Subgroups — C₂×#23776802 #35195110

Order 20224.

7022 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776802	C₂	1	32	yes	maximal, direct product factor
2	C₂×#19099192	C₂	1	12	yes	maximal, complemented
3	C₂×#19099189	C₂	1	3	yes	maximal
4	C₂×#19099115	C₂	1	12	yes	maximal, complemented
5	C₂²×#14579117	C₂	1	3	yes	maximal
6	C₂×#543×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂⁴×Dic₇₉	C₂²	1	1	yes	characteristic
8	#19099192	C₂²	1	192	yes	complemented
9	#19099189	C₂²	1	48	yes	—
10	#19099115	C₂²	1	192	yes	complemented
11	C₂×#14579160	C₂²	1	12	yes	complemented
12	C₂×#14579159	C₂²	1	12	yes	complemented
13	C₂×#14579151	C₂²	1	24	yes	complemented
14	C₂×#14579140	C₂²	1	12	yes	—
15	C₂×#14579138	C₂²	1	24	yes	—
16	C₂×Q₈×Dic₇₉	C₂²	1	4	yes	complemented
17	C₂×#14579126	C₂²	1	12	yes	complemented
18	C₂×#14579117	C₂²	1	12	yes	—
19	C₂×#14579117	C₂²	1	48	yes	—
20	C₂×#14579117	C₂²	1	12	yes	—
21	C₂²×Dic₇₉⋊C₄	C₂²	1	6	yes	—
22	C₂²×C₂²⋊Dic₇₉	C₂²	1	3	yes	—
23	C₂²×C₄×Dic₇₉	C₂²	1	6	yes	—
24	#543×C₇₉	C₂²	1	16	yes	—
25	C₂×C₂²⋊Q₈×C₇₉	C₂²	1	12	yes	—
26	C₂²×C₂²⋊C₄×C₇₉	C₂²	1	3	yes	—
27	C₂²×Dic₁₅₈	C₂³	1	12	yes	—
28	C₂³×Dic₇₉	Q₈	1	16	yes	complemented
29	C₂³×Dic₇₉	C₂³	1	6	yes	—
30	C₂³×Dic₇₉	C₂³	1	3	yes	—
31	C₂³×Dic₇₉	C₂³	1	3	yes	—
32	C₂³×Dic₇₉	C₂³	1	8	yes	—
33	#14579160	C₂³	1	96	yes	complemented
34	#14579159	C₂³	1	96	yes	complemented
35	#14579151	C₂³	1	192	yes	complemented
36	#14579140	C₂³	1	96	yes	—
37	#14579138	C₂³	1	192	yes	—
38	Q₈×Dic₇₉	C₂³	1	32	yes	complemented
39	#14579126	C₂³	1	96	yes	complemented
40	#14579117	C₂³	1	96	yes	—
41	#14579117	C₂³	1	96	yes	—
42	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
43	C₂×Dic₇₉⋊C₄	C₂³	1	48	yes	—
44	C₂×Dic₇₉⋊C₄	C₂³	1	24	yes	—
45	C₂×Dic₇₉⋊C₄	C₂³	1	24	yes	—
46	C₂×C₂²⋊Dic₇₉	C₂³	1	24	yes	—
47	C₂×C₂²⋊Dic₇₉	C₂³	1	12	yes	—
48	C₂×C₄×Dic₇₉	C₂³	1	12	yes	—
49	C₂×C₄×Dic₇₉	C₂³	1	12	yes	—
50	C₂×C₄×Dic₇₉	C₂³	1	48	yes	—
51	C₂⁵×C₇₉	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₇₉	C₂³	1	4	yes	—
53	C₂³×C₄×C₇₉	C₂³	1	6	yes	—
54	C₂²⋊Q₈×C₇₉	C₂³	1	96	yes	—
55	C₂×C₄⋊C₄×C₇₉	C₂³	1	12	yes	—
56	C₂×C₂²⋊C₄×C₇₉	C₂³	1	24	yes	—
57	C₂×C₂²⋊C₄×C₇₉	C₂³	1	12	yes	—
58	C₂×Dic₁₅₈	—	4	24	no	—
59	C₂×Dic₁₅₈	C₂⁴	1	48	yes	—
60	C₂²×Dic₇₉	—	4	12	no	—
61	C₂²×Dic₇₉	C₂×Q₈	1	24	yes	—
62	C₂²×Dic₇₉	C₂×Q₈	1	24	yes	—
63	C₂²×Dic₇₉	C₂×Q₈	1	64	yes	complemented
64	C₂²×Dic₇₉	C₂⁴	1	12	yes	—
65	C₂²×Dic₇₉	C₂⁴	1	24	yes	—
66	C₂²×Dic₇₉	C₂⁴	1	1	yes	characteristic
67	C₂²×Dic₇₉	C₂⁴	1	3	yes	—
68	C₂²×Dic₇₉	C₂⁴	1	12	yes	—
69	C₂²×Dic₇₉	C₂⁴	1	12	yes	—
70	C₄⋊Dic₇₉	C₂⁴	1	48	yes	—
