Subgroups — #31819610

Order 17792.

1660 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984888	C₂	1	1	yes	maximal, characteristic
2	#22581435	C₂	1	2	yes	maximal, complemented
3	#22581434	C₂	1	3	yes	maximal, complemented
4	#22581427	C₂	1	3	yes	maximal, complemented
5	#22581425	C₂	1	6	yes	maximal, complemented
6	#41×Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#14555786	C₂	1	2	yes	maximal, complemented
8	#22581421	C₂	1	3	yes	maximal, complemented
9	C₂×#17984804	C₂	1	3	yes	maximal, complemented
10	C₄×D₄×D₁₃₉	C₂	1	6	yes	maximal, complemented
11	#520×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#17984888	C₄	1	16	yes	complemented
13	C₂×#14555786	C₂²	1	2	yes	—
14	C₂×#14555785	C₂²	1	3	yes	—
15	C₂²×D₅₅₆	C₂²	1	3	yes	—
16	C₂×D₄×D₁₃₉	C₂²	1	6	yes	—
17	C₂²×C₄×D₁₃₉	C₂²	1	6	yes	complemented
18	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₁₃₉	C₂²	1	1	yes	characteristic, complemented
20	#17984807	C₂²	1	1	yes	characteristic, complemented
21	#17984807	C₂²	1	3	yes	complemented
22	C₄×D₅₅₆	C₂²	1	6	yes	complemented
23	C₄×D₅₅₆	C₂²	1	6	yes	complemented
24	D₄×Dic₁₃₉	C₂²	1	3	yes	complemented
25	D₄×Dic₁₃₉	C₂²	1	3	yes	complemented
26	#17984804	C₂²	1	6	yes	complemented
27	#17984804	C₂²	1	3	yes	complemented
28	#17984804	C₂²	1	6	yes	complemented
29	#17984804	C₂²	1	6	yes	complemented
30	#17984804	C₂²	1	3	yes	complemented
31	C₄×C₂²⋊D₁₃₉	C₂²	1	12	yes	complemented
32	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	6	yes	complemented
33	(C₂²⋊D₁₃₉)⋊C₄	C₂²	1	6	yes	complemented
34	#17984801	C₂²	1	3	yes	complemented
35	#17984800	C₂²	1	6	yes	complemented
36	#17984799	C₂²	1	6	yes	complemented
37	C₄⋊C₄×D₁₃₉	C₂²	1	6	yes	complemented
38	C₂×(C₂×D₁₃₉)⋊C₄	C₂²	1	6	yes	complemented
39	C₂²⋊C₄×D₁₃₉	C₂²	1	12	yes	complemented
40	C₂×C₄×Dic₁₃₉	C₂²	1	3	yes	complemented
41	C₄²×D₁₃₉	C₂²	1	6	yes	complemented
42	C₂×#41×C₁₃₉	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₁₃₉	C₂²	1	2	yes	complemented
44	C₄×D₄×C₁₃₉	C₂²	1	6	yes	complemented
45	#117×C₁₃₉	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	3	yes	complemented
47	C₂³×D₁₃₉	C₂³	1	6	yes	—
48	#14555786	C₂×C₄	1	16	yes	complemented
49	#14555785	C₂×C₄	1	24	yes	complemented
50	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
