉	—	2	4	no	—
89	C₂²×D₁₃₉	—	2	1	no	—
90	C₂²×D₁₃₉	—	2	2	no	—
91	C₂²×D₁₃₉	—	4	2	no	—
92	C₂²×D₁₃₉	—	4	2	no	—
93	Dic₂₇₈	—	2	1	no	—
94	Dic₂₇₈	C₂²×C₄	1	2	yes	complemented
95	D₅₅₆	—	2	4	no	—
96	D₅₅₆	—	2	2	no	—
97	D₅₅₆	—	2	1	no	—
98	D₅₅₆	C₂²×C₄	1	4	yes	—
99	D₅₅₆	C₂²×C₄	1	2	yes	complemented
100	C₂²⋊D₁₃₉	—	2	4	no	—
101	C₂×Dic₁₃₉	—	2	1	no	—
102	C₂×Dic₁₃₉	—	2	2	no	—
103	C₂×Dic₁₃₉	—	4	1	no	—
104	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
107	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
108	C₄×D₁₃₉	—	2	4	no	—
109	C₄×D₁₃₉	—	4	2	no	—
110	C₄×D₁₃₉	—	4	4	no	—
111	C₄×D₁₃₉	—	4	2	no	—
112	C₄×D₁₃₉	—	4	4	no	—
113	C₁₃₉⋊C₈	—	2	2	no	—
114	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃₉	—	2	1	no	—
116	C₂×C₄×C₁₃₉	—	2	1	no	—
117	C₂×C₄×C₁₃₉	—	2	2	no	—
118	C₂×C₄×C₁₃₉	—	4	2	no	—
119	C₂×C₄×C₁₃₉	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃₉	#41	1	2	yes	—
122	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
123	C₈×C₁₃₉	C₂²×C₄	1	4	yes	complemented
124	C₂²×C₁₃₉	—	2	1	no	—
125	C₂²×C₁₃₉	—	2	1	no	—
126	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
127	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×D₁₃₉	—	2	2	no	—
129	C₂×D₁₃₉	—	2	4	no	—
130	C₂×D₁₃₉	—	4	8	no	—
131	C₂×D₁₃₉	—	4	2	no	—
132	C₂×D₁₃₉	—	4	1	no	—
133	C₂×D₁₃₉	—	4	1	no	—
134	C₂×D₁₃₉	—	4	2	no	—
135	C₄×C₁₃₉	—	4	2	no	—
136	C₄×C₁₃₉	—	4	1	no	—
137	C₄×C₁₃₉	—	4	1	no	—
138	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
139	C₄×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
140	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
141	Dic₁₃₉	—	2	2	no	—
142	Dic₁₃₉	—	2	1	no	—
143	Dic₁₃₉	—	2	1	no	—
144	Dic₁₃₉	—	2	1	no	—
145	Dic₁₃₉	AGL(1,ℤ/8)	1	2	yes	complemented
146	C₂×C₁₃₉	—	2	2	no	—
147	C₂×C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₁₃₉	#575	1	1	yes	characteristic
150	D₁₃₉	—	4	2	no	—
151	D₁₃₉	—	4	4	no	—
152	C₁₃₉	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
153	C₄×AGL(1,ℤ/8)	—	139	1	no	maximal, Sylow
154	C₂×AGL(1,ℤ/8)	—	139	1	no	—
155	C₄×#41	—	139	1	no	—
156	C₂×C₄×D₄	—	139	1	no	—
157	#442	—	139	2	no	—
158	#440	—	139	2	no	—
159	C₄×SD₁₆	—	139	2	no	—
160	C₄×D₈	—	139	2	no	—
161	#428	D₁₃₉	1	1	yes	characteristic, complemented
162	#418	—	139	1	no	—
163	#417	—	139	1	no	—
164	C₄×M₁₆	—	139	1	no	—
165	C₂×#41	—	139	1	no	—
166	C₂²×D₄	—	139	1	no	—
167	C₂³×C₄	—	278	1	no	—
168	AGL(1,ℤ/8)	—	139	8	no	—
169	C₂×SD₁₆	—	139	2	no	—
170	C₂×D₈	—	139	2	no	—
171	C₂×M₁₆	—	139	1	no	—
172	C₂×M₁₆	C₂×D₁₃₉	1	1	yes	characteristic
