Subgroups — #34196995

Order 20224.

698 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771964	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098590	C₂	1	2	yes	maximal
3	C₂×#19098333	C₂	1	2	yes	maximal
4	#23763097	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11929608	C₂²	1	1	yes	characteristic
7	C₂²×#11929608	C₂²	1	2	yes	—
8	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
9	#19098590	—	2	4	no	—
10	#19098590	—	2	4	no	—
11	#19098590	—	2	4	no	—
12	#19098587	—	2	2	no	—
13	C₂×C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
14	#19098333	—	2	4	no	—
15	#19098318	—	2	2	no	—
16	C₂×#117×C₇₉	C₂²	1	2	yes	—
17	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
18	#342×C₇₉	C₄	1	4	yes	—
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	1	no	—
26	C₂×#11929608	—	2	2	no	—
27	C₂×#11929608	—	2	2	no	—
28	C₂×#11929608	Q₈	1	2	yes	—
29	C₂²×C₇₉⋊C₈	—	2	1	no	—
30	C₂²×C₇₉⋊C₈	—	2	2	no	—
31	C₂²×C₇₉⋊C₈	—	2	2	no	—
32	C₂²×C₇₉⋊C₈	—	2	1	no	—
33	C₂²×C₇₉⋊C₈	—	2	1	no	—
34	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
35	C₂²×C₇₉⋊C₈	D₄	1	4	yes	—
36	C₄⋊(C₇₉⋊C₈)	—	2	16	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
38	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
39	#117×C₇₉	—	2	2	no	—
40	#117×C₇₉	—	2	2	no	—
41	#117×C₇₉	—	2	2	no	—
42	#117×C₇₉	—	2	2	no	—
43	#117×C₇₉	D₄	1	4	yes	—
44	#117×C₇₉	Q₈	1	4	yes	—
45	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₉	Q₈	1	2	yes	—
47	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₇₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₇₉	—	2	2	no	—
50	C₂×C₄²×C₇₉	D₄	1	2	yes	—
51	#11929608	—	2	4	no	—
52	#11929608	—	4	2	no	—
53	#11929608	—	4	2	no	—
54	#11929608	—	4	2	no	—
55	#11929608	—	4	2	no	—
56	#11929608	—	4	2	no	—
57	C₂×C₇₉⋊C₈	—	2	4	no	—
58	C₂×C₇₉⋊C₈	—	2	4	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	2	2	no	—
61	C₂×C₇₉⋊C₈	—	2	4	no	—
62	C₂×C₇₉⋊C₈	—	2	2	no	—
63	C₂×C₇₉⋊C₈	—	4	1	no	—
64	C₂×C₇₉⋊C₈	—	4	2	no	—
65	C₂×C₇₉⋊C₈	—	4	1	no	—
66	C₂×C₇₉⋊C₈	—	4	1	no	—
67	C₂×C₇₉⋊C₈	—	4	1	no	—
68	C₂⁴×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	—	2	4	no	—
70	C₂²×C₄×C₇₉	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂×D₄	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×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	#41	1	2	yes	—
80	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₇₉	—	2	4	no	—
82	C₄⋊C₄×C₇₉	—	2	4	no	—
83	C₄⋊C₄×C₇₉	—	2	4	no	—
84	C₂²⋊C₄×C₇₉	—	2	4	no	—
85	C₂²⋊C₄×C₇₉	—	2	4	no	—
86	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	8	yes	—
87	C₄²×C₇₉	—	2	4	no	—
88	C₄²×C₇₉	—	2	2	no	—
89	C₄²×C₇₉	—	2	4	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	C₂×C₄⋊C₄	1	2	yes	—
94	C₂³×C₇₉	C₄×D₄	1	2	yes	—
95	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₇₉	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₇₉	—	2	4	no	—
98	C₂×C₄×C₇₉	—	2	8	no	—
99	C₂×C₄×C₇₉	—	4	8	no	—
