Subgroups — #34446023

Order 20224.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776390	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765638	C₂	1	2	yes	maximal
4	#23765627	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765548	C₂	1	1	yes	maximal, characteristic
6	#23765540	C₂	1	2	yes	maximal, complemented
7	#23765464	C₂	1	2	yes	maximal
8	#23765305	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765271	C₂	1	2	yes	maximal, complemented
10	C₄×#14579104	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₅₈	—	2	1	no	—
13	#19099287	—	2	1	no	—
14	C₂×#14579104	C₂²	1	1	yes	characteristic
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098967	C₂²	1	2	yes	—
17	#19098948	C₂²	1	2	yes	—
18	#19098858	—	2	1	no	—
19	#19098856	—	2	2	no	—
20	#19098787	—	2	2	no	—
21	#19098765	—	2	1	no	—
22	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
24	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
25	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
26	#19098391	C₂²	1	2	yes	—
27	#19098390	C₂²	1	1	yes	characteristic
28	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
29	#19098367	C₂²	1	1	yes	characteristic
30	#19098367	C₂²	1	2	yes	—
31	#19098367	C₂²	1	2	yes	—
32	#19098365	C₂²	1	2	yes	—
33	#19098363	C₂²	1	1	yes	characteristic
34	#19098338	C₂²	1	2	yes	—
35	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
36	C₂×Q₃₂×C₇₉	C₂²	1	2	yes	—
37	#494×C₇₉	C₂²	1	1	yes	characteristic
38	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#358×C₇₉	C₂²	1	2	yes	—
41	C₂×#11929611	—	2	2	no	—
42	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
43	#14579161	—	2	1	no	—
44	#14579161	—	4	1	no	—
45	#14579160	—	2	2	no	—
46	#14579160	—	4	2	no	—
47	#14579156	—	4	2	no	—
48	#14579150	—	2	2	no	—
49	#14579141	—	4	1	no	—
50	Q₈×Dic₇₉	—	2	2	no	—
51	Q₈×Dic₇₉	—	2	2	no	—
52	C₄×Dic₁₅₈	—	4	1	no	—
53	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
54	C₄×D₃₁₆	—	2	1	no	—
55	#14579119	—	4	1	no	—
56	#14579104	—	2	2	no	—
57	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
59	#14579095	—	2	1	no	—
60	#14579095	C₂³	1	1	yes	characteristic
61	#14579095	C₂³	1	2	yes	—
62	#14579094	—	2	2	no	—
63	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
64	Dic₇₉⋊C₈	D₄	1	2	yes	—
65	#14579085	—	2	2	no	—
66	#14579085	—	2	2	no	—
67	#14579083	—	2	1	no	—
68	C₈×Dic₇₉	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₇₉	—	2	1	no	—
70	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₇₉	—	2	1	no	—
73	C₄×Q₈×C₇₉	—	2	1	no	—
74	Q₃₂×C₇₉	—	2	4	no	—
75	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
76	#107×C₇₉	C₂³	1	1	yes	characteristic
77	#103×C₇₉	—	2	1	no	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅₈	—	4	2	no	—
80	C₂×Dic₁₅₈	—	4	1	no	—
81	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
82	C₂×D₃₁₆	—	2	1	no	—
83	C₂×C₄×D₇₉	—	4	1	no	—
84	#11929611	—	8	2	no	—
85	D₆₃₂	—	2	1	no	—
86	D₆₃₂	—	2	1	no	—
87	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
88	C₄⋊Dic₇₉	—	2	1	no	—
89	C₄⋊Dic₇₉	—	2	2	no	—
90	C₄⋊Dic₇₉	—	4	2	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	D₈	1	2	yes	—
96	(C₂×D₇₉)⋊C₄	—	4	2	no	—
97	(C₂×D₇₉)⋊C₄	—	4	2	no	—