51	C₂×D₅₅₆	C₂×C₄	1	24	yes	complemented
52	C₂×D₅₅₆	C₂³	1	3	yes	—
53	C₂×D₅₅₆	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₁₃₉	C₂³	1	6	yes	—
55	D₄×D₁₃₉	C₂×C₄	1	48	yes	complemented
56	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
57	C₂²×Dic₁₃₉	C₂³	1	3	yes	—
58	C₂×C₄×D₁₃₉	—	4	12	no	—
59	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
60	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
61	C₂×C₄×D₁₃₉	C₂³	1	12	yes	—
62	C₂×C₄×D₁₃₉	C₂³	1	6	yes	—
63	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
64	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
65	Dic₁₃₉⋊C₄	C₂³	1	6	yes	—
66	Dic₁₃₉⋊C₄	C₂³	1	2	yes	—
67	C₂²⋊Dic₁₃₉	C₂³	1	6	yes	—
68	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
69	(C₂×D₁₃₉)⋊C₄	C₂³	1	12	yes	—
70	(C₂×D₁₃₉)⋊C₄	C₂³	1	6	yes	—
71	C₄×Dic₁₃₉	C₂³	1	2	yes	—
72	C₄×Dic₁₃₉	C₂³	1	3	yes	—
73	C₄×Dic₁₃₉	C₂³	1	6	yes	—
74	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
75	C₄×Dic₁₃₉	C₂³	1	3	yes	—
76	C₄×Dic₁₃₉	C₂³	1	3	yes	—
77	#41×C₁₃₉	C₂×C₄	1	8	yes	complemented
78	C₂×Q₈×C₁₃₉	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₁₃₉	C₂³	1	3	yes	—
80	C₂²×C₄×C₁₃₉	C₂³	1	6	yes	—
81	C₂²×C₄×C₁₃₉	C₂³	1	3	yes	—
82	C₄⋊C₄×C₁₃₉	C₂³	1	6	yes	—
83	C₄⋊C₄×C₁₃₉	C₂³	1	3	yes	—
84	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₃₉	C₂³	1	6	yes	—
86	C₄²×C₁₃₉	C₂³	1	6	yes	—
87	C₂²×D₁₃₉	—	4	6	no	—
88	C₂²×D₁₃₉	—	4	6	no	—
89	C₂²×D₁₃₉	C₂²×C₄	1	24	yes	complemented
90	C₂²×D₁₃₉	C₂⁴	1	6	yes	—
91	Dic₂₇₈	C₂²×C₄	1	4	yes	complemented
92	D₅₅₆	C₂²×C₄	1	24	yes	complemented
93	D₅₅₆	C₂²×C₄	1	12	yes	complemented
94	C₂²⋊D₁₃₉	C₂²×C₄	1	24	yes	complemented
95	C₂×Dic₁₃₉	—	2	6	no	—
96	C₂×Dic₁₃₉	—	4	3	no	—
97	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
98	C₂×Dic₁₃₉	C₂⁴	1	3	yes	—
99	C₂×Dic₁₃₉	C₂⁴	1	3	yes	—
100	C₂×Dic₁₃₉	C₂⁴	1	2	yes	—
101	C₂×Dic₁₃₉	C₂⁴	1	1	yes	characteristic
102	C₄×D₁₃₉	—	2	12	no	—
103	C₄×D₁₃₉	—	4	12	no	—
104	C₄×D₁₃₉	—	4	6	no	—
105	C₄×D₁₃₉	C₂²×C₄	1	24	yes	complemented
106	C₂³×C₁₃₉	C₂⁴	1	3	yes	—
107	Q₈×C₁₃₉	C₂²×C₄	1	4	yes	complemented
108	D₄×C₁₃₉	C₂²×C₄	1	12	yes	complemented
109	C₂×C₄×C₁₃₉	—	4	6	no	—
110	C₂×C₄×C₁₃₉	C₂²×C₄	1	12	yes	complemented
111	C₂×C₄×C₁₃₉	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₃₉	C₂⁴	1	6	yes	—
113	C₂×C₄×C₁₃₉	C₂⁴	1	3	yes	—
