Subgroups — #34445897

Order 20224.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775832	C₂	1	1	yes	maximal, characteristic
2	#23775792	C₂	1	1	yes	maximal, characteristic
3	#23765680	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765610	C₂	1	2	yes	maximal, complemented
5	#23765596	C₂	1	2	yes	maximal, complemented
6	#23765543	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765507	C₂	1	2	yes	maximal, complemented
8	C₄×#14579107	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765231	C₂	1	1	yes	maximal, characteristic
10	#23765185	C₂	1	2	yes	maximal, complemented
11	C₇₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099286	—	2	1	no	—
13	#19099274	—	2	1	no	—
14	C₂×#14579107	C₂²	1	1	yes	characteristic
15	C₂×#14579100	C₂²	1	2	yes	—
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098903	—	2	2	no	—
18	#19098900	—	2	2	no	—
19	#19098855	—	2	1	no	—
20	#19098848	—	2	1	no	—
21	#19098805	C₂²	1	2	yes	—
22	#19098801	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	#19098391	C₂²	1	2	yes	—
28	#19098390	C₂²	1	2	yes	—
29	#19098390	C₂²	1	2	yes	—
30	#19098388	C₂²	1	1	yes	characteristic
31	#19098368	C₂²	1	2	yes	—
32	#19098367	C₂²	1	1	yes	characteristic
33	#19098366	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×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	#366×C₇₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#358×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	#14579156	—	4	2	no	—
45	#14579152	—	4	1	no	—
46	#14579149	—	4	1	no	—
47	#14579141	—	2	2	no	—
48	#14579137	—	2	2	no	—
49	#14579137	—	4	2	no	—
50	C₄×Dic₁₅₈	—	2	1	no	—
51	C₄×Dic₁₅₈	—	4	1	no	—
52	C₄×D₃₁₆	D₄	1	1	yes	characteristic
53	#14579123	—	4	1	no	—
54	#14579119	—	2	2	no	—
55	C₄⋊C₄×D₇₉	—	2	2	no	—
56	#14579107	—	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	#14579085	—	2	2	no	—
68	#14579084	—	2	2	no	—
69	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
70	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₇₉	—	2	1	no	—
72	C₄×Q₈×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	#103×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×C₄×D₇₉	—	4	2	no	—
83	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
84	Dic₃₁₆	—	2	1	no	—
85	Dic₃₁₆	—	2	1	no	—
86	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
87	C₈×D₇₉	—	2	2	no	—
88	C₈×D₇₉	—	2	2	no	—
89	C₄⋊Dic₇₉	—	2	2	no	—
90	C₄⋊Dic₇₉	—	2	1	no	—
91	C₄⋊Dic₇₉	—	4	2	no	—
92	C₄⋊Dic₇₉	—	4	1	no	—
93	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
94	Dic₇₉⋊C₄	—	4	2	no	—
95	Dic₇₉⋊C₄	—	4	2	no	—
96	Dic₇₉⋊C₄	—	4	1	no	—
97	(C₂×D₇₉)⋊C₄	—	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	—	2	1	no	—
106	C₂×Q₈×C₇₉	—	4	1	no	—
107	Q₁₆×C₇₉	—	2	1	no	—
108	Q₁₆×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₇₉	—	2	1	no	—
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₄⋊C₄×C₇₉	—	4	1	no	—
114	C₄⋊C₄×C₇₉	—	4	1	no	—
115	C₄²×C₇₉	—	4	1	no	—
