Subgroups — C₂×#23763253 #34286151

Order 20224.

1422 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₇₉⋊C₈)	C₂	1	2	yes	maximal
2	C₂²×#14579072	C₂	1	2	yes	maximal
3	#23763253	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098318	C₂	1	2	yes	maximal
5	C₂×C₄³×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₇₉⋊C₈	C₂²	1	4	yes	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	16	yes	complemented
8	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	8	yes	—
9	C₂×#14579072	—	2	8	no	—
10	C₂×#14579072	C₂²	1	8	yes	—
11	#19098318	C₂²	1	8	yes	—
12	C₂²×C₄²×C₇₉	C₄	1	2	yes	—
13	C₂²×C₄²×C₇₉	C₄	1	2	yes	—
14	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
17	C₄³×C₇₉	C₄	1	4	yes	—
18	C₄³×C₇₉	C₂²	1	4	yes	—
19	C₂²×C₇₉⋊C₈	—	2	8	no	—
20	C₂²×C₇₉⋊C₈	—	2	8	no	—
21	C₂²×C₇₉⋊C₈	C₂×C₄	1	16	yes	—
22	C₂²×C₇₉⋊C₈	C₂³	1	8	yes	—
23	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	32	yes	complemented
24	#14579072	—	2	16	no	—
25	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₇₉	—	2	8	no	—
31	C₂×C₄²×C₇₉	—	2	8	no	—
32	C₂×C₄²×C₇₉	—	2	4	no	—
33	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	—
34	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₇₉	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₇₉	D₄	1	4	yes	—
40	C₂×C₄²×C₇₉	Q₈	1	4	yes	—
41	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
42	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
44	C₂×C₇₉⋊C₈	—	2	32	no	—
45	C₂×C₇₉⋊C₈	—	2	8	no	—
46	C₂×C₇₉⋊C₈	—	2	8	no	—
47	C₂×C₇₉⋊C₈	C₂²×C₄	1	16	yes	—
48	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₇₉	—	2	4	no	—
50	C₂²×C₄×C₇₉	—	2	4	no	—
51	C₂²×C₄×C₇₉	—	2	4	no	—
52	C₂²×C₄×C₇₉	—	2	4	no	—
53	C₂²×C₄×C₇₉	—	2	4	no	—
54	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
55	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
56	C₂²×C₄×C₇₉	C₄⋊C₄	1	8	yes	—
57	C₂²×C₄×C₇₉	M₁₆	1	8	yes	—
58	C₂²×C₄×C₇₉	M₁₆	1	8	yes	—
59	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
61	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₇₉	#41	1	4	yes	—
68	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
69	C₄²×C₇₉	—	2	4	no	—
70	C₄²×C₇₉	—	2	16	no	—
71	C₄²×C₇₉	—	2	16	no	—
72	C₄²×C₇₉	—	2	4	no	—
73	C₄²×C₇₉	C₄²	1	8	yes	—
74	C₄²×C₇₉	C₄⋊C₄	1	8	yes	—
75	C₄²×C₇₉	C₂²×C₄	1	8	yes	—
76	C₄²×C₇₉	C₂×D₄	1	4	yes	—
77	C₄²×C₇₉	C₂×Q₈	1	4	yes	—
78	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
79	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₇₉	#117	1	2	yes	—
81	C₂³×C₇₉	C₂×M₁₆	1	4	yes	—
82	C₂³×C₇₉	C₂×M₁₆	1	4	yes	—
83	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	—	2	4	no	—
85	C₂×C₄×C₇₉	—	2	4	no	—
86	C₂×C₄×C₇₉	—	2	4	no	—
87	C₂×C₄×C₇₉	—	2	4	no	—
88	C₂×C₄×C₇₉	—	2	4	no	—
89	C₂×C₄×C₇₉	—	2	8	no	—
90	C₂×C₄×C₇₉	—	2	16	no	—
91	C₂×C₄×C₇₉	—	2	16	no	—
92	C₂×C₄×C₇₉	#97	1	16	yes	complemented
93	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
94	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
95	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
96	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
97	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₉	#117	1	2	yes	—
99	C₂×C₄×C₇₉	C₄×D₄	1	8	yes	—
100	C₂×C₄×C₇₉	C₄×Q₈	1	8	yes	—
101	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