71	Dic₇₉⋊C₄	C₂⁴	1	96	yes	—
72	Dic₇₉⋊C₄	C₂⁴	1	96	yes	—
73	C₂²⋊Dic₇₉	C₂⁴	1	48	yes	—
74	C₄×Dic₇₉	C₂⁴	1	48	yes	—
75	C₄×Dic₇₉	C₂⁴	1	48	yes	—
76	C₂⁴×C₇₉	—	4	4	no	—
77	C₂⁴×C₇₉	C₂×Q₈	1	8	yes	—
78	C₂⁴×C₇₉	C₂⁴	1	3	yes	—
79	C₂⁴×C₇₉	C₂⁴	1	4	yes	—
80	C₂×Q₈×C₇₉	—	4	8	no	—
81	C₂×Q₈×C₇₉	C₂⁴	1	16	yes	—
82	C₂²×C₄×C₇₉	—	4	12	no	—
83	C₂²×C₄×C₇₉	C₂⁴	1	12	yes	—
84	C₂²×C₄×C₇₉	C₂⁴	1	24	yes	—
85	C₄⋊C₄×C₇₉	C₂⁴	1	48	yes	—
86	C₂²⋊C₄×C₇₉	C₂⁴	1	48	yes	—
87	C₂³×C₇₉	—	2	12	no	—
88	C₂³×C₇₉	—	4	16	no	—
89	C₂³×C₇₉	—	4	8	no	—
90	C₂³×C₇₉	C₂²×Q₈	1	12	yes	—
91	C₂³×C₇₉	C₂²×Q₈	1	16	yes	—
92	C₂³×C₇₉	C₂⁵	1	6	yes	—
93	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
94	Q₈×C₇₉	—	4	16	no	—
95	C₂×C₄×C₇₉	—	2	24	no	—
96	C₂×C₄×C₇₉	—	4	24	no	—
97	C₂×C₄×C₇₉	C₂⁵	1	24	yes	—
98	C₂×Dic₇₉	—	2	24	no	—
99	C₂×Dic₇₉	—	4	24	no	—
100	C₂×Dic₇₉	C₂²×Q₈	1	4	yes	—
101	C₂×Dic₇₉	C₂²×Q₈	1	12	yes	—
102	C₂×Dic₇₉	C₂²×Q₈	1	48	yes	—
103	C₂×Dic₇₉	C₂²×Q₈	1	48	yes	—
104	C₂×Dic₇₉	C₂⁵	1	24	yes	—
105	C₂×Dic₇₉	C₂⁵	1	2	yes	—
106	C₂×Dic₇₉	C₂⁵	1	6	yes	—
107	Dic₁₅₈	—	4	48	no	—
108	C₂²×C₇₉	—	2	12	no	—
109	C₂²×C₇₉	—	2	24	no	—
110	C₂²×C₇₉	—	4	16	no	—
111	C₂²×C₇₉	C₂³×Q₈	1	12	yes	—
112	C₂²×C₇₉	C₂³×Q₈	1	2	yes	—
113	C₂²×C₇₉	#585	1	4	yes	—
114	C₂²×C₇₉	C₂⁶	1	1	yes	characteristic
115	C₄×C₇₉	—	2	24	no	—
116	Dic₇₉	—	2	24	no	—
117	Dic₇₉	C₂³×Q₈	1	4	yes	—
118	Dic₇₉	C₂³×Q₈	1	12	yes	—
119	C₂×#3458	—	79	1	no	maximal, Sylow
120	C₂×C₇₉	—	2	12	no	—
121	C₂×C₇₉	#3458	1	4	yes	complemented
122	C₂×C₇₉	C₂⁴×Q₈	1	1	yes	characteristic
123	C₂×C₇₉	C₂×#585	1	2	yes	—
124	#3458	—	79	32	no	—
125	C₂×#544	—	79	12	no	—
126	C₂×#543	—	79	3	no	—
127	C₂×#543	D₇₉	1	1	yes	characteristic
128	C₂×#533	—	79	12	no	—
129	C₂²×#117	—	79	3	no	—
130	C₇₉	C₂×#3458	1	1	yes	characteristic, complemented, Sylow
131	C₂⁴×C₄	—	79	1	no	—
132	#544	—	79	192	no	—
133	#543	—	79	48	no	—
134	#543	C₂×D₇₉	1	16	yes	—
135	#533	—	79	192	no	—
136	C₂×C₄⋊Q₈	—	79	12	no	—
137	C₂×C₄⋊Q₈	—	79	12	no	—
138	C₂×#125	—	79	24	no	—
139	C₂×C₂²⋊Q₈	—	79	12	no	—
140	C₂×C₂²⋊Q₈	—	79	24	no	—
141	C₂×C₂²⋊Q₈	C₂×D₇₉	1	12	yes	—
142	C₂×C₄×Q₈	—	79	4	no	—
143	C₂×C₄×Q₈	—	79	12	no	—
144	C₂×#117	—	79	12	no	—
145	C₂×#117	—	79	12	no	—
146	C₂×#117	—	79	48	no	—
147	C₂²×C₄⋊C₄	—	79	6	no	—
148	C₂²×C₂²⋊C₄	—	79	3	no	—
149	C₂²×C₂²⋊C₄	C₂×D₇₉	1	3	yes	—
150	C₂²×C₄²	—	79	6	no	—
151	C₂⁵	C₂²×D₇₉	1	1	yes	characteristic
152	C₂²×Q₈	—	79	12	no	—
153	C₂²×Q₈	C₂²×D₇₉	1	4	yes	—
154	C₂³×C₄	—	79	6	no	—
155	C₂³×C₄	—	79	16	no	—
156	C₂³×C₄	—	79	3	no	—
157	C₂³×C₄	—	79	8	no	—
158	C₂³×C₄	—	79	3	no	—
159	C₂³×C₄	C₂²×D₇₉	1	6	yes	—
