Subgroups — C₂³×#14579189 #35202063

Order 20224.

13310 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11929624	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14579189	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#11929613	C₂	1	2	yes	maximal, complemented
5	C₂³×D₆₃₂	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₉	—	2	1	no	—
8	C₂²×#11929624	—	2	14	no	—
9	C₂²×#11929624	C₂²	1	28	yes	complemented
10	C₂³×Dic₁₅₈	C₂²	1	1	yes	characteristic, complemented
11	C₂³×D₃₁₆	—	2	1	no	—
12	C₂³×D₃₁₆	—	2	14	no	—
13	C₂³×D₃₁₆	C₂²	1	2	yes	—
14	C₂³×D₃₁₆	C₂²	1	28	yes	complemented
15	C₂³×D₃₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂³×C₂²⋊D₇₉	—	2	1	no	—
17	C₂³×C₄×D₇₉	—	2	1	no	—
18	C₂×#14579189	C₂²	1	448	yes	direct product factor
19	C₂²×#11929613	C₂²	1	56	yes	complemented
20	C₂²×D₆₃₂	C₂²	1	56	yes	complemented
21	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
22	C₂²×M₁₆×C₇₉	C₂²	1	28	yes	complemented
23	C₂³×C₈×C₇₉	C₂²	1	2	yes	complemented
24	C₂⁴×D₇₉	—	2	2	no	—
25	C₂⁴×D₇₉	—	2	28	no	—
26	C₂⁴×D₇₉	—	2	1	no	—
27	C₂⁴×D₇₉	—	4	16	no	—
28	C₂×#11929624	—	2	168	no	—
29	C₂×#11929624	C₂³	1	112	yes	complemented
30	C₂²×Dic₁₅₈	—	2	7	no	—
31	C₂²×Dic₁₅₈	C₂³	1	14	yes	complemented
32	C₂²×D₃₁₆	—	2	14	no	—
33	C₂²×D₃₁₆	—	2	168	no	—
34	C₂²×D₃₁₆	—	2	28	no	—
35	C₂²×D₃₁₆	—	2	7	no	—
36	C₂²×D₃₁₆	C₂³	1	28	yes	—
37	C₂²×D₃₁₆	C₂³	1	112	yes	complemented
38	C₂²×D₃₁₆	C₂³	1	14	yes	complemented
39	C₂²×C₂²⋊D₇₉	—	2	28	no	—
40	C₂³×Dic₇₉	—	2	1	no	—
41	C₂²×C₄×D₇₉	—	2	28	no	—
42	#14579189	C₂³	1	512	yes	direct product factor
43	C₂×#11929613	C₂³	1	224	yes	complemented
44	C₂×D₆₃₂	C₂³	1	224	yes	complemented
45	C₂⁵×C₇₉	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₇₉	D₄	1	14	yes	—
48	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₇₉	C₂³	1	14	yes	—
50	C₂×M₁₆×C₇₉	C₂³	1	112	yes	complemented
51	C₂²×C₈×C₇₉	C₂³	1	28	yes	complemented
52	C₂³×D₇₉	—	2	28	no	—
53	C₂³×D₇₉	—	2	112	no	—
54	C₂³×D₇₉	—	2	14	no	—
55	C₂³×D₇₉	—	4	224	no	—
56	C₂³×D₇₉	—	4	8	no	—
57	C₂³×D₇₉	—	4	16	no	—
58	#11929624	—	2	224	no	—
59	#11929624	C₂⁴	1	64	yes	complemented
60	C₂×Dic₁₅₈	—	2	42	no	—
61	C₂×Dic₁₅₈	C₂⁴	1	28	yes	complemented
62	C₂×D₃₁₆	—	2	42	no	—
63	C₂×D₃₁₆	—	2	84	no	—
64	C₂×D₃₁₆	—	2	112	no	—
65	C₂×D₃₁₆	—	2	224	no	—
66	C₂×D₃₁₆	C₂⁴	1	56	yes	—
67	C₂×D₃₁₆	C₂⁴	1	64	yes	complemented
68	C₂×D₃₁₆	C₂⁴	1	28	yes	complemented
69	C₂×C₂²⋊D₇₉	—	2	112	no	—
70	C₂²×Dic₇₉	—	2	14	no	—
71	C₂×C₄×D₇₉	—	2	112	no	—
72	#11929613	C₂⁴	1	128	yes	complemented