102	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
103	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
104	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
107	C₇₉⋊C₈	—	2	16	no	—
108	C₂²×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
109	C₂²×C₇₉	C₂×#97	1	4	yes	—
110	C₂²×C₇₉	C₄×M₁₆	1	4	yes	—
111	C₂²×C₇₉	#423	1	8	yes	—
112	C₂²×C₇₉	#432	1	8	yes	—
113	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
114	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
116	C₂²×C₇₉	C₂²×M₁₆	1	2	yes	—
117	C₂²×C₇₉	C₂²×M₁₆	1	2	yes	—
118	C₄×C₇₉	—	2	4	no	—
119	C₄×C₇₉	—	2	8	no	—
120	C₄×C₇₉	—	2	8	no	—
121	C₄×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
122	C₄×C₇₉	C₂×#97	1	8	yes	complemented
123	C₄×C₇₉	C₂×C₄×D₄	1	2	yes	—
124	C₄×C₇₉	C₂×C₄×Q₈	1	2	yes	—
125	C₂×#1767	—	79	1	no	maximal, Sylow
126	C₂×C₇₉	#1767	1	8	yes	complemented
127	C₂×C₇₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₇₉	C₂²×#97	1	1	yes	characteristic
129	C₂×C₇₉	C₂×C₄×M₁₆	1	1	yes	characteristic
130	C₂×C₇₉	C₂×#423	1	2	yes	—
131	C₂×C₇₉	C₂×#432	1	2	yes	—
132	C₂²×C₄⋊C₈	—	79	2	no	—
133	C₂²×#97	—	79	2	no	—
134	C₂×C₄³	D₇₉	1	1	yes	characteristic
135	#1767	—	79	8	no	—
136	C₂×#336	—	79	2	no	—
137	C₇₉	C₂×#1767	1	1	yes	characteristic, complemented, Sylow
138	C₂³×C₈	—	79	4	no	—
139	C₂²×C₄²	Dic₇₉	1	2	yes	—
140	C₂²×C₄²	Dic₇₉	1	2	yes	—
141	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
142	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
143	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
144	C₂×C₄⋊C₈	—	79	8	no	—
145	C₂×C₄⋊C₈	—	79	16	no	—
146	C₂×#97	—	79	8	no	—
147	C₂×#97	—	158	8	no	—
148	C₄³	Dic₇₉	1	4	yes	—
149	C₄³	C₂×D₇₉	1	4	yes	—
150	#336	—	79	8	no	—
151	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
152	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
153	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
154	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
155	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
156	C₂²×C₈	—	79	16	no	—
157	C₂²×C₈	—	79	8	no	—
158	C₂²×C₈	—	158	8	no	—
159	C₂²×C₈	—	158	8	no	—
160	C₂×C₄²	—	2	8	no	—
161	C₂×C₄²	—	2	8	no	—
162	C₂×C₄²	—	2	4	no	—
163	C₂×C₄²	Dic₁₅₈	1	4	yes	—
164	C₂×C₄²	C₄×D₇₉	1	8	yes	—
165	C₂×C₄²	D₃₁₆	1	4	yes	—
166	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
167	C₂×C₄²	C₂×Dic₇₉	1	8	yes	—
168	C₂×C₄²	C₂×Dic₇₉	1	8	yes	—
169	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
170	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
171	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
172	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
173	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
174	C₄⋊C₈	—	79	32	no	—
175	#97	—	158	16	no	—
176	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	4	no	—
182	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
183	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
184	C₂²×C₄	C₄⋊Dic₇₉	1	8	yes	—
185	C₂²×C₄	#11929608	1	8	yes	—
186	C₂²×C₄	#11929608	1	8	yes	—
187	C₂²×C₄	C₂×C₄×D₇₉	1	4	yes	—
188	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
190	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
191	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
