Subgroups — C₂×#23763265 #34286200

Order 20224.

1438 conjugacy classes of subgroups.

The subgroup lattice has 296 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	1	yes	maximal, characteristic
2	C₂²×C₈×Dic₇₉	C₂	1	2	yes	maximal
3	C₂×C₄×C₄⋊Dic₇₉	C₂	1	1	yes	maximal, characteristic
4	#23763265	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19098317	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊Dic₇₉	C₄	1	2	yes	—
10	C₂²×C₄×Dic₇₉	C₂²	1	2	yes	—
11	C₂×C₄×C₇₉⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₄×C₇₉⋊C₈	C₂²	1	4	yes	—
13	C₂×C₈×Dic₇₉	—	2	8	no	—
14	C₂×C₈×Dic₇₉	C₂²	1	8	yes	—
15	C₄×C₄⋊Dic₇₉	C₄	1	4	yes	—
16	C₄×C₄⋊Dic₇₉	C₂²	1	4	yes	—
17	C₂×C₇₉⋊#95	C₄	1	2	yes	—
18	#19098317	C₂²	1	8	yes	—
19	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
20	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₈×C₇₉	C₄	1	8	yes	complemented
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	4	yes	—
23	C₂³×Dic₇₉	C₂×C₄	1	4	yes	—
24	C₂²×C₇₉⋊C₈	—	2	2	no	—
25	C₂²×C₇₉⋊C₈	—	2	2	no	—
26	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
28	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
29	C₂²×C₇₉⋊C₈	Q₈	1	4	yes	—
30	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
31	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
32	C₂×C₄⋊Dic₇₉	C₈	1	16	yes	complemented
33	C₂×C₄⋊Dic₇₉	C₂×C₄	1	8	yes	—
34	C₂×C₄×Dic₇₉	—	2	4	no	—
35	C₂×C₄×Dic₇₉	—	2	4	no	—
36	C₂×C₄×Dic₇₉	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₇₉	C₂³	1	4	yes	—
38	C₄×C₇₉⋊C₈	C₂×C₄	1	16	yes	complemented
39	C₈×Dic₇₉	—	2	16	no	—
40	C₇₉⋊#95	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	—	2	4	no	—
45	C₂²×C₈×C₇₉	—	2	4	no	—
46	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
47	C₂²×C₈×C₇₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₇₉	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
52	C₄⋊C₈×C₇₉	C₂×C₄	1	16	yes	complemented
53	C₂²×Dic₇₉	—	2	8	no	—
54	C₂²×Dic₇₉	—	2	8	no	—
55	C₂²×Dic₇₉	C₂×C₈	1	16	yes	—
56	C₂²×Dic₇₉	C₂²×C₄	1	8	yes	—
57	C₂×C₇₉⋊C₈	—	2	8	no	—
58	C₂×C₇₉⋊C₈	—	2	2	no	—
59	C₂×C₇₉⋊C₈	—	2	2	no	—
60	C₂×C₇₉⋊C₈	C₄⋊C₄	1	16	yes	complemented
61	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
62	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
63	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
64	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
65	C₄⋊Dic₇₉	C₂×C₈	1	32	yes	complemented
66	C₄×Dic₇₉	—	2	8	no	—
67	C₄×Dic₇₉	—	2	4	no	—
68	C₄×Dic₇₉	—	2	4	no	—
69	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	—	2	4	no	—
71	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
