Subgroups — #34196986

Order 20224.

666 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	#23771964	C₂	1	1	yes	maximal, characteristic
3	#23763139	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098328	C₂	1	2	yes	maximal
5	C₇₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂²×#11929608	C₂²	1	2	yes	—
8	#19098611	—	2	4	no	—
9	#19098611	—	2	4	no	—
10	C₂×#14579096	—	2	4	no	—
11	#19098590	—	2	4	no	—
12	#19098587	—	2	2	no	—
13	#19098332	C₄	1	4	yes	—
14	#19098328	—	2	4	no	—
15	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
18	#336×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	2	no	—
23	C₂×#11929608	—	2	2	no	—
24	C₂×#11929608	—	2	2	no	—
25	C₂×#11929608	—	2	2	no	—
26	C₂×#11929608	—	2	2	no	—
27	C₂×#11929608	C₂×C₄	1	4	yes	—
28	C₂×#11929608	D₄	1	4	yes	—
29	C₂×#11929608	D₄	1	4	yes	—
30	C₂×#11929608	Q₈	1	4	yes	—
31	C₂²×C₇₉⋊C₈	—	2	2	no	—
32	C₂²×C₇₉⋊C₈	—	2	2	no	—
33	#14579096	—	2	16	no	—
34	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
35	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₇₉	—	2	1	no	—
37	C₂×M₁₆×C₇₉	—	2	2	no	—
38	C₂×M₁₆×C₇₉	—	2	2	no	—
39	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
40	C₂²×C₈×C₇₉	—	2	1	no	—
41	C₂²×C₈×C₇₉	—	2	2	no	—
42	C₂²×C₈×C₇₉	—	2	1	no	—
43	C₂²×C₈×C₇₉	—	2	2	no	—
44	C₂²×C₈×C₇₉	—	2	1	no	—
45	C₂²×C₈×C₇₉	Q₈	1	2	yes	—
46	#117×C₇₉	—	2	2	no	—
47	#117×C₇₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₇₉	Q₈	1	2	yes	—
50	C₂×C₄²×C₇₉	D₄	1	2	yes	—
51	#11929608	—	2	4	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	—	4	2	no	—
58	#11929608	C₄⋊C₄	1	8	yes	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	2	4	no	—
61	C₂×C₇₉⋊C₈	—	2	4	no	—
62	C₂×C₇₉⋊C₈	—	2	4	no	—
63	C₂×C₇₉⋊C₈	—	4	2	no	—
64	C₂×C₇₉⋊C₈	—	4	2	no	—
65	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₇₉	C₂×D₄	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×Q₈	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	#41	1	2	yes	—
76	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
77	M₁₆×C₇₉	—	2	4	no	—
78	M₁₆×C₇₉	—	4	2	no	—
79	C₂×C₈×C₇₉	—	2	2	no	—
80	C₂×C₈×C₇₉	—	2	4	no	—
81	C₂×C₈×C₇₉	—	2	2	no	—
82	C₂×C₈×C₇₉	—	4	1	no	—
83	C₂×C₈×C₇₉	—	4	1	no	—
84	C₂×C₈×C₇₉	—	4	2	no	—
85	C₂×C₈×C₇₉	—	4	1	no	—
86	C₂×C₈×C₇₉	—	4	1	no	—
87	C₄⋊C₄×C₇₉	—	2	4	no	—
88	C₂²⋊C₄×C₇₉	—	2	4	no	—
89	C₄²×C₇₉	—	2	2	no	—
90	C₄²×C₇₉	—	2	2	no	—
91	C₂³×C₇₉	—	4	2	no	—
92	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
94	C₂³×C₇₉	#125	1	1	yes	characteristic
95	C₂³×C₇₉	C₄⋊Q₈	1	2	yes	—
96	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇₉	—	2	4	no	—
98	C₂×C₄×C₇₉	—	4	4	no	—
99	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
104	C₂×C₄×C₇₉	#120	1	2	yes	—
105	C₂×C₄×C₇₉	#121	1	2	yes	—
106	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#123	1	2	yes	—
113	C₂×C₄×C₇₉	#125	1	2	yes	—
114	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
117	C₈×C₇₉	—	4	2	no	—
118	C₈×C₇₉	—	4	2	no	—
119	C₇₉⋊C₈	—	2	8	no	—
120	C₇₉⋊C₈	—	4	4	no	—
121	C₂²×C₇₉	—	2	2	no	—
122	C₂²×C₇₉	—	2	4	no	—
123	C₂²×C₇₉	—	4	1	no	—
124	C₂²×C₇₉	—	4	2	no	—
125	C₂²×C₇₉	—	4	1	no	—
126	C₂²×C₇₉	#384	1	1	yes	characteristic
127	C₂²×C₇₉	#385	1	1	yes	characteristic
128	C₂²×C₇₉	#389	1	1	yes	characteristic
129	C₂²×C₇₉	#395	1	2	yes	—
130	C₂²×C₇₉	#398	1	2	yes	—
131	C₄×C₇₉	—	4	4	no	—
132	C₄×C₇₉	#385	1	2	yes	—
133	C₄×C₇₉	#389	1	2	yes	—
134	C₄×C₇₉	#392	1	1	yes	characteristic
135	C₄×C₇₉	#395	1	1	yes	characteristic
136	C₄×C₇₉	#398	1	1	yes	characteristic
137	C₄×C₇₉	#399	1	1	yes	characteristic
138	#11343	—	79	1	no	maximal, Sylow
139	C₂×C₇₉	—	2	1	no	—
140	C₂×C₇₉	—	2	1	no	—
141	C₂×C₇₉	—	2	2	no	—
142	C₂×C₇₉	—	2	2	no	—
143	C₂×C₇₉	#1444	1	1	yes	characteristic
144	C₂×C₇₉	#1858	1	2	yes	—
145	C₂×#430	—	79	2	no	—
146	#2905	—	79	1	no	—
147	#1742	—	79	1	no	—
148	C₂×#341	—	79	2	no	—
149	#1729	D₇₉	1	1	yes	characteristic
150	C₇₉	#11343	1	1	yes	characteristic, complemented, Sylow
