Subgroups — #34178951

Order 20224.

498 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	#23761782	C₂	1	1	yes	maximal, characteristic
2	#23761782	C₂	1	1	yes	maximal, characteristic
3	#23761692	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098285	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761238	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761238	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#131×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19098616	C₂²	1	1	yes	characteristic
9	#19098616	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
12	Q₈⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
13	Q₈⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
14	D₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
15	D₄⋊(C₇₉⋊C₈)	C₄	1	2	yes	—
16	#19098285	—	2	2	no	—
17	#19098285	—	2	1	no	—
18	#19098285	—	2	1	no	—
19	C₂×#131×C₇₉	C₄	1	2	yes	—
20	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
21	#405×C₇₉	—	2	2	no	—
22	#405×C₇₉	C₄	1	2	yes	—
23	C₄×M₁₆×C₇₉	—	2	1	no	—
24	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₉	—	2	1	no	—
26	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂²×C₇₉⋊C₈	—	2	1	no	—
28	C₂²×C₇₉⋊C₈	—	2	1	no	—
29	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
30	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
31	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
32	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
33	C₄×C₇₉⋊C₈	—	2	1	no	—
34	C₄×C₇₉⋊C₈	—	2	1	no	—
35	C₄×C₇₉⋊C₈	—	2	1	no	—
36	C₄×C₇₉⋊C₈	—	2	1	no	—
37	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
38	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
40	#131×C₇₉	—	2	4	no	—
41	#131×C₇₉	—	2	4	no	—
42	C₂×M₁₆×C₇₉	—	2	2	no	—
43	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
44	C₂²×C₈×C₇₉	—	2	2	no	—
45	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
46	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
47	C₄×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
48	C₄×D₄×C₇₉	—	2	1	no	—
49	C₄×D₄×C₇₉	—	2	1	no	—
50	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₄×D₄×C₇₉	C₂×C₄	1	1	yes	characteristic
52	#117×C₇₉	—	2	1	no	—
53	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₉	—	2	1	no	—
55	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
56	#97×C₇₉	—	2	2	no	—
57	#97×C₇₉	—	2	1	no	—
58	C₄×C₈×C₇₉	—	2	2	no	—
59	C₄×C₈×C₇₉	—	2	1	no	—
60	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₇₉	D₄	1	2	yes	—
63	C₂×C₇₉⋊C₈	—	2	4	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	2	no	—
66	C₂×C₇₉⋊C₈	—	2	2	no	—
67	C₂×C₇₉⋊C₈	—	2	2	no	—
68	C₂×C₇₉⋊C₈	—	4	2	no	—
69	#41×C₇₉	—	2	2	no	—
70	#41×C₇₉	M₁₆	1	4	yes	—
71	C₂×Q₈×C₇₉	C₄²	1	1	yes	characteristic
72	C₂×D₄×C₇₉	—	2	1	no	—
73	C₂×D₄×C₇₉	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
80	M₁₆×C₇₉	—	2	8	no	—
81	M₁₆×C₇₉	—	2	2	no	—
82	M₁₆×C₇₉	M₁₆	1	4	yes	—
83	C₂×C₈×C₇₉	—	2	8	no	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₂×C₈×C₇₉	—	2	2	no	—
86	C₂×C₈×C₇₉	—	2	2	no	—
87	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₇₉	—	2	1	no	—
90	C₄⋊C₄×C₇₉	—	2	1	no	—
91	C₄⋊C₄×C₇₉	C₄²	1	1	yes	characteristic
92	C₄⋊C₄×C₇₉	C₄²	1	1	yes	characteristic
93	C₂²⋊C₄×C₇₉	—	2	1	no	—
94	C₂²⋊C₄×C₇₉	—	2	1	no	—
95	C₂²⋊C₄×C₇₉	—	2	1	no	—
96	C₄²×C₇₉	—	2	1	no	—
97	C₄²×C₇₉	—	2	1	no	—
98	C₄²×C₇₉	—	2	1	no	—
99	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂³×C₇₉	—	2	1	no	—
104	C₂³×C₇₉	#117	1	1	yes	characteristic
105	Q₈×C₇₉	—	2	1	no	—
106	Q₈×C₇₉	#97	1	2	yes	—
107	D₄×C₇₉	—	2	4	no	—
108	D₄×C₇₉	—	2	1	no	—
109	D₄×C₇₉	#97	1	2	yes	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	4	no	—
112	C₂×C₄×C₇₉	—	2	2	no	—
113	C₂×C₄×C₇₉	—	2	2	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	4	2	no	—
117	C₂×C₄×C₇₉	—	4	2	no	—
118	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
126	C₈×C₇₉	—	2	2	no	—
127	C₈×C₇₉	—	2	4	no	—
128	C₈×C₇₉	—	2	2	no	—
129	C₇₉⋊C₈	—	2	4	no	—
130	C₇₉⋊C₈	—	4	2	no	—
131	C₇₉⋊C₈	—	4	2	no	—
132	C₇₉⋊C₈	—	4	2	no	—
133	C₂²×C₇₉	—	2	1	no	—
134	C₂²×C₇₉	—	2	1	no	—
135	C₂²×C₇₉	—	2	2	no	—
136	C₂²×C₇₉	—	4	1	no	—
137	C₂²×C₇₉	—	4	1	no	—
138	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₇₉	#432	1	2	yes	—
140	C₄×C₇₉	—	2	2	no	—
141	C₄×C₇₉	—	2	1	no	—
142	C₄×C₇₉	—	2	1	no	—
143	C₄×C₇₉	—	4	1	no	—
144	C₄×C₇₉	—	4	1	no	—
145	C₄×C₇₉	C₂×#97	1	1	yes	characteristic
146	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
147	C₄×C₇₉	#407	1	2	yes	—
148	C₄×C₇₉	#443	1	2	yes	—
149	C₄×C₇₉	#444	1	2	yes	—
150	#8186	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	2	2	no	—
152	C₂×C₇₉	—	4	2	no	—