100	C₂×C₄×C₇₉	—	4	4	no	—
101	C₂×C₄×C₇₉	#120	1	2	yes	—
102	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
103	C₂×C₄×C₇₉	#121	1	2	yes	—
104	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
106	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#123	1	2	yes	—
112	C₂×C₄×C₇₉	#125	1	1	yes	characteristic
113	C₂×C₄×C₇₉	#125	1	2	yes	—
114	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₄⋊Q₈	1	2	yes	—
119	C₇₉⋊C₈	—	4	2	no	—
120	C₇₉⋊C₈	—	4	4	no	—
121	C₇₉⋊C₈	—	4	2	no	—
122	C₂²×C₇₉	—	2	2	no	—
123	C₂²×C₇₉	—	2	4	no	—
124	C₂²×C₇₉	—	4	2	no	—
125	C₂²×C₇₉	—	4	1	no	—
126	C₂²×C₇₉	—	4	1	no	—
127	C₂²×C₇₉	#384	1	1	yes	characteristic
128	C₂²×C₇₉	#385	1	2	yes	—
129	C₂²×C₇₉	#385	1	1	yes	characteristic
130	C₂²×C₇₉	#389	1	2	yes	—
131	C₂²×C₇₉	#389	1	1	yes	characteristic
132	C₄×C₇₉	—	4	8	no	—
133	C₄×C₇₉	—	4	4	no	—
134	C₄×C₇₉	#392	1	1	yes	characteristic
135	C₄×C₇₉	#395	1	2	yes	—
136	C₄×C₇₉	#395	1	1	yes	characteristic
137	C₄×C₇₉	#398	1	2	yes	—
138	C₄×C₇₉	#398	1	1	yes	characteristic
139	C₄×C₇₉	#399	1	1	yes	characteristic
140	#11345	—	79	1	no	maximal, Sylow
141	C₂×C₇₉	—	2	2	no	—
142	C₂×C₇₉	—	2	1	no	—
143	C₂×C₇₉	—	2	1	no	—
144	C₂×C₇₉	—	2	2	no	—
145	C₂×C₇₉	#1444	1	1	yes	characteristic
146	C₂×C₇₉	#1868	1	2	yes	—
147	#2905	—	79	1	no	—
148	C₂×#424	—	79	2	no	—
149	C₂×#344	—	79	2	no	—
150	#1740	D₇₉	1	1	yes	characteristic
151	#1729	—	79	1	no	—
152	C₇₉	#11345	1	1	yes	characteristic, complemented, Sylow
153	C₂²×M₁₆	—	79	2	no	—
154	C₂²×M₁₆	—	79	1	no	—
155	C₂³×C₈	—	79	1	no	—
156	C₂×#117	C₂×D₇₉	1	2	yes	—
157	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
158	#424	—	158	4	no	—
159	#424	—	158	4	no	—
160	#424	—	158	4	no	—
161	#423	—	158	2	no	—
162	C₂×C₄⋊C₈	—	158	4	no	—
163	#344	—	158	4	no	—
164	#342	Dic₇₉	1	4	yes	—
165	#336	—	158	2	no	—
166	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
167	C₂×M₁₆	—	79	2	no	—
168	C₂×M₁₆	—	158	2	no	—
169	C₂×M₁₆	—	158	2	no	—
170	C₂×M₁₆	—	158	2	no	—
171	C₂×M₁₆	—	158	2	no	—
172	C₂×M₁₆	—	158	2	no	—
173	C₂×M₁₆	—	158	2	no	—
174	C₂×M₁₆	—	158	2	no	—
175	C₂×M₁₆	—	158	1	no	—
176	C₂×M₁₆	—	158	2	no	—
177	C₂²×C₈	—	79	2	no	—
178	C₂²×C₈	—	79	4	no	—
179	C₂²×C₈	—	158	2	no	—
180	C₂²×C₈	—	158	1	no	—
181	C₂²×C₈	—	158	2	no	—
182	C₂²×C₈	—	158	1	no	—
183	C₂²×C₈	—	158	1	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	Dic₁₅₈	1	4	yes	—
189	#117	D₃₁₆	1	4	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	Dic₁₅₈	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂×Dic₇₉	1	4	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	D₃₁₆	1	2	yes	—
196	C₄⋊C₈	—	158	16	no	—
197	C₄⋊C₈	—	158	8	no	—
198	C₂⁴	C₂²×Dic₇₉	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	C₄⋊Dic₇₉	1	4	yes	—
201	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
202	C₂²×C₄	D₄×D₇₉	1	2	yes	—
203	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
204	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
