Subgroups — #34204220

Order 20224.

482 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11929607	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098384	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098344	C₂	1	2	yes	maximal
4	#23761479	C₂	1	2	yes	maximal, complemented
5	C₂×#364×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11929607	C₄	1	1	yes	characteristic
7	C₂²×#11929607	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₇₉	C₄	1	1	yes	characteristic
9	C₂×#14579101	C₂²	1	2	yes	—
10	C₄×#11929607	—	2	2	no	—
11	C₄×#11929607	—	2	2	no	—
12	C₂×#14579071	C₄	1	1	yes	characteristic
13	C₂×#14579071	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇₉⋊C₈	C₄	1	1	yes	characteristic
15	#19098384	—	2	2	no	—
16	#19098344	—	2	4	no	—
17	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#364×C₇₉	—	2	2	no	—
20	C₂²×C₄×D₇₉	—	2	1	no	—
21	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×#11929607	—	2	2	no	—
23	C₂×#11929607	—	2	2	no	—
24	C₂×#11929607	—	2	1	no	—
25	C₂×#11929607	—	2	1	no	—
26	C₂×#11929607	—	2	1	no	—
27	C₂×#11929607	—	2	1	no	—
28	C₂×#11929607	C₈	1	4	yes	—
29	C₂×#11929607	D₄	1	2	yes	—
30	C₂×#11929607	Q₈	1	2	yes	—
31	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₇₉	—	2	1	no	—
34	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₇₉	—	2	2	no	—
36	C₄²×D₇₉	—	2	2	no	—
37	#14579101	—	2	2	no	—
38	#14579101	—	2	2	no	—
39	C₂×C₇₉⋊C₁₆	—	2	2	no	—
40	#14579071	—	2	2	no	—
41	#14579071	—	2	1	no	—
42	#14579071	—	2	1	no	—
43	C₄×C₇₉⋊C₈	—	2	2	no	—
44	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₇₉	—	2	2	no	—
48	M₃₂×C₇₉	—	2	2	no	—
49	C₂×C₁₆×C₇₉	—	2	2	no	—
50	C₄×C₈×C₇₉	D₄	1	2	yes	—
51	C₄×C₈×C₇₉	D₄	1	2	yes	—
52	C₂³×D₇₉	M₁₆	1	1	yes	characteristic
53	C₂²×Dic₇₉	—	2	1	no	—
54	C₂²×Dic₇₉	M₁₆	1	1	yes	characteristic
55	C₂×C₄×D₇₉	—	2	4	no	—
56	C₂×C₄×D₇₉	—	2	1	no	—
57	C₂×C₄×D₇₉	—	2	1	no	—
58	C₂×C₄×D₇₉	—	2	1	no	—
59	C₂×C₄×D₇₉	—	2	1	no	—
60	C₂×C₄×D₇₉	—	4	4	no	—
61	C₂×C₄×D₇₉	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₇₉	C₂×C₈	1	2	yes	—
63	#11929607	—	2	8	no	—
64	#11929607	—	2	2	no	—
65	#11929607	—	2	2	no	—
66	#11929607	—	2	2	no	—
67	#11929607	—	2	2	no	—
68	C₂×C₇₉⋊C₈	—	2	1	no	—
69	C₂×C₇₉⋊C₈	—	2	1	no	—
70	C₂×C₇₉⋊C₈	—	2	1	no	—
71	C₂×C₇₉⋊C₈	—	2	1	no	—
72	C₂×C₇₉⋊C₈	C₄⋊C₄	1	2	yes	—
73	C₂×C₇₉⋊C₈	C₂×C₈	1	2	yes	—
74	C₄×Dic₇₉	—	2	4	no	—
75	C₄×Dic₇₉	—	2	1	no	—
76	C₄×Dic₇₉	M₁₆	1	2	yes	—
77	C₇₉⋊C₁₆	—	4	2	no	—
78	C₂²×C₄×C₇₉	—	2	1	no	—
79	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₇₉	C₂×C₈	1	2	yes	—
82	C₂×C₈×C₇₉	M₁₆	1	2	yes	—
83	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
88	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
89	C₄²×C₇₉	—	2	1	no	—
90	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
91	C₁₆×C₇₉	—	4	2	no	—
92	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
93	C₂²×D₇₉	—	2	1	no	—
94	C₂²×D₇₉	—	2	1	no	—
95	C₂²×D₇₉	—	4	2	no	—
96	C₂²×D₇₉	C₄⋊C₈	1	2	yes	—