153	C₂×C₇₉	#1749	1	1	yes	characteristic
154	C₂×C₇₉	#1764	1	1	yes	characteristic
155	C₂×C₇₉	#1765	1	1	yes	characteristic
156	C₄×#131	D₇₉	1	1	yes	characteristic
157	#1476	—	79	1	no	—
158	#1476	—	79	1	no	—
159	#1452	—	79	1	no	—
160	C₂×#322	—	79	1	no	—
161	#1278	—	79	1	no	—
162	#1278	—	79	1	no	—
163	C₇₉	#8186	1	1	yes	characteristic, complemented, Sylow
164	C₂×#131	Dic₇₉	1	2	yes	—
165	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
166	#431	—	79	1	no	—
167	#431	—	79	1	no	—
168	#405	—	2	2	no	—
169	#405	Dic₇₉	1	2	yes	—
170	C₄×M₁₆	—	2	1	no	—
171	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
172	C₂×C₄×C₈	—	2	1	no	—
173	C₂×C₄×C₈	—	79	1	no	—
174	C₂×C₄×C₈	—	79	1	no	—
175	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
176	Q₈⋊C₈	—	79	2	no	—
177	Q₈⋊C₈	—	79	2	no	—
178	D₄⋊C₈	—	79	2	no	—
179	D₄⋊C₈	—	79	2	no	—
180	#322	—	158	2	no	—
181	#322	—	158	1	no	—
182	#322	—	158	1	no	—
183	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
184	#131	—	2	4	no	—
185	#131	—	2	4	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	C₂×Dic₇₉	1	2	yes	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	158	1	no	—
190	C₂²×C₈	—	158	1	no	—
191	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
192	C₄×Q₈	C₄×D₇₉	1	1	yes	characteristic
193	C₄×Q₈	C₄×D₇₉	1	1	yes	characteristic
194	C₄×D₄	—	2	1	no	—
195	C₄×D₄	—	2	1	no	—
196	C₄×D₄	C₄×D₇₉	1	1	yes	characteristic
197	C₄×D₄	C₄×D₇₉	1	1	yes	characteristic
198	#117	—	2	1	no	—
199	#117	C₂×Dic₇₉	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
202	C₄⋊C₈	—	79	2	no	—
203	C₄⋊C₈	—	79	2	no	—
204	C₂²⋊C₈	—	158	1	no	—
205	C₂²⋊C₈	—	158	1	no	—
206	#97	—	2	2	no	—
207	#97	—	2	1	no	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	79	2	no	—
211	C₄×C₈	—	79	2	no	—
212	C₄×C₈	—	158	1	no	—
213	C₄×C₈	—	158	1	no	—
214	C₄×C₈	—	158	1	no	—
215	C₄×C₈	—	158	1	no	—
216	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
217	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
219	#41	—	2	2	no	—
220	#41	#11929608	1	4	yes	—
221	C₂×Q₈	C₄×Dic₇₉	1	1	yes	characteristic
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄×Dic₇₉	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
228	C₂²×C₄	#11929624	1	1	yes	characteristic
229	C₂²×C₄	#11929624	1	1	yes	characteristic
230	M₁₆	—	2	8	no	—
231	M₁₆	—	2	2	no	—
232	M₁₆	#11929608	1	4	yes	—
233	C₂×C₈	—	2	8	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	316	2	no	—
238	C₂×C₈	—	158	2	no	—
239	C₂×C₈	—	158	2	no	—
240	C₂×C₈	—	158	2	no	—
241	C₂×C₈	—	158	4	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
244	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	C₄×Dic₇₉	1	1	yes	characteristic
248	C₄⋊C₄	C₄×Dic₇₉	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
256	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
257	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
258	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	C₂³	#14579120	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	#14579072	1	2	yes	—
263	D₄	—	2	4	no	—
264	D₄	—	2	1	no	—
265	D₄	#14579072	1	2	yes	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
276	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
277	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
278	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
279	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
280	C₂×C₄	C₂×#11929608	1	2	yes	—
281	C₂×C₄	C₂×#11929608	1	2	yes	—
282	C₈	—	2	2	no	—
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	316	2	no	—
286	C₈	—	316	2	no	—
287	C₈	—	316	2	no	—
288	C₈	—	158	4	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	1	no	—
294	C₂²	C₄×C₂²⋊Dic₇₉	1	1	yes	characteristic
295	C₂²	#19098620	1	2	yes	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	C₂×#14579072	1	1	yes	characteristic
302	C₄	C₄×#11929608	1	1	yes	characteristic
303	C₄	#19098527	1	2	yes	—
304	C₄	#19098678	1	2	yes	—
305	C₄	#19098683	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	4	2	no	—
308	C₂	#23763162	1	1	yes	characteristic
309	C₂	#23763235	1	1	yes	characteristic
310	C₂	#23763242	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#131×C₇₉	C₂
socle	C₂²×C₇₉	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#131	D₇₉
hypocenter	C₇₉	#8186
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
3. #23761692	C₂	semidirect
4. C₂×#19098285	C₂	semidirect
5. #23761238	C₂	semidirect
6. #23761238	C₂	semidirect
163. C₇₉	#8186	semidirect

Maximal subgroups

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

subgroup	index
#23761782	2
#23761692	2
C₂×#19098285	2
#23761238	2
C₄×#131×C₇₉	2
#8186	79