Subgroups — #34271386

Order 20224.

650 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774525	C₂	1	2	yes	maximal
2	C₂×#19098968	C₂	1	1	yes	maximal, characteristic, complemented
3	#23762923	C₂	1	4	yes	maximal, complemented
4	#23762799	C₂	1	4	yes	maximal, complemented
5	#23762148	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762082	C₂	1	2	yes	maximal, complemented
7	C₇₉×#2931	C₂	1	1	yes	maximal, characteristic
8	C₂×#14579161	C₂²	1	2	yes	—
9	#19098968	—	2	1	no	—
10	#19098968	—	2	1	no	—
11	#19098968	C₂²	1	2	yes	—
12	#19098968	C₂²	1	2	yes	—
13	#19098959	—	2	2	no	—
14	#19098909	—	2	2	no	—
15	#19098879	—	2	4	no	—
16	#19098853	C₂²	1	8	yes	—
17	#19098616	C₂²	1	2	yes	—
18	C₂×#14579095	—	2	1	no	—
19	C₂×#14579095	—	2	1	no	—
20	#19098316	C₂²	1	2	yes	—
21	#19098314	C₂²	1	2	yes	—
22	#19098311	C₂²	1	8	yes	—
23	C₂×#131×C₇₉	—	2	1	no	—
24	#532×C₇₉	C₂²	1	1	yes	characteristic
25	#436×C₇₉	—	2	1	no	—
26	#436×C₇₉	C₂²	1	2	yes	—
27	C₈×D₄×C₇₉	—	2	1	no	—
28	C₈×D₄×C₇₉	C₂²	1	2	yes	—
29	#423×C₇₉	C₂²	1	1	yes	characteristic
30	#408×C₇₉	—	2	1	no	—
31	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	C₂²×Dic₁₅₈	—	2	2	no	—
33	#14579161	—	2	2	no	—
34	#14579161	—	2	2	no	—
35	#14579161	—	2	2	no	—
36	#14579161	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₇₉	—	2	2	no	—
38	C₂×C₄⋊Dic₇₉	—	2	2	no	—
39	#14579095	—	2	2	no	—
40	#14579095	—	2	2	no	—
41	#14579095	—	2	2	no	—
42	#14579095	—	2	2	no	—
43	#14579092	—	2	8	no	—
44	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
45	#14579082	—	2	8	no	—
46	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
47	C₄×C₇₉⋊C₈	—	2	2	no	—
48	C₂×#41×C₇₉	—	2	1	no	—
49	#131×C₇₉	—	4	4	no	—
50	C₂×M₁₆×C₇₉	—	2	1	no	—
51	C₂×M₁₆×C₇₉	—	2	2	no	—
52	C₂²×C₈×C₇₉	—	2	2	no	—
53	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
55	C₄⋊Q₈×C₇₉	D₄	1	1	yes	characteristic
56	#127×C₇₉	D₄	1	1	yes	characteristic
57	#124×C₇₉	—	2	1	no	—
58	#121×C₇₉	—	2	1	no	—
59	#121×C₇₉	D₄	1	2	yes	—
60	C₄×D₄×C₇₉	—	2	1	no	—
61	C₄×D₄×C₇₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₇₉	—	2	1	no	—
64	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
65	C₂²⋊C₈×C₇₉	—	2	2	no	—
66	C₂²⋊C₈×C₇₉	—	2	2	no	—
67	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
68	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
69	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₅₈	—	2	4	no	—
72	C₂×Dic₁₅₈	—	4	2	no	—
73	C₂×Dic₁₅₈	—	8	4	no	—
74	C₂²×Dic₇₉	—	4	2	no	—
75	C₂×C₇₉⋊C₈	—	2	4	no	—
76	C₄⋊Dic₇₉	—	2	4	no	—
77	C₄⋊Dic₇₉	—	2	4	no	—
78	C₄⋊Dic₇₉	—	4	2	no	—
79	C₄⋊Dic₇₉	—	4	2	no	—
80	#41×C₇₉	—	4	2	no	—
81	#41×C₇₉	—	4	2	no	—
82	C₂×Q₈×C₇₉	—	2	1	no	—
83	C₂×D₄×C₇₉	—	2	2	no	—
84	C₂×D₄×C₇₉	—	2	1	no	—
85	C₂×D₄×C₇₉	—	2	1	no	—
86	C₂×D₄×C₇₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₇₉	—	2	2	no	—
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
91	M₁₆×C₇₉	—	4	4	no	—
92	M₁₆×C₇₉	—	4	2	no	—
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	—	2	2	no	—
95	C₂×C₈×C₇₉	—	4	4	no	—
96	C₂×C₈×C₇₉	SD₁₆	1	4	yes	—
97	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₄⋊C₄×C₇₉	—	2	1	no	—
102	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