205	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
208	C₂²×C₄	#11929623	1	2	yes	—
209	C₂²×C₄	#11929623	1	1	yes	characteristic
210	C₂²×C₄	#11929625	1	1	yes	characteristic
211	M₁₆	—	316	2	no	—
212	M₁₆	—	316	2	no	—
213	M₁₆	—	316	2	no	—
214	M₁₆	—	316	2	no	—
215	M₁₆	—	316	2	no	—
216	M₁₆	—	158	4	no	—
217	C₂×C₈	—	316	1	no	—
218	C₂×C₈	—	316	2	no	—
219	C₂×C₈	—	316	1	no	—
220	C₂×C₈	—	316	1	no	—
221	C₂×C₈	—	316	1	no	—
222	C₂×C₈	—	158	4	no	—
223	C₂×C₈	—	158	4	no	—
224	C₂×C₈	—	158	4	no	—
225	C₂×C₈	—	158	2	no	—
226	C₂×C₈	—	158	4	no	—
227	C₂×C₈	—	158	2	no	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	C₄⋊Dic₇₉	1	8	yes	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	2	no	—
236	C₄²	—	2	4	no	—
237	C₄²	—	2	2	no	—
238	C₂³	—	4	2	no	—
239	C₂³	C₂×C₄⋊Dic₇₉	1	1	yes	characteristic
240	C₂³	C₂×C₄⋊Dic₇₉	1	2	yes	—
241	C₂³	D₄×Dic₇₉	1	2	yes	—
242	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
243	C₂³	Q₈×Dic₇₉	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	4	8	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	#14579131	1	2	yes	—
249	C₂×C₄	#14579131	1	1	yes	characteristic
250	C₂×C₄	#14579136	1	2	yes	—
251	C₂×C₄	#14579136	1	1	yes	characteristic
252	C₂×C₄	#14579138	1	2	yes	—
253	C₂×C₄	#14579138	1	2	yes	—
254	C₂×C₄	#14579138	1	2	yes	—
255	C₂×C₄	#14579141	1	1	yes	characteristic
256	C₂×C₄	#14579141	1	2	yes	—
257	C₂×C₄	#14579145	1	1	yes	characteristic
258	C₂×C₄	#14579145	1	2	yes	—
259	C₂×C₄	#14579152	1	1	yes	characteristic
260	C₂×C₄	#14579152	1	2	yes	—
261	C₂×C₄	#14579158	1	1	yes	characteristic
262	C₂×C₄	#14579160	1	1	yes	characteristic
263	C₂×C₄	#14579160	1	2	yes	—
264	C₂×C₄	#14579161	1	2	yes	—
265	C₂×C₄	#14579161	1	1	yes	characteristic
266	C₈	—	316	2	no	—
267	C₈	—	316	4	no	—
268	C₈	—	316	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	#19098433	1	1	yes	characteristic
275	C₂²	#19098438	1	2	yes	—
276	C₂²	#19098438	1	1	yes	characteristic
277	C₂²	#19098462	1	2	yes	—
278	C₂²	#19098462	1	1	yes	characteristic
279	C₄	—	4	8	no	—
280	C₄	—	4	4	no	—
281	C₄	#19098473	1	1	yes	characteristic
282	C₄	#19098484	1	2	yes	—
283	C₄	#19098484	1	1	yes	characteristic
284	C₄	#19098495	1	2	yes	—
285	C₄	#19098495	1	1	yes	characteristic
286	C₄	#19098500	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	#23761665	1	1	yes	characteristic
292	C₂	#23763688	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14579131
commutator, prime radical	C₂⁴×C₇₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₉
Fitting	C₇₉×#1740	C₂
socle	C₂²×C₇₉	#385
cosocle	C₂³×C₄×C₇₉	C₂³
hypercenter	#1740	D₇₉
hypocenter	C₇₉	#11345
LCS	G ▷ C₂⁴×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
152. C₇₉	#11345	semidirect

Maximal subgroups

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

subgroup	index
#23771964	2
C₂×#19098590	2
C₂×#19098333	2
#23763097	2
C₇₉×#1740	2
#11345	79