Subgroups — C₂³×#14579167 #35202028

Order 20224.

11422 conjugacy classes of subgroups.

The subgroup lattice has 208 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
2	C₂²×#14579167	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11929608	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11929607	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×D₇₉	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11929624	C₄	1	28	yes	—
8	C₂²×#11929624	C₂²	1	28	yes	—
9	C₂³×Dic₁₅₈	C₄	1	1	yes	characteristic
10	C₂³×D₃₁₆	C₄	1	1	yes	characteristic
11	C₂³×C₂²⋊D₇₉	C₄	1	2	yes	—
12	C₂³×C₄×D₇₉	C₂²	1	2	yes	—
13	C₂×#14579167	C₂²	1	448	yes	direct product factor
14	C₂²×#11929608	C₂²	1	28	yes	complemented
15	C₂²×#11929607	C₂²	1	56	yes	complemented
16	C₂³×C₇₉⋊C₈	C₂²	1	2	yes	—
17	C₂²×C₈×D₇₉	C₂²	1	56	yes	complemented
18	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₇₉	C₂²	1	28	yes	complemented
22	C₂⁴×D₇₉	C₂×C₄	1	2	yes	—
23	C₂×#11929624	C₂×C₄	1	336	yes	—
24	C₂×#11929624	C₂³	1	112	yes	—
25	C₂²×Dic₁₅₈	C₂×C₄	1	28	yes	—
26	C₂²×D₃₁₆	C₂×C₄	1	28	yes	—
27	C₂²×C₂²⋊D₇₉	C₂×C₄	1	56	yes	—
28	C₂³×Dic₇₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×D₇₉	C₂×C₄	1	28	yes	—
30	C₂²×C₄×D₇₉	C₂³	1	28	yes	—
31	#14579167	C₂³	1	512	yes	direct product factor
32	C₂×#11929608	C₂³	1	112	yes	complemented
33	C₂×#11929607	C₂³	1	224	yes	complemented
34	C₂²×C₇₉⋊C₈	C₂³	1	28	yes	—
35	C₂×C₈×D₇₉	C₂³	1	224	yes	complemented
36	C₂⁵×C₇₉	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₇₉	C₂×C₄	1	14	yes	—
39	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₇₉	C₂³	1	14	yes	—
41	C₂²×C₈×C₇₉	C₂³	1	14	yes	—
42	C₂²×C₈×C₇₉	C₂³	1	14	yes	—
43	C₂²×C₈×C₇₉	C₂³	1	112	yes	complemented
44	C₂³×D₇₉	—	2	16	no	—
45	C₂³×D₇₉	C₂²×C₄	1	28	yes	—
46	#11929624	C₂²×C₄	1	448	yes	—
47	#11929624	C₂⁴	1	64	yes	—
48	C₂×Dic₁₅₈	C₂²×C₄	1	112	yes	—
49	C₂×D₃₁₆	C₂²×C₄	1	112	yes	—
50	C₂×C₂²⋊D₇₉	C₂²×C₄	1	224	yes	—
51	C₂²×Dic₇₉	C₂²×C₄	1	28	yes	—
52	C₂×C₄×D₇₉	C₂²×C₄	1	168	yes	—
53	C₂×C₄×D₇₉	C₂⁴	1	56	yes	—
54	#11929608	C₂⁴	1	64	yes	complemented
55	#11929607	C₂⁴	1	128	yes	complemented
56	C₂×C₇₉⋊C₈	C₂⁴	1	56	yes	—
57	C₈×D₇₉	C₂⁴	1	128	yes	complemented
58	C₂⁴×C₇₉	—	2	8	no	—
59	C₂⁴×C₇₉	C₂²×C₄	1	14	yes	—
60	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₉	C₂²×C₄	1	14	yes	—
62	C₂²×C₄×C₇₉	C₂²×C₄	1	84	yes	—
63	C₂²×C₄×C₇₉	C₂²×C₄	1	7	yes	—
64	C₂²×C₄×C₇₉	C₂⁴	1	28	yes	—
65	C₂²×C₄×C₇₉	C₂⁴	1	7	yes	—
66	C₂×C₈×C₇₉	C₂⁴	1	28	yes	—
67	C₂×C₈×C₇₉	C₂⁴	1	28	yes	—
68	C₂×C₈×C₇₉	C₂⁴	1	64	yes	complemented
69	C₂³×C₇₉	—	2	56	no	—
70	C₂³×C₇₉	#131	1	8	yes	complemented
71	C₂³×C₇₉	C₂³×C₄	1	28	yes	—
72	C₂³×C₇₉	C₂³×C₄	1	7	yes	—
73	C₂²×D₇₉	—	2	112	no	—
74	C₂²×D₇₉	C₂³×C₄	1	56	yes	—
75	C₂×C₄×C₇₉	C₂³×C₄	1	21	yes	—
