Subgroups — #34196944

Order 20224.

790 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771990	C₂	1	1	yes	maximal, characteristic
2	#23765883	C₂	1	2	yes	maximal
3	#23763111	C₂	1	2	yes	maximal
4	C₂×#19098320	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579120	C₂²	1	2	yes	—
7	C₂×#14579120	C₂²	1	2	yes	—
8	#19098605	—	2	4	no	—
9	#19098601	—	2	2	no	—
10	#19098461	—	2	4	no	—
11	#19098436	—	2	4	no	—
12	C₄×Dic₇₉⋊C₄	—	2	4	no	—
13	#19098324	—	2	4	no	—
14	#19098320	C₄	1	4	yes	complemented
15	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
18	#336×C₇₉	—	2	2	no	—
19	#14579120	—	2	2	no	—
20	#14579120	—	2	2	no	—
21	#14579120	—	2	2	no	—
22	#14579120	—	2	2	no	—
23	#14579120	—	2	2	no	—
24	#14579120	—	2	2	no	—
25	#14579120	C₂×C₄	1	4	yes	—
26	#14579120	Q₈	1	4	yes	—
27	C₂×C₄⋊Dic₇₉	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	D₄	1	4	yes	—
29	C₂×Dic₇₉⋊C₄	—	2	8	no	—
30	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
31	C₂×C₂²⋊Dic₇₉	—	2	2	no	—
32	C₂×C₄×Dic₇₉	—	2	2	no	—
33	C₂×C₄×Dic₇₉	D₄	1	4	yes	—
34	#14579095	—	2	4	no	—
35	#14579093	—	2	4	no	—
36	C₇₉⋊#95	—	2	8	no	—
37	C₂³×C₄×C₇₉	D₄	1	2	yes	—
38	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₇₉	—	2	2	no	—
40	C₂×M₁₆×C₇₉	—	2	2	no	—
41	C₂×M₁₆×C₇₉	—	2	1	no	—
42	C₂×M₁₆×C₇₉	—	2	2	no	—
43	C₂×M₁₆×C₇₉	—	2	2	no	—
44	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
46	C₂×M₁₆×C₇₉	Q₈	1	2	yes	—
47	C₂²×C₈×C₇₉	—	2	1	no	—
48	C₂²×C₈×C₇₉	D₄	1	2	yes	—
49	C₂×C₄²×C₇₉	—	2	2	no	—
50	C₂×C₄²×C₇₉	—	2	2	no	—
51	C₂×C₄²×C₇₉	—	2	1	no	—
52	C₂×C₄²×C₇₉	—	2	1	no	—
53	C₂²×Dic₇₉	—	2	4	no	—
54	C₂²×Dic₇₉	—	2	4	no	—
55	C₄⋊Dic₇₉	—	2	4	no	—
56	C₄⋊Dic₇₉	—	2	4	no	—
57	C₄⋊Dic₇₉	—	2	4	no	—
58	C₄⋊Dic₇₉	—	2	4	no	—
59	Dic₇₉⋊C₄	—	4	16	no	—
60	C₂²⋊Dic₇₉	—	2	4	no	—
61	C₂²⋊Dic₇₉	—	2	4	no	—
62	C₂²⋊Dic₇₉	—	4	2	no	—
63	C₂²⋊Dic₇₉	—	4	2	no	—
64	C₄×Dic₇₉	—	2	4	no	—
65	C₄×Dic₇₉	—	4	2	no	—
66	C₄×Dic₇₉	—	4	2	no	—
67	C₄×Dic₇₉	C₄⋊C₄	1	8	yes	complemented
68	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	—	2	4	no	—
70	C₂²×C₄×C₇₉	—	2	2	no	—
71	C₂²×C₄×C₇₉	—	4	4	no	—
72	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	#41	1	2	yes	—
80	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
82	M₁₆×C₇₉	—	2	4	no	—
83	M₁₆×C₇₉	—	2	4	no	—
84	M₁₆×C₇₉	—	2	4	no	—
85	M₁₆×C₇₉	—	4	2	no	—
86	C₂×C₈×C₇₉	—	2	2	no	—
87	C₂×C₈×C₇₉	—	2	2	no	—
88	C₂×C₈×C₇₉	—	2	2	no	—
89	C₂×C₈×C₇₉	—	4	1	no	—
90	C₂×C₈×C₇₉	—	4	1	no	—
91	C₂×C₈×C₇₉	C₄⋊C₄	1	4	yes	complemented
92	C₄²×C₇₉	—	4	1	no	—
93	C₄²×C₇₉	—	4	2	no	—
94	C₄²×C₇₉	—	4	1	no	—
95	C₂³×C₇₉	—	4	2	no	—
96	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
97	C₂³×C₇₉	#120	1	2	yes	—
98	C₂³×C₇₉	#121	1	1	yes	characteristic
99	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₇₉	—	2	8	no	—
102	C₂×C₄×C₇₉	—	2	4	no	—
103	C₂×C₄×C₇₉	—	4	4	no	—
