Subgroups — C₂×#23772038 #35152000

Order 20224.

2198 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772038	C₂	1	16	yes	maximal, direct product factor
2	C₂×#19098624	C₂	1	8	yes	maximal, complemented
3	C₂×#19098587	C₂	1	2	yes	maximal, complemented
4	C₂×#19098527	C₂	1	2	yes	maximal
5	C₂×C₄×#11929608	C₂	1	2	yes	maximal, complemented
6	C₂²×#117×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11929608	C₂²	1	4	yes	—
8	#19098624	C₂²	1	64	yes	complemented
9	#19098587	C₂²	1	16	yes	complemented
10	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	8	yes	complemented
11	#19098527	C₂²	1	16	yes	—
12	C₂×C₂²⋊(C₇₉⋊C₈)	C₂²	1	8	yes	—
13	C₄×#11929608	C₂²	1	16	yes	complemented
14	C₂×#14579072	C₂²	1	4	yes	complemented
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	complemented
16	C₂⁴×C₄×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₇₉	C₄	1	4	yes	—
18	C₂×#117×C₇₉	C₄	1	8	yes	—
19	C₂×#117×C₇₉	C₂²	1	4	yes	—
20	C₂×#117×C₇₉	C₂²	1	8	yes	—
21	C₂²×C₄⋊C₄×C₇₉	C₄	1	2	yes	—
22	C₂²×C₂²⋊C₄×C₇₉	C₄	1	2	yes	—
23	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×#11929608	—	2	16	no	—
26	C₂×#11929608	C₂³	1	16	yes	—
27	C₂²×C₇₉⋊C₈	C₂³	1	8	yes	—
28	C₄⋊(C₇₉⋊C₈)	C₂³	1	32	yes	complemented
29	C₂²⋊(C₇₉⋊C₈)	C₂³	1	32	yes	—
30	#14579072	C₂³	1	16	yes	complemented
31	C₄×C₇₉⋊C₈	C₂³	1	16	yes	complemented
32	C₂⁵×C₇₉	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₇₉	—	2	8	no	—
34	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₇₉	C₂×C₄	1	4	yes	—
37	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
38	C₂³×C₄×C₇₉	C₂×C₄	1	2	yes	—
39	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
40	C₂³×C₄×C₇₉	C₂³	1	4	yes	—
41	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
42	#117×C₇₉	C₂×C₄	1	16	yes	—
43	#117×C₇₉	C₂×C₄	1	32	yes	—
44	#117×C₇₉	C₂³	1	16	yes	—
45	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	16	yes	—
47	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	8	yes	—
48	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	16	yes	—
49	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
54	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
55	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₇₉	C₂³	1	4	yes	—
57	#11929608	—	2	32	no	—
58	C₂×C₇₉⋊C₈	—	2	16	no	—
59	C₂×C₇₉⋊C₈	C₂⁴	1	16	yes	—
60	C₂⁴×C₇₉	—	2	4	no	—
61	C₂⁴×C₇₉	—	4	4	no	—
62	C₂⁴×C₇₉	C₂²×C₄	1	2	yes	—
63	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
64	C₂⁴×C₇₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₇₉	—	2	4	no	—
66	C₂²×C₄×C₇₉	—	2	16	no	—
67	C₂²×C₄×C₇₉	—	2	8	no	—
68	C₂²×C₄×C₇₉	—	2	8	no	—
69	C₂²×C₄×C₇₉	—	2	16	no	—
70	C₂²×C₄×C₇₉	—	4	4	no	—
71	C₂²×C₄×C₇₉	—	4	4	no	—
72	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₇₉	C₂²×C₄	1	8	yes	—
80	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	#41	1	8	yes	—
83	C₂²×C₄×C₇₉	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
85	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₇₉	C₂²×C₄	1	32	yes	—
87	C₂²⋊C₄×C₇₉	C₂²×C₄	1	32	yes	—
88	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
89	C₄²×C₇₉	C₂²×C₄	1	8	yes	—
90	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
91	C₄²×C₇₉	C₂²×C₄	1	8	yes	—
92	C₄²×C₇₉	C₂⁴	1	4	yes	—
93	C₄²×C₇₉	C₂⁴	1	4	yes	—
94	C₂³×C₇₉	—	2	4	no	—
95	C₂³×C₇₉	—	2	8	no	—
96	C₂³×C₇₉	—	2	4	no	—
97	C₂³×C₇₉	—	2	8	no	—
98	C₂³×C₇₉	—	4	8	no	—
99	C₂³×C₇₉	—	4	16	no	—
100	C₂³×C₇₉	#117	1	4	yes	—
101	C₂³×C₇₉	C₂³×C₄	1	4	yes	—
102	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
103	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₇₉	—	2	8	no	—
