Subgroups — C₂×#23765573 #34443002

Order 20224.

526 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098681	C₂	1	1	yes	maximal, characteristic
2	#23765573	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19098385	C₂	1	2	yes	maximal, complemented
4	C₂×#19098384	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098372	C₂	1	2	yes	maximal
6	C₂×#499×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14579183	C₂²	1	1	yes	characteristic
8	C₂×#14579167	—	2	1	no	—
9	C₂×#14579101	C₂²	1	2	yes	—
10	#19098681	—	2	2	no	—
11	#19098681	C₂²	1	4	yes	—
12	C₂×#14579097	C₂²	1	1	yes	characteristic
13	C₂×#14579089	—	2	1	no	—
14	#19098385	C₂²	1	8	yes	complemented
15	#19098384	C₂²	1	4	yes	complemented
16	#19098372	C₂²	1	8	yes	—
17	C₂×#125×C₇₉	—	2	1	no	—
18	#499×C₇₉	—	2	2	no	—
19	#499×C₇₉	C₂²	1	4	yes	complemented
20	C₂×#107×C₇₉	—	2	1	no	—
21	C₂×#106×C₇₉	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×#11929624	—	2	1	no	—
24	#14579183	—	2	1	no	—
25	#14579183	—	2	1	no	—
26	#14579183	Q₈	1	2	yes	—
27	#14579183	C₂³	1	2	yes	—
28	C₂×Dic₃₁₆	Q₈	1	1	yes	characteristic
29	C₂×#11929613	—	2	1	no	—
30	C₂×D₆₃₂	Q₈	1	1	yes	characteristic
31	#14579167	—	2	4	no	—
32	#14579167	—	4	2	no	—
33	C₂×#11929608	—	2	1	no	—
34	C₂×#11929607	—	4	1	no	—
35	C₂×C₈×D₇₉	—	4	1	no	—
36	#14579101	—	2	2	no	—
37	#14579101	C₂³	1	4	yes	—
38	C₂×C₇₉⋊C₁₆	—	2	2	no	—
39	#14579097	—	2	1	no	—
40	#14579097	C₂³	1	2	yes	—
41	#14579089	—	2	4	no	—
42	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	#125×C₇₉	—	2	4	no	—
45	#125×C₇₉	—	4	2	no	—
46	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
48	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
49	#107×C₇₉	—	2	4	no	—
50	#106×C₇₉	—	2	2	no	—
51	#106×C₇₉	C₂³	1	4	yes	—
52	C₄×C₈×C₇₉	D₄	1	2	yes	—
53	C₄×C₈×C₇₉	C₂³	1	2	yes	—
54	#11929624	—	2	2	no	—
55	#11929624	—	2	2	no	—
56	#11929624	—	4	1	no	—
57	#11929624	—	4	1	no	—
58	C₂×Dic₁₅₈	—	2	1	no	—
59	C₂×D₃₁₆	—	2	1	no	—
60	C₂×C₂²⋊D₇₉	—	4	1	no	—
61	C₂×C₄×D₇₉	—	4	1	no	—
62	Dic₃₁₆	—	2	1	no	—
63	Dic₃₁₆	C₂×Q₈	1	2	yes	—
64	#11929613	—	2	2	no	—
65	#11929613	—	2	2	no	—
66	D₆₃₂	—	2	1	no	—
67	D₆₃₂	C₂×Q₈	1	2	yes	—
68	#11929608	—	2	2	no	—
69	#11929608	—	4	1	no	—
70	#11929607	—	4	4	no	—
71	C₂×C₇₉⋊C₈	—	4	1	no	—
72	C₈×D₇₉	—	4	4	no	—
73	C₇₉⋊C₁₆	—	2	4	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	4	2	no	—
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
78	C₂×C₈×C₇₉	SD₁₆	1	1	yes	characteristic
79	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
80	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
83	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
84	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₇₉	—	2	4	no	—
88	C₄⋊C₄×C₇₉	—	4	8	no	—
89	C₄⋊C₄×C₇₉	—	4	2	no	—
90	C₄²×C₇₉	—	2	1	no	—
91	C₄²×C₇₉	C₂×D₄	1	2	yes	—
92	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂²×D₇₉	—	4	1	no	—
94	C₂×C₄×C₇₉	—	2	2	no	—
95	C₂×C₄×C₇₉	—	4	2	no	—
96	C₂×C₄×C₇₉	—	4	4	no	—
97	C₂×C₄×C₇₉	—	8	4	no	—
98	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