151	C₂²×M₁₆	—	79	2	no	—
152	C₂²×M₁₆	—	79	2	no	—
153	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
154	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
155	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
156	#430	—	158	4	no	—
157	#430	—	158	4	no	—
158	C₂×#108	—	158	4	no	—
159	#424	—	158	4	no	—
160	#423	—	158	2	no	—
161	#343	—	79	4	no	—
162	#341	—	158	4	no	—
163	#336	—	2	2	no	—
164	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	1	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	79	4	no	—
169	C₂×M₁₆	—	79	4	no	—
170	C₂×M₁₆	—	79	4	no	—
171	C₂×M₁₆	—	79	4	no	—
172	C₂×M₁₆	—	158	2	no	—
173	C₂×M₁₆	—	158	2	no	—
174	C₂×M₁₆	—	158	2	no	—
175	C₂×M₁₆	—	158	2	no	—
176	C₂×M₁₆	—	158	2	no	—
177	C₂×M₁₆	—	158	2	no	—
178	C₂×M₁₆	—	158	2	no	—
179	C₂×M₁₆	—	158	2	no	—
180	C₂×M₁₆	D₃₁₆	1	2	yes	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	158	2	no	—
187	C₂²×C₈	—	158	2	no	—
188	C₂²×C₈	Dic₁₅₈	1	2	yes	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₄⋊C₄	C₄×D₇₉	1	2	yes	—
192	C₂×C₂²⋊C₄	Dic₁₅₈	1	2	yes	—
193	C₂×C₄²	C₂²⋊D₇₉	1	2	yes	—
194	#108	—	158	16	no	—
195	C₄⋊C₈	—	158	8	no	—
196	C₂⁴	C₂×Dic₁₅₈	1	1	yes	characteristic
197	C₂²×C₄	Dic₇₉⋊C₄	1	4	yes	—
198	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
199	C₂²×C₄	D₄×D₇₉	1	2	yes	—
200	C₂²×C₄	D₄×D₇₉	1	2	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
203	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
204	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
205	C₂²×C₄	#11929623	1	2	yes	—
206	C₂²×C₄	#11929624	1	1	yes	characteristic
207	C₂²×C₄	#11929624	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	316	2	no	—
211	M₁₆	—	316	2	no	—
212	M₁₆	—	79	8	no	—
213	M₁₆	—	158	4	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	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	316	2	no	—
227	C₂×C₈	—	316	2	no	—
228	C₂×C₈	—	158	4	no	—
229	C₂×C₈	—	158	4	no	—
230	C₂×C₈	—	158	4	no	—
231	C₂×C₈	—	158	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
238	C₂³	#14579138	1	2	yes	—
239	C₂³	#14579153	1	1	yes	characteristic
240	C₂³	#14579160	1	2	yes	—
241	C₂³	#14579161	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
245	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
246	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
247	C₂×C₄	C₄×Dic₁₅₈	1	1	yes	characteristic
248	C₂×C₄	#14579129	1	1	yes	characteristic
249	C₂×C₄	#14579131	1	2	yes	—
250	C₂×C₄	#14579134	1	2	yes	—
251	C₂×C₄	#14579135	1	1	yes	characteristic
252	C₂×C₄	#14579138	1	2	yes	—
253	C₂×C₄	#14579139	1	2	yes	—
254	C₂×C₄	#14579140	1	1	yes	characteristic
255	C₂×C₄	#14579141	1	2	yes	—
256	C₂×C₄	#14579142	1	1	yes	characteristic
257	C₂×C₄	#14579143	1	2	yes	—
258	C₂×C₄	#14579151	1	2	yes	—
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	—	4	2	no	—
263	C₈	—	4	2	no	—
264	C₈	—	316	4	no	—
265	C₈	—	158	8	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	#19098432	1	1	yes	characteristic
272	C₂²	#19098435	1	1	yes	characteristic
273	C₂²	#19098461	1	1	yes	characteristic
274	C₂²	#19098484	1	2	yes	—
275	C₂²	#19098497	1	2	yes	—
276	C₄	—	4	4	no	—
277	C₄	#19098437	1	2	yes	—
278	C₄	#19098460	1	2	yes	—
279	C₄	#19098472	1	1	yes	characteristic
280	C₄	#19098483	1	1	yes	characteristic
281	C₄	#19098493	1	1	yes	characteristic
282	C₄	#19098499	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	#23761664	1	1	yes	characteristic
288	C₂	#23763619	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579129
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1729	C₂
socle	C₂²×C₇₉	#385
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1729	D₇₉
hypocenter	C₇₉	#11343
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
150. C₇₉	#11343	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
#23771964	2
#23763139	2
C₂×#19098328	2
C₇₉×#1729	2
#11343	79