97	C₂×C₄×C₇₉	—	2	4	no	—
98	C₂×C₄×C₇₉	—	2	2	no	—
99	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
105	C₂×Dic₇₉	—	2	1	no	—
106	C₂×Dic₇₉	—	2	1	no	—
107	C₂×Dic₇₉	—	2	2	no	—
108	C₂×Dic₇₉	—	4	2	no	—
109	C₂×Dic₇₉	C₄⋊C₈	1	2	yes	—
110	C₄×D₇₉	—	2	2	no	—
111	C₄×D₇₉	—	2	2	no	—
112	C₄×D₇₉	—	2	2	no	—
113	C₄×D₇₉	—	2	2	no	—
114	C₄×D₇₉	—	4	8	no	—
115	C₈×C₇₉	C₂²⋊C₈	1	4	yes	—
116	C₈×C₇₉	#121	1	1	yes	characteristic
117	C₈×C₇₉	#121	1	1	yes	characteristic
118	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₇₉⋊C₈	—	2	2	no	—
121	C₇₉⋊C₈	—	2	2	no	—
122	C₂²×C₇₉	—	2	2	no	—
123	C₂²×C₇₉	#407	1	1	yes	characteristic
124	C₂²×C₇₉	C₂×C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₇₉	#432	1	1	yes	characteristic
126	C₂×D₇₉	—	2	2	no	—
127	C₂×D₇₉	—	4	2	no	—
128	C₂×D₇₉	—	4	2	no	—
129	C₂×D₇₉	—	4	2	no	—
130	C₄×C₇₉	—	2	4	no	—
131	C₄×C₇₉	#380	1	1	yes	characteristic
132	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₄×C₇₉	#423	1	1	yes	characteristic
134	C₄×C₇₉	#431	1	1	yes	characteristic
135	Dic₇₉	—	2	2	no	—
136	Dic₇₉	—	4	2	no	—
137	#12791	—	79	1	no	maximal, Sylow
138	C₂×C₇₉	—	2	1	no	—
139	C₂×C₇₉	—	2	1	no	—
140	C₂×C₇₉	#1797	1	1	yes	characteristic
141	C₂×C₇₉	#2165	1	2	yes	—
142	D₇₉	—	4	2	no	—
143	C₂×C₄×M₁₆	—	79	1	no	—
144	C₂×#364	—	79	1	no	—
145	C₂×#364	D₇₉	1	1	yes	characteristic, complemented
146	C₂×#349	—	79	2	no	—
147	#1375	—	79	2	no	—
148	C₇₉	#12791	1	1	yes	characteristic, complemented, Sylow
149	C₂²×M₁₆	—	79	1	no	—
150	C₂²×M₁₆	—	79	1	no	—
151	C₂²×C₄²	—	79	1	no	—
152	C₂×M₃₂	—	79	2	no	—
153	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
154	C₄×M₁₆	—	158	2	no	—
155	C₄×M₁₆	—	158	2	no	—
156	C₂×#97	—	79	1	no	—
157	C₂×#97	—	79	1	no	—
158	C₂×C₄×C₈	—	79	1	no	—
159	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
160	#364	—	2	2	no	—
161	#364	—	158	2	no	—
162	#349	—	158	4	no	—
163	C₂³×C₄	—	79	1	no	—
164	C₂³×C₄	—	158	1	no	—
165	C₂×M₁₆	—	79	4	no	—
166	C₂×M₁₆	—	79	2	no	—
167	C₂×M₁₆	—	79	2	no	—
168	C₂×M₁₆	—	158	2	no	—
169	C₂×M₁₆	—	158	2	no	—
170	C₂×M₁₆	—	158	1	no	—
171	C₂×M₁₆	—	158	1	no	—
172	C₂×M₁₆	—	158	1	no	—
173	C₂×M₁₆	—	158	1	no	—
174	C₂²×C₈	—	79	1	no	—
175	C₂²×C₈	—	79	1	no	—
176	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
178	C₂×C₄²	—	79	1	no	—
179	C₂×C₄²	—	158	2	no	—
180	C₂×C₄²	—	158	1	no	—
181	C₂×C₄²	—	158	2	no	—
182	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	158	2	no	—
186	M₃₂	—	158	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	158	2	no	—
189	#97	—	158	2	no	—
190	#97	—	158	1	no	—
191	#97	—	158	1	no	—
192	C₄×C₈	—	158	2	no	—
193	C₄×C₈	D₃₁₆	1	2	yes	—
194	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
195	C₂⁴	—	79	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	316	4	no	—
198	C₂²×C₄	—	79	1	no	—
199	C₂²×C₄	—	79	2	no	—
200	C₂²×C₄	—	79	2	no	—
201	C₂²×C₄	—	158	4	no	—
202	C₂²×C₄	—	158	1	no	—
203	C₂²×C₄	—	158	1	no	—
204	C₂²×C₄	—	158	1	no	—
205	C₂²×C₄	—	158	1	no	—
206	C₂²×C₄	—	158	1	no	—