76	C₂×C₄×C₇₉	C₂³×C₄	1	28	yes	—
77	C₂×C₄×C₇₉	C₂³×C₄	1	56	yes	—
78	C₂×C₄×C₇₉	C₂⁵	1	7	yes	—
79	C₂×C₄×C₇₉	C₂⁵	1	8	yes	—
80	C₂²⋊D₇₉	C₂³×C₄	1	128	yes	—
81	C₂×Dic₇₉	C₂³×C₄	1	56	yes	—
82	D₃₁₆	C₂³×C₄	1	64	yes	—
83	C₄×D₇₉	C₂³×C₄	1	112	yes	—
84	C₄×D₇₉	C₂⁵	1	16	yes	—
85	Dic₁₅₈	C₂³×C₄	1	64	yes	—
86	C₈×C₇₉	C₂⁵	1	8	yes	—
87	C₈×C₇₉	C₂⁵	1	8	yes	—
88	C₇₉⋊C₈	C₂⁵	1	16	yes	—
89	C₂²×C₇₉	—	2	56	no	—
90	C₂²×C₇₉	C₂×#131	1	28	yes	complemented
91	C₂²×C₇₉	C₂⁴×C₄	1	8	yes	—
92	C₂²×C₇₉	C₂⁴×C₄	1	7	yes	—
93	C₂×D₇₉	—	2	112	no	—
94	C₂×D₇₉	C₂⁴×C₄	1	16	yes	—
95	C₄×C₇₉	C₂⁴×C₄	1	8	yes	—
96	C₄×C₇₉	C₂⁴×C₄	1	7	yes	—
97	C₄×C₇₉	C₂⁶	1	1	yes	characteristic
98	Dic₇₉	C₂⁴×C₄	1	16	yes	—
99	C₂³×#131	—	79	1	no	maximal, Sylow
100	C₂×C₇₉	—	2	8	no	—
101	C₂×C₇₉	C₂²×#131	1	14	yes	complemented
102	C₂×C₇₉	C₂⁵×C₄	1	1	yes	characteristic
103	D₇₉	—	2	16	no	—
104	C₂³×#41	—	79	1	no	—
105	C₂²×#131	—	79	56	no	—
106	C₂³×M₁₆	—	79	1	no	—
107	C₂³×M₁₆	—	79	2	no	—
108	C₂⁴×C₈	—	79	2	no	—
109	C₂⁴×C₈	D₇₉	1	1	yes	characteristic, complemented
110	C₇₉	C₂³×#131	1	1	yes	characteristic, complemented, Sylow
111	C₂²×#41	—	79	28	no	—
112	C₂²×#41	—	79	28	no	—
113	C₂³×Q₈	—	79	1	no	—
114	C₂³×D₄	—	79	2	no	—
115	C₂³×D₄	—	79	1	no	—
116	C₂⁴×C₄	—	79	2	no	—
117	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
118	C₂×#131	—	79	448	no	—
119	C₂²×M₁₆	—	79	28	no	—
120	C₂²×M₁₆	—	79	56	no	—
121	C₂³×C₈	—	79	2	no	—
122	C₂³×C₈	—	79	56	no	—
123	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
124	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
125	C₂³×C₈	C₂×D₇₉	1	28	yes	complemented
126	C₂⁵	—	79	2	no	—
127	C₂⁵	C₄×D₇₉	1	1	yes	characteristic
128	C₂×#41	—	79	336	no	—
129	C₂×#41	—	79	112	no	—
130	C₂²×Q₈	—	79	28	no	—
131	C₂²×D₄	—	79	56	no	—
132	C₂²×D₄	—	79	28	no	—
133	C₂³×C₄	—	79	2	no	—
134	C₂³×C₄	—	79	28	no	—
135	C₂³×C₄	—	79	28	no	—
136	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
137	C₂³×C₄	C₄×D₇₉	1	14	yes	—
138	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
139	C₂³×C₄	C₂²×D₇₉	1	14	yes	—
140	#131	—	79	512	no	—
141	C₂×M₁₆	—	79	112	no	—
142	C₂×M₁₆	—	79	224	no	—
143	C₂²×C₈	—	79	28	no	—
144	C₂²×C₈	—	79	224	no	—
145	C₂²×C₈	C₂²×D₇₉	1	14	yes	—
146	C₂²×C₈	C₂²×D₇₉	1	14	yes	—
147	C₂²×C₈	C₂²×D₇₉	1	112	yes	complemented
148	C₂⁴	—	2	8	no	—
149	C₂⁴	—	79	28	no	—
150	C₂⁴	—	158	16	no	—
151	C₂⁴	C₂×C₄×D₇₉	1	14	yes	—
152	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
153	#41	—	79	64	no	—
154	#41	—	79	448	no	—
155	C₂×Q₈	—	79	112	no	—
156	C₂×D₄	—	79	224	no	—
157	C₂×D₄	—	79	112	no	—
158	C₂²×C₄	—	79	28	no	—
159	C₂²×C₄	—	79	56	no	—
160	C₂²×C₄	—	79	168	no	—
161	C₂²×C₄	C₂×C₄×D₇₉	1	7	yes	—
162	C₂²×C₄	C₂×C₄×D₇₉	1	14	yes	—
163	C₂²×C₄	C₂×C₄×D₇₉	1	84	yes	—
164	C₂²×C₄	C₂³×D₇₉	1	7	yes	—
165	C₂²×C₄	C₂³×D₇₉	1	28	yes	—