160	C₄⋊Q₈	—	79	96	no	—
161	C₄⋊Q₈	—	79	96	no	—
162	#125	—	79	192	no	—
163	C₂²⋊Q₈	—	79	96	no	—
164	C₂²⋊Q₈	—	79	192	no	—
165	C₂²⋊Q₈	C₂²×D₇₉	1	96	yes	—
166	C₄×Q₈	—	79	32	no	—
167	C₄×Q₈	—	79	96	no	—
168	#117	—	79	96	no	—
169	#117	—	79	96	no	—
170	C₂×C₄⋊C₄	—	79	24	no	—
171	C₂×C₄⋊C₄	—	79	48	no	—
172	C₂×C₄⋊C₄	—	79	24	no	—
173	C₂×C₄⋊C₄	—	79	12	no	—
174	C₂×C₄⋊C₄	C₂²×D₇₉	1	12	yes	—
175	C₂×C₂²⋊C₄	—	79	12	no	—
176	C₂×C₂²⋊C₄	—	79	24	no	—
177	C₂×C₂²⋊C₄	C₂²×D₇₉	1	12	yes	—
178	C₂×C₂²⋊C₄	C₂²×D₇₉	1	24	yes	—
179	C₂×C₄²	—	79	12	no	—
180	C₂×C₄²	—	79	12	no	—
181	C₂×C₄²	—	79	48	no	—
182	C₂⁴	—	4	4	no	—
183	C₂⁴	Q₈×D₇₉	1	8	yes	—
184	C₂⁴	C₂³×D₇₉	1	3	yes	—
185	C₂⁴	C₂³×D₇₉	1	4	yes	—
186	C₂×Q₈	—	4	8	no	—
187	C₂×Q₈	—	316	24	no	—
188	C₂×Q₈	—	79	48	no	—
189	C₂×Q₈	C₂³×D₇₉	1	16	yes	—
190	C₂²×C₄	—	4	12	no	—
191	C₂²×C₄	—	316	12	no	—
192	C₂²×C₄	—	79	24	no	—
193	C₂²×C₄	—	79	24	no	—
194	C₂²×C₄	—	79	24	no	—
195	C₂²×C₄	—	79	64	no	—
196	C₂²×C₄	—	79	12	no	—
197	C₂²×C₄	—	79	12	no	—
198	C₂²×C₄	—	79	1	no	—
199	C₂²×C₄	—	79	3	no	—
200	C₂²×C₄	—	79	12	no	—
201	C₂²×C₄	C₂³×D₇₉	1	12	yes	—
202	C₂²×C₄	C₂³×D₇₉	1	24	yes	—
203	C₄⋊C₄	—	79	96	no	—
204	C₄⋊C₄	—	79	96	no	—
205	C₄⋊C₄	—	79	48	no	—
206	C₄⋊C₄	C₂³×D₇₉	1	48	yes	—
207	C₂²⋊C₄	—	79	48	no	—
208	C₂²⋊C₄	C₂³×D₇₉	1	48	yes	—
209	C₄²	—	79	48	no	—
210	C₄²	—	79	48	no	—
211	C₂³	—	2	12	no	—
212	C₂³	—	4	16	no	—
213	C₂³	—	4	8	no	—
214	C₂³	C₂×Q₈×D₇₉	1	12	yes	—
215	C₂³	C₂×Q₈×D₇₉	1	16	yes	—
216	C₂³	C₂⁴×D₇₉	1	6	yes	—
217	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
218	Q₈	—	4	16	no	—
219	Q₈	—	316	48	no	—
220	C₂×C₄	—	2	24	no	—
221	C₂×C₄	—	4	24	no	—
222	C₂×C₄	—	316	24	no	—
223	C₂×C₄	—	79	2	no	—
224	C₂×C₄	—	79	6	no	—
225	C₂×C₄	—	79	48	no	—
226	C₂×C₄	—	79	48	no	—
227	C₂×C₄	—	79	4	no	—
228	C₂×C₄	—	79	12	no	—
229	C₂×C₄	—	79	24	no	—
230	C₂×C₄	—	158	24	no	—
231	C₂×C₄	C₂⁴×D₇₉	1	24	yes	—
232	C₂²	—	2	24	no	—
233	C₂²	—	2	12	no	—
234	C₂²	—	4	16	no	—
235	C₂²	C₂²×Q₈×D₇₉	1	12	yes	—
236	C₂²	C₂²×Q₈×D₇₉	1	2	yes	—
237	C₂²	#19099398	1	4	yes	—
238	C₂²	C₂⁵×D₇₉	1	1	yes	characteristic
239	C₄	—	2	24	no	—
240	C₄	—	79	4	no	—
241	C₄	—	79	12	no	—
242	C₄	—	158	24	no	—
243	C₂	—	2	12	no	—
244	C₂	#23776802	1	4	yes	direct product factor
245	C₂	C₂³×Q₈×D₇₉	1	1	yes	characteristic
246	C₂	C₂×#19099398	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂⁴×D₇₉
commutator, cosocle	C₂²×C₇₉	C₂⁶
Frattini	C₂²	C₂⁵×D₇₉
Fitting	C₂×#543×C₇₉	C₂
socle	C₂³×C₇₉	C₂⁵
prime radical	C₂⁵×C₇₉	C₂³
hypercenter	C₂×#543	D₇₉
hypocenter	C₇₉	C₂×#3458
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#543

