Subgroups — #35197151

Order 20224.

8650 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23777733	C₂	1	24	yes	maximal, complemented
2	#23777274	C₂	1	12	yes	maximal, complemented
3	#23777176	C₂	1	12	yes	maximal, complemented
4	#23777035	C₂	1	12	yes	maximal, complemented
5	#23776696	C₂	1	2	yes	maximal, complemented
6	C₂²×#127×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14579206	C₂²	1	12	yes	complemented
8	#19099300	C₂²	1	32	yes	complemented
9	#19099299	C₂²	1	24	yes	complemented
10	#19099233	C₂²	1	48	yes	complemented
11	#19099207	C₂²	1	96	yes	complemented
12	#19099198	C₂²	1	24	yes	complemented
13	D₄×C₂²⋊D₇₉	C₂²	1	96	yes	complemented
14	#19099167	C₂²	1	48	yes	complemented
15	#19099130	C₂²	1	24	yes	complemented
16	#19099126	C₂²	1	48	yes	complemented
17	#19099122	C₂²	1	48	yes	complemented
18	#19099120	C₂²	1	6	yes	complemented
19	#19099113	C₂²	1	6	yes	complemented
20	#19099111	C₂²	1	24	yes	complemented
21	#127×D₇₉	C₂²	1	8	yes	complemented
22	#19099097	C₂²	1	24	yes	complemented
23	#19099074	C₂²	1	12	yes	complemented
24	C₂×#14579130	C₂²	1	16	yes	complemented
25	#19099038	C₂²	1	12	yes	complemented
26	C₄×#11929624	C₂²	1	12	yes	complemented
27	C₂³×D₄×C₇₉	C₂²	1	6	yes	complemented
28	C₂×#127×C₇₉	C₂²	1	12	yes	complemented
29	C₂×#127×C₇₉	C₂²	1	12	yes	complemented
30	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic, complemented
31	#14579206	—	4	48	no	—
32	C₂×#11929624	C₂³	1	12	yes	—
33	C₂×#11929623	C₂³	1	48	yes	complemented
34	C₂²×C₂²⋊D₇₉	C₂³	1	48	yes	—
35	C₂×D₄×D₇₉	C₂³	1	48	yes	complemented
36	#14579161	C₂³	1	6	yes	complemented
37	#14579158	C₂³	1	2	yes	complemented
38	#14579157	C₂³	1	48	yes	complemented
39	#14579155	C₂³	1	16	yes	complemented
40	#14579153	C₂³	1	12	yes	complemented
41	#14579149	C₂³	1	12	yes	complemented
42	#14579148	C₂³	1	48	yes	complemented
43	#14579144	C₂³	1	24	yes	—
44	#14579142	C₂³	1	96	yes	complemented
45	#14579140	C₂³	1	24	yes	—
46	#14579135	C₂³	1	24	yes	—
47	#14579133	C₂³	1	192	yes	complemented
48	#14579132	C₂³	1	96	yes	complemented
49	#14579130	C₂³	1	96	yes	—
50	#14579130	C₂³	1	8	yes	—
51	#14579129	C₂³	1	96	yes	complemented
52	C₄×Dic₁₅₈	C₂³	1	24	yes	complemented
53	C₄×D₃₁₆	C₂³	1	24	yes	complemented
54	D₄×Dic₇₉	C₂³	1	96	yes	complemented
55	C₄×C₂²⋊D₇₉	C₂³	1	48	yes	—
56	#14579120	C₂³	1	12	yes	—
57	#14579118	C₂³	1	24	yes	complemented
58	C₂×C₂²⋊Dic₇₉	C₂³	1	48	yes	—
59	C₄²×D₇₉	C₂³	1	8	yes	complemented
60	C₂⁵×C₇₉	C₂³	1	4	yes	—
61	C₂²×D₄×C₇₉	—	4	24	no	—
62	C₂²×D₄×C₇₉	C₂³	1	12	yes	—
63	C₂²×D₄×C₇₉	C₂³	1	48	yes	—
64	C₂²×D₄×C₇₉	C₂³	1	12	yes	—
65	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
66	#127×C₇₉	C₂³	1	24	yes	—
67	#127×C₇₉	C₂³	1	24	yes	—