173	C₄×Q₈	—	139	1	no	—
174	C₄×Q₈	—	139	1	no	—
175	C₄×D₄	—	278	1	no	—
176	C₄×D₄	—	278	1	no	—
177	C₄×D₄	—	278	1	no	—
178	C₄×D₄	—	278	1	no	—
179	C₄×D₄	—	139	2	no	—
180	C₄×D₄	—	139	2	no	—
181	C₄×D₄	—	139	1	no	—
182	C₄×D₄	—	139	1	no	—
183	#117	—	278	1	no	—
184	#117	C₂×D₁₃₉	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂×D₁₃₉	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	278	1	no	—
187	C₂×C₄²	—	278	1	no	—
188	C₂×C₄²	—	139	1	no	—
189	#107	C₂×D₁₃₉	1	2	yes	complemented
190	#106	C₂×D₁₃₉	1	2	yes	complemented
191	#103	—	139	2	no	—
192	#102	—	139	2	no	—
193	#102	—	139	2	no	—
194	#102	—	139	2	no	—
195	#97	—	139	2	no	—
196	C₄×C₈	—	139	2	no	—
197	C₂⁴	—	278	1	no	—
198	#41	—	278	2	no	—
199	#41	—	139	4	no	—
200	C₂×Q₈	—	139	1	no	—
201	C₂×D₄	—	278	1	no	—
202	C₂×D₄	—	278	1	no	—
203	C₂×D₄	—	278	2	no	—
204	C₂×D₄	—	139	4	no	—
205	C₂×D₄	—	139	1	no	—
206	C₂×D₄	—	139	2	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	278	1	no	—
209	C₂²×C₄	—	278	1	no	—
210	C₂²×C₄	—	278	2	no	—
211	C₂²×C₄	—	278	2	no	—
212	C₂²×C₄	—	278	1	no	—
213	C₂²×C₄	—	556	4	no	—
214	C₂²×C₄	—	139	1	no	—
215	C₂²×C₄	—	139	1	no	—
216	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
217	SD₁₆	—	139	8	no	—
218	D₈	—	139	8	no	—
219	M₁₆	—	278	2	no	—
220	M₁₆	C₄×D₁₃₉	1	4	yes	complemented
221	C₂×C₈	—	139	2	no	—
222	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	278	1	no	—
225	C₄⋊C₄	—	278	1	no	—
226	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
227	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
228	C₄⋊C₄	C₂²×D₁₃₉	1	2	yes	—
229	C₂²⋊C₄	—	2	1	no	—
230	C₂²⋊C₄	—	278	2	no	—
231	C₂²⋊C₄	—	278	2	no	—
232	C₂²⋊C₄	—	278	1	no	—
233	C₂²⋊C₄	—	278	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	278	1	no	—
236	C₄²	—	278	1	no	—
237	C₄²	—	139	1	no	—
238	C₄²	—	139	1	no	—
239	C₄²	—	139	1	no	—
240	C₄²	—	139	1	no	—
241	C₂³	—	278	4	no	—
242	C₂³	—	278	1	no	—
243	C₂³	—	278	2	no	—
244	C₂³	—	556	2	no	—
245	C₂³	—	556	2	no	—
246	C₂³	D₄×D₁₃₉	1	1	yes	characteristic
247	Q₈	—	278	1	no	—
248	Q₈	—	139	2	no	—
249	D₄	—	278	4	no	—
250	D₄	—	278	4	no	—
251	D₄	—	278	2	no	—
252	D₄	—	278	1	no	—
253	D₄	—	139	2	no	—
254	D₄	—	139	4	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	278	1	no	—
260	C₂×C₄	—	278	2	no	—
261	C₂×C₄	—	278	4	no	—
262	C₂×C₄	—	556	4	no	—
263	C₂×C₄	—	556	2	no	—
264	C₂×C₄	—	556	4	no	—
265	C₂×C₄	—	556	2	no	—
266	C₂×C₄	—	556	1	no	—
267	C₂×C₄	—	139	1	no	—
268	C₂×C₄	—	139	1	no	—
269	C₂×C₄	—	139	1	no	—
270	C₂×C₄	—	139	1	no	—