105	C₂×C₄×C₇₉	—	2	4	no	—
106	C₂×C₄×C₇₉	—	2	8	no	—
107	C₂×C₄×C₇₉	—	2	8	no	—
108	C₂×C₄×C₇₉	—	2	8	no	—
109	C₂×C₄×C₇₉	—	2	16	no	—
110	C₂×C₄×C₇₉	—	2	8	no	—
111	C₂×C₄×C₇₉	—	4	8	no	—
112	C₂×C₄×C₇₉	—	4	8	no	—
113	C₂×C₄×C₇₉	#117	1	8	yes	—
114	C₂×C₄×C₇₉	#117	1	4	yes	—
115	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
116	C₂×C₄×C₇₉	C₂³×C₄	1	8	yes	—
117	C₂×C₄×C₇₉	C₂³×C₄	1	4	yes	—
118	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
119	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₂×#41	1	4	yes	—
121	C₂×C₄×C₇₉	C₂×#41	1	8	yes	—
122	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
123	C₇₉⋊C₈	—	2	16	no	—
124	C₂²×C₇₉	—	2	8	no	—
125	C₂²×C₇₉	—	2	4	no	—
126	C₂²×C₇₉	—	2	4	no	—
127	C₂²×C₇₉	—	2	8	no	—
128	C₂²×C₇₉	—	4	16	no	—
129	C₂²×C₇₉	—	4	8	no	—
130	C₂²×C₇₉	C₂×#117	1	4	yes	—
131	C₂²×C₇₉	C₂×#117	1	2	yes	—
132	C₂²×C₇₉	#568	1	4	yes	—
133	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
134	C₄×C₇₉	—	2	4	no	—
135	C₄×C₇₉	—	2	8	no	—
136	C₄×C₇₉	—	2	8	no	—
137	C₄×C₇₉	C₂×#117	1	2	yes	—
138	C₄×C₇₉	C₂×#117	1	4	yes	—
139	C₄×C₇₉	C₂²×#41	1	2	yes	—
140	C₂×#2921	—	79	1	no	maximal, Sylow
141	C₂×C₇₉	—	2	4	no	—
142	C₂×C₇₉	—	4	4	no	—
143	C₂×C₇₉	#2921	1	4	yes	complemented
144	C₂×C₇₉	C₂²×#117	1	1	yes	characteristic
145	C₂×C₇₉	C₂×#568	1	2	yes	—
146	C₂²×#117	D₇₉	1	1	yes	characteristic
147	#2921	—	79	16	no	—
148	C₂×#433	—	79	8	no	—
149	C₂×#423	—	79	2	no	—
150	C₂×#407	—	79	2	no	—
151	C₂×C₄×M₁₆	—	79	2	no	—
152	C₇₉	C₂×#2921	1	1	yes	characteristic, complemented, Sylow
153	C₂⁴×C₄	C₂×D₇₉	1	1	yes	characteristic
154	C₂²×M₁₆	—	79	4	no	—
155	C₂×#117	Dic₇₉	1	4	yes	—
156	C₂×#117	Dic₇₉	1	8	yes	—
157	C₂×#117	C₂×D₇₉	1	4	yes	—
158	C₂×#117	C₂×D₇₉	1	8	yes	—
159	C₂²×C₄⋊C₄	Dic₇₉	1	2	yes	—
160	C₂²×C₂²⋊C₄	Dic₇₉	1	2	yes	—
161	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
162	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
163	#433	—	79	64	no	—
164	#423	—	79	16	no	—
165	C₂×C₄⋊C₈	—	79	8	no	—
166	#407	—	79	16	no	—
167	C₂×C₂²⋊C₈	—	79	8	no	—
168	C₄×M₁₆	—	79	16	no	—
169	C₂×#97	—	79	4	no	—
170	C₂×C₄×C₈	—	79	4	no	—
171	C₂⁵	C₂×Dic₇₉	1	1	yes	characteristic
172	C₂³×C₄	—	2	8	no	—
173	C₂³×C₄	C₂×Dic₇₉	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
175	C₂³×C₄	C₂×Dic₇₉	1	4	yes	—
176	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
177	C₂³×C₄	C₂×Dic₇₉	1	2	yes	—
178	C₂³×C₄	C₂²×D₇₉	1	4	yes	—
179	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
180	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
181	C₂×M₁₆	—	79	16	no	—
182	C₂×M₁₆	—	158	16	no	—
183	C₂²×C₈	—	79	8	no	—
184	#117	C₂×Dic₇₉	1	16	yes	—
185	#117	C₂×Dic₇₉	1	32	yes	—
186	#117	C₂²×D₇₉	1	16	yes	—
187	C₂×C₄⋊C₄	C₂×Dic₇₉	1	8	yes	—
188	C₂×C₄⋊C₄	C₂×Dic₇₉	1	16	yes	—
189	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	8	yes	—
190	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	16	yes	—
191	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
192	C₂×C₄²	C₂×Dic₇₉	1	2	yes	—
193	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
194	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
195	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
196	C₂×C₄²	C₂²×D₇₉	1	4	yes	—
197	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
198	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
199	C₄⋊C₈	—	79	32	no	—
200	C₂²⋊C₈	—	79	32	no	—
201	#97	—	79	16	no	—
202	C₄×C₈	—	79	16	no	—
203	C₂⁴	—	2	4	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	C₂²×Dic₇₉	1	2	yes	—
206	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
207	C₂⁴	C₂²×Dic₇₉	1	4	yes	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	8	no	—
