Subgroups — C₂²×C₄×#11929607 #35151654

Order 20224.

3950 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11929607	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11929607	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14579071	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×C₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₇₉	C₄	1	2	yes	—
8	C₂³×C₄×D₇₉	C₂²	1	1	yes	characteristic
9	C₂²×#11929607	C₄	1	32	yes	direct product factor
10	C₂²×#11929607	C₂²	1	24	yes	—
11	C₂³×C₇₉⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₇₉	C₄	1	2	yes	—
13	C₂²×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₇₉	C₄	1	12	yes	—
15	C₂×C₄²×D₇₉	C₂²	1	12	yes	—
16	C₄×#11929607	C₂²	1	64	yes	direct product factor
17	C₂×#14579071	C₂²	1	24	yes	complemented
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	12	yes	complemented
19	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
20	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₉	C₂²	1	12	yes	complemented
22	C₂⁴×D₇₉	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₇₉	C₂×C₄	1	2	yes	—
24	C₂³×Dic₇₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×D₇₉	—	2	16	no	—
26	C₂²×C₄×D₇₉	C₂×C₄	1	24	yes	—
27	C₂²×C₄×D₇₉	C₂×C₄	1	6	yes	—
28	C₂²×C₄×D₇₉	C₂×C₄	1	16	yes	—
29	C₂²×C₄×D₇₉	C₂³	1	6	yes	—
30	C₂×#11929607	C₂×C₄	1	192	yes	direct product factor
31	C₂×#11929607	C₂³	1	32	yes	—
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	16	yes	complemented
33	C₂²×C₇₉⋊C₈	C₂³	1	12	yes	—
34	C₂×C₄×Dic₇₉	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₇₉	C₂×C₄	1	24	yes	—
36	C₂×C₄×Dic₇₉	C₂³	1	6	yes	—
37	C₄²×D₇₉	C₂×C₄	1	48	yes	—
38	C₄²×D₇₉	C₂³	1	16	yes	—
39	#14579071	C₂³	1	32	yes	complemented
40	C₄×C₇₉⋊C₈	C₂³	1	16	yes	complemented
41	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₇₉	C₂³	1	12	yes	—
45	C₂×C₄²×C₇₉	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₇₉	C₂³	1	6	yes	—
47	C₄×C₈×C₇₉	C₂³	1	16	yes	complemented
48	C₂³×D₇₉	—	2	8	no	—
49	C₂³×D₇₉	C₄²	1	8	yes	—
50	C₂³×D₇₉	C₂²×C₄	1	6	yes	—
51	C₂²×Dic₇₉	—	2	8	no	—
52	C₂²×Dic₇₉	C₄²	1	8	yes	—
53	C₂²×Dic₇₉	C₂²×C₄	1	12	yes	—
54	C₂²×Dic₇₉	C₂²×C₄	1	6	yes	—
55	C₂×C₄×D₇₉	—	2	96	no	—
56	C₂×C₄×D₇₉	C₄²	1	48	yes	—
57	C₂×C₄×D₇₉	C₂²×C₄	1	48	yes	—
58	C₂×C₄×D₇₉	C₂²×C₄	1	32	yes	—
59	C₂×C₄×D₇₉	C₂²×C₄	1	12	yes	—
60	C₂×C₄×D₇₉	C₂⁴	1	4	yes	—