114	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
115	C₂²×C₁₃₉	—	2	3	no	—
116	C₂²×C₁₃₉	—	2	3	no	—
117	C₂²×C₁₃₉	C₂³×C₄	1	6	yes	—
118	C₂²×C₁₃₉	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₃₉	—	2	6	no	—
120	C₂×D₁₃₉	—	2	6	no	—
121	C₂×D₁₃₉	—	4	24	no	—
122	C₂×D₁₃₉	C₂³×C₄	1	12	yes	—
123	C₄×C₁₃₉	—	2	6	no	—
124	C₄×C₁₃₉	—	2	2	no	—
125	C₄×C₁₃₉	C₂³×C₄	1	2	yes	—
126	C₄×C₁₃₉	C₂³×C₄	1	6	yes	—
127	Dic₁₃₉	—	2	3	no	—
128	Dic₁₃₉	—	2	3	no	—
129	Dic₁₃₉	—	2	1	no	—
130	Dic₁₃₉	C₂³×C₄	1	4	yes	—
131	Dic₁₃₉	H(5,𝔽₂)	1	2	yes	complemented
132	C₂×C₁₃₉	—	2	6	no	—
133	C₂×C₁₃₉	C₂⁴×C₄	1	1	yes	characteristic
134	C₂×C₁₃₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
135	C₂×C₁₃₉	#585	1	1	yes	characteristic
136	D₁₃₉	—	2	12	no	—
137	C₁₃₉	C₄×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
138	C₄×H(5,𝔽₂)	—	139	1	no	maximal, Sylow
139	C₂×H(5,𝔽₂)	—	139	1	no	—
140	#520	—	139	2	no	—
141	#520	—	139	3	no	—
142	#520	D₁₃₉	1	1	yes	characteristic, complemented
143	#518	—	139	6	no	—
144	#518	—	139	3	no	—
145	C₄×#41	—	139	1	no	—
146	C₄×#41	—	139	2	no	—
147	C₄×#41	—	139	3	no	—
148	C₂×C₄×D₄	—	139	6	no	—
149	C₂×C₄×D₄	—	139	3	no	—
150	H(5,𝔽₂)	—	139	16	no	—
151	C₂×#41	—	139	2	no	—
152	C₂×#41	—	139	3	no	—
153	C₂×#41	C₂×D₁₃₉	1	1	yes	characteristic
154	C₂²×D₄	—	139	6	no	—
155	C₂²×D₄	—	139	3	no	—
156	C₂³×C₄	—	139	6	no	—
157	C₄×Q₈	—	139	1	no	—
158	C₄×Q₈	—	139	1	no	—
159	C₄×Q₈	—	139	1	no	—
160	C₄×Q₈	—	139	3	no	—
161	C₄×Q₈	C₂×D₁₃₉	1	2	yes	complemented
162	C₄×D₄	—	139	3	no	—
163	C₄×D₄	—	139	3	no	—
164	C₄×D₄	—	139	12	no	—
165	C₄×D₄	—	139	6	no	—
166	C₄×D₄	—	139	3	no	—
167	C₄×D₄	—	139	6	no	—
168	C₄×D₄	—	139	6	no	—
169	C₄×D₄	—	139	3	no	—
170	C₄×D₄	—	139	6	no	—
171	C₄×D₄	—	139	6	no	—
172	C₄×D₄	—	139	6	no	—
173	C₄×D₄	—	139	6	no	—
174	C₄×D₄	C₂×D₁₃₉	1	6	yes	complemented
175	#117	—	139	3	no	—
176	#117	—	139	6	no	—
177	#117	—	139	6	no	—
178	#117	C₂×D₁₃₉	1	3	yes	complemented
179	C₂×C₄⋊C₄	—	139	6	no	—
180	C₂×C₄⋊C₄	C₂×D₁₃₉	1	3	yes	complemented
181	C₂×C₂²⋊C₄	—	139	6	no	—
182	C₂×C₂²⋊C₄	—	139	12	no	—
183	C₂×C₄²	—	139	3	no	—
184	C₂×C₄²	—	139	6	no	—
185	C₂⁴	—	139	6	no	—
186	#41	—	139	16	no	—
187	#41	—	139	24	no	—
188	#41	C₄×D₁₃₉	1	8	yes	complemented
189	C₂×Q₈	—	139	1	no	—
190	C₂×Q₈	C₂²×D₁₃₉	1	1	yes	characteristic
191	C₂×D₄	—	139	6	no	—
192	C₂×D₄	—	139	48	no	—
