Subgroups — #34196921

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098611	C₂	1	2	yes	maximal
2	C₂×#19098334	C₂	1	2	yes	maximal
3	C₂×#19098328	C₂	1	1	yes	maximal, characteristic
4	#23763096	C₂	1	1	yes	maximal, characteristic
5	C₂²×#108×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	1	yes	characteristic
7	C₂²×#11929608	C₂²	1	2	yes	—
8	C₂×#14579120	C₂²	1	1	yes	characteristic
9	#19098611	—	2	4	no	—
10	#19098611	—	2	4	no	—
11	C₂×#14579096	—	2	4	no	—
12	#19098334	—	2	4	no	—
13	#19098328	C₄	1	4	yes	complemented
14	#19098317	—	2	2	no	—
15	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
16	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#108×C₇₉	—	2	4	no	—
18	C₂×#108×C₇₉	—	2	2	no	—
19	C₂×#11929608	—	2	2	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	—	2	1	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	—	2	2	no	—
25	C₂×#11929608	C₂×C₄	1	4	yes	—
26	C₂×#11929608	D₄	1	2	yes	—
27	C₂×#11929608	D₄	1	4	yes	—
28	C₂×#11929608	Q₈	1	4	yes	—
29	C₂²×C₇₉⋊C₈	—	2	2	no	—
30	C₂²×C₇₉⋊C₈	—	2	1	no	—
31	#14579120	—	2	2	no	—
32	#14579120	—	2	2	no	—
33	C₂×C₄⋊Dic₇₉	—	2	1	no	—
34	C₂×C₂²⋊Dic₇₉	Q₈	1	2	yes	—
35	C₂×C₄×Dic₇₉	—	2	1	no	—
36	#14579096	—	2	16	no	—
37	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₉	—	2	2	no	—
39	C₂×M₁₆×C₇₉	—	2	2	no	—
40	C₂×M₁₆×C₇₉	—	2	2	no	—
41	C₂×M₁₆×C₇₉	—	2	2	no	—
42	C₂×M₁₆×C₇₉	—	2	2	no	—
43	C₂×M₁₆×C₇₉	D₄	1	4	yes	—
44	C₂²×C₈×C₇₉	—	2	2	no	—
45	C₂²×C₈×C₇₉	—	2	2	no	—
46	C₂²×C₈×C₇₉	—	2	2	no	—
47	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₇₉	D₄	1	2	yes	—
49	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
50	#108×C₇₉	—	2	8	no	—
51	C₂²×Dic₇₉	—	2	2	no	—
52	#11929608	—	2	4	no	—
53	#11929608	—	2	4	no	—
54	#11929608	—	2	4	no	—
55	#11929608	—	2	4	no	—
56	#11929608	—	4	2	no	—
57	#11929608	C₄⋊C₄	1	8	yes	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	2	2	no	—
61	C₂×C₇₉⋊C₈	—	2	4	no	—
62	C₂×C₇₉⋊C₈	—	4	1	no	—
63	C₂×C₇₉⋊C₈	—	4	1	no	—
64	C₄⋊Dic₇₉	—	4	2	no	—
65	C₂²⋊Dic₇₉	—	2	4	no	—
66	C₄×Dic₇₉	—	4	1	no	—
67	C₄×Dic₇₉	—	4	1	no	—
68	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	#41	1	2	yes	—
77	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
79	M₁₆×C₇₉	—	2	4	no	—
80	M₁₆×C₇₉	—	2	4	no	—
81	M₁₆×C₇₉	—	4	2	no	—
82	M₁₆×C₇₉	—	4	2	no	—
83	C₂×C₈×C₇₉	—	2	4	no	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₂×C₈×C₇₉	—	2	4	no	—
86	C₂×C₈×C₇₉	—	2	4	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	4	2	no	—
89	C₂×C₈×C₇₉	—	4	2	no	—
90	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	—
91	C₂³×C₇₉	—	4	2	no	—
92	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#120	1	2	yes	—
102	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