104	C₂×C₄×C₇₉	—	4	2	no	—
105	C₂×C₄×C₇₉	—	4	2	no	—
106	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#121	1	2	yes	—
112	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#123	1	2	yes	—
118	C₂×C₄×C₇₉	#125	1	2	yes	—
119	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
124	C₂×Dic₇₉	—	2	8	no	—
125	C₂×Dic₇₉	—	4	8	no	—
126	C₂×Dic₇₉	—	4	8	no	—
127	C₂×Dic₇₉	—	4	4	no	—
128	C₈×C₇₉	—	2	4	no	—
129	C₈×C₇₉	—	4	2	no	—
130	C₂²×C₇₉	—	2	4	no	—
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	—	4	1	no	—
133	C₂²×C₇₉	—	4	2	no	—
134	C₂²×C₇₉	—	4	1	no	—
135	C₂²×C₇₉	#385	1	2	yes	—
136	C₂²×C₇₉	#389	1	1	yes	characteristic
137	C₂²×C₇₉	#392	1	1	yes	characteristic
138	C₂²×C₇₉	#395	1	2	yes	—
139	C₂²×C₇₉	#398	1	1	yes	characteristic
140	C₄×C₇₉	—	4	4	no	—
141	C₄×C₇₉	#384	1	1	yes	characteristic
142	C₄×C₇₉	#385	1	1	yes	characteristic
143	C₄×C₇₉	#389	1	2	yes	—
144	C₄×C₇₉	#395	1	1	yes	characteristic
145	C₄×C₇₉	#398	1	2	yes	—
146	C₄×C₇₉	#399	1	1	yes	characteristic
147	Dic₇₉	—	4	8	no	—
148	Dic₇₉	—	8	4	no	—
149	#11335	—	79	1	no	maximal, Sylow
150	C₂×C₇₉	—	2	2	no	—
151	C₂×C₇₉	—	2	1	no	—
152	C₂×C₇₉	—	2	2	no	—
153	C₂×C₇₉	—	2	1	no	—
154	C₂×C₇₉	#1444	1	1	yes	characteristic
155	C₂×C₇₉	#2010	1	2	yes	—
156	#2911	—	79	1	no	—
157	#2296	—	79	2	no	—
158	#1733	—	79	2	no	—
159	C₂×H(3,ℤ/4)	—	79	1	no	—
160	#1728	D₇₉	1	1	yes	characteristic
161	C₇₉	#11335	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
163	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
164	C₂×#117	—	79	2	no	—
165	C₂×#117	—	79	2	no	—
166	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
167	#428	—	158	4	no	—
168	#427	—	158	2	no	—
169	#389	—	158	4	no	—
170	#385	—	158	4	no	—
171	C₄×C₄⋊C₄	—	158	4	no	—
172	#339	—	158	4	no	—
173	H(3,ℤ/4)	—	79	4	no	—
174	#336	—	2	2	no	—
175	C₂³×C₄	D₃₁₆	1	2	yes	—
176	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
183	C₂×M₁₆	Dic₁₅₈	1	2	yes	—
184	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
187	#117	—	79	4	no	—
188	#117	—	79	4	no	—
189	#117	—	158	2	no	—
190	#117	—	158	2	no	—
191	#117	—	158	2	no	—
192	#117	—	158	2	no	—
193	#117	—	158	2	no	—
194	#117	—	158	2	no	—
195	C₂×C₄⋊C₄	—	79	4	no	—
196	C₂×C₄⋊C₄	—	158	8	no	—
197	C₂×C₄⋊C₄	—	158	2	no	—
198	C₂×C₂²⋊C₄	—	158	2	no	—
199	C₂×C₂²⋊C₄	—	158	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	—	79	4	no	—
205	C₂×C₄²	—	158	2	no	—
206	#107	—	158	4	no	—
207	#106	—	158	4	no	—
208	#95	—	158	8	no	—
209	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	158	4	no	—
214	C₂²×C₄	—	158	4	no	—
215	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₇₉	1	2	yes	—
217	C₂²×C₄	D₄×D₇₉	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
219	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
220	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