72	C₂²×C₄×C₇₉	C₄²	1	4	yes	—
73	C₂²×C₄×C₇₉	C₂×C₈	1	8	yes	—
74	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₇₉	#41	1	2	yes	—
82	C₂²×C₄×C₇₉	#41	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₇₉	—	2	16	no	—
85	C₂×C₈×C₇₉	—	2	4	no	—
86	C₂×C₈×C₇₉	—	2	4	no	—
87	C₂×C₈×C₇₉	C₂²×C₄	1	8	yes	—
88	C₄²×C₇₉	C₄²	1	8	yes	—
89	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
90	C₄²×C₇₉	C₂²×C₄	1	4	yes	—
91	C₂³×C₇₉	C₂×C₄²	1	2	yes	—
92	C₂³×C₇₉	C₄×D₄	1	2	yes	—
93	C₂³×C₇₉	C₄×Q₈	1	2	yes	—
94	C₂³×C₇₉	C₂²×C₈	1	4	yes	—
95	C₂³×C₇₉	#131	1	4	yes	—
96	C₂³×C₇₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₇₉	—	2	4	no	—
98	C₂×C₄×C₇₉	—	2	8	no	—
99	C₂×C₄×C₇₉	C₄×C₈	1	16	yes	complemented
100	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
101	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₇₉	C₂×C₄²	1	4	yes	—
103	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
104	C₂×C₄×C₇₉	#117	1	4	yes	—
105	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
107	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₇₉	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₇₉	C₂²×C₈	1	8	yes	—
110	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
116	C₂×Dic₇₉	—	2	8	no	—
117	C₂×Dic₇₉	—	2	8	no	—
118	C₂×Dic₇₉	—	2	32	no	—
119	C₂×Dic₇₉	C₂²×C₈	1	16	yes	—
120	C₈×C₇₉	—	2	8	no	—
121	C₇₉⋊C₈	—	2	4	no	—
122	C₇₉⋊C₈	C₂×C₄⋊C₄	1	8	yes	complemented
123	C₂²×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
124	C₂²×C₇₉	C₂×C₄×C₈	1	4	yes	—
125	C₂²×C₇₉	#405	1	4	yes	—
126	C₂²×C₇₉	C₈×D₄	1	8	yes	—
127	C₂²×C₇₉	C₈×Q₈	1	8	yes	—
128	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
129	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
131	C₂²×C₇₉	C₂³×C₈	1	2	yes	—
132	C₂²×C₇₉	C₂×#131	1	2	yes	—
133	C₄×C₇₉	—	2	4	no	—
134	C₄×C₇₉	C₄×C₄⋊C₄	1	4	yes	—
135	C₄×C₇₉	C₂×C₄×C₈	1	8	yes	complemented
136	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
138	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
140	Dic₇₉	—	2	16	no	—
141	C₂×C₈×C₄⋊C₄	—	79	1	no	maximal, Sylow
142	C₂×C₇₉	C₈×C₄⋊C₄	1	8	yes	complemented
143	C₂×C₇₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₇₉	C₂²×C₄×C₈	1	1	yes	characteristic
145	C₂×C₇₉	C₂×#405	1	1	yes	characteristic
146	C₂×C₇₉	C₂×C₈×D₄	1	2	yes	—
147	C₂×C₇₉	C₂×C₈×Q₈	1	2	yes	—
148	C₂²×C₄⋊C₈	D₇₉	1	1	yes	characteristic
149	C₂²×C₄×C₈	—	79	1	no	—
150	C₂²×C₄×C₈	—	79	2	no	—
151	C₂×C₄×C₄⋊C₄	—	79	1	no	—
152	C₈×C₄⋊C₄	—	79	8	no	—
153	C₂×#336	—	79	2	no	—
154	C₇₉	C₂×C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₈	—	79	1	no	—
156	C₂³×C₈	—	79	1	no	—
157	C₂³×C₈	C₂×D₇₉	1	2	yes	—