104	C₂²⋊D₇₉	—	4	2	no	—
105	C₂²⋊D₇₉	—	4	2	no	—
106	C₂×Dic₇₉	—	4	1	no	—
107	D₃₁₆	—	2	2	no	—
108	D₃₁₆	—	4	1	no	—
109	C₄×D₇₉	—	4	2	no	—
110	C₄×D₇₉	—	4	2	no	—
111	Dic₁₅₈	—	2	2	no	—
112	Dic₁₅₈	—	4	1	no	—
113	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
116	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
117	C₈×C₇₉	C₂²×Q₈	1	1	yes	characteristic
118	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
119	C₇₉⋊C₈	—	4	2	no	—
120	C₂²×C₇₉	—	2	2	no	—
121	C₂²×C₇₉	#476	1	1	yes	characteristic
122	C₂²×C₇₉	#476	1	1	yes	characteristic
123	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×D₇₉	—	4	2	no	—
125	C₂×D₇₉	—	8	2	no	—
126	C₄×C₇₉	—	4	2	no	—
127	C₄×C₇₉	—	8	4	no	—
128	C₄×C₇₉	#476	1	1	yes	characteristic
129	C₄×C₇₉	#476	1	1	yes	characteristic
130	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
131	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
132	Dic₇₉	—	4	2	no	—
133	C₂×#2231	—	79	1	no	maximal, Sylow
134	C₂×C₇₉	—	2	1	no	—
135	C₂×C₇₉	—	2	1	no	—
136	C₂×C₇₉	#2231	1	2	yes	complemented
137	C₂×C₇₉	C₂×#476	1	1	yes	characteristic
138	D₇₉	—	8	2	no	—
139	C₂×#499	D₇₉	1	1	yes	characteristic, complemented
140	C₂×#443	—	79	1	no	—
141	#2231	—	79	8	no	—
142	C₂×#365	—	79	2	no	—
143	C₂×#364	—	79	1	no	—
144	C₂×#360	—	79	2	no	—
145	C₇₉	C₂×#2231	1	1	yes	characteristic, complemented, Sylow
146	C₂×#135	—	79	1	no	—
147	C₂×#131	—	158	1	no	—
148	C₂×#125	—	2	1	no	—
149	C₂×M₃₂	—	79	2	no	—
150	#499	—	2	2	no	—
151	#499	C₂×D₇₉	1	4	yes	complemented
152	#443	—	79	4	no	—
153	#443	—	158	2	no	—
154	C₂×#108	—	79	1	no	—
155	C₂×#107	—	2	1	no	—
156	C₂×#106	C₂×D₇₉	1	2	yes	—
157	C₂×#104	—	158	1	no	—
158	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
159	#365	—	79	8	no	—
160	#364	—	79	4	no	—
161	#360	—	79	8	no	—
162	C₂×#41	—	158	1	no	—
163	#135	—	79	2	no	—
164	#135	—	79	2	no	—
165	#135	—	158	1	no	—
166	#135	—	158	1	no	—
167	C₂×Q₁₆	—	79	1	no	—
168	C₂×SD₁₆	—	158	1	no	—
169	C₂×D₈	—	79	1	no	—
170	#131	—	316	2	no	—
171	#131	—	158	4	no	—
172	C₂×M₁₆	—	316	1	no	—
173	C₂×M₁₆	—	158	1	no	—
174	C₂²×C₈	—	316	1	no	—
175	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
177	#125	—	2	4	no	—
178	#125	—	4	2	no	—
179	C₂×C₄⋊C₄	—	2	2	no	—
180	C₂×C₄⋊C₄	—	4	2	no	—
181	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
182	M₃₂	—	79	4	no	—
183	M₃₂	—	158	2	no	—
184	C₂×C₁₆	—	158	2	no	—
185	#108	—	79	2	no	—
186	#108	—	158	1	no	—
187	#107	—	2	4	no	—
188	#106	—	2	2	no	—
189	#106	C₂²×D₇₉	1	4	yes	—
190	#104	—	158	4	no	—
191	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
192	C₄×C₈	C₂²×D₇₉	1	2	yes	—
193	#41	—	316	1	no	—
194	#41	—	316	1	no	—
195	#41	—	158	2	no	—
196	#41	—	158	2	no	—
197	C₂×Q₈	—	158	1	no	—
198	C₂×D₄	—	316	1	no	—
199	C₂×D₄	—	158	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	—	316	1	no	—
203	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
204	Q₁₆	—	79	2	no	—
205	Q₁₆	—	158	1	no	—
206	SD₁₆	—	158	2	no	—
207	SD₁₆	—	158	2	no	—
208	D₈	—	79	2	no	—
