Subgroups — #34288739

Order 20224.

954 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098612	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14579096	C₂	1	2	yes	maximal
3	#23771963	C₂	1	2	yes	maximal
4	C₂×#19098521	C₂	1	1	yes	maximal, characteristic
5	#23763573	C₂	1	4	yes	maximal, complemented
6	#23763570	C₂	1	4	yes	maximal, complemented
7	C₂×#430×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11929608	C₂²	1	2	yes	—
9	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#14579120	C₂²	1	1	yes	characteristic
11	#19098612	—	2	4	no	—
12	C₂×#14579097	C₂²	1	4	yes	—
13	C₂×#14579096	—	2	8	no	—
14	C₂×#14579096	C₂²	1	4	yes	—
15	C₂×#14579096	C₂²	1	4	yes	—
16	#19098591	—	2	4	no	—
17	#19098589	—	2	4	no	—
18	#19098586	C₂²	1	8	yes	—
19	#19098521	C₄	1	8	yes	—
20	C₂×#14579071	C₂²	1	2	yes	—
21	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
22	C₂²×M₁₆×C₇₉	C₂²	1	2	yes	—
23	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	#430×C₇₉	—	2	4	no	—
25	C₂×#108×C₇₉	C₂²	1	4	yes	—
26	C₂×#11929608	—	2	2	no	—
27	C₂×#11929608	—	2	4	no	—
28	C₂×#11929608	—	2	4	no	—
29	C₂×#11929608	C₂³	1	4	yes	—
30	C₂²×C₇₉⋊C₈	—	2	2	no	—
31	C₂²×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
33	C₂²×C₇₉⋊C₈	Q₈	1	4	yes	—
34	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
35	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
36	#14579120	C₂×C₄	1	4	yes	—
37	#14579120	D₄	1	4	yes	—
38	C₂×C₄⋊Dic₇₉	Q₈	1	2	yes	—
39	C₂×C₂²⋊Dic₇₉	Q₈	1	2	yes	—
40	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
41	#14579097	—	2	8	no	—
42	#14579096	—	2	8	no	—
43	#14579096	—	2	8	no	—
44	Dic₇₉⋊C₈	—	2	16	no	—
45	#14579071	C₂×C₄	1	8	yes	—
46	C₈×Dic₇₉	C₂×C₄	1	8	yes	—
47	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₇₉	—	2	2	no	—
49	C₂×M₁₆×C₇₉	—	2	4	no	—
50	C₂×M₁₆×C₇₉	—	2	4	no	—
51	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₇₉	D₄	1	4	yes	—
53	C₂×M₁₆×C₇₉	Q₈	1	4	yes	—
54	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
55	C₂²×C₈×C₇₉	—	2	2	no	—
56	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
57	#108×C₇₉	—	2	8	no	—
58	C₂²×Dic₇₉	C₂×Q₈	1	4	yes	—
59	#11929608	—	2	4	no	—
60	#11929608	—	2	4	no	—
61	#11929608	—	4	4	no	—
62	C₂×C₇₉⋊C₈	—	2	4	no	—
63	C₂×C₇₉⋊C₈	—	4	4	no	—
64	C₂×C₇₉⋊C₈	C₄⋊C₄	1	16	yes	—
65	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
67	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
68	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
69	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
70	C₂×C₇₉⋊C₈	C₂×Q₈	1	2	yes	—
71	C₄⋊Dic₇₉	C₄⋊C₄	1	8	yes	—
72	C₂²⋊Dic₇₉	C₄⋊C₄	1	8	yes	—
73	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
74	C₄×Dic₇₉	C₂×D₄	1	4	yes	—
75	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₇₉	—	2	4	no	—
85	M₁₆×C₇₉	—	2	4	no	—
86	M₁₆×C₇₉	—	4	4	no	—
87	M₁₆×C₇₉	C₄⋊C₄	1	16	yes	—
88	C₂×C₈×C₇₉	—	2	4	no	—
89	C₂×C₈×C₇₉	—	4	4	no	—
90	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
93	C₂³×C₇₉	—	4	2	no	—
94	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
99	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₄×C₇₉	#127	1	4	yes	—
102	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₇₉	C₄⋊Q₈	1	4	yes	—
104	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
105	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
111	C₂×Dic₇₉	—	4	4	no	—
112	C₂×Dic₇₉	C₂×C₄⋊C₄	1	8	yes	—
113	C₈×C₇₉	—	4	4	no	—
114	C₈×C₇₉	C₂×C₄⋊C₄	1	8	yes	—
115	C₇₉⋊C₈	—	4	4	no	—
116	C₇₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
117	C₇₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
118	C₂²×C₇₉	—	2	4	no	—
119	C₂²×C₇₉	—	2	2	no	—
120	C₂²×C₇₉	—	4	2	no	—
121	C₂²×C₇₉	—	4	2	no	—
122	C₂²×C₇₉	#384	1	4	yes	—
123	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
124	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
126	C₄×C₇₉	#384	1	4	yes	—
127	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
128	C₄×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
130	C₄×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
131	Dic₇₉	—	2	8	no	—
132	#20807	—	79	1	no	maximal, Sylow
133	C₂×C₇₉	—	2	1	no	—
134	C₂×C₇₉	—	2	1	no	—
135	C₂×C₇₉	—	2	4	no	—
136	C₂×C₇₉	C₂×#384	1	1	yes	characteristic
137	C₂×C₇₉	#2916	1	2	yes	—
138	C₂×#430	—	79	1	no	—
139	C₂×#430	D₇₉	1	1	yes	characteristic
140	C₂²×#108	—	79	2	no	—