98	(C₂×D₇₉)⋊C₄	—	4	1	no	—
99	C₄×Dic₇₉	—	4	2	no	—
100	C₄×Dic₇₉	—	4	2	no	—
101	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
102	C₇₉⋊C₁₆	—	2	1	no	—
103	C₇₉⋊C₁₆	—	2	1	no	—
104	C₂×Q₈×C₇₉	—	2	1	no	—
105	C₂×Q₈×C₇₉	—	2	2	no	—
106	Q₁₆×C₇₉	—	2	1	no	—
107	Q₁₆×C₇₉	—	2	2	no	—
108	Q₁₆×C₇₉	—	2	2	no	—
109	Q₁₆×C₇₉	—	8	1	no	—
110	Q₁₆×C₇₉	#41	1	2	yes	—
111	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₉	—	2	1	no	—
114	C₄⋊C₄×C₇₉	—	4	1	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	—	4	1	no	—
118	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₇₉	—	2	2	no	—
120	C₂²×D₇₉	—	4	1	no	—
121	Q₈×C₇₉	—	2	2	no	—
122	Q₈×C₇₉	—	4	1	no	—
123	Q₈×C₇₉	—	4	2	no	—
124	Q₈×C₇₉	—	4	2	no	—
125	C₂×C₄×C₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₇₉	—	4	1	no	—
132	C₂×Dic₇₉	—	4	2	no	—
133	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
134	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
135	D₃₁₆	—	4	1	no	—
136	D₃₁₆	—	4	1	no	—
137	C₄×D₇₉	—	8	2	no	—
138	Dic₁₅₈	—	2	2	no	—
139	Dic₁₅₈	—	16	2	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	C₂²×D₈	1	1	yes	characteristic
146	C₂×D₇₉	—	8	1	no	—
147	C₂×D₇₉	—	8	1	no	—
148	C₂×D₇₉	—	8	1	no	—
149	C₄×C₇₉	—	2	1	no	—
150	C₄×C₇₉	—	4	2	no	—
151	C₄×C₇₉	—	8	2	no	—
152	C₄×C₇₉	—	8	1	no	—
153	C₄×C₇₉	#510	1	2	yes	—
154	C₄×C₇₉	#548	1	1	yes	characteristic
155	C₄×C₇₉	#577	1	1	yes	characteristic
156	Dic₇₉	—	2	2	no	—
157	Dic₇₉	—	2	2	no	—
158	Dic₇₉	—	8	1	no	—
159	Dic₇₉	—	8	2	no	—
160	#33936	—	79	1	no	maximal, Sylow
161	C₂×C₇₉	#3358	1	1	yes	characteristic
162	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
163	C₂×C₇₉	#3416	1	1	yes	characteristic
164	D₇₉	—	16	1	no	—
165	#3393	—	79	1	no	—
166	Q₈×Q₁₆	—	79	1	no	—
167	#2243	—	79	2	no	—
168	#2241	—	79	1	no	—
169	#2226	—	79	1	no	—
170	#2225	—	79	2	no	—
171	#2211	—	79	2	no	—
172	#2209	D₇₉	1	1	yes	characteristic, complemented
173	#2183	—	79	1	no	—
174	#2176	—	79	2	no	—
175	C₄×SD₃₂	—	79	1	no	—
176	C₇₉	#33936	1	1	yes	characteristic, complemented, Sylow
177	Q₈²	—	158	1	no	—
178	#556	—	158	1	no	—
179	C₂×Q₃₂	C₂×D₇₉	1	2	yes	—
180	C₂×SD₃₂	—	79	1	no	—
181	#500	—	79	2	no	—
182	#500	—	79	1	no	—
183	#495	—	79	2	no	—
184	#494	C₂×D₇₉	1	1	yes	characteristic
185	#477	—	158	2	no	—
186	#477	—	158	1	no	—
187	#464	—	158	2	no	—
188	#461	—	158	1	no	—
189	C₈×Q₈	—	79	1	no	—
190	C₄×Q₁₆	—	79	2	no	—
191	C₄×Q₁₆	—	79	2	no	—
192	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
193	C₄×D₈	—	79	1	no	—
194	#367	—	79	2	no	—
195	#367	—	79	1	no	—
196	C₄⋊C₁₆	—	79	2	no	—
197	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
198	#358	—	79	1	no	—
199	#358	—	79	2	no	—
200	#358	—	79	2	no	—
201	#358	C₂×D₇₉	1	2	yes	—
202	#357	—	79	2	no	—
203	#357	—	79	1	no	—
204	#346	—	79	2	no	—
205	C₄×C₁₆	—	79	1	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
208	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
209	C₂×SD₁₆	—	158	2	no	—
210	C₂×D₈	—	79	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	316	1	no	—
213	C₄⋊Q₈	—	316	2	no	—
214	C₄⋊Q₈	—	158	2	no	—
215	C₄⋊Q₈	—	158	1	no	—
216	#126	—	316	2	no	—
217	#124	—	158	2	no	—
