Subgroups — #35197287

Order 20224.

7130 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23777203	C₂	1	16	yes	maximal, complemented
2	#23777198	C₂	1	32	yes	maximal, complemented
3	#23776805	C₂	1	4	yes	maximal, complemented
4	C₂×#19099189	C₂	1	8	yes	maximal, complemented
5	C₂×#19099038	C₂	1	2	yes	maximal
6	C₂×#518×C₇₉	C₂	1	1	yes	maximal, characteristic
7	#19099272	C₂²	1	128	yes	complemented
8	#19099270	C₂²	1	64	yes	complemented
9	D₄×Dic₁₅₈	C₂²	1	64	yes	complemented
10	#19099194	C₂²	1	16	yes	complemented
11	#19099191	C₂²	1	32	yes	complemented
12	#19099189	C₂²	1	16	yes	—
13	#19099189	C₂²	1	64	yes	complemented
14	#19099189	C₂²	1	32	yes	complemented
15	#19099120	C₂²	1	16	yes	complemented
16	#19099116	C₂²	1	32	yes	complemented
17	C₂×#14579140	C₂²	1	32	yes	complemented
18	C₂×#14579138	C₂²	1	64	yes	complemented
19	#19099038	C₂²	1	16	yes	—
20	#19099038	C₂²	1	16	yes	—
21	C₂×D₄×Dic₇₉	C₂²	1	16	yes	—
22	C₂×#14579120	C₂²	1	4	yes	—
23	C₂²×C₂²⋊Dic₇₉	C₂²	1	8	yes	—
24	C₂³×D₄×C₇₉	C₂²	1	1	yes	characteristic
25	#518×C₇₉	C₂²	1	16	yes	—
26	C₂×C₄×D₄×C₇₉	C₂²	1	8	yes	—
27	C₂×#117×C₇₉	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₇₉	C₂²	1	4	yes	—
29	C₂²×Dic₁₅₈	C₂³	1	32	yes	complemented
30	C₂³×Dic₇₉	C₂³	1	16	yes	—
31	#14579161	C₂³	1	32	yes	complemented
32	#14579160	C₂³	1	64	yes	complemented
33	#14579153	C₂³	1	32	yes	complemented
34	#14579152	C₂³	1	64	yes	complemented
35	#14579140	C₂³	1	64	yes	—
36	#14579140	C₂³	1	128	yes	complemented
37	#14579138	C₂³	1	128	yes	—
38	#14579138	C₂³	1	256	yes	complemented
39	C₄×Dic₁₅₈	C₂³	1	64	yes	complemented
40	D₄×Dic₇₉	C₂³	1	128	yes	—
41	#14579120	C₂³	1	16	yes	—
42	#14579120	C₂³	1	16	yes	—
43	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
44	C₂×C₄⋊Dic₇₉	C₂³	1	32	yes	complemented
45	C₂×Dic₇₉⋊C₄	C₂³	1	64	yes	complemented
46	C₂×C₂²⋊Dic₇₉	C₂³	1	8	yes	—
47	C₂×C₂²⋊Dic₇₉	C₂³	1	16	yes	—
48	C₂×C₂²⋊Dic₇₉	C₂³	1	32	yes	—
49	C₂×C₂²⋊Dic₇₉	C₂³	1	32	yes	—
50	C₂×C₄×Dic₇₉	C₂³	1	8	yes	—
51	C₂⁵×C₇₉	C₂³	1	2	yes	—
52	C₂²×D₄×C₇₉	Q₈	1	16	yes	complemented
53	C₂²×D₄×C₇₉	C₂³	1	4	yes	—
54	C₂²×D₄×C₇₉	C₂³	1	8	yes	—
55	C₂³×C₄×C₇₉	C₂³	1	8	yes	—
56	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
57	C₄×D₄×C₇₉	C₂³	1	64	yes	—
58	#117×C₇₉	C₂³	1	16	yes	—
59	C₂×C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
60	C₂×C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₇₉	C₂³	1	8	yes	—
62	C₂×C₂²⋊C₄×C₇₉	C₂³	1	32	yes	—
63	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
64	C₂×Dic₁₅₈	—	8	32	no	—