Complements

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

N	H ≅ G/N	type
1. #23776802	C₂	direct (N × H)
2. C₂×#19099192	C₂	semidirect
4. C₂×#19099115	C₂	semidirect
8. #19099192	C₂²	semidirect
10. #19099115	C₂²	semidirect
11. C₂×#14579160	C₂²	semidirect
12. C₂×#14579159	C₂²	semidirect
13. C₂×#14579151	C₂²	semidirect
16. C₂×Q₈×Dic₇₉	C₂²	semidirect
17. C₂×#14579126	C₂²	semidirect
28. C₂³×Dic₇₉	Q₈	semidirect
33. #14579160	C₂³	semidirect
34. #14579159	C₂³	semidirect
35. #14579151	C₂³	semidirect
38. Q₈×Dic₇₉	C₂³	semidirect
39. #14579126	C₂³	semidirect
63. C₂²×Dic₇₉	C₂×Q₈	semidirect
121. C₂×C₇₉	#3458	semidirect
130. C₇₉	C₂×#3458	semidirect
244. C₂	#23776802	direct (N × H)

Maximal subgroups

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

subgroup	index
#23776802	2
C₂×#19099192	2
C₂×#19099189	2
C₂×#19099115	2
C₂²×#14579117	2
C₂×#543×C₇₉	2
C₂×#3458	79

Subgroups — C2×#23776802 #35195110

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23776802 #35195110