103	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#121	1	2	yes	—
105	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#123	1	2	yes	—
110	C₂×C₄×C₇₉	#125	1	2	yes	—
111	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
115	C₂×Dic₇₉	—	4	4	no	—
116	C₂×Dic₇₉	—	4	2	no	—
117	C₈×C₇₉	—	2	4	no	—
118	C₈×C₇₉	—	4	4	no	—
119	C₇₉⋊C₈	—	2	8	no	—
120	C₇₉⋊C₈	—	4	2	no	—
121	C₂²×C₇₉	—	2	2	no	—
122	C₂²×C₇₉	—	2	2	no	—
123	C₂²×C₇₉	—	2	2	no	—
124	C₂²×C₇₉	—	4	2	no	—
125	C₂²×C₇₉	—	4	2	no	—
126	C₂²×C₇₉	#389	1	2	yes	—
127	C₂²×C₇₉	#389	1	1	yes	characteristic
128	C₂²×C₇₉	#395	1	1	yes	characteristic
129	C₂²×C₇₉	#395	1	2	yes	—
130	C₂²×C₇₉	#399	1	1	yes	characteristic
131	C₄×C₇₉	#384	1	1	yes	characteristic
132	C₄×C₇₉	#385	1	2	yes	—
133	C₄×C₇₉	#385	1	1	yes	characteristic
134	C₄×C₇₉	#392	1	1	yes	characteristic
135	C₄×C₇₉	#398	1	1	yes	characteristic
136	C₄×C₇₉	#398	1	2	yes	—
137	Dic₇₉	—	8	2	no	—
138	#11332	—	79	1	no	maximal, Sylow
139	C₂×C₇₉	—	2	1	no	—
140	C₂×C₇₉	—	2	2	no	—
141	C₂×C₇₉	—	2	1	no	—
142	C₂×C₇₉	—	2	2	no	—
143	C₂×C₇₉	#1444	1	1	yes	characteristic
144	C₂×C₇₉	#1953	1	2	yes	—
145	C₂×#430	—	79	2	no	—
146	C₂²×#108	D₇₉	1	1	yes	characteristic
147	C₂×#344	—	79	2	no	—
148	C₂×#341	—	79	1	no	—
149	#1729	—	79	1	no	—
150	C₇₉	#11332	1	1	yes	characteristic, complemented, Sylow
151	C₂²×M₁₆	—	79	2	no	—
152	C₂²×M₁₆	—	79	1	no	—
153	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
154	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
155	C₂×#117	—	79	1	no	—
156	#430	—	158	4	no	—
157	#430	—	158	4	no	—
158	C₂×#108	—	2	4	no	—
159	C₂×#108	—	2	2	no	—
160	C₂×#108	—	158	4	no	—
161	#344	—	158	4	no	—
162	#341	—	79	4	no	—
163	#336	—	158	2	no	—
164	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	79	4	no	—
171	C₂×M₁₆	—	79	4	no	—
172	C₂×M₁₆	—	79	4	no	—
173	C₂×M₁₆	—	79	2	no	—
174	C₂×M₁₆	—	158	2	no	—
175	C₂×M₁₆	—	158	2	no	—
176	C₂×M₁₆	—	158	1	no	—
177	C₂×M₁₆	—	158	2	no	—
178	C₂×M₁₆	—	158	2	no	—
179	C₂×M₁₆	—	158	2	no	—
180	C₂×M₁₆	C₂²⋊D₇₉	1	4	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	158	1	no	—
185	C₂²×C₈	—	158	2	no	—
186	C₂²×C₈	Dic₁₅₈	1	2	yes	—
187	C₂²×C₈	C₄×D₇₉	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
189	#117	—	158	2	no	—
190	#117	—	158	2	no	—
191	C₂×C₄⋊C₄	—	158	1	no	—
192	C₂×C₂²⋊C₄	—	79	2	no	—
193	C₂×C₄²	—	158	1	no	—
194	#108	—	2	8	no	—
195	#108	—	158	16	no	—
196	C₂⁴	Q₈×D₇₉	1	1	yes	characteristic
197	C₂²×C₄	—	158	2	no	—
198	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₇₉	1	2	yes	—
200	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
203	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
204	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