65	C₂×Dic₁₅₈	C₂⁴	1	64	yes	—
66	C₂²×Dic₇₉	—	8	16	no	—
67	C₂²×Dic₇₉	C₂⁴	1	64	yes	—
68	C₂²×Dic₇₉	C₂⁴	1	16	yes	—
69	C₄⋊Dic₇₉	C₂⁴	1	32	yes	—
70	C₄⋊Dic₇₉	C₂⁴	1	64	yes	—
71	Dic₇₉⋊C₄	C₂⁴	1	128	yes	—
72	C₂²⋊Dic₇₉	C₂⁴	1	16	yes	—
73	C₂²⋊Dic₇₉	C₂⁴	1	16	yes	—
74	C₂²⋊Dic₇₉	C₂⁴	1	64	yes	—
75	C₄×Dic₇₉	C₂⁴	1	32	yes	—
76	C₂⁴×C₇₉	—	8	4	no	—
77	C₂⁴×C₇₉	C₂×Q₈	1	16	yes	complemented
78	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
79	C₂⁴×C₇₉	C₂⁴	1	8	yes	—
80	C₂⁴×C₇₉	C₂⁴	1	4	yes	—
81	C₂×D₄×C₇₉	C₂×Q₈	1	32	yes	complemented
82	C₂×D₄×C₇₉	C₂×Q₈	1	64	yes	complemented
83	C₂×D₄×C₇₉	C₂⁴	1	16	yes	—
84	C₂²×C₄×C₇₉	—	8	8	no	—
85	C₂²×C₄×C₇₉	C₂×Q₈	1	8	yes	complemented
86	C₂²×C₄×C₇₉	C₂⁴	1	32	yes	—
87	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
88	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
90	C₄⋊C₄×C₇₉	C₂⁴	1	16	yes	—
91	C₂²⋊C₄×C₇₉	C₂⁴	1	16	yes	—
92	C₂²⋊C₄×C₇₉	C₂⁴	1	32	yes	—
93	C₄²×C₇₉	C₂⁴	1	16	yes	—
94	C₂³×C₇₉	—	4	16	no	—
95	C₂³×C₇₉	—	4	8	no	—
96	C₂³×C₇₉	—	4	2	no	—
97	C₂³×C₇₉	—	8	16	no	—
98	C₂³×C₇₉	C₂²×Q₈	1	4	yes	—
99	C₂³×C₇₉	C₂²×Q₈	1	32	yes	—
100	C₂³×C₇₉	C₂²×Q₈	1	16	yes	—
101	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
102	C₂³×C₇₉	C₂⁵	1	2	yes	—
103	C₂³×C₇₉	C₂⁵	1	8	yes	—
104	D₄×C₇₉	C₂²×Q₈	1	64	yes	—
105	C₂×C₄×C₇₉	—	4	32	no	—
106	C₂×C₄×C₇₉	—	4	8	no	—
107	C₂×C₄×C₇₉	C₂²×Q₈	1	8	yes	—
108	C₂×C₄×C₇₉	C₂²×Q₈	1	16	yes	—
109	C₂×C₄×C₇₉	C₂⁵	1	16	yes	—
110	C₂×C₄×C₇₉	C₂⁵	1	4	yes	—
111	C₂×Dic₇₉	—	4	16	no	—
112	C₂×Dic₇₉	—	4	64	no	—
113	C₂×Dic₇₉	C₂⁵	1	32	yes	—
114	Dic₁₅₈	—	4	64	no	—
115	C₂²×C₇₉	—	2	2	no	—
116	C₂²×C₇₉	—	2	16	no	—
117	C₂²×C₇₉	—	2	4	no	—
118	C₂²×C₇₉	—	4	4	no	—
119	C₂²×C₇₉	—	4	32	no	—
120	C₂²×C₇₉	—	4	16	no	—
121	C₂²×C₇₉	C₂³×Q₈	1	2	yes	—
122	C₂²×C₇₉	C₂³×Q₈	1	4	yes	—
123	C₂²×C₇₉	C₂³×Q₈	1	16	yes	—
124	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
125	C₄×C₇₉	—	2	16	no	—
126	C₄×C₇₉	C₂³×Q₈	1	8	yes	—
127	Dic₇₉	—	2	32	no	—
128	#62843	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	—	2	2	no	—
130	C₂×C₇₉	—	2	4	no	—
131	C₂×C₇₉	—	2	16	no	—
132	C₂×C₇₉	C₂⁴×Q₈	1	1	yes	characteristic
133	C₂×C₇₉	H(7,𝔽₂)	1	2	yes	—
134	#3501	—	79	32	no	—
135	#3501	—	79	16	no	—
136	#3458	—	79	4	no	—
137	C₂×#543	—	79	8	no	—
138	C₂×#518	—	79	2	no	—
139	C₂×#518	D₇₉	1	1	yes	characteristic
140	C₇₉	#62843	1	1	yes	characteristic, complemented, Sylow
141	C₂³×D₄	C₂×D₇₉	1	1	yes	characteristic
142	#554	—	79	64	no	—
143	#554	—	79	128	no	—
144	D₄×Q₈	—	79	64	no	—
145	#544	—	79	32	no	—
146	#544	—	79	16	no	—
147	#543	—	79	16	no	—
148	#543	—	79	64	no	—