61	#11929607	C₂²×C₄	1	128	yes	direct product factor
62	C₂×C₇₉⋊C₈	C₂²×C₄	1	48	yes	complemented
63	C₂×C₇₉⋊C₈	C₂⁴	1	8	yes	—
64	C₄×Dic₇₉	C₂²×C₄	1	12	yes	—
65	C₄×Dic₇₉	C₂²×C₄	1	32	yes	—
66	C₄×Dic₇₉	C₂⁴	1	4	yes	—
67	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	M₁₆	1	16	yes	complemented
69	C₂²×C₄×C₇₉	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
72	C₂²×C₄×C₇₉	C₂⁴	1	3	yes	—
73	C₂×C₈×C₇₉	C₂²×C₄	1	48	yes	complemented
74	C₂×C₈×C₇₉	C₂⁴	1	8	yes	—
75	C₄²×C₇₉	C₂²×C₄	1	12	yes	—
76	C₄²×C₇₉	C₂⁴	1	4	yes	—
77	C₂³×C₇₉	C₂×C₄²	1	4	yes	—
78	C₂³×C₇₉	C₂×M₁₆	1	8	yes	—
79	C₂³×C₇₉	C₂³×C₄	1	3	yes	—
80	C₂²×D₇₉	—	2	24	no	—
81	C₂²×D₇₉	—	2	32	no	—
82	C₂²×D₇₉	C₂×C₄²	1	24	yes	—
83	C₂²×D₇₉	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₇₉	C₂×C₄²	1	12	yes	—
85	C₂×C₄×C₇₉	C₂×M₁₆	1	48	yes	complemented
86	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
87	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₇₉	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
90	C₂×Dic₇₉	—	2	24	no	—
91	C₂×Dic₇₉	C₂×C₄²	1	24	yes	—
92	C₂×Dic₇₉	C₂³×C₄	1	8	yes	—
93	C₂×Dic₇₉	C₂³×C₄	1	4	yes	—
94	C₄×D₇₉	—	2	64	no	—
95	C₄×D₇₉	C₂×C₄²	1	48	yes	—
96	C₄×D₇₉	C₂³×C₄	1	16	yes	—
97	C₈×C₇₉	C₂³×C₄	1	16	yes	complemented
98	C₇₉⋊C₈	C₂³×C₄	1	16	yes	complemented
99	C₂²×C₇₉	C₄×M₁₆	1	16	yes	complemented
100	C₂²×C₇₉	C₂²×C₄²	1	6	yes	—
101	C₂²×C₇₉	C₂²×M₁₆	1	12	yes	—
102	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
103	C₂×D₇₉	—	2	48	no	—
104	C₂×D₇₉	—	2	8	no	—
105	C₂×D₇₉	C₂²×C₄²	1	8	yes	—
106	C₄×C₇₉	C₂²×C₄²	1	6	yes	—
107	C₄×C₇₉	C₂²×M₁₆	1	16	yes	complemented
108	C₄×C₇₉	C₂⁴×C₄	1	2	yes	—
109	Dic₇₉	—	2	8	no	—
110	Dic₇₉	C₂²×C₄²	1	8	yes	—
111	C₂²×C₄×M₁₆	—	79	1	no	maximal, Sylow
112	C₂×C₇₉	C₂×C₄×M₁₆	1	12	yes	complemented
113	C₂×C₇₉	C₂³×C₄²	1	1	yes	characteristic
114	C₂×C₇₉	C₂³×M₁₆	1	2	yes	—
115	D₇₉	—	2	8	no	—
116	C₂³×M₁₆	—	79	2	no	—
117	C₂³×C₄²	—	79	1	no	—
118	C₂×C₄×M₁₆	—	79	24	no	—
119	C₂²×#97	—	79	2	no	—
120	C₂²×C₄×C₈	—	79	1	no	—
121	C₂²×C₄×C₈	D₇₉	1	1	yes	characteristic, complemented
122	C₇₉	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
123	C₂⁴×C₄	—	79	1	no	—
124	C₂⁴×C₄	—	79	2	no	—
125	C₂²×M₁₆	—	79	24	no	—
126	C₂²×M₁₆	—	79	32	no	—
127	C₂³×C₈	—	79	2	no	—
128	C₂³×C₈	C₂×D₇₉	1	2	yes	—
129	C₂²×C₄²	—	79	1	no	—
130	C₂²×C₄²	—	79	2	no	—
131	C₂²×C₄²	—	79	12	no	—
132	C₂²×C₄²	—	79	12	no	—
133	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