68	#127×C₇₉	C₂³	1	8	yes	—
69	#127×C₇₉	C₂³	1	8	yes	—
70	C₂×C₄²×C₇₉	C₂³	1	6	yes	—
71	C₂×C₄²×C₇₉	C₂³	1	6	yes	—
72	C₂³×D₇₉	C₂⁴	1	16	yes	—
73	#11929624	—	2	48	no	—
74	#11929623	—	4	96	no	—
75	C₂×Dic₁₅₈	C₂⁴	1	12	yes	—
76	C₂×D₃₁₆	C₂⁴	1	12	yes	—
77	C₂×C₂²⋊D₇₉	—	4	96	no	—
78	C₂×C₂²⋊D₇₉	C₂⁴	1	24	yes	—
79	C₂×C₂²⋊D₇₉	C₂⁴	1	96	yes	—
80	D₄×D₇₉	—	4	96	no	—
81	C₂²×Dic₇₉	C₂⁴	1	48	yes	—
82	C₂×C₄×D₇₉	C₂⁴	1	24	yes	—
83	C₄⋊Dic₇₉	C₂⁴	1	24	yes	—
84	Dic₇₉⋊C₄	C₂⁴	1	48	yes	—
85	C₂²⋊Dic₇₉	C₂⁴	1	24	yes	—
86	C₂²⋊Dic₇₉	C₂⁴	1	96	yes	—
87	(C₂×D₇₉)⋊C₄	C₂⁴	1	48	yes	—
88	C₄×Dic₇₉	C₂⁴	1	24	yes	—
89	C₂⁴×C₇₉	—	4	24	no	—
90	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
91	C₂⁴×C₇₉	C₂⁴	1	24	yes	—
92	C₂×D₄×C₇₉	—	2	48	no	—
93	C₂×D₄×C₇₉	—	4	96	no	—
94	C₂×D₄×C₇₉	—	4	24	no	—
95	C₂×D₄×C₇₉	C₂⁴	1	48	yes	—
96	C₂×D₄×C₇₉	C₂⁴	1	48	yes	—
97	C₂²×C₄×C₇₉	—	2	12	no	—
98	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
99	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
100	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
101	C₄²×C₇₉	C₂⁴	1	2	yes	—
102	C₄²×C₇₉	C₂⁴	1	6	yes	—
103	C₄²×C₇₉	C₂⁴	1	6	yes	—
104	C₄²×C₇₉	C₂⁴	1	2	yes	—
105	C₂³×C₇₉	—	2	6	no	—
106	C₂³×C₇₉	—	2	48	no	—
107	C₂³×C₇₉	—	4	48	no	—
108	C₂³×C₇₉	—	8	32	no	—
109	C₂³×C₇₉	C₂⁵	1	16	yes	—
110	C₂³×C₇₉	C₂⁵	1	3	yes	—
111	C₂²×D₇₉	—	4	48	no	—
112	C₂²×D₇₉	C₂⁵	1	8	yes	—
113	D₄×C₇₉	—	2	96	no	—
114	D₄×C₇₉	—	4	48	no	—
115	C₂×C₄×C₇₉	—	2	6	no	—
116	C₂×C₄×C₇₉	—	2	24	no	—
117	C₂×C₄×C₇₉	H(5,𝔽₂)	1	12	yes	complemented
118	C₂×C₄×C₇₉	C₂⁵	1	6	yes	—
119	C₂×C₄×C₇₉	C₂⁵	1	6	yes	—
120	C₂²⋊D₇₉	—	2	48	no	—
121	C₂²⋊D₇₉	—	4	96	no	—
122	C₂×Dic₇₉	—	4	48	no	—
123	C₂×Dic₇₉	C₂⁵	1	24	yes	—
124	D₃₁₆	—	2	24	no	—
125	C₄×D₇₉	—	2	48	no	—
126	Dic₁₅₈	—	2	24	no	—
127	C₂²×C₇₉	—	2	8	no	—
128	C₂²×C₇₉	—	2	6	no	—
129	C₂²×C₇₉	—	2	48	no	—
130	C₂²×C₇₉	—	8	48	no	—
131	C₂²×C₇₉	C₂×H(5,𝔽₂)	1	6	yes	—
132	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
133	C₂×D₇₉	—	2	24	no	—
134	C₂×D₇₉	—	16	16	no	—
135	C₄×C₇₉	—	2	6	no	—
136	C₄×C₇₉	C₂×H(5,𝔽₂)	1	12	yes	—
137	Dic₇₉	—	2	24	no	—
138	#62832	—	79	1	no	maximal, Sylow
139	C₂×C₇₉	—	2	6	no	—
140	C₂×C₇₉	—	4	16	no	—
141	C₂×C₇₉	C₂²×H(5,𝔽₂)	1	3	yes	—
142	D₇₉	—	4	8	no	—
143	#3543	—	79	24	no	—
144	#3508	—	79	12	no	—
145	#3498	—	79	12	no	—
146	#3484	—	79	12	no	—
147	#3444	—	79	2	no	—
148	C₂²×#127	D₇₉	1	1	yes	characteristic, complemented
149	C₇₉	#62832	1	1	yes	characteristic, complemented, Sylow
150	C₂×H(5,𝔽₂)	—	79	12	no	—
151	C₂³×D₄	C₂×D₇₉	1	6	yes	complemented
152	#560	—	79	32	no	—
153	#559	—	79	24	no	—
154	#550	—	79	48	no	—
155	#546	—	79	96	no	—
156	D₄²	—	79	96	no	—
157	D₄²	—	79	24	no	—
158	#540	—	79	48	no	—
159	#535	—	79	24	no	—
160	#535	—	79	48	no	—