210	C₂²×C₄	—	2	16	no	—
211	C₂²×C₄	—	2	16	no	—
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
216	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
217	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
218	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
219	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
220	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
221	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
222	C₂²×C₄	C₂²×Dic₇₉	1	8	yes	—
223	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
225	C₂²×C₄	#11929624	1	8	yes	—
226	C₂²×C₄	C₂³×D₇₉	1	4	yes	—
227	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
228	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
229	M₁₆	—	158	32	no	—
230	C₂×C₈	—	79	16	no	—
231	C₂×C₈	—	158	16	no	—
232	C₄⋊C₄	C₂²×Dic₇₉	1	32	yes	—
233	C₂²⋊C₄	C₂²×Dic₇₉	1	32	yes	—
234	C₄²	C₂²×Dic₇₉	1	8	yes	—
235	C₄²	C₂²×Dic₇₉	1	8	yes	—
236	C₄²	C₂²×Dic₇₉	1	4	yes	—
237	C₄²	C₂²×Dic₇₉	1	4	yes	—
238	C₄²	C₂³×D₇₉	1	4	yes	—
239	C₄²	C₂³×D₇₉	1	4	yes	—
240	C₂³	—	2	4	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	2	8	no	—
244	C₂³	—	4	8	no	—
245	C₂³	—	4	16	no	—
246	C₂³	#14579120	1	4	yes	—
247	C₂³	C₂³×Dic₇₉	1	4	yes	—
248	C₂³	C₂³×Dic₇₉	1	2	yes	—
249	C₂³	C₂³×Dic₇₉	1	1	yes	characteristic
250	C₂×C₄	—	2	8	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	16	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	8	no	—
259	C₂×C₄	#14579120	1	8	yes	—
260	C₂×C₄	#14579120	1	4	yes	—
261	C₂×C₄	C₂³×Dic₇₉	1	8	yes	—
262	C₂×C₄	C₂³×Dic₇₉	1	8	yes	—
263	C₂×C₄	C₂³×Dic₇₉	1	4	yes	—
264	C₂×C₄	C₂³×Dic₇₉	1	2	yes	—
265	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#11929624	1	4	yes	—
267	C₂×C₄	C₂×#11929624	1	8	yes	—
268	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
269	C₈	—	158	16	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	8	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	4	16	no	—
275	C₂²	—	4	8	no	—
276	C₂²	C₂×#14579120	1	4	yes	—
277	C₂²	C₂×#14579120	1	2	yes	—
278	C₂²	#19099330	1	4	yes	—
279	C₂²	C₂⁴×Dic₇₉	1	1	yes	characteristic
280	C₄	—	2	4	no	—
281	C₄	—	2	8	no	—
282	C₄	—	2	8	no	—
283	C₄	C₂×#14579120	1	2	yes	—
284	C₄	C₂×#14579120	1	4	yes	—
285	C₄	C₂²×#11929624	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	—	4	4	no	—
288	C₂	#23772038	1	4	yes	direct product factor
289	C₂	C₂²×#14579120	1	1	yes	characteristic
290	C₂	C₂×#19099330	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₇₉
commutator	C₂²×C₇₉	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂²×#117×C₇₉	C₂
socle	C₂³×C₇₉	C₂³×C₄
prime radical	C₂⁵×C₇₉	C₂×C₄
cosocle	C₂×C₄×C₇₉	C₂⁵
hypercenter	C₂²×#117	D₇₉
hypocenter	C₇₉	C₂×#2921
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
1. #23772038	C₂	direct (N × H)
2. C₂×#19098624	C₂	semidirect
3. C₂×#19098587	C₂	semidirect
5. C₂×C₄×#11929608	C₂	semidirect
8. #19098624	C₂²	semidirect
9. #19098587	C₂²	semidirect
10. C₂×C₄⋊(C₇₉⋊C₈)	C₂²	semidirect
13. C₄×#11929608	C₂²	semidirect
14. C₂×#14579072	C₂²	semidirect
15. C₂×C₄×C₇₉⋊C₈	C₂²	semidirect
28. C₄⋊(C₇₉⋊C₈)	C₂³	semidirect
30. #14579072	C₂³	semidirect
31. C₄×C₇₉⋊C₈	C₂³	semidirect
143. C₂×C₇₉	#2921	semidirect
152. C₇₉	C₂×#2921	semidirect
288. C₂	#23772038	direct (N × H)

Maximal subgroups

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

subgroup	index
#23772038	2
C₂×#19098624	2
C₂×#19098587	2
C₂×#19098527	2
C₂×C₄×#11929608	2
C₂²×#117×C₇₉	2
C₂×#2921	79

Subgroups — C2×#23772038 #35152000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23772038 #35152000