Subgroups — C₂³×#14579186 #35202060

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	C₂⁴×D₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14579186	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11929611	C₂	1	2	yes	maximal, complemented
4	C₂³×#11929609	C₂	1	2	yes	maximal, complemented
5	C₂³×#11929608	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#41×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₉	—	2	1	no	—
8	C₂³×D₃₁₆	—	2	1	no	—
9	C₂³×D₃₁₆	—	2	14	no	—
10	C₂³×D₃₁₆	C₂²	1	2	yes	—
11	C₂³×D₃₁₆	C₂²	1	28	yes	complemented
12	C₂×#14579186	C₂²	1	448	yes	direct product factor
13	C₂²×#11929611	C₂²	1	56	yes	complemented
14	C₂²×#11929609	C₂²	1	56	yes	complemented
15	C₂²×#11929608	C₂²	1	28	yes	complemented
16	C₂³×C₇₉⋊C₈	C₂²	1	2	yes	complemented
17	C₂²×#41×C₇₉	—	2	14	no	—
18	C₂²×#41×C₇₉	C₂²	1	28	yes	complemented
19	C₂³×Q₈×C₇₉	C₂²	1	1	yes	characteristic, complemented
20	C₂³×D₄×C₇₉	—	2	1	no	—
21	C₂³×D₄×C₇₉	C₂²	1	1	yes	characteristic, complemented
22	C₂⁴×C₄×C₇₉	—	2	1	no	—
23	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
24	C₂⁴×D₇₉	—	2	2	no	—
25	C₂⁴×D₇₉	—	2	28	no	—
26	C₂⁴×D₇₉	—	4	16	no	—
27	C₂²×D₃₁₆	—	2	14	no	—
28	C₂²×D₃₁₆	—	2	168	no	—
29	C₂²×D₃₁₆	—	2	28	no	—
30	C₂²×D₃₁₆	C₂³	1	28	yes	—
31	C₂²×D₃₁₆	C₂³	1	112	yes	complemented
32	#14579186	C₂³	1	512	yes	direct product factor
33	C₂×#11929611	C₂³	1	224	yes	complemented
34	C₂×#11929609	C₂³	1	224	yes	complemented
35	C₂×#11929608	C₂³	1	112	yes	complemented
36	C₂²×C₇₉⋊C₈	C₂³	1	28	yes	complemented
37	C₂⁵×C₇₉	—	2	1	no	—
38	C₂⁵×C₇₉	D₄	1	1	yes	characteristic
39	C₂×#41×C₇₉	—	2	168	no	—
40	C₂×#41×C₇₉	C₂³	1	112	yes	complemented
41	C₂²×Q₈×C₇₉	—	2	7	no	—
42	C₂²×Q₈×C₇₉	C₂³	1	14	yes	complemented
43	C₂²×D₄×C₇₉	—	2	28	no	—
44	C₂²×D₄×C₇₉	—	2	7	no	—
45	C₂²×D₄×C₇₉	C₂³	1	14	yes	complemented
46	C₂³×C₄×C₇₉	—	2	1	no	—
47	C₂³×C₄×C₇₉	—	2	28	no	—
48	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₇₉	D₄	1	14	yes	—
50	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₇₉	C₂³	1	14	yes	—
52	C₂³×D₇₉	—	2	28	no	—
53	C₂³×D₇₉	—	2	112	no	—
54	C₂³×D₇₉	—	4	224	no	—
55	C₂³×D₇₉	—	4	16	no	—
56	C₂×D₃₁₆	—	2	84	no	—
57	C₂×D₃₁₆	—	2	112	no	—
58	C₂×D₃₁₆	—	2	224	no	—
59	C₂×D₃₁₆	C₂⁴	1	56	yes	—
60	C₂×D₃₁₆	C₂⁴	1	64	yes	complemented
61	#11929611	C₂⁴	1	128	yes	complemented
62	#11929609	C₂⁴	1	128	yes	complemented
63	#11929608	C₂⁴	1	64	yes	complemented
64	C₂×C₇₉⋊C₈	C₂⁴	1	56	yes	complemented
65	C₂⁴×C₇₉	—	2	8	no	—
66	C₂⁴×C₇₉	—	2	14	no	—
67	C₂⁴×C₇₉	—	4	8	no	—
68	C₂⁴×C₇₉	C₂×D₄	1	14	yes	—
