Subgroups — C₂×#23761693 #34253873

Order 20224.

902 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098616	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#23761693	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19098285	C₂	1	2	yes	maximal, complemented
5	C₂×#19098284	C₂	1	2	yes	maximal, complemented
6	C₂×#431×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₇₉⋊C₈	C₄	1	2	yes	—
8	#19098616	C₄	1	4	yes	—
9	#19098616	C₂²	1	4	yes	—
10	C₂×C₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
11	C₂×C₂²⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
12	C₂×C₄×C₇₉⋊C₈	C₄	1	4	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	2	yes	—
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
18	#19098285	C₂²	1	8	yes	complemented
19	#19098284	C₂²	1	8	yes	complemented
20	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	#431×C₇₉	C₄	1	4	yes	—
22	#431×C₇₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₇₉	C₄	1	2	yes	—
24	C₂×C₂²⋊C₈×C₇₉	C₄	1	2	yes	—
25	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
26	C₂²×C₇₉⋊C₈	—	2	8	no	—
27	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
30	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
31	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
32	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	8	yes	—
33	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
34	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
35	C₄×C₇₉⋊C₈	C₂×C₄	1	8	yes	—
36	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
37	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
38	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
39	C₄×C₇₉⋊C₈	C₂³	1	4	yes	—
40	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
53	C₂²⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₇₉	C₂³	1	4	yes	—
56	C₂×C₇₉⋊C₈	—	2	16	no	—
57	C₂×C₇₉⋊C₈	—	2	4	no	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	2	8	no	—
60	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
61	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
62	C₂×C₇₉⋊C₈	C₂²×C₄	1	8	yes	—
63	C₂⁴×C₇₉	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₇₉	—	2	4	no	—
65	C₂²×C₄×C₇₉	—	2	4	no	—
66	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
69	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
70	C₂²×C₄×C₇₉	C₄²	1	2	yes	—
71	C₂²×C₄×C₇₉	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	M₁₆	1	8	yes	—
73	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	—	2	8	no	—
80	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
81	C₄²×C₇₉	C₄²	1	2	yes	—
82	C₄²×C₇₉	C₄²	1	2	yes	—
83	C₄²×C₇₉	C₄²	1	2	yes	—
84	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
85	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
88	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₇₉	—	2	2	no	—
92	C₂³×C₇₉	—	2	2	no	—
93	C₂³×C₇₉	#97	1	4	yes	—
94	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
95	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₇₉	—	2	2	no	—
97	C₂×C₄×C₇₉	—	2	2	no	—
98	C₂×C₄×C₇₉	—	2	4	no	—
99	C₂×C₄×C₇₉	—	2	4	no	—
100	C₂×C₄×C₇₉	—	2	4	no	—
101	C₂×C₄×C₇₉	—	2	4	no	—
102	C₂×C₄×C₇₉	#97	1	8	yes	—
103	C₂×C₄×C₇₉	#97	1	4	yes	—
104	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
106	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂×M₁₆	1	8	yes	—
111	C₂×C₄×C₇₉	C₂×M₁₆	1	4	yes	—
112	C₂×C₄×C₇₉	#131	1	4	yes	—
113	C₂×C₄×C₇₉	#131	1	4	yes	—
114	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
117	C₈×C₇₉	—	2	8	no	—
118	C₇₉⋊C₈	—	2	4	no	—
119	C₇₉⋊C₈	—	2	4	no	—
120	C₇₉⋊C₈	—	2	8	no	—
121	C₂²×C₇₉	—	2	2	no	—
122	C₂²×C₇₉	—	2	2	no	—
123	C₂²×C₇₉	—	2	8	no	—
124	C₂²×C₇₉	C₂×#97	1	4	yes	—
125	C₂²×C₇₉	C₂×#97	1	2	yes	—
126	C₂²×C₇₉	#405	1	2	yes	—
127	C₂²×C₇₉	#405	1	2	yes	—
128	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
129	C₄×C₇₉	—	2	2	no	—
130	C₄×C₇₉	—	2	2	no	—
131	C₄×C₇₉	C₂×#97	1	2	yes	—
132	C₄×C₇₉	C₂×#97	1	4	yes	—
133	C₄×C₇₉	#405	1	2	yes	—
134	C₄×C₇₉	#405	1	2	yes	—
135	C₄×C₇₉	C₂²×M₁₆	1	2	yes	—
136	C₄×C₇₉	C₂×#131	1	2	yes	—
137	C₄×C₇₉	C₂×#131	1	2	yes	—
138	C₂×#1452	—	79	1	no	maximal, Sylow
139	C₂×C₇₉	—	2	4	no	—
140	C₂×C₇₉	#1452	1	4	yes	complemented
141	C₂×C₇₉	C₂²×#97	1	1	yes	characteristic
142	C₂×C₇₉	C₂×#405	1	1	yes	characteristic
143	C₂×C₇₉	C₂×#405	1	1	yes	characteristic
144	C₂×#431	—	79	1	no	—
145	C₂×#431	D₇₉	1	1	yes	characteristic
146	C₂²×C₄×C₈	—	79	1	no	—
147	#1452	—	79	8	no	—
148	C₂×#322	—	79	2	no	—
149	C₂×#322	—	79	2	no	—
150	C₇₉	C₂×#1452	1	1	yes	characteristic, complemented, Sylow
151	C₂³×C₈	—	79	2	no	—
152	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
153	#431	—	79	4	no	—
154	#431	—	79	4	no	—
155	#431	Dic₇₉	1	4	yes	—
156	#431	C₂×D₇₉	1	4	yes	—
157	C₂×C₄⋊C₈	—	79	2	no	—
158	C₂×C₄⋊C₈	Dic₇₉	1	2	yes	—
159	C₂×C₂²⋊C₈	—	79	2	no	—
160	C₂×C₂²⋊C₈	Dic₇₉	1	2	yes	—
