Subgroups — #34196538

Order 20224.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098587	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098328	C₂	1	2	yes	maximal
3	C₂×#19098328	C₂	1	2	yes	maximal
4	#23763094	C₂	1	1	yes	maximal, characteristic
5	C₂×#336×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	2	yes	—
7	C₂²×#11929608	C₂²	1	1	yes	characteristic
8	C₂²×#11929608	C₂²	1	1	yes	characteristic
9	#19098587	—	2	4	no	—
10	C₂×C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
11	#19098328	—	2	4	no	—
12	#19098328	—	2	4	no	—
13	#19098318	—	2	2	no	—
14	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₇₉	C₂²	1	1	yes	characteristic
16	#336×C₇₉	C₄	1	2	yes	—
17	#336×C₇₉	C₄	1	2	yes	—
18	C₂×#11929608	—	2	2	no	—
19	C₂×#11929608	—	2	1	no	—
20	C₂×#11929608	—	2	1	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	—	2	2	no	—
28	C₂×#11929608	—	2	2	no	—
29	C₂×#11929608	D₄	1	2	yes	—
30	C₂×#11929608	D₄	1	4	yes	—
31	C₂×#11929608	Q₈	1	2	yes	—
32	C₂²×C₇₉⋊C₈	—	2	1	no	—
33	C₂²×C₇₉⋊C₈	—	2	1	no	—
34	C₂²×C₇₉⋊C₈	—	2	2	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
36	C₂³×C₄×C₇₉	Q₈	1	2	yes	—
37	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	—	2	2	no	—
39	C₂²×C₈×C₇₉	—	2	2	no	—
40	C₂²×C₈×C₇₉	—	2	2	no	—
41	C₂²×C₈×C₇₉	—	2	2	no	—
42	C₂²×C₈×C₇₉	—	2	2	no	—
43	C₂²×C₈×C₇₉	—	2	2	no	—
44	C₂²×C₈×C₇₉	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₇₉	—	2	2	no	—
46	C₂×C₄²×C₇₉	—	2	2	no	—
47	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	D₄	1	2	yes	—
50	#11929608	—	2	4	no	—
51	#11929608	—	2	4	no	—
52	#11929608	—	2	4	no	—
53	#11929608	—	2	4	no	—
54	#11929608	—	4	2	no	—
55	#11929608	—	4	2	no	—
56	#11929608	—	4	2	no	—
57	#11929608	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	2	2	no	—
60	C₂×C₇₉⋊C₈	—	2	2	no	—
61	C₂×C₇₉⋊C₈	—	4	1	no	—
62	C₂×C₇₉⋊C₈	—	4	2	no	—
63	C₂×C₇₉⋊C₈	—	4	1	no	—
64	C₂×C₇₉⋊C₈	—	4	1	no	—
65	C₂×C₇₉⋊C₈	—	4	1	no	—
66	C₂×C₇₉⋊C₈	—	4	2	no	—
67	C₂⁴×C₇₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	—	2	4	no	—
69	C₂²×C₄×C₇₉	—	2	4	no	—
70	C₂²×C₄×C₇₉	—	4	2	no	—
71	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	#41	1	2	yes	—
79	C₂²×C₄×C₇₉	#41	1	2	yes	—
80	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	#41	1	2	yes	—
82	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₇₉	—	2	4	no	—
84	C₂×C₈×C₇₉	—	2	4	no	—
85	C₂×C₈×C₇₉	—	2	4	no	—
86	C₂×C₈×C₇₉	—	4	2	no	—
87	C₂×C₈×C₇₉	—	4	2	no	—
88	C₂×C₈×C₇₉	—	4	2	no	—
89	C₂×C₈×C₇₉	—	4	2	no	—
90	C₄²×C₇₉	—	2	2	no	—
91	C₄²×C₇₉	—	2	2	no	—
92	C₄²×C₇₉	—	2	2	no	—
93	C₄²×C₇₉	—	2	2	no	—
94	C₂³×C₇₉	—	2	2	no	—
95	C₂³×C₇₉	—	2	2	no	—
96	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₇₉	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₇₉	#125	1	1	yes	characteristic
99	C₂³×C₇₉	#125	1	2	yes	—
100	C₂³×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	2	2	no	—
106	C₂×C₄×C₇₉	—	4	8	no	—
107	C₂×C₄×C₇₉	#108	1	4	yes	—
108	C₂×C₄×C₇₉	#117	1	2	yes	—