158	C₂²×C₄⋊C₄	—	79	2	no	—
159	C₂²×C₄²	—	79	2	no	—
160	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
161	C₂×C₄⋊C₈	Dic₇₉	1	8	yes	complemented
162	C₂×C₄⋊C₈	C₂×D₇₉	1	4	yes	—
163	C₂×C₄×C₈	—	79	4	no	—
164	C₂×C₄×C₈	—	79	8	no	—
165	C₂×C₄×C₈	—	79	8	no	—
166	C₂×C₄×C₈	—	158	8	no	—
167	C₄×C₄⋊C₄	—	79	4	no	—
168	C₄×C₄⋊C₄	—	79	4	no	—
169	C₂×#95	—	79	2	no	—
170	#336	—	79	8	no	—
171	C₂³×C₄	—	79	4	no	—
172	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
173	C₂³×C₄	C₄×D₇₉	1	1	yes	characteristic
174	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
175	C₂²×C₈	—	2	4	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	79	4	no	—
178	C₂²×C₈	—	79	4	no	—
179	C₂²×C₈	—	79	4	no	—
180	C₂²×C₈	—	79	2	no	—
181	C₂²×C₈	—	79	4	no	—
182	C₂²×C₈	—	79	2	no	—
183	C₂²×C₈	—	158	2	no	—
184	C₂²×C₈	—	158	2	no	—
185	C₂²×C₈	C₂×Dic₇₉	1	8	yes	—
186	C₂²×C₈	C₂²×D₇₉	1	4	yes	—
187	C₂×C₄⋊C₄	—	79	16	no	—
188	C₂×C₄⋊C₄	—	79	8	no	—
189	C₂×C₄²	—	79	4	no	—
190	C₂×C₄²	—	79	4	no	—
191	C₂×C₄²	—	158	4	no	—
192	C₂×C₄²	—	158	4	no	—
193	C₂×C₄²	C₄×D₇₉	1	2	yes	—
194	C₂×C₄²	C₄×D₇₉	1	4	yes	—
195	C₂×C₄²	C₂×Dic₇₉	1	4	yes	—
196	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
197	C₄⋊C₈	C₂×Dic₇₉	1	16	yes	complemented
198	C₄×C₈	—	79	16	no	—
199	C₄×C₈	—	158	16	no	—
200	#95	—	79	8	no	—
201	C₂⁴	C₂×C₄×D₇₉	1	1	yes	characteristic
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	79	16	no	—
204	C₂²×C₄	—	79	8	no	—
205	C₂²×C₄	—	158	8	no	—
206	C₂²×C₄	—	158	8	no	—
207	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
208	C₂²×C₄	C₄×Dic₇₉	1	4	yes	—
209	C₂²×C₄	C₈×D₇₉	1	8	yes	—
210	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
211	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₇₉	1	2	yes	—
215	C₂²×C₄	D₄×D₇₉	1	2	yes	—
216	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
217	C₂²×C₄	#11929623	1	2	yes	—
218	C₂²×C₄	#11929625	1	2	yes	—
219	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	79	4	no	—
224	C₂×C₈	—	79	16	no	—
225	C₂×C₈	—	79	4	no	—
226	C₂×C₈	—	79	4	no	—
227	C₂×C₈	—	79	4	no	—
228	C₂×C₈	—	158	2	no	—
229	C₂×C₈	—	158	2	no	—
230	C₂×C₈	—	158	8	no	—
231	C₂×C₈	C₂²×Dic₇₉	1	8	yes	—
232	C₄⋊C₄	—	79	32	no	—
233	C₄²	—	158	4	no	—
234	C₄²	—	158	8	no	—
235	C₄²	—	158	4	no	—
236	C₄²	C₄×Dic₇₉	1	8	yes	—
237	C₄²	C₂×C₄×D₇₉	1	4	yes	—
238	C₄²	C₂²×Dic₇₉	1	4	yes	—
239	C₂³	C₂×C₄×Dic₇₉	1	2	yes	—
240	C₂³	#14579123	1	2	yes	—
241	C₂³	#14579126	1	2	yes	—
242	C₂³	C₂×C₈×D₇₉	1	4	yes	—
243	C₂³	#14579169	1	4	yes	—
244	C₂³	C₂²×C₄×D₇₉	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	79	16	no	—
248	C₂×C₄	—	158	32	no	—