193	C₂×D₄	—	139	24	no	—
194	C₂×D₄	—	139	3	no	—
195	C₂×D₄	—	139	6	no	—
196	C₂×D₄	C₂²×D₁₃₉	1	3	yes	—
197	C₂²×C₄	—	556	12	no	—
198	C₂²×C₄	—	139	3	no	—
199	C₂²×C₄	—	139	6	no	—
200	C₂²×C₄	—	139	3	no	—
201	C₂²×C₄	—	139	6	no	—
202	C₂²×C₄	—	139	12	no	—
203	C₂²×C₄	—	139	6	no	—
204	C₂²×C₄	C₂²×D₁₃₉	1	6	yes	—
205	C₂²×C₄	C₂²×D₁₃₉	1	3	yes	—
206	C₄⋊C₄	—	139	6	no	—
207	C₄⋊C₄	—	139	3	no	—
208	C₄⋊C₄	—	139	2	no	—
209	C₄⋊C₄	—	139	3	no	—
210	C₄⋊C₄	C₂²×D₁₃₉	1	6	yes	—
211	C₄⋊C₄	C₂²×D₁₃₉	1	3	yes	—
212	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
213	C₂²⋊C₄	—	139	6	no	—
214	C₂²⋊C₄	—	139	12	no	—
215	C₂²⋊C₄	—	139	6	no	—
216	C₂²⋊C₄	—	139	6	no	—
217	C₂²⋊C₄	C₂²×D₁₃₉	1	6	yes	—
218	C₄²	—	139	1	no	—
219	C₄²	—	139	3	no	—
220	C₄²	—	139	2	no	—
221	C₄²	—	139	3	no	—
222	C₄²	—	139	3	no	—
223	C₄²	—	139	6	no	—
224	C₄²	C₂²×D₁₃₉	1	6	yes	—
225	C₂³	—	556	6	no	—
226	C₂³	—	556	6	no	—
227	C₂³	—	139	6	no	—
228	C₂³	—	139	24	no	—
229	C₂³	C₂³×D₁₃₉	1	3	yes	—
230	Q₈	—	139	4	no	—
231	Q₈	C₂×C₄×D₁₃₉	1	4	yes	complemented
232	D₄	—	139	24	no	—
233	D₄	—	139	24	no	—
234	D₄	—	139	12	no	—
235	D₄	C₂×C₄×D₁₃₉	1	12	yes	complemented
236	C₂×C₄	—	4	6	no	—
237	C₂×C₄	—	278	12	no	—
238	C₂×C₄	—	278	6	no	—
239	C₂×C₄	—	556	12	no	—
240	C₂×C₄	—	556	6	no	—
241	C₂×C₄	—	556	3	no	—
242	C₂×C₄	—	139	3	no	—
243	C₂×C₄	—	139	1	no	—
244	C₂×C₄	—	139	3	no	—
245	C₂×C₄	—	139	1	no	—
246	C₂×C₄	—	139	2	no	—
247	C₂×C₄	—	139	24	no	—
248	C₂×C₄	C₂×C₄×D₁₃₉	1	12	yes	complemented
249	C₂×C₄	C₂³×D₁₃₉	1	2	yes	—
250	C₂×C₄	C₂³×D₁₃₉	1	6	yes	—
251	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₁₃₉	1	3	yes	—
253	C₂²	—	2	3	no	—
254	C₂²	—	2	3	no	—
255	C₂²	—	278	6	no	—
256	C₂²	—	278	6	no	—
257	C₂²	—	556	24	no	—
258	C₂²	—	139	12	no	—
259	C₂²	C₂²×C₄×D₁₃₉	1	6	yes	—
260	C₂²	C₂⁴×D₁₃₉	1	1	yes	characteristic
261	C₄	—	2	6	no	—
262	C₄	—	2	2	no	—
263	C₄	—	278	3	no	—
264	C₄	—	278	3	no	—
265	C₄	—	278	1	no	—
266	C₄	—	139	4	no	—
267	C₄	—	139	2	no	—
268	C₄	C₂²×C₄×D₁₃₉	1	6	yes	—
269	C₄	C₂²×C₄×D₁₃₉	1	2	yes	—
270	C₂	—	2	6	no	—
271	C₂	—	278	12	no	—
272	C₂	C₂³×C₄×D₁₃₉	1	1	yes	characteristic
273	C₂	H(5,𝔽₂)×D₁₃₉	1	1	yes	characteristic
274	C₂	#22581790	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₉
commutator	C₂×C₁₃₉	C₂⁴×C₄
Fitting	#520×C₁₃₉	C₂
socle, cosocle	C₂²×C₁₃₉	C₂⁵
prime radical	C₂×#17984888	C₂