161	C₂×C₄×C₈	—	79	4	no	—
162	C₂×C₄×C₈	—	79	1	no	—
163	C₂×C₄×C₈	—	79	1	no	—
164	C₂×C₄×C₈	—	79	2	no	—
165	C₂×C₄×C₈	—	79	2	no	—
166	C₂×C₄×C₈	—	79	4	no	—
167	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
168	#322	—	79	8	no	—
169	#322	—	79	8	no	—
170	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
171	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
172	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
173	C₂²×C₈	—	79	8	no	—
174	C₂²×C₈	—	79	2	no	—
175	C₂²×C₈	—	79	2	no	—
176	C₂²×C₈	—	79	4	no	—
177	C₂²×C₈	—	158	8	no	—
178	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
179	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
180	C₂×C₄²	C₄×D₇₉	1	2	yes	—
181	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₇₉	1	2	yes	—
183	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
184	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
185	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
186	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
187	C₄⋊C₈	—	79	8	no	—
188	C₄⋊C₈	C₂×Dic₇₉	1	8	yes	—
189	C₂²⋊C₈	—	79	8	no	—
190	C₂²⋊C₈	C₂×Dic₇₉	1	8	yes	—
191	C₄×C₈	—	79	8	no	—
192	C₄×C₈	—	79	2	no	—
193	C₄×C₈	—	79	2	no	—
194	C₄×C₈	—	79	4	no	—
195	C₄×C₈	—	79	2	no	—
196	C₄×C₈	—	79	2	no	—
197	C₄×C₈	—	79	4	no	—
198	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
199	C₄×C₈	C₂²×D₇₉	1	4	yes	—
200	C₂⁴	C₄×Dic₇₉	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
204	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
205	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
206	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
207	C₂²×C₄	C₄×Dic₇₉	1	2	yes	—
208	C₂²×C₄	C₄×Dic₇₉	1	1	yes	characteristic
209	C₂²×C₄	#11929607	1	8	yes	—
210	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
212	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
216	C₂×C₈	—	2	8	no	—
217	C₂×C₈	—	79	4	no	—
218	C₂×C₈	—	79	4	no	—
219	C₂×C₈	—	79	8	no	—
220	C₂×C₈	—	158	16	no	—
221	C₂×C₈	—	158	4	no	—
222	C₂×C₈	—	158	4	no	—
223	C₂×C₈	—	158	8	no	—
224	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
225	C₄²	C₄×Dic₇₉	1	2	yes	—
226	C₄²	C₄×Dic₇₉	1	2	yes	—
227	C₄²	C₄×Dic₇₉	1	2	yes	—
228	C₄²	C₂×C₄×D₇₉	1	2	yes	—
229	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
230	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₇₉	1	2	yes	—
232	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
233	C₄²	C₂²×Dic₇₉	1	2	yes	—
234	C₄²	C₂³×D₇₉	1	1	yes	characteristic
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	#14579071	1	4	yes	—
238	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
239	C₂³	C₂×C₄×Dic₇₉	1	1	yes	characteristic
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	#14579071	1	8	yes	—
247	C₂×C₄	#14579071	1	4	yes	—
248	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
249	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
250	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
252	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
254	C₂×C₄	C₂×#11929607	1	8	yes	—
255	C₂×C₄	C₂×#11929607	1	4	yes	—
256	C₂×C₄	#14579168	1	4	yes	—
257	C₂×C₄	#14579169	1	4	yes	—
258	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
260	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
261	C₈	—	2	8	no	—
262	C₈	—	158	4	no	—
263	C₈	—	158	4	no	—
264	C₈	—	158	8	no	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	C₂×#14579071	1	4	yes	—
269	C₂²	C₂×#14579071	1	2	yes	—
270	C₂²	#19098519	1	2	yes	—
271	C₂²	#19098521	1	2	yes	—
272	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	C₂×#14579071	1	2	yes	—
276	C₄	C₂×#14579071	1	4	yes	—
277	C₄	#19098519	1	2	yes	—
278	C₄	#19098521	1	2	yes	—
279	C₄	C₂²×#11929607	1	2	yes	—
280	C₄	C₂×#14579168	1	2	yes	—
281	C₄	C₂×#14579169	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	#23761693	1	4	yes	direct product factor
284	C₂	C₂²×#14579071	1	1	yes	characteristic
285	C₂	C₂×#19098519	1	1	yes	characteristic
286	C₂	C₂×#19098521	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₉
commutator	C₂²×C₇₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#431×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₄²
prime radical	C₂⁴×C₇₉	C₄²
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₂×#431	D₇₉
hypocenter	C₇₉	C₂×#1452
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #23761693	C₂	direct (N × H)
4. C₂×#19098285	C₂	semidirect
5. C₂×#19098284	C₂	semidirect
18. #19098285	C₂²	semidirect
19. #19098284	C₂²	semidirect
140. C₂×C₇₉	#1452	semidirect
150. C₇₉	C₂×#1452	semidirect
283. C₂	#23761693	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×#19098616	2
C₂²×C₄×C₇₉⋊C₈	2
#23761693	2
C₂×#19098285	2
C₂×#19098284	2
C₂×#431×C₇₉	2
C₂×#1452	79

Subgroups — C2×#23761693 #34253873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761693 #34253873