161	#534	—	79	48	no	—
162	#533	—	79	6	no	—
163	#532	—	79	24	no	—
164	#532	—	79	6	no	—
165	C₂×#127	—	79	8	no	—
166	C₂×#127	C₂×D₇₉	1	12	yes	complemented
167	C₂×#127	C₂×D₇₉	1	12	yes	complemented
168	#529	—	79	24	no	—
169	#525	—	79	12	no	—
170	C₂×#120	—	79	16	no	—
171	#518	—	79	12	no	—
172	C₄×#41	—	79	12	no	—
173	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic, complemented
174	C₂⁵	C₂²×D₇₉	1	4	yes	—
175	H(5,𝔽₂)	—	316	48	no	—
176	C₂×#41	—	79	48	no	—
177	C₂×#41	—	79	12	no	—
178	C₂²×D₄	—	4	24	no	—
179	C₂²×D₄	—	79	48	no	—
180	C₂²×D₄	—	79	48	no	—
181	C₂²×D₄	C₂²×D₇₉	1	12	yes	—
182	C₂²×D₄	C₂²×D₇₉	1	12	yes	—
183	C₂²×D₄	C₂²×D₇₉	1	48	yes	—
184	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
185	C₄⋊Q₈	—	79	6	no	—
186	#127	—	79	48	no	—
187	#127	—	79	2	no	—
188	#127	C₂²×D₇₉	1	8	yes	—
189	#127	C₂²×D₇₉	1	24	yes	—
190	#127	C₂²×D₇₉	1	24	yes	—
191	#127	C₂²×D₇₉	1	8	yes	—
192	#126	—	79	16	no	—
193	#125	—	79	12	no	—
194	#124	—	79	48	no	—
195	#124	—	79	12	no	—
196	#123	—	79	96	no	—
197	#123	—	79	24	no	—
198	C₂²⋊Q₈	—	79	24	no	—
199	#121	—	79	96	no	—
200	#121	—	79	192	no	—
201	#121	—	79	24	no	—
202	#120	—	79	8	no	—
203	#120	—	79	96	no	—
204	#120	—	79	96	no	—
205	C₄×Q₈	—	79	24	no	—
206	C₄×D₄	—	79	96	no	—
207	C₄×D₄	—	79	48	no	—
208	C₄×D₄	—	79	24	no	—
209	#117	—	79	12	no	—
210	#117	—	79	24	no	—
211	C₂×C₂²⋊C₄	—	79	48	no	—
212	C₂×C₄²	—	79	8	no	—
213	C₂×C₄²	C₂²×D₇₉	1	6	yes	—
214	C₂×C₄²	C₂²×D₇₉	1	6	yes	—
215	C₂⁴	—	4	24	no	—
216	C₂⁴	—	79	16	no	—
217	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
218	C₂⁴	C₂³×D₇₉	1	24	yes	—
219	#41	—	316	96	no	—
220	#41	—	158	48	no	—
221	C₂×Q₈	—	79	12	no	—
222	C₂×D₄	—	2	48	no	—
223	C₂×D₄	—	4	96	no	—
224	C₂×D₄	—	4	24	no	—
225	C₂×D₄	—	316	96	no	—
226	C₂×D₄	—	316	96	no	—
227	C₂×D₄	—	79	96	no	—
228	C₂×D₄	—	79	24	no	—
229	C₂×D₄	—	79	12	no	—
230	C₂×D₄	C₂³×D₇₉	1	48	yes	—
231	C₂×D₄	C₂³×D₇₉	1	48	yes	—
232	C₂²×C₄	—	2	12	no	—
233	C₂²×C₄	—	79	48	no	—
234	C₂²×C₄	—	79	24	no	—
235	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
236	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
237	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
238	C₄⋊C₄	—	79	48	no	—
239	C₄⋊C₄	—	79	24	no	—
240	C₂²⋊C₄	—	79	24	no	—
241	C₂²⋊C₄	—	79	96	no	—
242	C₂²⋊C₄	—	79	48	no	—
243	C₄²	—	79	24	no	—
244	C₄²	C₂³×D₇₉	1	2	yes	—
245	C₄²	C₂³×D₇₉	1	6	yes	—
246	C₄²	C₂³×D₇₉	1	6	yes	—
247	C₄²	C₂³×D₇₉	1	2	yes	—
248	C₂³	—	2	6	no	—
249	C₂³	—	2	48	no	—
250	C₂³	—	4	48	no	—
251	C₂³	—	8	32	no	—
252	C₂³	—	316	48	no	—
253	C₂³	—	79	8	no	—
254	C₂³	C₂⁴×D₇₉	1	16	yes	—
255	C₂³	C₂⁴×D₇₉	1	3	yes	—
256	Q₈	—	158	24	no	—
257	D₄	—	2	96	no	—
258	D₄	—	4	48	no	—
259	D₄	—	316	96	no	—
260	D₄	—	158	48	no	—