209	D₈	—	158	1	no	—
210	M₁₆	—	316	1	no	—
211	M₁₆	—	316	4	no	—
212	M₁₆	—	158	2	no	—
213	C₂×C₈	—	316	1	no	—
214	C₂×C₈	—	316	4	no	—
215	C₂×C₈	#11929611	1	1	yes	characteristic
216	C₂×C₈	#11929611	1	1	yes	characteristic
217	C₂×C₈	#11929611	1	2	yes	—
218	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
219	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
220	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
221	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
222	C₂×C₈	#11929625	1	1	yes	characteristic
223	C₂×C₈	#11929625	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	8	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
230	C₁₆	—	158	4	no	—
231	C₂³	—	316	1	no	—
232	C₂³	#14579187	1	1	yes	characteristic
233	Q₈	—	316	1	no	—
234	Q₈	—	158	2	no	—
235	D₄	—	316	2	no	—
236	D₄	—	316	2	no	—
237	D₄	—	316	1	no	—
238	D₄	—	158	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	8	4	no	—
243	C₂×C₄	—	316	1	no	—
244	C₂×C₄	—	316	2	no	—
245	C₂×C₄	—	316	2	no	—
246	C₂×C₄	#14579137	1	1	yes	characteristic
247	C₂×C₄	#14579137	1	1	yes	characteristic
248	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
249	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
250	C₂×C₄	#14579187	1	1	yes	characteristic
251	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
252	C₈	—	316	2	no	—
253	C₈	#14579137	1	1	yes	characteristic
254	C₈	#14579137	1	1	yes	characteristic
255	C₈	C₂×#11929611	1	2	yes	—
256	C₈	C₂×#11929611	1	2	yes	—
257	C₈	C₂×Q₈×D₇₉	1	1	yes	characteristic
258	C₈	C₂×#11929625	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	316	2	no	—
261	C₂²	—	632	2	no	—
262	C₂²	#19098850	1	1	yes	characteristic
263	C₂²	#19098850	1	1	yes	characteristic
264	C₂²	C₂×#14579187	1	1	yes	characteristic
265	C₄	—	4	2	no	—
266	C₄	—	8	4	no	—
267	C₄	—	316	2	no	—
268	C₄	#19098850	1	1	yes	characteristic
269	C₄	#19098850	1	1	yes	characteristic
270	C₄	C₂×#14579137	1	1	yes	characteristic
271	C₄	C₂²×#11929611	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	632	2	no	—
275	C₂	#23765573	1	2	yes	direct product factor
276	C₂	C₂×#19098850	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098850
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#499×C₇₉	C₂
socle	C₂²×C₇₉	#476
prime radical	C₂×#14579183	C₂²
hypercenter	C₂×#499	D₇₉
hypocenter	C₇₉	C₂×#2231
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
2. #23765573	C₂	direct (N × H)
3. C₂×#19098385	C₂	semidirect
4. C₂×#19098384	C₂	semidirect
6. C₂×#499×C₇₉	C₂	semidirect
14. #19098385	C₂²	semidirect
15. #19098384	C₂²	semidirect
19. #499×C₇₉	C₂²	semidirect
136. C₂×C₇₉	#2231	semidirect
139. C₂×#499	D₇₉	semidirect
145. C₇₉	C₂×#2231	semidirect
151. #499	C₂×D₇₉	semidirect
275. C₂	#23765573	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098681	2
#23765573	2
C₂×#19098385	2
C₂×#19098384	2
C₂×#19098372	2
C₂×#499×C₇₉	2
C₂×#2231	79

Subgroups — C2×#23765573 #34443002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765573 #34443002