Subgroups — C₂×#23762910 #34270795

Order 20224.

758 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098765	C₂	1	2	yes	maximal
2	C₂×#19098642	C₂	1	1	yes	maximal, characteristic
3	#23762910	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098314	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098309	C₂	1	2	yes	maximal, complemented
6	C₂×#500×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11929611	—	2	2	no	—
8	C₂²×#11929607	—	2	1	no	—
9	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
10	#19098765	—	2	4	no	—
11	#19098765	C₂²	1	8	yes	—
12	#19098642	—	2	2	no	—
13	#19098642	C₂²	1	4	yes	—
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
16	C₂×#14579085	—	2	2	no	—
17	C₂×(C₂×D₇₉)⋊C₈	—	2	1	no	—
18	#19098314	C₂²	1	4	yes	complemented
19	#19098309	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
21	#500×C₇₉	—	2	2	no	—
22	#500×C₇₉	C₂²	1	4	yes	complemented
23	C₂×#107×C₇₉	—	2	1	no	—
24	C₂×#107×C₇₉	—	2	1	no	—
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₁₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₇₉	—	2	1	no	—
28	C₂×#11929611	—	2	8	no	—
29	C₂×#11929611	—	4	8	no	—
30	C₂×#11929607	—	2	4	no	—
31	C₂×#11929607	—	4	4	no	—
32	C₂²×C₇₉⋊C₈	—	2	1	no	—
33	C₂²×C₇₉⋊C₈	—	2	2	no	—
34	C₄×D₃₁₆	—	2	1	no	—
35	C₄×D₃₁₆	—	2	1	no	—
36	C₄×D₃₁₆	D₄	1	2	yes	—
37	C₄×D₃₁₆	C₂³	1	2	yes	—
38	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
39	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
40	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
41	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
43	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
44	#14579085	—	2	8	no	—
45	(C₂×D₇₉)⋊C₈	—	2	4	no	—
46	C₂²×Q₈×C₇₉	—	2	2	no	—
47	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
49	C₄⋊Q₈×C₇₉	—	2	2	no	—
50	C₄⋊Q₈×C₇₉	—	2	2	no	—
51	C₄⋊Q₈×C₇₉	—	2	2	no	—
52	C₄⋊Q₈×C₇₉	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	#107×C₇₉	—	2	4	no	—
57	#107×C₇₉	—	2	4	no	—
58	C₄×C₈×C₇₉	D₄	1	2	yes	—
59	C₄×C₈×C₇₉	C₂³	1	2	yes	—
60	C₂³×D₇₉	—	2	1	no	—
61	C₂×D₃₁₆	—	2	1	no	—
62	C₂×D₃₁₆	—	2	4	no	—
63	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
64	C₂²×Dic₇₉	—	2	1	no	—
65	C₂×C₄×D₇₉	—	2	2	no	—
66	C₂×C₄×D₇₉	—	2	2	no	—
67	C₂×C₄×D₇₉	—	4	2	no	—
68	C₂×C₄×D₇₉	—	4	2	no	—
69	#11929611	—	4	16	no	—
70	#11929607	—	4	8	no	—
71	C₂×C₇₉⋊C₈	—	2	2	no	—
72	C₂×C₇₉⋊C₈	—	2	4	no	—
73	C₂×C₇₉⋊C₈	—	4	2	no	—
74	C₂×C₇₉⋊C₈	—	4	4	no	—
75	C₄⋊Dic₇₉	—	2	1	no	—
76	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
77	(C₂×D₇₉)⋊C₄	—	2	2	no	—
78	(C₂×D₇₉)⋊C₄	—	2	2	no	—
79	C₂×Q₈×C₇₉	—	2	4	no	—
80	C₂×Q₈×C₇₉	—	4	2	no	—
81	C₂×Q₈×C₇₉	—	4	4	no	—
82	C₂×Q₈×C₇₉	—	4	4	no	—
83	C₂²×C₄×C₇₉	—	4	2	no	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	SD₁₆	1	4	yes	complemented
89	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₇₉	—	2	4	no	—
92	C₄⋊C₄×C₇₉	—	2	4	no	—
93	C₄⋊C₄×C₇₉	—	4	2	no	—
94	C₄⋊C₄×C₇₉	—	4	2	no	—
95	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₉	#41	1	1	yes	characteristic
98	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
99	C₂³×C₇₉	#121	1	1	yes	characteristic
100	C₂²×D₇₉	—	2	2	no	—
101	C₂²×D₇₉	—	4	2	no	—
102	C₂²×D₇₉	—	4	2	no	—
103	C₂²×D₇₉	—	4	2	no	—
104	Q₈×C₇₉	—	4	4	no	—
105	Q₈×C₇₉	—	4	4	no	—
106	Q₈×C₇₉	—	8	4	no	—
107	C₂×C₄×C₇₉	—	2	2	no	—