hypercenter	#520	D₁₃₉
hypocenter	C₁₃₉	C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #22581435	C₂	semidirect
3. #22581434	C₂	semidirect
4. #22581427	C₂	semidirect
5. #22581425	C₂	semidirect
6. #41×Dic₁₃₉	C₂	semidirect
7. C₄×#14555786	C₂	semidirect
8. #22581421	C₂	semidirect
9. C₂×#17984804	C₂	semidirect
10. C₄×D₄×D₁₃₉	C₂	semidirect
11. #520×C₁₃₉	C₂	semidirect
12. #17984888	C₄	semidirect
17. C₂²×C₄×D₁₃₉	C₂²	semidirect
18. Q₈×Dic₁₃₉	C₂²	semidirect
19. Q₈×Dic₁₃₉	C₂²	semidirect
20. #17984807	C₂²	semidirect
21. #17984807	C₂²	semidirect
22. C₄×D₅₅₆	C₂²	semidirect
23. C₄×D₅₅₆	C₂²	semidirect
24. D₄×Dic₁₃₉	C₂²	semidirect
25. D₄×Dic₁₃₉	C₂²	semidirect
26. #17984804	C₂²	semidirect
27. #17984804	C₂²	semidirect
28. #17984804	C₂²	semidirect
29. #17984804	C₂²	semidirect
30. #17984804	C₂²	semidirect
31. C₄×C₂²⋊D₁₃₉	C₂²	semidirect
32. (C₂²⋊D₁₃₉)⋊C₄	C₂²	semidirect
33. (C₂²⋊D₁₃₉)⋊C₄	C₂²	semidirect
34. #17984801	C₂²	semidirect
35. #17984800	C₂²	semidirect
36. #17984799	C₂²	semidirect
37. C₄⋊C₄×D₁₃₉	C₂²	semidirect
38. C₂×(C₂×D₁₃₉)⋊C₄	C₂²	semidirect
39. C₂²⋊C₄×D₁₃₉	C₂²	semidirect
40. C₂×C₄×Dic₁₃₉	C₂²	semidirect
41. C₄²×D₁₃₉	C₂²	semidirect
43. C₄×Q₈×C₁₃₉	C₂²	semidirect
44. C₄×D₄×C₁₃₉	C₂²	semidirect
45. #117×C₁₃₉	C₂²	semidirect
46. C₂×C₄⋊C₄×C₁₃₉	C₂²	semidirect
48. #14555786	C₂×C₄	semidirect
49. #14555785	C₂×C₄	semidirect
51. C₂×D₅₅₆	C₂×C₄	semidirect
55. D₄×D₁₃₉	C₂×C₄	semidirect
77. #41×C₁₃₉	C₂×C₄	semidirect
89. C₂²×D₁₃₉	C₂²×C₄	semidirect
91. Dic₂₇₈	C₂²×C₄	semidirect
92. D₅₅₆	C₂²×C₄	semidirect
93. D₅₅₆	C₂²×C₄	semidirect
94. C₂²⋊D₁₃₉	C₂²×C₄	semidirect
105. C₄×D₁₃₉	C₂²×C₄	semidirect
107. Q₈×C₁₃₉	C₂²×C₄	semidirect
108. D₄×C₁₃₉	C₂²×C₄	semidirect
110. C₂×C₄×C₁₃₉	C₂²×C₄	semidirect
131. Dic₁₃₉	H(5,𝔽₂)	semidirect
137. C₁₃₉	C₄×H(5,𝔽₂)	semidirect
142. #520	D₁₃₉	semidirect
161. C₄×Q₈	C₂×D₁₃₉	semidirect
174. C₄×D₄	C₂×D₁₃₉	semidirect
178. #117	C₂×D₁₃₉	semidirect
180. C₂×C₄⋊C₄	C₂×D₁₃₉	semidirect
188. #41	C₄×D₁₃₉	semidirect
231. Q₈	C₂×C₄×D₁₃₉	semidirect
235. D₄	C₂×C₄×D₁₃₉	semidirect
248. C₂×C₄	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×#17984888	2
#22581435	2
#22581434	2
#22581427	2
#22581425	2
#41×Dic₁₃₉	2
C₄×#14555786	2
#22581421	2
C₂×#17984804	2
C₄×D₄×D₁₃₉	2
#520×C₁₃₉	2
C₄×H(5,𝔽₂)	139