109	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#121	1	2	yes	—
114	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#123	1	2	yes	—
118	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
119	C₂×C₄×C₇₉	#123	1	2	yes	—
120	C₂×C₄×C₇₉	#124	1	2	yes	—
121	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#125	1	2	yes	—
124	C₂×C₄×C₇₉	#126	1	2	yes	—
125	C₂×C₄×C₇₉	#126	1	2	yes	—
126	C₈×C₇₉	—	4	4	no	—
127	C₇₉⋊C₈	—	4	2	no	—
128	C₇₉⋊C₈	—	4	4	no	—
129	C₇₉⋊C₈	—	4	2	no	—
130	C₂²×C₇₉	—	2	2	no	—
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	—	2	4	no	—
133	C₂²×C₇₉	—	2	2	no	—
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	#382	1	1	yes	characteristic
136	C₂²×C₇₉	#385	1	1	yes	characteristic
137	C₂²×C₇₉	#389	1	1	yes	characteristic
138	C₂²×C₇₉	#398	1	2	yes	—
139	C₂²×C₇₉	#400	1	2	yes	—
140	C₂²×C₇₉	C₂×#108	1	2	yes	—
141	C₂²×C₇₉	#430	1	2	yes	—
142	C₄×C₇₉	—	2	4	no	—
143	C₄×C₇₉	—	4	2	no	—
144	C₄×C₇₉	#387	1	2	yes	—
145	C₄×C₇₉	#388	1	2	yes	—
146	C₄×C₇₉	#393	1	1	yes	characteristic
147	C₄×C₇₉	#394	1	1	yes	characteristic
148	C₄×C₇₉	#397	1	1	yes	characteristic
149	C₄×C₇₉	#400	1	1	yes	characteristic
150	#11269	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	2	2	no	—
152	C₂×C₇₉	—	2	2	no	—
153	C₂×C₇₉	#1442	1	1	yes	characteristic
154	C₂×C₇₉	#1837	1	1	yes	characteristic
155	C₂×C₇₉	#1838	1	1	yes	characteristic
156	C₂×C₇₉	#1855	1	2	yes	—
157	C₂×C₇₉	#1950	1	2	yes	—
158	C₂×#423	—	79	1	no	—
159	C₂×#341	—	79	2	no	—
160	C₂×#341	—	79	2	no	—
161	#1728	—	79	1	no	—
162	C₂×#336	D₇₉	1	1	yes	characteristic
163	C₇₉	#11269	1	1	yes	characteristic, complemented, Sylow
164	C₂²×M₁₆	—	79	2	no	—
165	C₂²×M₁₆	—	79	1	no	—
166	C₂²×M₁₆	—	79	1	no	—
167	C₂³×C₈	C₂×D₇₉	1	2	yes	—
168	C₂²×C₄²	C₂×D₇₉	1	1	yes	characteristic
169	#423	—	158	4	no	—
170	C₂×C₄⋊C₈	—	158	2	no	—
171	#341	—	158	4	no	—
172	#341	—	158	4	no	—
173	#336	—	158	2	no	—
174	#336	Dic₇₉	1	2	yes	—
175	#336	Dic₇₉	1	2	yes	—
176	C₂³×C₄	Dic₁₅₈	1	2	yes	—
177	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
178	C₂×M₁₆	—	79	2	no	—
179	C₂×M₁₆	—	79	4	no	—
180	C₂×M₁₆	—	79	2	no	—
181	C₂×M₁₆	—	158	2	no	—
182	C₂×M₁₆	—	158	2	no	—
183	C₂×M₁₆	—	158	2	no	—
184	C₂×M₁₆	—	158	2	no	—
185	C₂×M₁₆	—	158	2	no	—
186	C₂×M₁₆	—	158	2	no	—
187	C₂×M₁₆	—	158	2	no	—
188	C₂×M₁₆	—	158	2	no	—
189	C₂×M₁₆	—	158	2	no	—
190	C₂×M₁₆	—	158	1	no	—
191	C₂×M₁₆	—	158	1	no	—
192	C₂²×C₈	—	2	2	no	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	158	1	no	—
199	C₂²×C₈	—	158	1	no	—
200	C₂²×C₈	—	158	2	no	—
201	C₂²×C₈	C₂×Dic₇₉	1	4	yes	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	D₃₁₆	1	2	yes	—
205	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
206	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
207	C₄⋊C₈	—	158	8	no	—
208	C₂⁴	C₂×Dic₁₅₈	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
213	C₂²×C₄	C₄⋊Dic₇₉	1	2	yes	—
214	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₇₉	1	2	yes	—