134	C₄×M₁₆	—	79	64	no	—
135	C₂×#97	—	79	24	no	—
136	C₂×C₄×C₈	—	79	12	no	—
137	C₂×C₄×C₈	C₂×D₇₉	1	12	yes	complemented
138	C₂⁵	—	79	1	no	—
139	C₂³×C₄	—	79	1	no	—
140	C₂³×C₄	—	79	16	no	—
141	C₂³×C₄	—	79	6	no	—
142	C₂³×C₄	—	79	6	no	—
143	C₂³×C₄	—	79	2	no	—
144	C₂³×C₄	—	79	24	no	—
145	C₂³×C₄	—	158	16	no	—
146	C₂³×C₄	C₄×D₇₉	1	2	yes	—
147	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
148	C₂×M₁₆	—	79	192	no	—
149	C₂×M₁₆	—	79	32	no	—
150	C₂²×C₈	—	79	16	no	—
151	C₂²×C₈	—	79	12	no	—
152	C₂²×C₈	C₄×D₇₉	1	16	yes	complemented
153	C₂²×C₈	C₂²×D₇₉	1	12	yes	—
154	C₂×C₄²	—	79	24	no	—
155	C₂×C₄²	—	79	6	no	—
156	C₂×C₄²	—	79	6	no	—
157	C₂×C₄²	—	79	48	no	—
158	C₂×C₄²	—	79	16	no	—
159	C₂×C₄²	C₄×D₇₉	1	6	yes	—
160	C₂×C₄²	C₂²×D₇₉	1	6	yes	—
161	#97	—	79	32	no	—
162	C₄×C₈	—	79	16	no	—
163	C₄×C₈	C₂²×D₇₉	1	16	yes	complemented
164	C₂⁴	—	79	8	no	—
165	C₂⁴	—	79	6	no	—
166	C₂⁴	—	158	8	no	—
167	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
168	C₂²×C₄	—	79	8	no	—
169	C₂²×C₄	—	79	6	no	—
170	C₂²×C₄	—	79	4	no	—
171	C₂²×C₄	—	79	48	no	—
172	C₂²×C₄	—	79	12	no	—
173	C₂²×C₄	—	79	48	no	—
174	C₂²×C₄	—	79	12	no	—
175	C₂²×C₄	—	79	32	no	—
176	C₂²×C₄	—	158	8	no	—
177	C₂²×C₄	—	158	96	no	—
178	C₂²×C₄	#11929607	1	16	yes	direct product factor
179	C₂²×C₄	C₂×C₄×D₇₉	1	3	yes	—
180	C₂²×C₄	C₂×C₄×D₇₉	1	8	yes	—
181	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
182	C₂²×C₄	C₂³×D₇₉	1	3	yes	—
183	M₁₆	—	79	128	no	—
184	C₂×C₈	—	79	8	no	—
185	C₂×C₈	—	79	48	no	—
186	C₂×C₈	C₂×C₄×D₇₉	1	48	yes	complemented
187	C₂×C₈	C₂³×D₇₉	1	8	yes	—
188	C₄²	—	79	4	no	—
189	C₄²	—	79	32	no	—
190	C₄²	—	79	12	no	—
191	C₄²	C₂×C₄×D₇₉	1	12	yes	—
192	C₄²	C₂³×D₇₉	1	4	yes	—
193	C₂³	—	79	24	no	—
194	C₂³	—	79	4	no	—
195	C₂³	—	158	32	no	—
196	C₂³	—	158	24	no	—
197	C₂³	C₄²×D₇₉	1	4	yes	—
198	C₂³	C₂×#11929607	1	8	yes	—
199	C₂³	C₂²×C₄×D₇₉	1	3	yes	—
200	C₂×C₄	—	79	24	no	—
201	C₂×C₄	—	79	4	no	—
202	C₂×C₄	—	79	48	no	—
203	C₂×C₄	—	79	16	no	—
204	C₂×C₄	—	79	8	no	—
205	C₂×C₄	—	158	24	no	—
206	C₂×C₄	—	158	64	no	—
207	C₂×C₄	C₄²×D₇₉	1	12	yes	—
208	C₂×C₄	C₂×#11929607	1	48	yes	direct product factor
209	C₂×C₄	C₂²×C₄×D₇₉	1	3	yes	—
210	C₂×C₄	C₂²×C₄×D₇₉	1	12	yes	—
211	C₂×C₄	C₂²×C₄×D₇₉	1	8	yes	—
212	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
213	C₈	—	79	16	no	—
214	C₈	C₂²×C₄×D₇₉	1	16	yes	complemented
215	C₂²	—	79	8	no	—