192	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
193	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
194	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
195	C₂²×C₄	#11929624	1	4	yes	—
196	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
197	C₂×C₈	—	79	16	no	—
198	C₂×C₈	—	158	8	no	—
199	C₂×C₈	—	158	32	no	—
200	C₂×C₈	—	158	8	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	16	no	—
204	C₄²	—	2	16	no	—
205	C₄²	C₄×Dic₇₉	1	8	yes	—
206	C₄²	C₄⋊Dic₇₉	1	8	yes	—
207	C₄²	C₂×C₄×D₇₉	1	8	yes	—
208	C₄²	C₂×D₃₁₆	1	4	yes	—
209	C₄²	C₂×Dic₁₅₈	1	4	yes	—
210	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
211	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
212	C₂³	#14579120	1	2	yes	—
213	C₂³	C₂×#11929608	1	4	yes	—
214	C₂³	C₂×#11929608	1	4	yes	—
215	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	8	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	16	no	—
223	C₂×C₄	—	2	16	no	—
224	C₂×C₄	#14579072	1	16	yes	complemented
225	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
226	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
227	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₇₉	1	8	yes	—
229	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
230	C₂×C₄	#14579120	1	2	yes	—
231	C₂×C₄	C₄×D₃₁₆	1	8	yes	—
232	C₂×C₄	C₄×Dic₁₅₈	1	8	yes	—
233	C₂×C₄	C₂×#11929608	1	8	yes	—
234	C₂×C₄	C₂×#11929608	1	8	yes	—
235	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
236	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
237	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
238	C₂×C₄	C₂×#11929624	1	2	yes	—
239	C₈	—	158	16	no	—
240	C₂²	C₄×C₄⋊Dic₇₉	1	4	yes	—
241	C₂²	C₂×#14579072	1	4	yes	—
242	C₂²	C₄×#11929608	1	4	yes	—
243	C₂²	#19098587	1	8	yes	—
244	C₂²	#19098620	1	8	yes	—
245	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
246	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
247	C₂²	C₂×#14579120	1	1	yes	characteristic
248	C₂²	C₂²×#11929608	1	2	yes	—
249	C₂²	C₂²×#11929608	1	2	yes	—
250	C₄	—	2	8	no	—
251	C₄	—	2	8	no	—
252	C₄	—	2	4	no	—
253	C₄	C₄×C₄⋊Dic₇₉	1	4	yes	—
254	C₄	C₂×#14579072	1	8	yes	complemented
255	C₄	C₂×C₄×D₃₁₆	1	2	yes	—
256	C₄	C₂×C₄×Dic₁₅₈	1	2	yes	—
257	C₂	#23763253	1	8	yes	direct product factor
258	C₂	C₂×C₄×C₄⋊Dic₇₉	1	1	yes	characteristic
259	C₂	C₂²×#14579072	1	1	yes	characteristic
260	C₂	C₂×C₄×#11929608	1	1	yes	characteristic
261	C₂	C₂×#19098587	1	2	yes	—
262	C₂	C₂×#19098620	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₄³×C₇₉	C₂
socle, prime radical	C₂⁴×C₇₉	C₂²×C₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄³	D₇₉
hypocenter	C₇₉	C₂×#1767
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
3. #23763253	C₂	direct (N × H)
7. C₂×C₄⋊(C₇₉⋊C₈)	C₄	semidirect
23. C₄⋊(C₇₉⋊C₈)	C₂×C₄	semidirect
92. C₂×C₄×C₇₉	#97	semidirect
122. C₄×C₇₉	C₂×#97	semidirect
126. C₂×C₇₉	#1767	semidirect
137. C₇₉	C₂×#1767	semidirect
224. C₂×C₄	#14579072	semidirect
254. C₄	C₂×#14579072	semidirect
257. C₂	#23763253	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₇₉⋊C₈)	2
C₂²×#14579072	2
#23763253	2
C₂×#19098318	2
C₂×C₄³×C₇₉	2
C₂×#1767	79

Subgroups — C2×#23763253 #34286151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763253 #34286151