Subgroups — #34445905

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	#23775823	C₂	1	1	yes	maximal, characteristic
2	#23765673	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765605	C₂	1	2	yes	maximal, complemented
4	#23765580	C₂	1	2	yes	maximal, complemented
5	#23765552	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765544	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765515	C₂	1	2	yes	maximal, complemented
8	C₄×#14579104	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765232	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765186	C₂	1	2	yes	maximal, complemented
11	C₇₉×#3338	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099278	—	2	1	no	—
13	C₂×#14579104	C₂²	1	1	yes	characteristic
14	C₂×#14579101	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098888	—	2	2	no	—
17	#19098854	—	2	1	no	—
18	#19098799	C₂²	1	2	yes	—
19	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
20	#19098601	C₂²	1	2	yes	—
21	#19098390	C₂²	1	1	yes	characteristic
22	#19098390	C₂²	1	2	yes	—
23	#19098388	C₂²	1	2	yes	—
24	#19098388	C₂²	1	2	yes	—
25	#19098388	C₂²	1	2	yes	—
26	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
27	#19098367	C₂²	1	1	yes	characteristic
28	#19098367	C₂²	1	2	yes	—
29	#19098363	C₂²	1	1	yes	characteristic
30	#19098363	C₂²	1	2	yes	—
31	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	#554×C₇₉	—	2	1	no	—
34	#500×C₇₉	C₂²	1	1	yes	characteristic
35	#484×C₇₉	—	2	2	no	—
36	#477×C₇₉	—	2	1	no	—
37	#467×C₇₉	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×#107×C₇₉	C₂²	1	2	yes	—
41	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
42	#14579161	—	2	1	no	—
43	#14579147	—	4	2	no	—
44	#14579135	—	2	2	no	—
45	#14579132	—	4	1	no	—
46	C₄×D₃₁₆	—	2	1	no	—
47	#14579120	—	2	2	no	—
48	#14579119	—	4	1	no	—
49	#14579104	—	2	2	no	—
50	#14579101	—	2	4	no	—
51	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
54	#14579095	—	2	1	no	—
55	#14579095	C₂³	1	2	yes	—
56	#14579095	C₂³	1	1	yes	characteristic
57	#14579093	—	2	2	no	—
58	#14579083	—	2	1	no	—
59	#14579083	—	2	2	no	—
60	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₇₉	—	2	1	no	—
63	#125×C₇₉	—	4	1	no	—
64	C₂²⋊Q₈×C₇₉	—	2	2	no	—
65	C₂²⋊Q₈×C₇₉	—	4	2	no	—
66	C₄×Q₈×C₇₉	—	2	1	no	—
67	C₄×D₄×C₇₉	—	4	1	no	—
68	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
70	#107×C₇₉	—	2	2	no	—
71	#107×C₇₉	—	2	1	no	—
72	#107×C₇₉	C₂³	1	1	yes	characteristic
73	#107×C₇₉	C₂³	1	2	yes	—
74	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
75	#103×C₇₉	—	2	1	no	—
76	#103×C₇₉	—	2	2	no	—
77	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅₈	—	4	1	no	—
80	C₂×D₃₁₆	—	2	1	no	—
81	C₂×C₂²⋊D₇₉	—	4	2	no	—
82	C₂×C₄×D₇₉	—	4	1	no	—
83	D₆₃₂	—	2	1	no	—
84	D₆₃₂	—	2	1	no	—
85	C₄⋊Dic₇₉	—	2	1	no	—
86	C₄⋊Dic₇₉	—	2	2	no	—
87	C₄⋊Dic₇₉	—	4	1	no	—
88	C₂²⋊Dic₇₉	—	4	2	no	—
89	(C₂×D₇₉)⋊C₄	—	4	2	no	—
90	(C₂×D₇₉)⋊C₄	—	4	1	no	—
91	C₄×Dic₇₉	—	4	2	no	—
92	C₇₉⋊C₁₆	—	2	2	no	—
93	C₇₉⋊C₁₆	—	2	1	no	—
94	C₇₉⋊C₁₆	#41	1	2	yes	—
95	C₂×Q₈×C₇₉	—	2	1	no	—
96	C₂×Q₈×C₇₉	—	4	1	no	—
97	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₇₉	—	4	2	no	—
99	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
100	Q₁₆×C₇₉	—	2	1	no	—
101	Q₁₆×C₇₉	—	2	1	no	—