108	C₂×C₄×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	—	8	4	no	—
110	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₇₉	#137	1	2	yes	—
116	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
119	C₂×Dic₇₉	—	2	2	no	—
120	C₂×Dic₇₉	—	4	2	no	—
121	D₃₁₆	—	2	4	no	—
122	D₃₁₆	—	4	2	no	—
123	C₄×D₇₉	—	4	2	no	—
124	C₄×D₇₉	—	4	2	no	—
125	C₄×D₇₉	—	4	4	no	—
126	C₈×C₇₉	—	2	2	no	—
127	C₈×C₇₉	C₂×SD₁₆	1	4	yes	complemented
128	C₇₉⋊C₈	—	4	2	no	—
129	C₇₉⋊C₈	—	4	4	no	—
130	C₂²×C₇₉	#461	1	2	yes	—
131	C₂²×C₇₉	#469	1	2	yes	—
132	C₂²×C₇₉	#473	1	2	yes	—
133	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
134	C₂×D₇₉	—	4	2	no	—
135	C₂×D₇₉	—	4	2	no	—
136	C₂×D₇₉	—	4	2	no	—
137	C₂×D₇₉	—	8	4	no	—
138	C₄×C₇₉	—	2	1	no	—
139	C₄×C₇₉	—	2	1	no	—
140	C₄×C₇₉	—	8	4	no	—
141	C₄×C₇₉	#461	1	2	yes	—
142	C₄×C₇₉	#469	1	2	yes	—
143	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
144	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
145	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
146	Dic₇₉	—	4	2	no	—
147	C₂×#1693	—	79	1	no	maximal, Sylow
148	C₂×C₇₉	#1693	1	4	yes	complemented
149	C₂×C₇₉	C₂×#461	1	1	yes	characteristic
150	C₂×C₇₉	C₂×#469	1	1	yes	characteristic
151	C₂×C₇₉	C₂×#473	1	1	yes	characteristic
152	D₇₉	—	8	2	no	—
153	C₂×#500	D₇₉	1	1	yes	characteristic, complemented
154	C₂×#461	—	79	2	no	—
155	C₂×#436	—	79	1	no	—
156	#1693	—	79	8	no	—
157	C₂×#334	—	79	1	no	—
158	C₂×#332	—	79	2	no	—
159	C₇₉	C₂×#1693	1	1	yes	characteristic, complemented, Sylow
160	C₂²×SD₁₆	—	158	2	no	—
161	C₂²×M₁₆	—	158	1	no	—
162	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
163	C₂×C₄×D₄	—	79	1	no	—
164	#500	—	2	2	no	—
165	#500	C₂×D₇₉	1	4	yes	complemented
166	#461	—	79	8	no	—
167	#461	—	158	4	no	—
168	#436	—	79	4	no	—
169	#436	—	158	2	no	—
170	C₂×#107	—	2	1	no	—
171	C₂×#107	—	2	1	no	—
172	C₂×C₄⋊C₈	—	79	1	no	—
173	C₂×C₄⋊C₈	—	79	2	no	—
174	C₂×#103	—	158	2	no	—
175	C₂×C₂²⋊C₈	—	158	1	no	—
176	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
177	#334	—	79	4	no	—
178	#332	—	79	8	no	—
179	C₂²×Q₈	—	2	2	no	—
180	C₂²×D₄	—	79	1	no	—
181	C₂³×C₄	—	158	1	no	—
182	C₂×SD₁₆	—	316	8	no	—
183	C₂×SD₁₆	—	158	8	no	—
184	C₂×M₁₆	—	316	4	no	—
185	C₂×M₁₆	—	158	4	no	—
186	C₂²×C₈	—	158	2	no	—
187	C₂²×C₈	—	158	1	no	—
188	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
189	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
190	C₄⋊Q₈	—	2	2	no	—
191	C₄⋊Q₈	—	2	2	no	—
192	C₄⋊Q₈	—	2	2	no	—
193	C₄⋊Q₈	C₂²×D₇₉	1	4	yes	—
194	C₄×D₄	—	79	2	no	—
195	C₄×D₄	—	79	2	no	—
196	C₄×D₄	—	158	1	no	—
197	C₄×D₄	—	158	1	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	79	1	no	—
201	C₂×C₂²⋊C₄	—	158	1	no	—
202	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
203	#107	—	2	4	no	—
204	#107	—	2	4	no	—
205	C₄⋊C₈	—	79	2	no	—
206	C₄⋊C₈	—	79	4	no	—
207	C₄⋊C₈	—	158	2	no	—
208	C₄⋊C₈	—	158	1	no	—
209	#103	—	158	8	no	—
210	C₂²⋊C₈	—	158	4	no	—
211	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
212	C₄×C₈	C₂²×D₇₉	1	2	yes	—
213	C₂⁴	—	158	1	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	2	no	—
216	C₂×Q₈	—	4	4	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×D₄	—	79	2	no	—
219	C₂×D₄	—	158	1	no	—
220	C₂×D₄	—	158	4	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	316	2	no	—