222	C₂²×C₄	#11929623	1	2	yes	—
223	C₂²×C₄	#11929624	1	1	yes	characteristic
224	C₂²×C₄	#11929624	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	Dic₇₉⋊C₄	1	4	yes	complemented
235	C₄⋊C₄	—	316	16	no	—
236	C₄⋊C₄	—	158	4	no	—
237	C₄⋊C₄	—	158	4	no	—
238	C₄⋊C₄	—	158	4	no	—
239	C₄⋊C₄	—	158	4	no	—
240	C₂²⋊C₄	—	316	2	no	—
241	C₂²⋊C₄	—	316	2	no	—
242	C₂²⋊C₄	—	158	4	no	—
243	C₂²⋊C₄	—	158	4	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	316	2	no	—
248	C₄²	—	316	2	no	—
249	C₄²	—	79	8	no	—
250	C₄²	—	158	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
253	C₂³	#14579131	1	2	yes	—
254	C₂³	#14579135	1	1	yes	characteristic
255	C₂³	#14579141	1	2	yes	—
256	C₂³	#14579161	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	316	8	no	—
263	C₂×C₄	—	316	8	no	—
264	C₂×C₄	—	316	4	no	—
265	C₂×C₄	—	158	8	no	—
266	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
267	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
268	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
269	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
270	C₂×C₄	#14579129	1	1	yes	characteristic
271	C₂×C₄	#14579134	1	2	yes	—
272	C₂×C₄	#14579138	1	2	yes	—
273	C₂×C₄	#14579138	1	2	yes	—
274	C₂×C₄	#14579139	1	2	yes	—
275	C₂×C₄	#14579140	1	1	yes	characteristic
276	C₂×C₄	#14579142	1	1	yes	characteristic
277	C₂×C₄	#14579143	1	2	yes	—
278	C₂×C₄	#14579151	1	2	yes	—
279	C₂×C₄	#14579153	1	1	yes	characteristic
280	C₂×C₄	#14579157	1	1	yes	characteristic
281	C₂×C₄	#14579159	1	1	yes	characteristic
282	C₂×C₄	#14579160	1	2	yes	—
283	C₂×C₄	#14579160	1	2	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	4	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	#19098437	1	2	yes	—
292	C₂²	#19098461	1	1	yes	characteristic
293	C₂²	#19098472	1	1	yes	characteristic
294	C₂²	#19098484	1	2	yes	—
295	C₂²	#19098493	1	1	yes	characteristic
296	C₄	—	4	4	no	—
297	C₄	—	316	8	no	—
298	C₄	—	632	4	no	—
299	C₄	#19098432	1	1	yes	characteristic
300	C₄	#19098435	1	1	yes	characteristic
301	C₄	#19098460	1	2	yes	—
302	C₄	#19098483	1	1	yes	characteristic
303	C₄	#19098497	1	2	yes	—
304	C₄	#19098499	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	#23761664	1	1	yes	characteristic
310	C₂	#23764467	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579157
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1728	C₂
socle	C₂²×C₇₉	#392
prime radical	C₂⁴×C₇₉	C₂×D₄
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1728	D₇₉
hypocenter	C₇₉	#11335
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
14. #19098320	C₄	semidirect
67. C₄×Dic₇₉	C₄⋊C₄	semidirect
91. C₂×C₈×C₇₉	C₄⋊C₄	semidirect
161. C₇₉	#11335	semidirect
234. C₂×C₈	Dic₇₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
#23771990	2
#23765883	2
#23763111	2
C₂×#19098320	2
C₇₉×#1728	2
#11335	79