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	—	2	2	no	—
104	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₇₉	—	2	1	no	—
108	C₄⋊C₄×C₇₉	—	2	2	no	—
109	C₄⋊C₄×C₇₉	—	4	2	no	—
110	C₄⋊C₄×C₇₉	—	4	1	no	—
111	C₄⋊C₄×C₇₉	—	4	1	no	—
112	C₄⋊C₄×C₇₉	—	4	2	no	—
113	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
114	C₂²⋊C₄×C₇₉	—	4	2	no	—
115	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
116	C₄²×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₇₉	C₂×D₈	1	2	yes	—
119	C₂²×D₇₉	—	4	1	no	—
120	Q₈×C₇₉	—	4	1	no	—
121	Q₈×C₇₉	—	4	1	no	—
122	Q₈×C₇₉	—	16	1	no	—
123	D₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	2	2	no	—
125	C₂×C₄×C₇₉	—	2	2	no	—
126	C₂×C₄×C₇₉	—	4	1	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	2	no	—
129	C₂×C₄×C₇₉	—	16	2	no	—
130	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₇₉	—	16	2	no	—
135	C₂×Dic₇₉	—	4	1	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	D₃₁₆	—	4	1	no	—
138	D₃₁₆	—	4	1	no	—
139	C₄×D₇₉	—	8	2	no	—
140	Dic₁₅₈	—	16	1	no	—
141	C₈×C₇₉	—	2	2	no	—
142	C₈×C₇₉	—	2	1	no	—
143	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²×C₇₉	—	2	2	no	—
146	C₂²×C₇₉	—	2	2	no	—
147	C₂²×C₇₉	—	2	2	no	—
148	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₇₉	—	8	1	no	—
150	C₂×D₇₉	—	8	1	no	—
151	C₂×D₇₉	—	8	1	no	—
152	C₄×C₇₉	—	2	2	no	—
153	C₄×C₇₉	—	2	1	no	—
154	C₄×C₇₉	—	8	1	no	—
155	C₄×C₇₉	—	8	1	no	—
156	C₄×C₇₉	—	8	2	no	—
157	C₄×C₇₉	#510	1	2	yes	—
158	C₄×C₇₉	#547	1	1	yes	characteristic
159	C₄×C₇₉	#578	1	1	yes	characteristic
160	Dic₇₉	—	8	1	no	—
161	Dic₇₉	—	8	2	no	—
162	#33924	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	—	4	2	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	—	16	1	no	—
168	#3341	—	79	1	no	—
169	#3338	D₇₉	1	1	yes	characteristic, complemented
170	#2252	—	79	1	no	—
171	#2237	—	79	2	no	—
172	#2232	—	79	2	no	—
173	#2227	—	79	1	no	—
174	#2226	—	79	1	no	—
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	—	2	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	—	2	2	no	—
187	#482	—	158	2	no	—
188	#477	—	2	1	no	—
189	#476	—	158	1	no	—
190	#467	C₂×D₇₉	1	2	yes	—
191	#466	—	79	2	no	—
192	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
193	C₄×D₈	—	79	1	no	—
194	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
195	#427	—	79	2	no	—
196	C₂×#107	C₂×D₇₉	1	2	yes	—
197	#367	—	79	1	no	—
198	#367	—	79	2	no	—
199	#366	—	79	2	no	—
200	#366	—	79	2	no	—
201	#366	—	79	2	no	—
202	C₄⋊C₁₆	—	79	1	no	—
203	#358	—	79	1	no	—
204	#358	—	79	2	no	—
205	#357	—	79	1	no	—
206	#357	—	79	2	no	—
207	C₂²⋊C₁₆	—	79	2	no	—
208	C₄×C₁₆	—	79	1	no	—
209	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
210	C₂×D₈	—	79	1	no	—
211	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	158	1	no	—
214	#125	—	4	1	no	—
215	#124	—	316	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	#121	—	316	1	no	—
219	#121	—	158	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×D₄	—	4	1	no	—
222	C₄×D₄	—	158	1	no	—
223	C₄×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
224	#117	—	316	1	no	—
225	#117	—	158	2	no	—
226	C₂×C₄⋊C₄	—	2	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	—	79	2	no	—
232	#107	—	2	2	no	—