218	C₂²⋊Q₈	—	316	1	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	316	1	no	—
221	C₄×Q₈	—	79	1	no	—
222	C₄×Q₈	—	158	2	no	—
223	C₄×Q₈	—	158	2	no	—
224	C₄×D₄	—	158	1	no	—
225	#117	—	316	1	no	—
226	Q₃₂	—	2	4	no	—
227	SD₃₂	—	158	2	no	—
228	C₂×C₁₆	—	79	1	no	—
229	C₂×C₁₆	—	79	1	no	—
230	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
231	#107	—	79	2	no	—
232	#107	—	79	1	no	—
233	#107	—	158	2	no	—
234	#107	—	158	1	no	—
235	#107	C₂²×D₇₉	1	1	yes	characteristic
236	C₄⋊C₈	—	79	2	no	—
237	C₄⋊C₈	—	79	1	no	—
238	#103	—	2	1	no	—
239	#103	—	158	2	no	—
240	#103	—	158	2	no	—
241	#102	—	158	1	no	—
242	C₄×C₈	—	79	2	no	—
243	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	316	2	no	—
247	C₂×Q₈	—	316	1	no	—
248	C₂×Q₈	—	79	1	no	—
249	C₂×D₄	—	158	1	no	—
250	C₂²×C₄	—	316	1	no	—
251	Q₁₆	—	2	2	no	—
252	Q₁₆	—	2	1	no	—
253	Q₁₆	—	2	2	no	—
254	Q₁₆	—	8	1	no	—
255	Q₁₆	#11929623	1	2	yes	—
256	SD₁₆	—	632	2	no	—
257	D₈	—	158	1	no	—
258	D₈	—	158	1	no	—
259	C₂×C₈	—	79	2	no	—
260	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	316	2	no	—
267	C₄⋊C₄	—	316	2	no	—
268	C₄⋊C₄	—	316	2	no	—
269	C₄⋊C₄	—	316	1	no	—
270	C₄⋊C₄	—	79	2	no	—
271	C₄⋊C₄	—	79	1	no	—
272	C₄⋊C₄	—	158	1	no	—
273	C₄⋊C₄	—	158	2	no	—
274	C₂²⋊C₄	—	316	2	no	—
275	C₂²⋊C₄	—	316	2	no	—
276	C₂²⋊C₄	—	316	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	316	2	no	—
279	C₄²	—	316	2	no	—
280	C₄²	—	79	2	no	—
281	C₄²	D₄×D₇₉	1	1	yes	characteristic
282	C₁₆	—	2	2	no	—
283	C₁₆	—	158	1	no	—
284	C₁₆	—	158	1	no	—
285	C₂³	—	316	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	158	2	no	—
291	Q₈	—	1264	1	no	—
292	Q₈	—	1264	2	no	—
293	D₄	—	316	1	no	—
294	D₄	—	316	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	316	1	no	—
299	C₂×C₄	—	316	2	no	—
300	C₂×C₄	—	79	2	no	—
301	C₂×C₄	—	79	2	no	—
302	C₂×C₄	—	632	2	no	—
303	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
304	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
306	C₈	—	2	1	no	—
307	C₈	—	158	2	no	—
308	C₈	C₂×#11929623	1	1	yes	characteristic
309	C₈	#14579208	1	1	yes	characteristic
310	C₂²	—	632	1	no	—
311	C₂²	—	632	1	no	—
312	C₂²	—	632	1	no	—
313	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	1	no	—
318	C₄	—	632	1	no	—
319	C₄	—	632	2	no	—
320	C₄	—	158	2	no	—
321	C₄	—	158	2	no	—
322	C₄	#19099005	1	2	yes	—
323	C₄	#19099219	1	1	yes	characteristic
324	C₄	#19099373	1	1	yes	characteristic
325	C₂	—	1264	1	no	—
326	C₂	#23776030	1	1	yes	characteristic
327	C₂	C₂×#19099005	1	1	yes	characteristic
328	C₂	#23776545	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×#2209	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₆₃₂	C₂³
hypercenter	#2209	D₇₉
hypocenter	C₇₉	#33936
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₅₈	C₂	semidirect
4. #23765627	C₂	semidirect
6. #23765540	C₂	semidirect
8. #23765305	C₂	semidirect
9. #23765271	C₂	semidirect
11. C₇₉×#2209	C₂	semidirect
172. #2209	D₇₉	semidirect
176. C₇₉	#33936	semidirect

Maximal subgroups

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

subgroup	index
#23776390	2
Q₁₆×Dic₁₅₈	2
#23765638	2
#23765627	2
#23765548	2
#23765540	2
#23765464	2
#23765305	2
#23765271	2
C₄×#14579104	2
C₇₉×#2209	2
#33936	79