205	C₂²×C₄	#11929623	1	1	yes	characteristic
206	C₂²×C₄	#11929624	1	2	yes	—
207	C₂²×C₄	#11929625	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	316	2	no	—
213	M₁₆	—	79	8	no	—
214	M₁₆	—	158	4	no	—
215	M₁₆	—	158	4	no	—
216	M₁₆	—	158	4	no	—
217	M₁₆	—	158	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	316	1	no	—
226	C₂×C₈	—	316	1	no	—
227	C₂×C₈	—	158	2	no	—
228	C₂×C₈	—	158	4	no	—
229	C₂×C₈	—	158	4	no	—
230	C₂×C₈	—	158	4	no	—
231	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	—
232	C₄⋊C₄	—	316	2	no	—
233	C₂²⋊C₄	—	158	4	no	—
234	C₄²	—	316	1	no	—
235	C₄²	—	316	1	no	—
236	C₂³	—	4	2	no	—
237	C₂³	#14579126	1	1	yes	characteristic
238	C₂³	#14579137	1	1	yes	characteristic
239	C₂³	#14579139	1	2	yes	—
240	C₂³	#14579159	1	2	yes	—
241	C₂³	#14579160	1	1	yes	characteristic
242	C₂×C₄	—	316	4	no	—
243	C₂×C₄	—	316	2	no	—
244	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
245	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
246	C₂×C₄	C₄×C₂²⋊D₇₉	1	2	yes	—
247	C₂×C₄	#14579123	1	1	yes	characteristic
248	C₂×C₄	#14579129	1	2	yes	—
249	C₂×C₄	#14579130	1	1	yes	characteristic
250	C₂×C₄	#14579132	1	1	yes	characteristic
251	C₂×C₄	#14579133	1	2	yes	—
252	C₂×C₄	#14579137	1	2	yes	—
253	C₂×C₄	#14579139	1	2	yes	—
254	C₂×C₄	#14579140	1	2	yes	—
255	C₂×C₄	#14579144	1	1	yes	characteristic
256	C₂×C₄	#14579146	1	2	yes	—
257	C₂×C₄	#14579151	1	2	yes	—
258	C₂×C₄	#14579152	1	1	yes	characteristic
259	C₂×C₄	#14579157	1	1	yes	characteristic
260	C₂×C₄	#14579159	1	1	yes	characteristic
261	C₂×C₄	#14579160	1	2	yes	—
262	C₈	—	2	4	no	—
263	C₈	—	4	4	no	—
264	C₈	—	316	2	no	—
265	C₈	—	158	8	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	#19098458	1	2	yes	—
272	C₂²	#19098459	1	1	yes	characteristic
273	C₂²	#19098482	1	1	yes	characteristic
274	C₂²	#19098483	1	2	yes	—
275	C₂²	#19098498	1	1	yes	characteristic
276	C₄	—	632	2	no	—
277	C₄	#19098432	1	1	yes	characteristic
278	C₄	#19098434	1	2	yes	—
279	C₄	#19098436	1	1	yes	characteristic
280	C₄	#19098471	1	1	yes	characteristic
281	C₄	#19098494	1	1	yes	characteristic
282	C₄	#19098496	1	2	yes	—
283	C₂	—	2	1	no	—
284	C₂	—	2	2	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	#23761663	1	1	yes	characteristic
288	C₂	#23764188	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579132
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂²×#108×C₇₉	C₂
socle	C₂²×C₇₉	#389
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂²×#108	D₇₉
hypocenter	C₇₉	#11332
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
13. #19098328	C₄	semidirect
150. C₇₉	#11332	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098611	2
C₂×#19098334	2
C₂×#19098328	2
#23763096	2
C₂²×#108×C₇₉	2
#11332	79