207	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
208	M₁₆	—	158	8	no	—
209	M₁₆	—	158	2	no	—
210	M₁₆	—	158	2	no	—
211	M₁₆	—	158	2	no	—
212	M₁₆	—	158	2	no	—
213	C₂×C₈	—	79	2	no	—
214	C₂×C₈	—	79	2	no	—
215	C₂×C₈	—	158	1	no	—
216	C₂×C₈	—	158	1	no	—
217	C₂×C₈	—	158	1	no	—
218	C₂×C₈	—	158	1	no	—
219	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
220	C₂×C₈	C₈×D₇₉	1	2	yes	—
221	C₂×C₈	#11929607	1	2	yes	—
222	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
223	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
224	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
225	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
226	C₂×C₈	#11929623	1	1	yes	characteristic
227	C₂×C₈	#11929625	1	1	yes	characteristic
228	C₄²	—	2	1	no	—
229	C₄²	—	79	2	no	—
230	C₄²	—	158	4	no	—
231	C₄²	—	158	1	no	—
232	C₄²	(C₂×D₇₉)⋊C₄	1	2	yes	—
233	C₁₆	—	4	2	no	—
234	C₁₆	—	316	2	no	—
235	C₂³	—	316	2	no	—
236	C₂³	—	79	2	no	—
237	C₂³	—	158	1	no	—
238	C₂³	—	158	1	no	—
239	C₂³	M₁₆×D₇₉	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	316	8	no	—
243	C₂×C₄	—	316	2	no	—
244	C₂×C₄	—	79	2	no	—
245	C₂×C₄	—	158	2	no	—
246	C₂×C₄	—	158	1	no	—
247	C₂×C₄	—	158	1	no	—
248	C₂×C₄	—	158	2	no	—
249	C₂×C₄	—	158	2	no	—
250	C₂×C₄	—	158	2	no	—
251	C₂×C₄	—	158	2	no	—
252	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
254	C₂×C₄	#14579119	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₇₉	1	1	yes	characteristic
256	C₂×C₄	M₁₆×D₇₉	1	1	yes	characteristic
257	C₂×C₄	C₂×#11929607	1	1	yes	characteristic
258	C₈	—	158	2	no	—
259	C₈	—	158	2	no	—
260	C₈	(C₂×D₇₉)⋊C₈	1	4	yes	—
261	C₈	#14579132	1	1	yes	characteristic
262	C₈	#14579136	1	1	yes	characteristic
263	C₈	#14579137	1	1	yes	characteristic
264	C₈	#14579141	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	316	2	no	—
267	C₂²	—	316	2	no	—
268	C₂²	—	316	2	no	—
269	C₂²	—	158	2	no	—
270	C₂²	#19098526	1	1	yes	characteristic
271	C₂²	C₄⋊C₈×D₇₉	1	1	yes	characteristic
272	C₂²	#19098619	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	316	2	no	—
275	C₄	—	158	2	no	—
276	C₄	#19098420	1	1	yes	characteristic
277	C₄	C₂×(C₂×D₇₉)⋊C₈	1	1	yes	characteristic
278	C₄	#19098585	1	1	yes	characteristic
279	C₄	#19098615	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	316	2	no	—
283	C₂	#23763377	1	1	yes	characteristic
284	C₂	#23765211	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579119
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₂×#364×C₇₉	C₂
socle	C₂²×C₇₉	#432
prime radical	C₂³×D₇₉	M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#364	D₇₉
hypocenter	C₇₉	#12791
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#19098384	C₂	semidirect
4. #23761479	C₂	semidirect
5. C₂×#364×C₇₉	C₂	semidirect
145. C₂×#364	D₇₉	semidirect
148. C₇₉	#12791	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11929607	2
C₂×#19098384	2
C₂×#19098344	2
#23761479	2
C₂×#364×C₇₉	2
#12791	79