Subgroups — #34445908

Order 20224.

482 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775824	C₂	1	1	yes	maximal, characteristic
2	#23775792	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765673	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765611	C₂	1	2	yes	maximal, complemented
5	#23765579	C₂	1	2	yes	maximal, complemented
6	#23765553	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765512	C₂	1	2	yes	maximal, complemented
8	C₄×#14579105	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765231	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765187	C₂	1	2	yes	maximal, complemented
11	C₇₉×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099278	—	2	1	no	—
13	#19099274	—	2	1	no	—
14	C₂×#14579105	C₂²	1	1	yes	characteristic
15	C₂×#14579100	C₂²	1	2	yes	—
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098900	—	2	2	no	—
18	#19098885	—	2	2	no	—
19	#19098855	—	2	1	no	—
20	#19098853	—	2	1	no	—
21	#19098805	C₂²	1	2	yes	—
22	#19098796	C₂²	1	2	yes	—
23	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
24	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
25	#19098600	C₂²	1	2	yes	—
26	#107×D₇₉	C₂²	1	2	yes	—
27	#19098390	C₂²	1	1	yes	characteristic
28	#19098389	C₂²	1	2	yes	—
29	#19098388	C₂²	1	2	yes	—
30	#19098388	C₂²	1	2	yes	—
31	#19098368	C₂²	1	2	yes	—
32	#19098366	C₂²	1	1	yes	characteristic
33	#19098364	C₂²	1	1	yes	characteristic
34	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
36	#500×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
38	#367×C₇₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#357×C₇₉	C₂²	1	2	yes	—
41	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₇₉	C₂³	1	2	yes	—
43	#14579161	—	2	1	no	—
44	#14579152	—	4	1	no	—
45	#14579147	—	4	2	no	—
46	#14579141	—	2	2	no	—
47	#14579137	—	4	2	no	—
48	#14579135	—	4	1	no	—
49	#14579132	—	2	2	no	—
50	C₄×Dic₁₅₈	—	2	1	no	—
51	C₄×D₃₁₆	D₄	1	1	yes	characteristic
52	#14579123	—	4	1	no	—
53	#14579120	—	4	1	no	—
54	#14579119	—	2	2	no	—
55	C₄⋊C₄×D₇₉	—	2	2	no	—
56	#14579105	—	2	2	no	—
57	#14579100	—	2	4	no	—
58	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
60	#14579095	—	2	1	no	—
61	#14579095	C₂³	1	2	yes	—
62	#14579095	C₂³	1	1	yes	characteristic
63	#14579094	—	2	2	no	—
64	#14579092	—	2	2	no	—
65	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
66	#14579086	—	2	1	no	—
67	#14579084	—	2	2	no	—
68	#14579082	—	2	2	no	—
69	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
70	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₇₉	—	2	1	no	—
72	C₄×D₄×C₇₉	—	2	1	no	—
73	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
74	#107×C₇₉	—	2	1	no	—
75	#107×C₇₉	C₂³	1	1	yes	characteristic
76	#107×C₇₉	C₂³	1	2	yes	—
77	#102×C₇₉	—	2	1	no	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅₈	—	2	1	no	—
80	C₂×Dic₁₅₈	—	4	1	no	—
81	C₂×D₃₁₆	D₈	1	1	yes	characteristic
82	C₂×C₂²⋊D₇₉	—	4	2	no	—
83	C₂×C₄×D₇₉	—	4	2	no	—
84	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
85	Dic₃₁₆	—	2	1	no	—
86	Dic₃₁₆	—	2	1	no	—
87	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
88	C₈×D₇₉	—	2	2	no	—
89	C₈×D₇₉	—	2	2	no	—
90	C₄⋊Dic₇₉	—	2	2	no	—
91	C₄⋊Dic₇₉	—	2	1	no	—
92	C₄⋊Dic₇₉	—	4	2	no	—
93	C₄⋊Dic₇₉	—	4	1	no	—
94	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
95	Dic₇₉⋊C₄	—	4	2	no	—
96	Dic₇₉⋊C₄	—	4	1	no	—
97	C₂²⋊Dic₇₉	—	4	2	no	—
98	(C₂×D₇₉)⋊C₄	—	4	2	no	—
99	(C₂×D₇₉)⋊C₄	—	4	2	no	—