216	C₂²	—	158	8	no	—
217	C₂²	—	158	48	no	—
218	C₂²	C₄×#11929607	1	16	yes	direct product factor
219	C₂²	C₂×C₄²×D₇₉	1	6	yes	—
220	C₂²	C₂²×#11929607	1	12	yes	—
221	C₂²	C₂³×C₄×D₇₉	1	1	yes	characteristic
222	C₄	—	79	8	no	—
223	C₄	—	158	8	no	—
224	C₄	C₂×C₄²×D₇₉	1	6	yes	—
225	C₄	C₂²×#11929607	1	16	yes	direct product factor
226	C₄	C₂³×C₄×D₇₉	1	2	yes	—
227	C₂	—	158	8	no	—
228	C₂	C₂×C₄×#11929607	1	12	yes	direct product factor
229	C₂	C₂²×C₄²×D₇₉	1	1	yes	characteristic
230	C₂	C₂³×#11929607	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄²	C₂×D₇₉
commutator	C₂×C₇₉	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂²×C₄×C₈×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂²×C₄
prime radical	C₂⁴×D₇₉	C₂×C₄
cosocle	C₂×C₄×C₇₉	C₂⁵
hypercenter	C₂²×C₄×C₈	D₇₉
hypocenter	C₇₉	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#11929607	C₂	direct (N × H)
4. C₂²×#14579071	C₂	semidirect
5. C₂²×C₄×C₇₉⋊C₈	C₂	semidirect
6. C₂²×C₄×C₈×C₇₉	C₂	semidirect
9. C₂²×#11929607	C₄	direct (N × H)
16. C₄×#11929607	C₂²	direct (N × H)
17. C₂×#14579071	C₂²	semidirect
18. C₂×C₄×C₇₉⋊C₈	C₂²	semidirect
21. C₂×C₄×C₈×C₇₉	C₂²	semidirect
30. C₂×#11929607	C₂×C₄	direct (N × H)
32. C₂²×C₇₉⋊C₈	C₂×C₄	semidirect
39. #14579071	C₂³	semidirect
40. C₄×C₇₉⋊C₈	C₂³	semidirect
43. C₂²×C₈×C₇₉	C₂×C₄	semidirect
47. C₄×C₈×C₇₉	C₂³	semidirect
61. #11929607	C₂²×C₄	direct (N × H)
62. C₂×C₇₉⋊C₈	C₂²×C₄	semidirect
68. C₂²×C₄×C₇₉	M₁₆	semidirect
73. C₂×C₈×C₇₉	C₂²×C₄	semidirect
85. C₂×C₄×C₇₉	C₂×M₁₆	semidirect
97. C₈×C₇₉	C₂³×C₄	semidirect
98. C₇₉⋊C₈	C₂³×C₄	semidirect
99. C₂²×C₇₉	C₄×M₁₆	semidirect
107. C₄×C₇₉	C₂²×M₁₆	semidirect
112. C₂×C₇₉	C₂×C₄×M₁₆	semidirect
121. C₂²×C₄×C₈	D₇₉	semidirect
122. C₇₉	C₂²×C₄×M₁₆	semidirect
137. C₂×C₄×C₈	C₂×D₇₉	semidirect
152. C₂²×C₈	C₄×D₇₉	semidirect
163. C₄×C₈	C₂²×D₇₉	semidirect
178. C₂²×C₄	#11929607	direct (N × H)
186. C₂×C₈	C₂×C₄×D₇₉	semidirect
208. C₂×C₄	C₂×#11929607	direct (N × H)
214. C₈	C₂²×C₄×D₇₉	semidirect
218. C₂²	C₄×#11929607	direct (N × H)
225. C₄	C₂²×#11929607	direct (N × H)
228. C₂	C₂×C₄×#11929607	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×#11929607	2
C₂²×C₄²×D₇₉	2
C₂×C₄×#11929607	2
C₂²×#14579071	2
C₂²×C₄×C₇₉⋊C₈	2
C₂²×C₄×C₈×C₇₉	2
C₂²×C₄×M₁₆	79

Subgroups — C22×C4×#11929607 #35151654

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×#11929607 #35151654