166	M₁₆	—	79	64	no	—
167	M₁₆	—	79	128	no	—
168	C₂×C₈	—	79	56	no	—
169	C₂×C₈	—	79	128	no	—
170	C₂×C₈	C₂³×D₇₉	1	28	yes	—
171	C₂×C₈	C₂³×D₇₉	1	28	yes	—
172	C₂×C₈	C₂³×D₇₉	1	64	yes	complemented
173	C₂³	—	2	56	no	—
174	C₂³	—	79	56	no	—
175	C₂³	—	158	112	no	—
176	C₂³	#14579167	1	8	yes	direct product factor
177	C₂³	C₂²×C₄×D₇₉	1	28	yes	—
178	C₂³	C₂²×C₄×D₇₉	1	7	yes	—
179	Q₈	—	79	64	no	—
180	D₄	—	79	128	no	—
181	D₄	—	79	64	no	—
182	C₂×C₄	—	79	56	no	—
183	C₂×C₄	—	79	16	no	—
184	C₂×C₄	—	79	112	no	—
185	C₂×C₄	C₂²×C₄×D₇₉	1	21	yes	—
186	C₂×C₄	C₂²×C₄×D₇₉	1	28	yes	—
187	C₂×C₄	C₂²×C₄×D₇₉	1	56	yes	—
188	C₂×C₄	C₂⁴×D₇₉	1	7	yes	—
189	C₂×C₄	C₂⁴×D₇₉	1	8	yes	—
190	C₈	—	79	16	no	—
191	C₈	C₂⁴×D₇₉	1	8	yes	—
192	C₈	C₂⁴×D₇₉	1	8	yes	—
193	C₂²	—	2	56	no	—
194	C₂²	—	79	16	no	—
195	C₂²	—	158	112	no	—
196	C₂²	C₂×#14579167	1	28	yes	direct product factor
197	C₂²	C₂³×C₄×D₇₉	1	8	yes	—
198	C₂²	C₂³×C₄×D₇₉	1	7	yes	—
199	C₄	—	79	16	no	—
200	C₄	C₂³×C₄×D₇₉	1	7	yes	—
201	C₄	C₂³×C₄×D₇₉	1	8	yes	—
202	C₄	C₂⁵×D₇₉	1	1	yes	characteristic
203	C₂	—	2	8	no	—
204	C₂	—	158	16	no	—
205	C₂	C₂²×#14579167	1	14	yes	direct product factor
206	C₂	C₂⁴×C₄×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂×D₇₉
commutator	C₂×C₇₉	C₂⁵×C₄
Frattini	C₄	C₂⁵×D₇₉
Fitting	C₂⁴×C₈×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂²×C₄
prime radical	C₂³×#11929624	C₂
cosocle	C₄×C₇₉	C₂⁶
hypercenter	C₂⁴×C₈	D₇₉
hypocenter	C₇₉	C₂³×#131
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³×C₈ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×#14579167	C₂	direct (N × H)
3. C₂³×#11929608	C₂	semidirect
4. C₂³×#11929607	C₂	semidirect
5. C₂³×C₈×D₇₉	C₂	semidirect
6. C₂⁴×C₈×C₇₉	C₂	semidirect
13. C₂×#14579167	C₂²	direct (N × H)
14. C₂²×#11929608	C₂²	semidirect
15. C₂²×#11929607	C₂²	semidirect
17. C₂²×C₈×D₇₉	C₂²	semidirect
21. C₂³×C₈×C₇₉	C₂²	semidirect
31. #14579167	C₂³	direct (N × H)
32. C₂×#11929608	C₂³	semidirect
33. C₂×#11929607	C₂³	semidirect
35. C₂×C₈×D₇₉	C₂³	semidirect
43. C₂²×C₈×C₇₉	C₂³	semidirect
54. #11929608	C₂⁴	semidirect
55. #11929607	C₂⁴	semidirect
57. C₈×D₇₉	C₂⁴	semidirect
68. C₂×C₈×C₇₉	C₂⁴	semidirect
70. C₂³×C₇₉	#131	semidirect
90. C₂²×C₇₉	C₂×#131	semidirect
101. C₂×C₇₉	C₂²×#131	semidirect
109. C₂⁴×C₈	D₇₉	semidirect
110. C₇₉	C₂³×#131	semidirect
125. C₂³×C₈	C₂×D₇₉	semidirect
147. C₂²×C₈	C₂²×D₇₉	semidirect
172. C₂×C₈	C₂³×D₇₉	semidirect
176. C₂³	#14579167	direct (N × H)
196. C₂²	C₂×#14579167	direct (N × H)
205. C₂	C₂²×#14579167	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
C₂²×#14579167	2
C₂³×#11929608	2
C₂³×#11929607	2
C₂³×C₈×D₇₉	2
C₂⁴×C₈×C₇₉	2
C₂³×#131	79

Subgroups — C23×#14579167 #35202028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14579167 #35202028