100	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
101	C₄×Dic₇₉	—	4	2	no	—
102	C₄×Dic₇₉	—	4	1	no	—
103	C₇₉⋊C₁₆	—	2	1	no	—
104	C₇₉⋊C₁₆	#41	1	2	yes	—
105	C₂×Q₈×C₇₉	—	4	1	no	—
106	C₂×D₄×C₇₉	—	2	1	no	—
107	C₂²×C₄×C₇₉	—	4	1	no	—
108	D₈×C₇₉	—	2	1	no	—
109	D₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₇₉	—	2	1	no	—
113	C₄⋊C₄×C₇₉	—	2	2	no	—
114	C₄⋊C₄×C₇₉	—	4	1	no	—
115	C₂²⋊C₄×C₇₉	—	4	1	no	—
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₉	—	2	2	no	—
118	C₂³×C₇₉	—	4	1	no	—
119	C₂²×D₇₉	C₂×D₈	1	2	yes	—
120	Q₈×C₇₉	—	16	1	no	—
121	D₄×C₇₉	—	4	1	no	—
122	D₄×C₇₉	—	4	1	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	8	2	no	—
126	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₇₉	—	16	2	no	—
130	C₂×Dic₇₉	—	4	1	no	—
131	C₂×Dic₇₉	—	4	2	no	—
132	C₂×Dic₇₉	—	4	1	no	—
133	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
134	D₃₁₆	—	2	2	no	—
135	C₄×D₇₉	—	2	2	no	—
136	C₄×D₇₉	—	2	2	no	—
137	C₄×D₇₉	—	16	2	no	—
138	Dic₁₅₈	—	4	1	no	—
139	Dic₁₅₈	—	4	1	no	—
140	Dic₁₅₈	—	16	1	no	—
141	C₈×C₇₉	—	2	1	no	—
142	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₇₉⋊C₈	—	2	2	no	—
145	C₂²×C₇₉	—	8	1	no	—
146	C₂²×C₇₉	—	8	1	no	—
147	C₂²×C₇₉	—	8	1	no	—
148	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₇₉	—	2	2	no	—
150	C₂×D₇₉	—	2	2	no	—
151	C₂×D₇₉	—	2	2	no	—
152	C₄×C₇₉	—	2	1	no	—
153	C₄×C₇₉	—	8	1	no	—
154	C₄×C₇₉	—	8	2	no	—
155	C₄×C₇₉	#510	1	2	yes	—
156	C₄×C₇₉	#547	1	1	yes	characteristic
157	C₄×C₇₉	#578	1	1	yes	characteristic
158	Dic₇₉	—	2	2	no	—
159	Dic₇₉	—	8	1	no	—
160	Dic₇₉	—	8	2	no	—
161	Dic₇₉	—	8	1	no	—
162	#33924	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	—	16	1	no	—
164	C₂×C₇₉	#3334	1	1	yes	characteristic
165	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
166	C₂×C₇₉	#3415	1	1	yes	characteristic
167	D₇₉	—	4	2	no	—
168	#3341	—	79	1	no	—
169	#3338	—	79	1	no	—
170	#2252	—	79	1	no	—
171	#2237	—	79	2	no	—
172	#2232	—	79	2	no	—
173	#2227	—	79	1	no	—
174	#2226	D₇₉	1	1	yes	characteristic, complemented
175	#2220	—	79	2	no	—
176	C₄×SD₃₂	—	79	1	no	—
177	#2169	—	79	1	no	—
178	#2159	—	79	2	no	—
179	C₇₉	#33924	1	1	yes	characteristic, complemented, Sylow
180	#555	—	158	1	no	—
181	#554	—	158	1	no	—
182	C₂×SD₃₂	—	79	1	no	—
183	C₂×M₃₂	—	79	2	no	—
184	#500	—	79	1	no	—
185	#500	C₂×D₇₉	1	1	yes	characteristic
186	#484	—	158	2	no	—
187	#482	—	158	2	no	—
188	#477	—	158	1	no	—
189	#476	—	158	1	no	—
190	#467	—	79	2	no	—
191	#466	—	79	2	no	—
192	C₄×Q₁₆	—	79	1	no	—
193	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
194	C₈×D₄	—	79	1	no	—
195	#427	—	79	2	no	—
196	C₂×#107	—	79	2	no	—
197	#367	—	79	1	no	—
198	#367	C₂×D₇₉	1	2	yes	—
199	#366	—	79	2	no	—
200	#366	—	79	2	no	—
201	#366	—	79	2	no	—
202	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
203	#358	—	79	2	no	—
204	#358	—	79	1	no	—
205	#357	—	79	1	no	—
206	#357	C₂×D₇₉	1	2	yes	—
207	C₂²⋊C₁₆	—	79	2	no	—
208	C₄×C₁₆	—	79	1	no	—
209	C₂×Q₁₆	—	79	1	no	—
210	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
211	C₂²×C₈	—	79	2	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	158	1	no	—
214	#125	—	316	1	no	—
215	#124	—	316	2	no	—
216	C₂²⋊Q₈	—	316	2	no	—
217	C₂²⋊Q₈	—	158	2	no	—
218	#121	—	316	1	no	—
219	#121	—	158	2	no	—
220	C₄×Q₈	—	158	1	no	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	316	1	no	—
223	C₄×D₄	—	79	1	no	—
224	#117	—	316	1	no	—
225	#117	—	158	2	no	—
226	C₂×C₄⋊C₄	—	158	2	no	—