149	#543	—	79	32	no	—
150	#533	—	79	32	no	—
151	#533	—	79	16	no	—
152	C₂×C₂²⋊Q₈	—	79	32	no	—
153	C₂×C₂²⋊Q₈	—	79	64	no	—
154	#518	—	79	16	no	—
155	#518	—	79	16	no	—
156	#518	C₂×D₇₉	1	16	yes	—
157	C₂×C₄×D₄	—	79	16	no	—
158	C₂×C₄×D₄	C₂×D₇₉	1	8	yes	—
159	C₂×#117	—	79	4	no	—
160	C₂×#117	C₂×D₇₉	1	2	yes	—
161	C₂²×C₂²⋊C₄	—	79	8	no	—
162	C₂²×C₂²⋊C₄	C₂×D₇₉	1	4	yes	—
163	C₂⁵	C₂²×D₇₉	1	2	yes	—
164	C₂²×Q₈	—	79	32	no	—
165	C₂²×D₄	Dic₁₅₈	1	16	yes	complemented
166	C₂²×D₄	C₂²×D₇₉	1	8	yes	—
167	C₂²×D₄	C₂²×D₇₉	1	4	yes	—
168	C₂³×C₄	—	79	16	no	—
169	C₂³×C₄	C₂²×D₇₉	1	8	yes	—
170	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
171	C₄⋊Q₈	—	79	64	no	—
172	C₄⋊Q₈	—	79	32	no	—
173	#125	—	79	64	no	—
174	#125	—	79	32	no	—
175	C₂²⋊Q₈	—	79	64	no	—
176	C₂²⋊Q₈	—	79	128	no	—
177	C₂²⋊Q₈	—	79	128	no	—
178	C₂²⋊Q₈	—	79	256	no	—
179	C₄×Q₈	—	79	64	no	—
180	C₄×D₄	—	79	128	no	—
181	C₄×D₄	C₂²×D₇₉	1	64	yes	—
182	#117	—	79	16	no	—
183	#117	—	79	16	no	—
184	#117	C₂²×D₇₉	1	16	yes	—
185	C₂×C₄⋊C₄	—	79	8	no	—
186	C₂×C₄⋊C₄	—	79	64	no	—
187	C₂×C₄⋊C₄	—	79	32	no	—
188	C₂×C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
189	C₂×C₂²⋊C₄	—	79	8	no	—
190	C₂×C₂²⋊C₄	—	79	16	no	—
191	C₂×C₂²⋊C₄	—	79	32	no	—
192	C₂×C₂²⋊C₄	—	79	32	no	—
193	C₂×C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂²×D₇₉	1	8	yes	—
195	C₂×C₂²⋊C₄	C₂²×D₇₉	1	32	yes	—
196	C₂×C₄²	—	79	8	no	—
197	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
198	C₂⁴	—	8	4	no	—
199	C₂⁴	C₂×Dic₁₅₈	1	16	yes	complemented
200	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
201	C₂⁴	C₂³×D₇₉	1	4	yes	—
202	C₂⁴	C₂³×D₇₉	1	8	yes	—
203	C₂×Q₈	—	79	64	no	—
204	C₂×Q₈	—	632	32	no	—
205	C₂×D₄	C₂×Dic₁₅₈	1	64	yes	complemented
206	C₂×D₄	C₂×Dic₁₅₈	1	32	yes	complemented
207	C₂×D₄	C₂³×D₇₉	1	16	yes	—
208	C₂²×C₄	—	8	8	no	—
209	C₂²×C₄	—	79	16	no	—
210	C₂²×C₄	—	79	64	no	—
211	C₂²×C₄	—	632	16	no	—
212	C₂²×C₄	C₂×Dic₁₅₈	1	8	yes	complemented
213	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
214	C₂²×C₄	C₂³×D₇₉	1	32	yes	—
215	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
216	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
217	C₄⋊C₄	—	79	128	no	—
218	C₄⋊C₄	—	79	64	no	—
219	C₄⋊C₄	—	79	32	no	—
220	C₄⋊C₄	C₂³×D₇₉	1	16	yes	—
221	C₂²⋊C₄	—	79	16	no	—
222	C₂²⋊C₄	—	79	16	no	—
223	C₂²⋊C₄	—	79	64	no	—
224	C₂²⋊C₄	C₂³×D₇₉	1	16	yes	—
225	C₂²⋊C₄	C₂³×D₇₉	1	32	yes	—
226	C₄²	—	79	32	no	—
227	C₄²	C₂³×D₇₉	1	16	yes	—
228	C₂³	—	4	8	no	—
229	C₂³	—	4	16	no	—
230	C₂³	—	4	2	no	—
231	C₂³	—	8	16	no	—
232	C₂³	C₂²×Dic₁₅₈	1	4	yes	—