69	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
70	#41×C₇₉	—	2	224	no	—
71	#41×C₇₉	C₂⁴	1	64	yes	complemented
72	C₂×Q₈×C₇₉	—	2	42	no	—
73	C₂×Q₈×C₇₉	C₂⁴	1	28	yes	complemented
74	C₂×D₄×C₇₉	—	2	112	no	—
75	C₂×D₄×C₇₉	—	2	42	no	—
76	C₂×D₄×C₇₉	C₂⁴	1	28	yes	complemented
77	C₂²×C₄×C₇₉	—	2	14	no	—
78	C₂²×C₄×C₇₉	—	2	112	no	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	14	yes	—
80	C₂²×C₄×C₇₉	C₂×D₄	1	7	yes	—
81	C₂²×C₄×C₇₉	C₂×D₄	1	84	yes	—
82	C₂²×C₄×C₇₉	C₂⁴	1	7	yes	—
83	C₂²×C₄×C₇₉	C₂⁴	1	28	yes	—
84	C₂³×C₇₉	—	2	56	no	—
85	C₂³×C₇₉	—	2	28	no	—
86	C₂³×C₇₉	—	4	56	no	—
87	C₂³×C₇₉	AGL(1,ℤ/8)	1	8	yes	complemented
88	C₂³×C₇₉	C₂²×D₄	1	28	yes	—
89	C₂³×C₇₉	C₂²×D₄	1	7	yes	—
90	C₂²×D₇₉	—	2	56	no	—
91	C₂²×D₇₉	—	2	64	no	—
92	C₂²×D₇₉	—	4	448	no	—
93	C₂²×D₇₉	—	4	112	no	—
94	Q₈×C₇₉	—	2	28	no	—
95	Q₈×C₇₉	C₂⁵	1	8	yes	complemented
96	D₄×C₇₉	—	2	64	no	—
97	D₄×C₇₉	—	2	28	no	—
98	D₄×C₇₉	C₂⁵	1	8	yes	complemented
99	C₂×C₄×C₇₉	—	2	28	no	—
100	C₂×C₄×C₇₉	—	2	64	no	—
101	C₂×C₄×C₇₉	C₂²×D₄	1	28	yes	—
102	C₂×C₄×C₇₉	C₂²×D₄	1	21	yes	—
103	C₂×C₄×C₇₉	C₂²×D₄	1	56	yes	—
104	C₂×C₄×C₇₉	C₂⁵	1	7	yes	—
105	C₂×C₄×C₇₉	C₂⁵	1	8	yes	—
106	D₃₁₆	—	2	56	no	—
107	D₃₁₆	—	2	64	no	—
108	D₃₁₆	C₂⁵	1	16	yes	—
109	C₇₉⋊C₈	C₂⁵	1	16	yes	complemented
110	C₂²×C₇₉	—	2	56	no	—
111	C₂²×C₇₉	—	2	8	no	—
112	C₂²×C₇₉	—	4	56	no	—
113	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	28	yes	complemented
114	C₂²×C₇₉	C₂³×D₄	1	8	yes	—
115	C₂²×C₇₉	C₂³×D₄	1	7	yes	—
116	C₂×D₇₉	—	2	16	no	—
117	C₂×D₇₉	—	4	128	no	—
118	C₂×D₇₉	—	4	112	no	—
119	C₄×C₇₉	—	2	8	no	—
120	C₄×C₇₉	C₂³×D₄	1	8	yes	—
121	C₄×C₇₉	C₂³×D₄	1	7	yes	—
122	C₄×C₇₉	C₂⁶	1	1	yes	characteristic
123	C₂³×AGL(1,ℤ/8)	—	79	1	no	maximal, Sylow
124	C₂×C₇₉	—	2	8	no	—
125	C₂×C₇₉	—	4	8	no	—
126	C₂×C₇₉	C₂²×AGL(1,ℤ/8)	1	14	yes	complemented
127	C₂×C₇₉	C₂⁴×D₄	1	1	yes	characteristic
128	D₇₉	—	4	16	no	—
129	C₂³×#41	D₇₉	1	1	yes	characteristic, complemented
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	—	79	1	no	—
135	C₇₉	C₂³×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
136	C₂⁶	—	158	1	no	—
137	C₂²×#41	—	2	14	no	—
138	C₂²×#41	C₂×D₇₉	1	28	yes	complemented
139	C₂³×Q₈	C₂×D₇₉	1	1	yes	characteristic, complemented
140	C₂³×D₄	—	2	1	no	—
141	C₂³×D₄	—	79	2	no	—
142	C₂³×D₄	—	79	28	no	—
143	C₂³×D₄	—	158	14	no	—
144	C₂³×D₄	—	158	1	no	—
145	C₂³×D₄	C₂×D₇₉	1	1	yes	characteristic, complemented
146	C₂⁴×C₄	—	2	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	—	79	28	no	—
152	C₂³×C₈	—	79	2	no	—
153	C₂⁵	—	2	1	no	—