227	SD₃₂	—	158	2	no	—
228	M₃₂	—	158	4	no	—
229	C₂×C₁₆	—	79	1	no	—
230	C₂×C₁₆	—	79	1	no	—
231	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
232	#107	—	2	1	no	—
233	#107	—	79	1	no	—
234	#107	—	79	2	no	—
235	#107	—	158	2	no	—
236	#107	—	158	1	no	—
237	#107	C₂²×D₇₉	1	2	yes	—
238	#107	C₂²×D₇₉	1	1	yes	characteristic
239	#106	—	158	2	no	—
240	C₄⋊C₈	—	79	1	no	—
241	#103	—	158	2	no	—
242	#103	—	158	1	no	—
243	#102	—	2	1	no	—
244	#102	—	158	2	no	—
245	C₂²⋊C₈	—	79	2	no	—
246	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	316	1	no	—
249	C₂×Q₈	—	158	1	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	316	2	no	—
252	C₂×D₄	—	79	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	316	2	no	—
255	C₂²×C₄	—	79	2	no	—
256	Q₁₆	—	158	1	no	—
257	Q₁₆	—	158	1	no	—
258	D₈	—	2	1	no	—
259	D₈	—	2	1	no	—
260	C₂×C₈	—	79	2	no	—
261	C₂×C₈	—	158	2	no	—
262	C₂×C₈	—	158	2	no	—
263	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	316	2	no	—
269	C₄⋊C₄	—	316	2	no	—
270	C₄⋊C₄	—	316	1	no	—
271	C₄⋊C₄	—	316	1	no	—
272	C₄⋊C₄	—	79	1	no	—
273	C₄⋊C₄	—	158	2	no	—
274	C₄⋊C₄	—	158	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	316	2	no	—
277	C₂²⋊C₄	—	316	2	no	—
278	C₂²⋊C₄	—	316	2	no	—
279	C₂²⋊C₄	—	79	2	no	—
280	C₄²	—	316	2	no	—
281	C₄²	—	316	1	no	—
282	C₄²	D₄×D₇₉	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	79	2	no	—
285	C₁₆	—	158	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	—	79	2	no	—
288	Q₈	—	16	1	no	—
289	Q₈	—	316	1	no	—
290	Q₈	—	316	1	no	—
291	Q₈	—	1264	1	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	158	2	no	—
295	D₄	—	1264	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	316	1	no	—
300	C₂×C₄	—	316	2	no	—
301	C₂×C₄	—	316	1	no	—
302	C₂×C₄	—	79	2	no	—
303	C₂×C₄	—	158	2	no	—
304	C₂×C₄	—	158	2	no	—
305	C₂×C₄	—	1264	2	no	—
306	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
307	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
308	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	158	2	no	—
311	C₈	#41×D₇₉	1	1	yes	characteristic
312	C₈	#14579209	1	1	yes	characteristic
313	C₂²	—	8	1	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	158	2	no	—
317	C₂²	—	158	2	no	—
318	C₂²	—	158	2	no	—
319	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	632	1	no	—
324	C₄	—	632	1	no	—
325	C₄	—	632	2	no	—
326	C₄	—	158	2	no	—
327	C₄	#19099002	1	2	yes	—
328	C₄	#19099215	1	1	yes	characteristic
329	C₄	#19099379	1	1	yes	characteristic
330	C₂	—	16	1	no	—
331	C₂	—	316	2	no	—
332	C₂	#23775746	1	1	yes	characteristic
333	C₂	C₂×#19099002	1	1	yes	characteristic
334	C₂	#23776540	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2226	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23775824	C₂
hypercenter	#2226	D₇₉
hypocenter	C₇₉	#33924
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
2. #23775792	C₂	semidirect
3. #23765673	C₂	semidirect
4. #23765611	C₂	semidirect
5. #23765579	C₂	semidirect
6. #23765553	C₂	semidirect
7. #23765512	C₂	semidirect
8. C₄×#14579105	C₂	semidirect
9. #23765231	C₂	semidirect
10. #23765187	C₂	semidirect
11. C₇₉×#2226	C₂	semidirect
174. #2226	D₇₉	semidirect
179. C₇₉	#33924	semidirect

Maximal subgroups

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

subgroup	index
#23775824	2
#23775792	2
#23765673	2
#23765611	2
#23765579	2
#23765553	2
#23765512	2
C₄×#14579105	2
#23765231	2
#23765187	2
C₇₉×#2226	2
#33924	79