73	D₆₃₂	C₂⁴	1	128	yes	complemented
74	C₂⁴×C₇₉	—	2	8	no	—
75	C₂⁴×C₇₉	C₂×D₄	1	14	yes	—
76	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×D₄	1	7	yes	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	84	yes	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	14	yes	—
80	C₂²×C₄×C₇₉	C₂⁴	1	7	yes	—
81	C₂²×C₄×C₇₉	C₂⁴	1	28	yes	—
82	M₁₆×C₇₉	C₂⁴	1	64	yes	complemented
83	C₂×C₈×C₇₉	C₂⁴	1	56	yes	complemented
84	C₂³×C₇₉	—	2	56	no	—
85	C₂³×C₇₉	AGL(1,ℤ/8)	1	8	yes	complemented
86	C₂³×C₇₉	C₂²×D₄	1	28	yes	—
87	C₂³×C₇₉	C₂²×D₄	1	7	yes	—
88	C₂²×D₇₉	—	2	28	no	—
89	C₂²×D₇₉	—	2	56	no	—
90	C₂²×D₇₉	—	2	64	no	—
91	C₂²×D₇₉	—	4	448	no	—
92	C₂²×D₇₉	—	4	56	no	—
93	C₂²×D₇₉	—	4	112	no	—
94	C₂×C₄×C₇₉	C₂²×D₄	1	28	yes	—
95	C₂×C₄×C₇₉	C₂²×D₄	1	21	yes	—
96	C₂×C₄×C₇₉	C₂²×D₄	1	56	yes	—
97	C₂×C₄×C₇₉	C₂⁵	1	7	yes	—
98	C₂×C₄×C₇₉	C₂⁵	1	8	yes	—
99	C₂²⋊D₇₉	—	2	64	no	—
100	C₂×Dic₇₉	—	2	28	no	—
101	D₃₁₆	—	2	56	no	—
102	D₃₁₆	—	2	64	no	—
103	D₃₁₆	—	2	28	no	—
104	D₃₁₆	C₂⁵	1	16	yes	—
105	D₃₁₆	C₂⁵	1	8	yes	complemented
106	C₄×D₇₉	—	2	64	no	—
107	Dic₁₅₈	—	2	28	no	—
108	Dic₁₅₈	C₂⁵	1	8	yes	complemented
109	C₈×C₇₉	C₂⁵	1	16	yes	complemented
110	C₂²×C₇₉	—	2	56	no	—
111	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	28	yes	complemented
112	C₂²×C₇₉	C₂³×D₄	1	8	yes	—
113	C₂²×C₇₉	C₂³×D₄	1	7	yes	—
114	C₂×D₇₉	—	2	8	no	—
115	C₂×D₇₉	—	2	16	no	—
116	C₂×D₇₉	—	4	128	no	—
117	C₂×D₇₉	—	4	56	no	—
118	C₂×D₇₉	—	4	112	no	—
119	C₄×C₇₉	C₂³×D₄	1	7	yes	—
120	C₄×C₇₉	C₂³×D₄	1	8	yes	—
121	C₄×C₇₉	C₂⁶	1	1	yes	characteristic
122	Dic₇₉	—	2	8	no	—
123	C₂³×AGL(1,ℤ/8)	—	79	1	no	maximal, Sylow
124	C₂×C₇₉	—	2	8	no	—
125	C₂×C₇₉	C₂²×AGL(1,ℤ/8)	1	14	yes	complemented
126	C₂×C₇₉	C₂⁴×D₄	1	1	yes	characteristic
127	D₇₉	—	4	8	no	—
128	D₇₉	—	4	16	no	—
129	C₂³×#41	—	79	1	no	—
130	C₂⁴×D₄	—	79	1	no	—
131	C₂²×AGL(1,ℤ/8)	—	79	56	no	—
132	C₂³×SD₁₆	—	79	2	no	—
133	C₂³×D₈	—	79	2	no	—
134	C₂³×M₁₆	D₇₉	1	1	yes	characteristic, complemented
135	C₇₉	C₂³×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
136	C₂⁶	—	158	1	no	—
137	C₂²×#41	—	79	28	no	—
138	C₂²×#41	—	158	14	no	—
139	C₂³×Q₈	—	79	1	no	—
140	C₂³×D₄	—	79	2	no	—
141	C₂³×D₄	—	79	28	no	—
142	C₂³×D₄	—	79	1	no	—
143	C₂³×D₄	—	158	1	no	—
144	C₂³×D₄	—	158	14	no	—
145	C₂³×D₄	—	158	1	no	—
146	C₂⁴×C₄	—	158	1	no	—
147	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
148	C₂×AGL(1,ℤ/8)	—	79	448	no	—
149	C₂²×SD₁₆	—	79	56	no	—
150	C₂²×D₈	—	79	56	no	—
151	C₂²×M₁₆	C₂×D₇₉	1	28	yes	complemented
152	C₂³×C₈	C₂×D₇₉	1	2	yes	complemented