154	C₂⁵	—	316	16	no	—
155	C₂⁵	—	158	28	no	—
156	C₂⁵	—	158	2	no	—
157	C₂⁵	C₂²⋊D₇₉	1	1	yes	characteristic
158	C₂×#41	—	2	168	no	—
159	C₂×#41	C₂²×D₇₉	1	112	yes	complemented
160	C₂²×Q₈	—	2	7	no	—
161	C₂²×Q₈	C₂²×D₇₉	1	14	yes	complemented
162	C₂²×D₄	—	2	28	no	—
163	C₂²×D₄	—	2	7	no	—
164	C₂²×D₄	—	79	28	no	—
165	C₂²×D₄	—	79	112	no	—
166	C₂²×D₄	—	158	28	no	—
167	C₂²×D₄	—	158	14	no	—
168	C₂²×D₄	—	158	168	no	—
169	C₂²×D₄	C₂²×D₇₉	1	14	yes	complemented
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	—	2	28	no	—
172	C₂³×C₄	C₂²⋊D₇₉	1	14	yes	—
173	C₂³×C₄	C₂²⋊D₇₉	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	—	79	112	no	—
180	C₂²×C₈	—	79	28	no	—
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	14	no	—
183	C₂⁴	—	4	8	no	—
184	C₂⁴	—	316	224	no	—
185	C₂⁴	—	316	16	no	—
186	C₂⁴	—	158	112	no	—
187	C₂⁴	—	158	28	no	—
188	C₂⁴	C₂×C₂²⋊D₇₉	1	14	yes	—
189	C₂⁴	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
190	#41	—	2	224	no	—
191	#41	C₂³×D₇₉	1	64	yes	complemented
192	C₂×Q₈	—	2	42	no	—
193	C₂×Q₈	C₂³×D₇₉	1	28	yes	complemented
194	C₂×D₄	—	2	112	no	—
195	C₂×D₄	—	2	42	no	—
196	C₂×D₄	—	79	56	no	—
197	C₂×D₄	—	79	64	no	—
198	C₂×D₄	—	158	112	no	—
199	C₂×D₄	—	158	84	no	—
200	C₂×D₄	—	158	224	no	—
201	C₂×D₄	C₂³×D₇₉	1	28	yes	complemented
202	C₂²×C₄	—	2	14	no	—
203	C₂²×C₄	—	2	112	no	—
204	C₂²×C₄	C₂×C₂²⋊D₇₉	1	14	yes	—
205	C₂²×C₄	C₂×C₂²⋊D₇₉	1	7	yes	—
206	C₂²×C₄	C₂×C₂²⋊D₇₉	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	—	79	64	no	—
212	C₂×C₈	—	79	56	no	—
213	C₂³	—	2	56	no	—
214	C₂³	—	2	28	no	—
215	C₂³	—	4	56	no	—
216	C₂³	—	316	448	no	—
217	C₂³	—	316	112	no	—
218	C₂³	—	158	64	no	—
219	C₂³	—	158	56	no	—
220	C₂³	#14579186	1	8	yes	direct product factor
221	C₂³	C₂²×C₂²⋊D₇₉	1	28	yes	—
222	C₂³	C₂²×C₂²⋊D₇₉	1	7	yes	—
223	Q₈	—	2	28	no	—
224	Q₈	C₂⁴×D₇₉	1	8	yes	complemented
225	D₄	—	2	64	no	—
226	D₄	—	2	28	no	—
227	D₄	—	79	16	no	—
228	D₄	—	158	64	no	—
229	D₄	—	158	56	no	—
230	D₄	C₂⁴×D₇₉	1	8	yes	complemented
231	C₂×C₄	—	2	28	no	—
232	C₂×C₄	—	2	64	no	—
233	C₂×C₄	C₂²×C₂²⋊D₇₉	1	28	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₇₉	1	21	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₇₉	1	56	yes	—
236	C₂×C₄	C₂⁴×D₇₉	1	7	yes	—
237	C₂×C₄	C₂⁴×D₇₉	1	8	yes	—
238	C₈	—	79	16	no	—
239	C₂²	—	2	56	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	4	56	no	—
242	C₂²	—	316	128	no	—
243	C₂²	—	316	112	no	—
244	C₂²	—	158	16	no	—
245	C₂²	C₂×#14579186	1	28	yes	direct product factor
246	C₂²	C₂³×C₂²⋊D₇₉	1	8	yes	—