249	C₂×C₄	—	158	8	no	—
250	C₂×C₄	—	158	8	no	—
251	C₂×C₄	C₈×Dic₇₉	1	16	yes	complemented
252	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
253	C₂×C₄	C₂×C₄×Dic₇₉	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₇₉	1	4	yes	—
255	C₂×C₄	C₄⋊C₄×D₇₉	1	4	yes	—
256	C₂×C₄	#14579119	1	4	yes	—
257	C₂×C₄	#14579123	1	2	yes	—
258	C₂×C₄	D₄×Dic₇₉	1	4	yes	—
259	C₂×C₄	#14579126	1	2	yes	—
260	C₂×C₄	Q₈×Dic₇₉	1	4	yes	—
261	C₂×C₄	C₂×C₈×D₇₉	1	8	yes	—
262	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
263	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
264	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
265	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
267	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
268	C₈	—	2	8	no	—
269	C₈	—	79	8	no	—
270	C₈	—	158	4	no	—
271	C₂²	C₄⋊C₄×Dic₇₉	1	4	yes	—
272	C₂²	C₂×C₈×Dic₇₉	1	4	yes	—
273	C₂²	#19098521	1	4	yes	—
274	C₂²	#19098628	1	8	yes	—
275	C₂²	#19098687	1	8	yes	—
276	C₂²	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
277	C₂²	C₂×#14579123	1	1	yes	characteristic
278	C₂²	C₂×#14579126	1	1	yes	characteristic
279	C₂²	C₂²×C₈×D₇₉	1	2	yes	—
280	C₂²	C₂×#14579169	1	2	yes	—
281	C₄	—	2	4	no	—
282	C₄	—	158	16	no	—
283	C₄	C₄⋊C₄×Dic₇₉	1	4	yes	—
284	C₄	C₂×C₈×Dic₇₉	1	8	yes	complemented
285	C₄	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
286	C₄	C₂×#14579119	1	1	yes	characteristic
287	C₄	C₂×D₄×Dic₇₉	1	1	yes	characteristic
288	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
289	C₂	#23763265	1	8	yes	direct product factor
290	C₂	C₂×C₄⋊C₄×Dic₇₉	1	1	yes	characteristic
291	C₂	C₂²×C₈×Dic₇₉	1	1	yes	characteristic
292	C₂	C₂×#19098521	1	1	yes	characteristic
293	C₂	C₂×#19098628	1	2	yes	—
294	C₂	C₂×#19098687	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #23763265	C₂	direct (N × H)
11. C₂×C₄×C₇₉⋊C₈	C₄	semidirect
21. C₂×C₄⋊C₈×C₇₉	C₄	semidirect
32. C₂×C₄⋊Dic₇₉	C₈	semidirect
38. C₄×C₇₉⋊C₈	C₂×C₄	semidirect
52. C₄⋊C₈×C₇₉	C₂×C₄	semidirect
60. C₂×C₇₉⋊C₈	C₄⋊C₄	semidirect
65. C₄⋊Dic₇₉	C₂×C₈	semidirect
99. C₂×C₄×C₇₉	C₄×C₈	semidirect
122. C₇₉⋊C₈	C₂×C₄⋊C₄	semidirect
135. C₄×C₇₉	C₂×C₄×C₈	semidirect
142. C₂×C₇₉	C₈×C₄⋊C₄	semidirect
154. C₇₉	C₂×C₈×C₄⋊C₄	semidirect
161. C₂×C₄⋊C₈	Dic₇₉	semidirect
197. C₄⋊C₈	C₂×Dic₇₉	semidirect
251. C₂×C₄	C₈×Dic₇₉	semidirect
284. C₄	C₂×C₈×Dic₇₉	semidirect
289. C₂	#23763265	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×C₈×Dic₇₉	2
C₂×C₄×C₄⋊Dic₇₉	2
#23763265	2
C₂×#19098317	2
C₂²×C₄⋊C₈×C₇₉	2
C₂×C₈×C₄⋊C₄	79

Subgroups — C2×#23763265 #34286200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23763265 #34286200