233	C₂³	C₂²×Dic₁₅₈	1	16	yes	—
234	C₂³	C₂²×Dic₁₅₈	1	32	yes	—
235	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
236	C₂³	C₂⁴×D₇₉	1	2	yes	—
237	C₂³	C₂⁴×D₇₉	1	8	yes	—
238	Q₈	—	316	64	no	—
239	D₄	C₂²×Dic₁₅₈	1	64	yes	—
240	C₂×C₄	—	4	8	no	—
241	C₂×C₄	—	4	32	no	—
242	C₂×C₄	—	316	16	no	—
243	C₂×C₄	—	316	64	no	—
244	C₂×C₄	—	79	32	no	—
245	C₂×C₄	C₂²×Dic₁₅₈	1	16	yes	—
246	C₂×C₄	C₂²×Dic₁₅₈	1	8	yes	—
247	C₂×C₄	C₂⁴×D₇₉	1	16	yes	—
248	C₂×C₄	C₂⁴×D₇₉	1	4	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	16	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	16	no	—
254	C₂²	—	4	32	no	—
255	C₂²	C₂³×Dic₁₅₈	1	2	yes	—
256	C₂²	C₂³×Dic₁₅₈	1	4	yes	—
257	C₂²	C₂³×Dic₁₅₈	1	16	yes	—
258	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
259	C₄	—	2	16	no	—
260	C₄	—	158	32	no	—
261	C₄	C₂³×Dic₁₅₈	1	8	yes	—
262	C₂	—	2	2	no	—
263	C₂	—	2	4	no	—
264	C₂	—	2	16	no	—
265	C₂	C₂⁴×Dic₁₅₈	1	1	yes	characteristic
266	C₂	#23778210	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×D₇₉
commutator, socle, cosocle	C₂²×C₇₉	C₂⁶
Fitting	C₂×#518×C₇₉	C₂
prime radical	C₂³×D₄×C₇₉	C₂²
hypercenter	C₂×#518	D₇₉
hypocenter	C₇₉	#62843
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×#518

Complements

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

N	H ≅ G/N	type
1. #23777203	C₂	semidirect
2. #23777198	C₂	semidirect
3. #23776805	C₂	semidirect
4. C₂×#19099189	C₂	semidirect
7. #19099272	C₂²	semidirect
8. #19099270	C₂²	semidirect
9. D₄×Dic₁₅₈	C₂²	semidirect
10. #19099194	C₂²	semidirect
11. #19099191	C₂²	semidirect
13. #19099189	C₂²	semidirect
14. #19099189	C₂²	semidirect
15. #19099120	C₂²	semidirect
16. #19099116	C₂²	semidirect
17. C₂×#14579140	C₂²	semidirect
18. C₂×#14579138	C₂²	semidirect
29. C₂²×Dic₁₅₈	C₂³	semidirect
31. #14579161	C₂³	semidirect
32. #14579160	C₂³	semidirect
33. #14579153	C₂³	semidirect
34. #14579152	C₂³	semidirect
36. #14579140	C₂³	semidirect
38. #14579138	C₂³	semidirect
39. C₄×Dic₁₅₈	C₂³	semidirect
44. C₂×C₄⋊Dic₇₉	C₂³	semidirect
45. C₂×Dic₇₉⋊C₄	C₂³	semidirect
52. C₂²×D₄×C₇₉	Q₈	semidirect
77. C₂⁴×C₇₉	C₂×Q₈	semidirect
81. C₂×D₄×C₇₉	C₂×Q₈	semidirect
82. C₂×D₄×C₇₉	C₂×Q₈	semidirect
85. C₂²×C₄×C₇₉	C₂×Q₈	semidirect
140. C₇₉	#62843	semidirect
165. C₂²×D₄	Dic₁₅₈	semidirect
199. C₂⁴	C₂×Dic₁₅₈	semidirect
205. C₂×D₄	C₂×Dic₁₅₈	semidirect
206. C₂×D₄	C₂×Dic₁₅₈	semidirect
212. C₂²×C₄	C₂×Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23777203	2
#23777198	2
#23776805	2
C₂×#19099189	2
C₂×#19099038	2
C₂×#518×C₇₉	2
#62843	79