247	C₂²	C₂³×C₂²⋊D₇₉	1	7	yes	—
248	C₄	—	2	8	no	—
249	C₄	C₂³×C₂²⋊D₇₉	1	8	yes	—
250	C₄	C₂³×C₂²⋊D₇₉	1	7	yes	—
251	C₄	C₂⁵×D₇₉	1	1	yes	characteristic
252	C₂	—	2	8	no	—
253	C₂	—	4	8	no	—
254	C₂	—	316	16	no	—
255	C₂	C₂²×#14579186	1	14	yes	direct product factor
256	C₂	C₂⁴×C₂²⋊D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₇₉
commutator, cosocle	C₄×C₇₉	C₂⁶
Frattini	C₄	C₂⁵×D₇₉
Fitting	C₂³×#41×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂×D₄
hypercenter	C₂³×#41	D₇₉
hypocenter	C₇₉	C₂³×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂⁴×D₃₁₆	C₂	semidirect
2. C₂²×#14579186	C₂	direct (N × H)
3. C₂³×#11929611	C₂	semidirect
4. C₂³×#11929609	C₂	semidirect
5. C₂³×#11929608	C₂	semidirect
6. C₂³×#41×C₇₉	C₂	semidirect
11. C₂³×D₃₁₆	C₂²	semidirect
12. C₂×#14579186	C₂²	direct (N × H)
13. C₂²×#11929611	C₂²	semidirect
14. C₂²×#11929609	C₂²	semidirect
15. C₂²×#11929608	C₂²	semidirect
16. C₂³×C₇₉⋊C₈	C₂²	semidirect
18. C₂²×#41×C₇₉	C₂²	semidirect
19. C₂³×Q₈×C₇₉	C₂²	semidirect
21. C₂³×D₄×C₇₉	C₂²	semidirect
31. C₂²×D₃₁₆	C₂³	semidirect
32. #14579186	C₂³	direct (N × H)
33. C₂×#11929611	C₂³	semidirect
34. C₂×#11929609	C₂³	semidirect
35. C₂×#11929608	C₂³	semidirect
36. C₂²×C₇₉⋊C₈	C₂³	semidirect
40. C₂×#41×C₇₉	C₂³	semidirect
42. C₂²×Q₈×C₇₉	C₂³	semidirect
45. C₂²×D₄×C₇₉	C₂³	semidirect
60. C₂×D₃₁₆	C₂⁴	semidirect
61. #11929611	C₂⁴	semidirect
62. #11929609	C₂⁴	semidirect
63. #11929608	C₂⁴	semidirect
64. C₂×C₇₉⋊C₈	C₂⁴	semidirect
71. #41×C₇₉	C₂⁴	semidirect
73. C₂×Q₈×C₇₉	C₂⁴	semidirect
76. C₂×D₄×C₇₉	C₂⁴	semidirect
87. C₂³×C₇₉	AGL(1,ℤ/8)	semidirect
95. Q₈×C₇₉	C₂⁵	semidirect
98. D₄×C₇₉	C₂⁵	semidirect
109. C₇₉⋊C₈	C₂⁵	semidirect
113. C₂²×C₇₉	C₂×AGL(1,ℤ/8)	semidirect
126. C₂×C₇₉	C₂²×AGL(1,ℤ/8)	semidirect
129. C₂³×#41	D₇₉	semidirect
135. C₇₉	C₂³×AGL(1,ℤ/8)	semidirect
138. C₂²×#41	C₂×D₇₉	semidirect
139. C₂³×Q₈	C₂×D₇₉	semidirect
145. C₂³×D₄	C₂×D₇₉	semidirect
159. C₂×#41	C₂²×D₇₉	semidirect
161. C₂²×Q₈	C₂²×D₇₉	semidirect
169. C₂²×D₄	C₂²×D₇₉	semidirect
191. #41	C₂³×D₇₉	semidirect
193. C₂×Q₈	C₂³×D₇₉	semidirect
201. C₂×D₄	C₂³×D₇₉	semidirect
220. C₂³	#14579186	direct (N × H)
224. Q₈	C₂⁴×D₇₉	semidirect
230. D₄	C₂⁴×D₇₉	semidirect
245. C₂²	C₂×#14579186	direct (N × H)
255. C₂	C₂²×#14579186	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₃₁₆	2
C₂²×#14579186	2
C₂³×#11929611	2
C₂³×#11929609	2
C₂³×#11929608	2
C₂³×#41×C₇₉	2
C₂³×AGL(1,ℤ/8)	79

Subgroups — C23×#14579186 #35202060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14579186 #35202060