261	D₄	—	158	24	no	—
262	C₂×C₄	—	2	6	no	—
263	C₂×C₄	—	2	24	no	—
264	C₂×C₄	—	316	48	no	—
265	C₂×C₄	—	79	24	no	—
266	C₂×C₄	—	158	48	no	—
267	C₂×C₄	#14579206	1	12	yes	complemented
268	C₂×C₄	C₂⁴×D₇₉	1	6	yes	—
269	C₂×C₄	C₂⁴×D₇₉	1	6	yes	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	6	no	—
272	C₂²	—	2	48	no	—
273	C₂²	—	8	48	no	—
274	C₂²	—	158	24	no	—
275	C₂²	—	1264	16	no	—
276	C₂²	C₂×#14579206	1	6	yes	—
277	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
278	C₄	—	2	6	no	—
279	C₄	—	158	24	no	—
280	C₄	C₂×#14579206	1	12	yes	—
281	C₂	—	2	6	no	—
282	C₂	—	4	16	no	—
283	C₂	—	316	8	no	—
284	C₂	C₂²×#14579206	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁶
Fitting	C₂²×#127×C₇₉	C₂
hypercenter	C₂²×#127	D₇₉
hypocenter	C₇₉	#62832
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. #23777733	C₂	semidirect
2. #23777274	C₂	semidirect
3. #23777176	C₂	semidirect
4. #23777035	C₂	semidirect
5. #23776696	C₂	semidirect
6. C₂²×#127×C₇₉	C₂	semidirect
7. C₂×#14579206	C₂²	semidirect
8. #19099300	C₂²	semidirect
9. #19099299	C₂²	semidirect
10. #19099233	C₂²	semidirect
11. #19099207	C₂²	semidirect
12. #19099198	C₂²	semidirect
13. D₄×C₂²⋊D₇₉	C₂²	semidirect
14. #19099167	C₂²	semidirect
15. #19099130	C₂²	semidirect
16. #19099126	C₂²	semidirect
17. #19099122	C₂²	semidirect
18. #19099120	C₂²	semidirect
19. #19099113	C₂²	semidirect
20. #19099111	C₂²	semidirect
21. #127×D₇₉	C₂²	semidirect
22. #19099097	C₂²	semidirect
23. #19099074	C₂²	semidirect
24. C₂×#14579130	C₂²	semidirect
25. #19099038	C₂²	semidirect
26. C₄×#11929624	C₂²	semidirect
27. C₂³×D₄×C₇₉	C₂²	semidirect
28. C₂×#127×C₇₉	C₂²	semidirect
29. C₂×#127×C₇₉	C₂²	semidirect
30. C₂²×C₄²×C₇₉	C₂²	semidirect
33. C₂×#11929623	C₂³	semidirect
35. C₂×D₄×D₇₉	C₂³	semidirect
36. #14579161	C₂³	semidirect
37. #14579158	C₂³	semidirect
38. #14579157	C₂³	semidirect
39. #14579155	C₂³	semidirect
40. #14579153	C₂³	semidirect
41. #14579149	C₂³	semidirect
42. #14579148	C₂³	semidirect
44. #14579142	C₂³	semidirect
47. #14579133	C₂³	semidirect
48. #14579132	C₂³	semidirect
51. #14579129	C₂³	semidirect
52. C₄×Dic₁₅₈	C₂³	semidirect
53. C₄×D₃₁₆	C₂³	semidirect
54. D₄×Dic₇₉	C₂³	semidirect
57. #14579118	C₂³	semidirect
59. C₄²×D₇₉	C₂³	semidirect
117. C₂×C₄×C₇₉	H(5,𝔽₂)	semidirect
148. C₂²×#127	D₇₉	semidirect
149. C₇₉	#62832	semidirect
151. C₂³×D₄	C₂×D₇₉	semidirect
166. C₂×#127	C₂×D₇₉	semidirect
167. C₂×#127	C₂×D₇₉	semidirect
173. C₂²×C₄²	C₂×D₇₉	semidirect
267. C₂×C₄	#14579206	semidirect

Maximal subgroups

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

subgroup	index
#23777733	2
#23777274	2
#23777176	2
#23777035	2
#23776696	2
C₂²×#127×C₇₉	2
#62832	79