271	C₂×C₄	C₂×C₄×D₁₃₉	1	2	yes	—
272	C₂×C₄	D₄×D₁₃₉	1	1	yes	characteristic
273	C₂×C₄	#14555786	1	2	yes	—
274	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
275	C₈	—	278	2	no	—
276	C₈	C₂×C₄×D₁₃₉	1	4	yes	complemented
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	278	2	no	—
280	C₂²	—	278	4	no	—
281	C₂²	—	556	8	no	—
282	C₂²	—	556	2	no	—
283	C₂²	—	556	1	no	—
284	C₂²	—	556	1	no	—
285	C₂²	—	556	2	no	—
286	C₂²	#17984804	1	2	yes	—
287	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
288	C₄	—	4	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	278	2	no	—
292	C₄	—	278	1	no	—
293	C₄	—	278	1	no	—
294	C₄	—	278	1	no	—
295	C₄	—	139	2	no	—
296	C₄	#17984804	1	2	yes	—
297	C₄	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
298	C₄	C₂×#14555786	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	556	4	no	—
301	C₂	—	556	2	no	—
302	C₂	C₂×#17984804	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₁₃₉	1	1	yes	characteristic
304	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	#428×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984870	C₂
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#428	D₁₃₉
hypocenter	C₁₃₉	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
2. #22581421	C₂	semidirect
3. C₂×#17984804	C₂	semidirect
4. #22581064	C₂	semidirect
5. #22581052	C₂	semidirect
6. #22581040	C₂	semidirect
7. #22581035	C₂	semidirect
8. #22580957	C₂	semidirect
9. #22580952	C₂	semidirect
10. M₁₆×Dic₁₃₉	C₂	semidirect
11. #428×C₁₃₉	C₂	semidirect
15. #17984870	C₄	semidirect
19. #17984807	C₂²	semidirect
20. #17984807	C₂²	semidirect
25. #17984804	C₂²	semidirect
26. #17984804	C₂²	semidirect
33. #17984765	C₂²	semidirect
36. #17984762	C₂²	semidirect
37. #17984752	C₂²	semidirect
38. C₈×Dic₁₃₉	C₂²	semidirect
42. #107×C₁₃₉	C₂²	semidirect
43. #106×C₁₃₉	C₂²	semidirect
46. #14555785	C₂×C₄	semidirect
50. C₂×D₅₅₆	C₂×C₄	semidirect
62. #14555774	C₂×C₄	semidirect
63. D₁₁₁₂	C₂×C₄	semidirect
80. M₁₆×C₁₃₉	C₂×C₄	semidirect
94. Dic₂₇₈	C₂²×C₄	semidirect
99. D₅₅₆	C₂²×C₄	semidirect
123. C₈×C₁₃₉	C₂²×C₄	semidirect
145. Dic₁₃₉	AGL(1,ℤ/8)	semidirect
152. C₁₃₉	C₄×AGL(1,ℤ/8)	semidirect
161. #428	D₁₃₉	semidirect
189. #107	C₂×D₁₃₉	semidirect
190. #106	C₂×D₁₃₉	semidirect
220. M₁₆	C₄×D₁₃₉	semidirect
276. C₈	C₂×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×#17984870	2
#22581421	2
C₂×#17984804	2
#22581064	2
#22581052	2
#22581040	2
#22581035	2
#22580957	2
#22580952	2
M₁₆×Dic₁₃₉	2
#428×C₁₃₉	2
C₄×AGL(1,ℤ/8)	139