103	C₂²⋊C₄×C₇₉	—	2	2	no	—
104	C₂²⋊C₄×C₇₉	—	2	2	no	—
105	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇₉	#41	1	1	yes	characteristic
108	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
109	C₂³×C₇₉	—	2	2	no	—
110	C₂³×C₇₉	#121	1	1	yes	characteristic
111	Q₈×C₇₉	—	4	2	no	—
112	D₄×C₇₉	—	2	4	no	—
113	D₄×C₇₉	—	4	4	no	—
114	D₄×C₇₉	—	4	2	no	—
115	D₄×C₇₉	—	4	2	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	2	no	—
119	C₂×C₄×C₇₉	—	4	2	no	—
120	C₂×C₄×C₇₉	—	4	2	no	—
121	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₇₉	#137	1	2	yes	—
126	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
129	C₂×Dic₇₉	—	4	4	no	—
130	C₂×Dic₇₉	—	8	4	no	—
131	Dic₁₅₈	—	8	4	no	—
132	Dic₁₅₈	—	8	4	no	—
133	C₈×C₇₉	—	2	1	no	—
134	C₈×C₇₉	—	2	1	no	—
135	C₈×C₇₉	—	4	2	no	—
136	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
137	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
138	C₇₉⋊C₈	—	4	4	no	—
139	C₂²×C₇₉	—	2	1	no	—
140	C₂²×C₇₉	—	2	1	no	—
141	C₂²×C₇₉	—	4	2	no	—
142	C₂²×C₇₉	—	4	2	no	—
143	C₂²×C₇₉	—	4	2	no	—
144	C₂²×C₇₉	#461	1	2	yes	—
145	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
146	C₄×C₇₉	—	2	1	no	—
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	4	2	no	—
149	C₄×C₇₉	#461	1	2	yes	—
150	C₄×C₇₉	#472	1	2	yes	—
151	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
152	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
153	C₄×C₇₉	#578	1	2	yes	—
154	Dic₇₉	—	8	4	no	—
155	#19193	—	79	1	no	maximal, Sylow
156	C₂×C₇₉	—	2	2	no	—
157	C₂×C₇₉	—	8	2	no	—
158	C₂×C₇₉	C₂×#461	1	1	yes	characteristic
159	C₂×C₇₉	#3059	1	1	yes	characteristic
160	C₂×C₇₉	C₂×#472	1	1	yes	characteristic
161	#3227	—	79	2	no	—
162	C₂×#500	—	79	1	no	—
163	#2931	D₇₉	1	1	yes	characteristic
164	#1695	—	79	4	no	—
165	#1675	—	79	4	no	—
166	#1564	—	79	1	no	—
167	#1548	—	79	2	no	—
168	C₇₉	#19193	1	1	yes	characteristic, complemented, Sylow
169	C₂×#131	—	2	1	no	—
170	#532	C₂×D₇₉	1	1	yes	characteristic
171	C₂×C₄⋊Q₈	—	79	2	no	—
172	#500	—	79	2	no	—
173	#500	—	79	2	no	—
174	#500	—	158	1	no	—
175	#500	—	158	1	no	—
176	#498	—	158	2	no	—
177	#486	—	158	2	no	—
178	#481	—	158	4	no	—
179	#476	—	79	8	no	—
180	#436	—	2	1	no	—
181	#436	C₂×D₇₉	1	2	yes	—
182	C₈×D₄	—	2	1	no	—
183	C₈×D₄	C₂×D₇₉	1	2	yes	—
184	#431	—	79	2	no	—
185	C₂×#107	—	158	1	no	—
186	C₂×#107	—	158	1	no	—
187	#423	C₂×D₇₉	1	1	yes	characteristic
188	#408	—	2	1	no	—
189	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
190	#335	—	79	2	no	—
191	#334	—	79	2	no	—
192	#333	—	79	8	no	—
193	C₂×#41	—	2	1	no	—
194	C₂²×Q₈	—	158	2	no	—
195	#131	—	4	4	no	—
196	C₂×M₁₆	—	2	1	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
201	C₄⋊Q₈	—	79	4	no	—
202	C₄⋊Q₈	—	158	2	no	—
203	C₄⋊Q₈	—	158	2	no	—
204	C₄⋊Q₈	—	158	2	no	—
205	C₄⋊Q₈	D₃₁₆	1	1	yes	characteristic
206	#127	D₃₁₆	1	1	yes	characteristic
207	#124	—	2	1	no	—
208	#121	—	2	1	no	—
209	#121	D₃₁₆	1	2	yes	—
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	C₂²×D₇₉	1	2	yes	—
212	C₂×C₄⋊C₄	—	158	2	no	—
213	C₂×C₄⋊C₄	—	158	2	no	—
214	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
215	#107	—	158	2	no	—
216	#107	—	158	2	no	—
217	#107	—	158	2	no	—