233	#107	—	2	1	no	—
234	#107	—	79	2	no	—
235	#107	—	79	1	no	—
236	#107	—	158	1	no	—
237	#107	C₂²×D₇₉	1	1	yes	characteristic
238	#107	C₂²×D₇₉	1	2	yes	—
239	#106	—	158	2	no	—
240	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
241	#103	—	2	1	no	—
242	#103	—	2	2	no	—
243	#102	—	158	1	no	—
244	#102	—	158	2	no	—
245	C₂²⋊C₈	C₂²⋊D₇₉	1	2	yes	—
246	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
247	C₂×Q₈	—	2	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	316	1	no	—
250	C₂×D₄	—	316	2	no	—
251	C₂×D₄	—	158	1	no	—
252	C₂×D₄	#11929609	1	1	yes	characteristic
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	—	316	1	no	—
255	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	1	no	—
258	D₈	—	158	1	no	—
259	D₈	—	158	1	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
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	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	316	1	no	—
272	C₄⋊C₄	—	158	1	no	—
273	C₄⋊C₄	—	158	2	no	—
274	C₄⋊C₄	#11929609	1	1	yes	characteristic
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	316	2	no	—
277	C₂²⋊C₄	—	316	2	no	—
278	C₂²⋊C₄	—	316	1	no	—
279	C₂²⋊C₄	#11929609	1	2	yes	—
280	C₄²	—	4	1	no	—
281	C₄²	—	316	2	no	—
282	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
283	C₁₆	—	79	2	no	—
284	C₁₆	—	158	2	no	—
285	C₁₆	—	158	1	no	—
286	C₂³	—	316	1	no	—
287	C₂³	C₂×#11929609	1	2	yes	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	1264	1	no	—
292	D₄	—	2	2	no	—
293	D₄	—	316	1	no	—
294	D₄	—	316	1	no	—
295	D₄	—	1264	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	16	2	no	—
302	C₂×C₄	—	316	1	no	—
303	C₂×C₄	—	316	2	no	—
304	C₂×C₄	—	632	2	no	—
305	C₂×C₄	C₂×#11929609	1	2	yes	—
306	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
307	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	2	1	no	—
311	C₈	#41×D₇₉	1	1	yes	characteristic
312	C₈	#14579208	1	1	yes	characteristic
313	C₂²	—	2	2	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	632	1	no	—
317	C₂²	—	632	1	no	—
318	C₂²	—	632	1	no	—
319	C₂²	C₂²×#11929609	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	632	1	no	—
326	C₄	—	632	2	no	—
327	C₄	#19099004	1	2	yes	—
328	C₄	#19099212	1	1	yes	characteristic
329	C₄	#19099378	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	1264	1	no	—
332	C₂	#23775743	1	1	yes	characteristic
333	C₂	C₂×#19099004	1	1	yes	characteristic
334	C₂	#23776539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3338	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23775823	C₂
hypercenter	#3338	D₇₉
hypocenter	C₇₉	#33924
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3338

Complements

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

N	H ≅ G/N	type
2. #23765673	C₂	semidirect
3. #23765605	C₂	semidirect
4. #23765580	C₂	semidirect
5. #23765552	C₂	semidirect
6. #23765544	C₂	semidirect
7. #23765515	C₂	semidirect
8. C₄×#14579104	C₂	semidirect
9. #23765232	C₂	semidirect
10. #23765186	C₂	semidirect
11. C₇₉×#3338	C₂	semidirect
169. #3338	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
#23775823	2
#23765673	2
#23765605	2
#23765580	2
#23765552	2
#23765544	2
#23765515	2
C₄×#14579104	2
#23765232	2
#23765186	2
C₇₉×#3338	2
#33924	79