216	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
218	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
219	C₂²×C₄	#11929623	1	1	yes	characteristic
220	C₂²×C₄	#11929623	1	2	yes	—
221	C₂²×C₄	#11929624	1	2	yes	—
222	C₂²×C₄	#11929624	1	2	yes	—
223	C₂²×C₄	#11929625	1	1	yes	characteristic
224	M₁₆	—	316	2	no	—
225	M₁₆	—	316	2	no	—
226	M₁₆	—	316	2	no	—
227	M₁₆	—	316	2	no	—
228	M₁₆	—	158	4	no	—
229	M₁₆	—	158	4	no	—
230	M₁₆	—	158	4	no	—
231	M₁₆	—	158	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	316	1	no	—
240	C₂×C₈	—	316	2	no	—
241	C₂×C₈	—	316	1	no	—
242	C₂×C₈	—	316	1	no	—
243	C₂×C₈	—	316	2	no	—
244	C₂×C₈	—	316	1	no	—
245	C₂×C₈	—	158	4	no	—
246	C₂×C₈	—	158	2	no	—
247	C₂×C₈	—	158	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
255	C₂³	#14579138	1	2	yes	—
256	C₂³	#14579152	1	1	yes	characteristic
257	C₂³	#14579153	1	2	yes	—
258	C₂³	#14579160	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#14579097	1	4	yes	—
266	C₂×C₄	#14579120	1	2	yes	—
267	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
268	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
269	C₂×C₄	Q₈×Dic₇₉	1	1	yes	characteristic
270	C₂×C₄	#14579131	1	1	yes	characteristic
271	C₂×C₄	#14579134	1	2	yes	—
272	C₂×C₄	#14579136	1	1	yes	characteristic
273	C₂×C₄	#14579139	1	2	yes	—
274	C₂×C₄	#14579141	1	1	yes	characteristic
275	C₂×C₄	#14579143	1	2	yes	—
276	C₂×C₄	#14579145	1	1	yes	characteristic
277	C₂×C₄	#14579146	1	2	yes	—
278	C₂×C₄	#14579147	1	2	yes	—
279	C₂×C₄	#14579149	1	1	yes	characteristic
280	C₂×C₄	#14579149	1	1	yes	characteristic
281	C₂×C₄	#14579151	1	2	yes	—
282	C₂×C₄	#14579154	1	2	yes	—
283	C₂×C₄	#14579156	1	2	yes	—
284	C₈	—	4	4	no	—
285	C₈	—	316	2	no	—
286	C₈	—	316	4	no	—
287	C₈	—	316	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#19098429	1	1	yes	characteristic
294	C₂²	#19098438	1	1	yes	characteristic
295	C₂²	#19098462	1	1	yes	characteristic
296	C₂²	#19098497	1	2	yes	—
297	C₂²	#19098505	1	2	yes	—
298	C₂²	C₂×#14579097	1	2	yes	—
299	C₂²	#19098612	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	#19098449	1	2	yes	—
303	C₄	#19098456	1	2	yes	—
304	C₄	#19098476	1	1	yes	characteristic
305	C₄	#19098481	1	1	yes	characteristic
306	C₄	#19098490	1	1	yes	characteristic
307	C₄	#19098503	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#23761657	1	1	yes	characteristic
311	C₂	#23763538	1	1	yes	characteristic
312	C₂	#23763541	1	1	yes	characteristic
313	C₂	#23763607	1	2	yes	—
314	C₂	#23764176	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₂×#336×C₇₉	C₂
socle	C₂³×C₇₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₉	C₂×Q₈
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	C₂×#336	D₇₉
hypocenter	C₇₉	#11269
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
163. C₇₉	#11269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098587	2
C₂×#19098328	2
#23763094	2
C₂×#336×C₇₉	2
#11269	79