218	#107	—	158	2	no	—
219	#106	—	158	8	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	79	4	no	—
222	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
223	#102	—	158	8	no	—
224	C₂²⋊C₈	—	2	2	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	C₂²⋊C₈	—	158	2	no	—
227	C₄×C₈	—	158	2	no	—
228	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
229	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
232	#41	—	4	2	no	—
233	#41	—	4	2	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	316	2	no	—
236	C₂×Q₈	—	632	4	no	—
237	C₂×Q₈	—	158	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	1	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	C₂×D₃₁₆	1	2	yes	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	316	2	no	—
244	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
246	C₂²×C₄	#11929624	1	1	yes	characteristic
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	4	4	no	—
252	C₂×C₈	—	158	4	no	—
253	C₂×C₈	#11929612	1	4	yes	—
254	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	316	2	no	—
260	C₄⋊C₄	—	316	2	no	—
261	C₄⋊C₄	—	158	4	no	—
262	C₄⋊C₄	—	158	4	no	—
263	C₄⋊C₄	C₂×D₃₁₆	1	2	yes	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
267	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
268	C₄²	#11929624	1	1	yes	characteristic
269	C₄²	C₂³×D₇₉	1	1	yes	characteristic
270	C₂³	—	2	2	no	—
271	C₂³	#14579135	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	632	4	no	—
274	Q₈	—	632	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	4	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	316	4	no	—
285	C₂×C₄	—	632	4	no	—
286	C₂×C₄	#14579135	1	1	yes	characteristic
287	C₂×C₄	#14579135	1	1	yes	characteristic
288	C₂×C₄	#14579135	1	1	yes	characteristic
289	C₂×C₄	C₂×#11929612	1	2	yes	—
290	C₂×C₄	#14579191	1	2	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
293	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	4	2	no	—
297	C₈	—	316	4	no	—
298	C₈	C₂×#11929612	1	2	yes	—
299	C₈	C₂×#11929612	1	2	yes	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	#19098768	1	2	yes	—
306	C₂²	C₂×#14579135	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	632	4	no	—
311	C₄	#19098768	1	2	yes	—
312	C₄	#19098833	1	2	yes	—
313	C₄	C₂²×#11929612	1	1	yes	characteristic
314	C₄	C₂×#14579191	1	1	yes	characteristic
315	C₄	#19099380	1	2	yes	—
316	C₂	—	2	2	no	—
317	C₂	—	8	2	no	—
318	C₂	C₂×#19098768	1	1	yes	characteristic
319	C₂	#23773055	1	1	yes	characteristic
320	C₂	C₂×#19098833	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579135
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#2931	C₂
socle	C₂²×C₇₉	C₂×#121
prime radical	#532×C₇₉	C₂²
hypercenter	#2931	D₇₉
hypocenter	C₇₉	#19193
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
2. C₂×#19098968	C₂	semidirect
3. #23762923	C₂	semidirect
4. #23762799	C₂	semidirect
5. #23762148	C₂	semidirect
6. #23762082	C₂	semidirect
168. C₇₉	#19193	semidirect

Maximal subgroups

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

subgroup	index
#23774525	2
C₂×#19098968	2
#23762923	2
#23762799	2
#23762148	2
#23762082	2
C₇₉×#2931	2
#19193	79