153	C₂⁵	—	316	16	no	—
154	C₂⁵	—	158	28	no	—
155	C₂⁵	—	158	1	no	—
156	C₂⁵	—	158	2	no	—
157	C₂⁵	D₃₁₆	1	1	yes	characteristic
158	C₂×#41	—	79	112	no	—
159	C₂×#41	—	158	168	no	—
160	C₂²×Q₈	—	79	14	no	—
161	C₂²×Q₈	—	158	7	no	—
162	C₂²×D₄	—	79	14	no	—
163	C₂²×D₄	—	79	28	no	—
164	C₂²×D₄	—	79	112	no	—
165	C₂²×D₄	—	158	28	no	—
166	C₂²×D₄	—	158	28	no	—
167	C₂²×D₄	—	158	14	no	—
168	C₂²×D₄	—	158	168	no	—
169	C₂²×D₄	—	158	7	no	—
170	C₂³×C₄	—	158	1	no	—
171	C₂³×C₄	—	158	28	no	—
172	C₂³×C₄	D₃₁₆	1	14	yes	—
173	C₂³×C₄	D₃₁₆	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₇₉	1	14	yes	—
175	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
176	AGL(1,ℤ/8)	—	79	512	no	—
177	C₂×SD₁₆	—	79	224	no	—
178	C₂×D₈	—	79	224	no	—
179	C₂×M₁₆	C₂²×D₇₉	1	112	yes	complemented
180	C₂²×C₈	C₂²×D₇₉	1	28	yes	complemented
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	316	224	no	—
183	C₂⁴	—	316	16	no	—
184	C₂⁴	—	316	8	no	—
185	C₂⁴	—	158	14	no	—
186	C₂⁴	—	158	112	no	—
187	C₂⁴	—	158	28	no	—
188	C₂⁴	C₂×D₃₁₆	1	14	yes	—
189	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
190	#41	—	79	64	no	—
191	#41	—	158	224	no	—
192	C₂×Q₈	—	79	28	no	—
193	C₂×Q₈	—	158	42	no	—
194	C₂×D₄	—	79	56	no	—
195	C₂×D₄	—	79	28	no	—
196	C₂×D₄	—	79	64	no	—
197	C₂×D₄	—	158	112	no	—
198	C₂×D₄	—	158	112	no	—
199	C₂×D₄	—	158	84	no	—
200	C₂×D₄	—	158	224	no	—
201	C₂×D₄	—	158	42	no	—
202	C₂²×C₄	—	158	14	no	—
203	C₂²×C₄	—	158	112	no	—
204	C₂²×C₄	C₂×D₃₁₆	1	14	yes	—
205	C₂²×C₄	C₂×D₃₁₆	1	7	yes	—
206	C₂²×C₄	C₂×D₃₁₆	1	84	yes	—
207	C₂²×C₄	C₂³×D₇₉	1	7	yes	—
208	C₂²×C₄	C₂³×D₇₉	1	28	yes	—
209	SD₁₆	—	79	128	no	—
210	D₈	—	79	128	no	—
211	M₁₆	C₂³×D₇₉	1	64	yes	complemented
212	C₂×C₈	C₂³×D₇₉	1	56	yes	complemented
213	C₂³	—	2	56	no	—
214	C₂³	—	316	448	no	—
215	C₂³	—	316	112	no	—
216	C₂³	—	316	56	no	—
217	C₂³	—	158	28	no	—
218	C₂³	—	158	64	no	—
219	C₂³	—	158	56	no	—
220	C₂³	#14579189	1	8	yes	direct product factor
221	C₂³	C₂²×D₃₁₆	1	28	yes	—
222	C₂³	C₂²×D₃₁₆	1	7	yes	—
223	Q₈	—	79	8	no	—
224	Q₈	—	158	28	no	—
225	D₄	—	79	8	no	—
226	D₄	—	79	16	no	—
227	D₄	—	158	64	no	—
228	D₄	—	158	64	no	—
229	D₄	—	158	56	no	—
230	D₄	—	158	28	no	—
231	C₂×C₄	—	158	28	no	—
232	C₂×C₄	—	158	64	no	—
233	C₂×C₄	C₂²×D₃₁₆	1	28	yes	—
234	C₂×C₄	C₂²×D₃₁₆	1	21	yes	—
235	C₂×C₄	C₂²×D₃₁₆	1	56	yes	—
236	C₂×C₄	C₂⁴×D₇₉	1	7	yes	—
237	C₂×C₄	C₂⁴×D₇₉	1	8	yes	—
238	C₈	C₂⁴×D₇₉	1	16	yes	complemented
239	C₂²	—	2	56	no	—
240	C₂²	—	316	128	no	—
241	C₂²	—	316	112	no	—
242	C₂²	—	316	56	no	—