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₉	—	2	2	no	—
118	C₂²×D₇₉	C₂×D₈	1	2	yes	—
119	Q₈×C₇₉	—	4	1	no	—
120	Q₈×C₇₉	—	4	1	no	—
121	Q₈×C₇₉	—	16	1	no	—
122	C₂×C₄×C₇₉	—	4	1	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₇₉	—	4	2	no	—
129	C₂×Dic₇₉	—	4	1	no	—
130	C₂×Dic₇₉	—	4	1	no	—
131	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
132	D₃₁₆	—	2	2	no	—
133	C₄×D₇₉	—	2	2	no	—
134	C₄×D₇₉	—	2	2	no	—
135	C₄×D₇₉	—	16	2	no	—
136	Dic₁₅₈	—	4	1	no	—
137	Dic₁₅₈	—	4	1	no	—
138	Dic₁₅₈	—	8	2	no	—
139	C₈×C₇₉	—	2	1	no	—
140	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
141	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₇₉⋊C₈	—	2	2	no	—
143	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₇₉	—	2	2	no	—
145	C₂×D₇₉	—	2	2	no	—
146	C₂×D₇₉	—	2	2	no	—
147	C₄×C₇₉	—	2	1	no	—
148	C₄×C₇₉	—	8	1	no	—
149	C₄×C₇₉	—	8	1	no	—
150	C₄×C₇₉	—	8	2	no	—
151	C₄×C₇₉	#510	1	2	yes	—
152	C₄×C₇₉	#547	1	1	yes	characteristic
153	C₄×C₇₉	#578	1	1	yes	characteristic
154	Dic₇₉	—	2	2	no	—
155	Dic₇₉	—	8	1	no	—
156	Dic₇₉	—	8	2	no	—
157	Dic₇₉	—	8	1	no	—
158	#33923	—	79	1	no	maximal, Sylow
159	C₂×C₇₉	#3334	1	1	yes	characteristic
160	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
161	C₂×C₇₉	#3416	1	1	yes	characteristic
162	D₇₉	—	4	2	no	—
163	#3342	—	79	1	no	—
164	#3338	—	79	1	no	—
165	#2254	—	79	1	no	—
166	#2237	—	79	2	no	—
167	#2235	—	79	2	no	—
168	#2227	D₇₉	1	1	yes	characteristic, complemented
169	#2225	—	79	1	no	—
170	#2219	—	79	2	no	—
171	C₄×Q₃₂	—	79	1	no	—
172	#2169	—	79	1	no	—
173	#2159	—	79	2	no	—
174	C₇₉	#33923	1	1	yes	characteristic, complemented, Sylow
175	#556	—	158	1	no	—
176	#554	—	158	1	no	—
177	C₂×Q₃₂	—	79	1	no	—
178	C₂×M₃₂	—	79	2	no	—
179	#500	—	79	1	no	—
180	#500	C₂×D₇₉	1	1	yes	characteristic
181	#485	—	158	2	no	—
182	#484	—	158	2	no	—
183	#477	—	158	1	no	—
184	#475	—	158	1	no	—
185	#467	—	79	2	no	—
186	#467	—	79	2	no	—
187	C₄×Q₁₆	—	79	1	no	—
188	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
189	C₈×D₄	—	79	1	no	—
190	#427	—	79	2	no	—
191	C₂×#107	—	79	2	no	—
192	#367	—	79	2	no	—
193	#367	—	79	2	no	—
194	#367	—	79	2	no	—
195	#366	—	79	1	no	—
196	#366	C₂×D₇₉	1	2	yes	—
197	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
198	#358	—	79	1	no	—
199	#358	—	79	2	no	—
200	#358	—	79	1	no	—
201	#358	C₂×D₇₉	1	2	yes	—
202	C₂²⋊C₁₆	—	79	2	no	—
203	C₄×C₁₆	—	79	1	no	—
204	C₂×Q₁₆	—	79	1	no	—
205	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
206	C₂²×C₈	—	79	2	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	158	1	no	—
209	#126	—	316	2	no	—
210	#125	—	316	1	no	—
211	#124	—	316	1	no	—
212	C₂²⋊Q₈	—	316	2	no	—
213	C₂²⋊Q₈	—	158	2	no	—
214	C₂²⋊Q₈	—	158	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	316	1	no	—
217	C₄×Q₈	—	158	1	no	—
218	C₄×D₄	—	316	1	no	—
219	C₄×D₄	—	79	1	no	—
220	#117	—	158	2	no	—
221	C₂×C₄⋊C₄	—	158	2	no	—
222	Q₃₂	—	158	2	no	—
223	M₃₂	—	158	4	no	—
224	C₂×C₁₆	—	79	1	no	—
225	C₂×C₁₆	—	79	1	no	—
226	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