223	C₂²×C₄	—	316	2	no	—
224	C₂²×C₄	—	158	1	no	—
225	C₂²×C₄	—	158	2	no	—
226	C₂²×C₄	—	158	2	no	—
227	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
229	C₂²×C₄	#11929623	1	1	yes	characteristic
230	SD₁₆	—	316	16	no	—
231	M₁₆	—	316	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	316	2	no	—
234	C₂×C₈	—	316	4	no	—
235	C₂×C₈	—	158	4	no	—
236	C₂×C₈	—	158	2	no	—
237	C₂×C₈	#11929611	1	4	yes	complemented
238	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	79	2	no	—
245	C₄⋊C₄	—	158	1	no	—
246	C₂²⋊C₄	—	158	2	no	—
247	C₂²⋊C₄	—	158	2	no	—
248	C₄²	D₄×D₇₉	1	1	yes	characteristic
249	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
250	C₄²	#11929623	1	1	yes	characteristic
251	C₄²	C₂³×D₇₉	1	1	yes	characteristic
252	C₂³	—	316	2	no	—
253	C₂³	—	316	2	no	—
254	C₂³	—	316	2	no	—
255	C₂³	—	158	2	no	—
256	C₂³	#14579132	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	4	4	no	—
259	Q₈	—	8	4	no	—
260	D₄	—	316	2	no	—
261	D₄	—	158	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	316	2	no	—
266	C₂×C₄	—	316	4	no	—
267	C₂×C₄	—	316	2	no	—
268	C₂×C₄	—	316	2	no	—
269	C₂×C₄	—	158	2	no	—
270	C₂×C₄	#14579132	1	1	yes	characteristic
271	C₂×C₄	#14579132	1	1	yes	characteristic
272	C₂×C₄	#14579132	1	1	yes	characteristic
273	C₂×C₄	C₂×#11929611	1	2	yes	—
274	C₂×C₄	#14579185	1	4	yes	—
275	C₂×C₄	#14579192	1	2	yes	—
276	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
277	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
278	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	316	2	no	—
281	C₈	—	316	4	no	—
282	C₈	C₂×#11929611	1	4	yes	complemented
283	C₂²	—	316	2	no	—
284	C₂²	—	316	2	no	—
285	C₂²	—	316	2	no	—
286	C₂²	—	632	4	no	—
287	C₂²	#19098765	1	2	yes	—
288	C₂²	#19098813	1	2	yes	—
289	C₂²	#19098835	1	2	yes	—
290	C₂²	C₂×#14579132	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	8	4	no	—
294	C₄	—	316	2	no	—
295	C₄	#19098765	1	2	yes	—
296	C₄	#19098813	1	2	yes	—
297	C₄	C₂²×#11929611	1	1	yes	characteristic
298	C₄	C₂×#14579185	1	2	yes	—
299	C₄	C₂×#14579192	1	1	yes	characteristic
300	C₂	—	632	2	no	—
301	C₂	#23762910	1	4	yes	direct product factor
302	C₂	C₂×#19098765	1	1	yes	characteristic
303	C₂	C₂×#19098813	1	1	yes	characteristic
304	C₂	C₂×#19098835	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579132
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#500×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂²×D₃₁₆	D₄
hypercenter	C₂×#500	D₇₉
hypocenter	C₇₉	C₂×#1693
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #23762910	C₂	direct (N × H)
4. C₂×#19098314	C₂	semidirect
5. C₂×#19098309	C₂	semidirect
6. C₂×#500×C₇₉	C₂	semidirect
18. #19098314	C₂²	semidirect
19. #19098309	C₂²	semidirect
22. #500×C₇₉	C₂²	semidirect
88. C₂×C₈×C₇₉	SD₁₆	semidirect
127. C₈×C₇₉	C₂×SD₁₆	semidirect
148. C₂×C₇₉	#1693	semidirect
153. C₂×#500	D₇₉	semidirect
159. C₇₉	C₂×#1693	semidirect
165. #500	C₂×D₇₉	semidirect
237. C₂×C₈	#11929611	semidirect
282. C₈	C₂×#11929611	semidirect
301. C₂	#23762910	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×#19098765	2
C₂×#19098642	2
#23762910	2
C₂×#19098314	2
C₂×#19098309	2
C₂×#500×C₇₉	2
C₂×#1693	79

Subgroups — C2×#23762910 #34270795

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23762910 #34270795