141	#2905	—	79	2	no	—
142	C₂×#405	—	79	1	no	—
143	#1849	—	79	4	no	—
144	#1848	—	79	4	no	—
145	C₇₉	#20807	1	1	yes	characteristic, complemented, Sylow
146	C₂²×M₁₆	—	79	2	no	—
147	C₂²×M₁₆	C₂×D₇₉	1	2	yes	—
148	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
149	C₂³×C₈	—	79	1	no	—
150	C₂×#117	—	79	1	no	—
151	#430	—	2	4	no	—
152	#430	—	158	4	no	—
153	C₂×#108	—	79	4	no	—
154	C₂×#108	—	79	4	no	—
155	C₂×#108	—	79	4	no	—
156	C₂×#108	—	158	8	no	—
157	C₂×#108	C₂×D₇₉	1	4	yes	—
158	#424	—	158	4	no	—
159	#424	—	158	4	no	—
160	#423	—	79	8	no	—
161	#405	—	79	8	no	—
162	C₂×#97	—	79	2	no	—
163	C₂×C₄×C₈	—	79	2	no	—
164	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	4	no	—
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	79	4	no	—
169	C₂×M₁₆	—	158	4	no	—
170	C₂×M₁₆	—	158	4	no	—
171	C₂×M₁₆	—	158	2	no	—
172	C₂×M₁₆	Dic₁₅₈	1	4	yes	—
173	C₂×M₁₆	C₄×D₇₉	1	4	yes	—
174	C₂×M₁₆	C₂²⋊D₇₉	1	4	yes	—
175	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	79	4	no	—
178	C₂²×C₈	—	79	4	no	—
179	C₂²×C₈	—	79	4	no	—
180	C₂²×C₈	—	79	1	no	—
181	C₂²×C₈	—	79	1	no	—
182	C₂²×C₈	—	158	2	no	—
183	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
184	#117	—	79	4	no	—
185	#117	—	79	4	no	—
186	C₂×C₄⋊C₄	—	79	2	no	—
187	C₂×C₂²⋊C₄	—	79	2	no	—
188	C₂×C₄²	—	79	2	no	—
189	#108	—	2	8	no	—
190	#108	—	158	8	no	—
191	#108	—	158	8	no	—
192	#108	—	158	8	no	—
193	C₄⋊C₈	—	158	16	no	—
194	#97	—	79	8	no	—
195	C₄×C₈	—	79	8	no	—
196	C₂⁴	Q₈×D₇₉	1	1	yes	characteristic
197	C₂²×C₄	—	79	4	no	—
198	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
199	C₂²×C₄	D₄×D₇₉	1	2	yes	—
200	C₂²×C₄	D₄×D₇₉	1	2	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
202	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
203	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
205	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
206	M₁₆	—	2	4	no	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	4	no	—
209	M₁₆	—	316	4	no	—
210	M₁₆	—	158	4	no	—
211	M₁₆	—	158	4	no	—
212	M₁₆	Dic₇₉⋊C₄	1	16	yes	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	4	no	—
215	C₂×C₈	—	316	4	no	—
216	C₂×C₈	—	79	2	no	—
217	C₂×C₈	—	79	2	no	—
218	C₂×C₈	—	79	16	no	—
219	C₂×C₈	—	79	2	no	—
220	C₂×C₈	—	79	2	no	—
221	C₂×C₈	—	79	2	no	—
222	C₂×C₈	—	79	2	no	—
223	C₂×C₈	—	158	4	no	—
224	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
225	C₂×C₈	C₂×C₂²⋊D₇₉	1	4	yes	—
226	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
227	C₄⋊C₄	—	79	8	no	—
228	C₂²⋊C₄	—	79	8	no	—
229	C₄²	—	79	4	no	—
230	C₄²	—	79	4	no	—
231	C₂³	—	4	2	no	—
232	C₂³	C₄⋊C₄×D₇₉	1	2	yes	—
233	C₂³	#14579159	1	2	yes	—
234	C₂³	#14579160	1	2	yes	—
235	C₂³	C₂×Q₈×D₇₉	1	1	yes	characteristic
236	C₂×C₄	—	316	4	no	—
237	C₂×C₄	—	79	8	no	—
238	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
239	C₂×C₄	C₄⋊C₄×D₇₉	1	4	yes	—
240	C₂×C₄	C₂×Dic₇₉⋊C₄	1	4	yes	—
241	C₂×C₄	#14579157	1	4	yes	—
242	C₂×C₄	#14579159	1	2	yes	—
243	C₂×C₄	#14579160	1	2	yes	—
244	C₂×C₄	#14579160	1	4	yes	—
245	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
247	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
250	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
251	C₈	—	4	4	no	—
252	C₈	—	316	4	no	—
253	C₈	—	79	4	no	—
254	C₈	—	79	4	no	—
255	C₈	C₂×Dic₇₉⋊C₄	1	8	yes	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	#19098432	1	4	yes	—
261	C₂²	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
262	C₂²	C₂×#14579159	1	1	yes	characteristic
263	C₂²	C₂×#14579160	1	1	yes	characteristic
264	C₄	—	158	8	no	—
265	C₄	#19098432	1	4	yes	—
266	C₄	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
267	C₄	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic
268	C₄	C₂×#14579157	1	1	yes	characteristic
269	C₄	C₂×#14579160	1	1	yes	characteristic
270	C₂	—	2	4	no	—
271	C₂	—	2	1	no	—
272	C₂	—	2	1	no	—
273	C₂	C₂×#19098432	1	1	yes	characteristic
274	C₂	#23772010	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#430×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#430	D₇₉
hypocenter	C₇₉	#20807
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
5. #23763573	C₂	semidirect
6. #23763570	C₂	semidirect
145. C₇₉	#20807	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098612	2
C₂²×#14579096	2
#23771963	2
C₂×#19098521	2
#23763573	2
#23763570	2
C₂×#430×C₇₉	2
#20807	79