227	#107	—	2	1	no	—
228	#107	—	79	1	no	—
229	#107	—	79	2	no	—
230	#107	—	158	2	no	—
231	#107	—	158	1	no	—
232	#107	C₂²×D₇₉	1	1	yes	characteristic
233	#107	C₂²×D₇₉	1	2	yes	—
234	#106	—	158	2	no	—
235	C₄⋊C₈	—	79	1	no	—
236	#103	—	2	1	no	—
237	#103	—	158	2	no	—
238	#103	—	158	2	no	—
239	#103	—	158	1	no	—
240	C₂²⋊C₈	—	79	2	no	—
241	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	316	1	no	—
245	C₂×Q₈	—	158	1	no	—
246	C₂×D₄	—	79	1	no	—
247	C₂²×C₄	—	316	2	no	—
248	C₂²×C₄	—	79	2	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	158	1	no	—
252	Q₁₆	—	158	1	no	—
253	C₂×C₈	—	79	2	no	—
254	C₂×C₈	—	158	2	no	—
255	C₂×C₈	—	158	2	no	—
256	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	316	2	no	—
263	C₄⋊C₄	—	316	2	no	—
264	C₄⋊C₄	—	316	2	no	—
265	C₄⋊C₄	—	316	1	no	—
266	C₄⋊C₄	—	316	1	no	—
267	C₄⋊C₄	—	79	1	no	—
268	C₄⋊C₄	—	158	2	no	—
269	C₄⋊C₄	—	158	1	no	—
270	C₂²⋊C₄	—	316	2	no	—
271	C₂²⋊C₄	—	316	2	no	—
272	C₂²⋊C₄	—	316	2	no	—
273	C₂²⋊C₄	—	79	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	316	2	no	—
276	C₄²	—	316	1	no	—
277	C₄²	D₄×D₇₉	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	79	2	no	—
280	C₁₆	—	158	1	no	—
281	C₂³	—	79	2	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	16	1	no	—
285	Q₈	—	316	1	no	—
286	Q₈	—	316	1	no	—
287	Q₈	—	632	2	no	—
288	D₄	—	158	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	316	1	no	—
293	C₂×C₄	—	316	2	no	—
294	C₂×C₄	—	316	1	no	—
295	C₂×C₄	—	79	2	no	—
296	C₂×C₄	—	158	2	no	—
297	C₂×C₄	—	158	2	no	—
298	C₂×C₄	—	1264	2	no	—
299	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
300	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	158	2	no	—
304	C₈	#41×D₇₉	1	1	yes	characteristic
305	C₈	#14579209	1	1	yes	characteristic
306	C₂²	—	158	2	no	—
307	C₂²	—	158	2	no	—
308	C₂²	—	158	2	no	—
309	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	1	no	—
314	C₄	—	632	1	no	—
315	C₄	—	632	2	no	—
316	C₄	—	632	1	no	—
317	C₄	—	158	2	no	—
318	C₄	#19099005	1	2	yes	—
319	C₄	#19099215	1	1	yes	characteristic
320	C₄	#19099379	1	1	yes	characteristic
321	C₂	—	316	2	no	—
322	C₂	#23775746	1	1	yes	characteristic
323	C₂	C₂×#19099005	1	1	yes	characteristic
324	C₂	#23776546	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×#2227	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₃₁₆	D₈
hypercenter	#2227	D₇₉
hypocenter	C₇₉	#33923
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
3. #23765680	C₂	semidirect
4. #23765610	C₂	semidirect
5. #23765596	C₂	semidirect
6. #23765543	C₂	semidirect
7. #23765507	C₂	semidirect
8. C₄×#14579107	C₂	semidirect
10. #23765185	C₂	semidirect
11. C₇₉×#2227	C₂	semidirect
168. #2227	D₇₉	semidirect
174. C₇₉	#33923	semidirect

Maximal subgroups

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

subgroup	index
#23775832	2
#23775792	2
#23765680	2
#23765610	2
#23765596	2
#23765543	2
#23765507	2
C₄×#14579107	2
#23765231	2
#23765185	2
C₇₉×#2227	2
#33923	79