243	C₂²	—	158	8	no	—
244	C₂²	—	158	16	no	—
245	C₂²	C₂×#14579189	1	28	yes	direct product factor
246	C₂²	C₂³×D₃₁₆	1	8	yes	—
247	C₂²	C₂³×D₃₁₆	1	7	yes	—
248	C₄	—	158	8	no	—
249	C₄	C₂³×D₃₁₆	1	8	yes	—
250	C₄	C₂³×D₃₁₆	1	7	yes	—
251	C₄	C₂⁵×D₇₉	1	1	yes	characteristic
252	C₂	—	2	8	no	—
253	C₂	—	316	16	no	—
254	C₂	—	316	8	no	—
255	C₂	C₂²×#14579189	1	14	yes	direct product factor
256	C₂	C₂⁴×D₃₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×D₃₁₆
commutator, cosocle	C₄×C₇₉	C₂⁶
Frattini	C₄	C₂⁵×D₇₉
Fitting	C₂³×M₁₆×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂×D₄
hypercenter	C₂³×M₁₆	D₇₉
hypocenter	C₇₉	C₂³×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂³×#11929624	C₂	semidirect
2. C₂⁴×D₃₁₆	C₂	semidirect
3. C₂²×#14579189	C₂	direct (N × H)
4. C₂³×#11929613	C₂	semidirect
5. C₂³×D₆₃₂	C₂	semidirect
6. C₂³×M₁₆×C₇₉	C₂	semidirect
9. C₂²×#11929624	C₂²	semidirect
10. C₂³×Dic₁₅₈	C₂²	semidirect
14. C₂³×D₃₁₆	C₂²	semidirect
15. C₂³×D₃₁₆	C₂²	semidirect
18. C₂×#14579189	C₂²	direct (N × H)
19. C₂²×#11929613	C₂²	semidirect
20. C₂²×D₆₃₂	C₂²	semidirect
22. C₂²×M₁₆×C₇₉	C₂²	semidirect
23. C₂³×C₈×C₇₉	C₂²	semidirect
29. C₂×#11929624	C₂³	semidirect
31. C₂²×Dic₁₅₈	C₂³	semidirect
37. C₂²×D₃₁₆	C₂³	semidirect
38. C₂²×D₃₁₆	C₂³	semidirect
42. #14579189	C₂³	direct (N × H)
43. C₂×#11929613	C₂³	semidirect
44. C₂×D₆₃₂	C₂³	semidirect
50. C₂×M₁₆×C₇₉	C₂³	semidirect
51. C₂²×C₈×C₇₉	C₂³	semidirect
59. #11929624	C₂⁴	semidirect
61. C₂×Dic₁₅₈	C₂⁴	semidirect
67. C₂×D₃₁₆	C₂⁴	semidirect
68. C₂×D₃₁₆	C₂⁴	semidirect
72. #11929613	C₂⁴	semidirect
73. D₆₃₂	C₂⁴	semidirect
82. M₁₆×C₇₉	C₂⁴	semidirect
83. C₂×C₈×C₇₉	C₂⁴	semidirect
85. C₂³×C₇₉	AGL(1,ℤ/8)	semidirect
105. D₃₁₆	C₂⁵	semidirect
108. Dic₁₅₈	C₂⁵	semidirect
109. C₈×C₇₉	C₂⁵	semidirect
111. C₂²×C₇₉	C₂×AGL(1,ℤ/8)	semidirect
125. C₂×C₇₉	C₂²×AGL(1,ℤ/8)	semidirect
134. C₂³×M₁₆	D₇₉	semidirect
135. C₇₉	C₂³×AGL(1,ℤ/8)	semidirect
151. C₂²×M₁₆	C₂×D₇₉	semidirect
152. C₂³×C₈	C₂×D₇₉	semidirect
179. C₂×M₁₆	C₂²×D₇₉	semidirect
180. C₂²×C₈	C₂²×D₇₉	semidirect
211. M₁₆	C₂³×D₇₉	semidirect
212. C₂×C₈	C₂³×D₇₉	semidirect
220. C₂³	#14579189	direct (N × H)
238. C₈	C₂⁴×D₇₉	semidirect
245. C₂²	C₂×#14579189	direct (N × H)
255. C₂	C₂²×#14579189	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11929624	2
C₂⁴×D₃₁₆	2
C₂²×#14579189	2
C₂³×#11929613	2
C₂³×D₆₃₂	2
C₂³×M₁₆×C₇₉	2
C₂³×AGL(1,ℤ/8)	79

Subgroups — C23×#14579189 #35202063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14579189 #35202063