Subgroups — #34205213

Order 20224.

582 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098601	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098364	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761491	C₂	1	2	yes	maximal, complemented
5	C₂×#466×C₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579120	—	2	1	no	—
7	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
8	#19098601	—	2	4	no	—
9	C₂×#14579095	C₂²	1	2	yes	—
10	C₂×#14579093	—	2	1	no	—
11	#19098364	—	2	4	no	—
12	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
13	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#121×C₇₉	—	2	1	no	—
16	#466×C₇₉	—	2	2	no	—
17	#466×C₇₉	C₄	1	4	yes	complemented
18	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
19	C₂×#102×C₇₉	—	2	1	no	—
20	#14579120	—	2	4	no	—
21	#14579120	—	4	2	no	—
22	C₂×C₄⋊Dic₇₉	—	2	2	no	—
23	C₂×C₂²⋊Dic₇₉	—	4	1	no	—
24	C₂×C₄×Dic₇₉	—	4	1	no	—
25	C₂×C₇₉⋊C₁₆	—	2	2	no	—
26	C₂×C₇₉⋊C₁₆	—	2	2	no	—
27	C₂×C₇₉⋊C₁₆	—	2	2	no	—
28	#14579095	—	2	2	no	—
29	#14579095	D₄	1	4	yes	complemented
30	#14579093	—	2	4	no	—
31	C₂²×D₄×C₇₉	—	2	1	no	—
32	C₂²×D₄×C₇₉	—	4	1	no	—
33	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
34	C₂×D₈×C₇₉	—	2	2	no	—
35	C₂×D₈×C₇₉	—	2	1	no	—
36	C₂×D₈×C₇₉	—	2	2	no	—
37	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
38	C₂²×C₈×C₇₉	—	2	2	no	—
39	C₂²×C₈×C₇₉	—	2	2	no	—
40	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
41	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	D₄	1	2	yes	—
43	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
44	#121×C₇₉	—	2	4	no	—
45	#121×C₇₉	—	4	2	no	—
46	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
47	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
48	#107×C₇₉	—	2	1	no	—
49	#107×C₇₉	C₂×C₄	1	2	yes	—
50	#102×C₇₉	—	2	4	no	—
51	C₂²×Dic₇₉	—	4	2	no	—
52	C₄⋊Dic₇₉	—	2	4	no	—
53	C₄⋊Dic₇₉	—	4	2	no	—
54	C₂²⋊Dic₇₉	—	4	4	no	—
55	C₄×Dic₇₉	—	4	4	no	—
56	C₇₉⋊C₁₆	—	4	2	no	—
57	C₂⁴×C₇₉	—	4	1	no	—
58	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
59	C₂×D₄×C₇₉	—	2	2	no	—
60	C₂×D₄×C₇₉	—	4	4	no	—
61	C₂×D₄×C₇₉	—	4	1	no	—
62	C₂×D₄×C₇₉	—	4	1	no	—
63	C₂×D₄×C₇₉	—	4	1	no	—
64	C₂×D₄×C₇₉	—	4	1	no	—
65	C₂×D₄×C₇₉	—	4	1	no	—
66	C₂×D₄×C₇₉	—	8	4	no	—
67	C₂²×C₄×C₇₉	—	2	1	no	—
68	C₂²×C₄×C₇₉	—	2	1	no	—
69	C₂²×C₄×C₇₉	—	2	1	no	—
70	C₂²×C₄×C₇₉	—	2	1	no	—
71	C₂²×C₄×C₇₉	—	4	1	no	—
72	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	SD₁₆	1	2	yes	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	D₈×C₇₉	—	2	4	no	—
77	D₈×C₇₉	—	4	2	no	—
78	C₂×C₈×C₇₉	—	2	4	no	—
79	C₂×C₈×C₇₉	—	2	4	no	—
80	C₂×C₈×C₇₉	—	2	1	no	—
81	C₂×C₈×C₇₉	—	2	1	no	—
82	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₇₉	#41	1	2	yes	—
87	C₄⋊C₄×C₇₉	—	2	2	no	—
88	C₄⋊C₄×C₇₉	—	4	1	no	—
89	C₂²⋊C₄×C₇₉	—	4	4	no	—
90	C₂³×C₇₉	—	2	1	no	—
91	C₂³×C₇₉	—	2	1	no	—
92	C₂³×C₇₉	—	2	2	no	—
93	C₂³×C₇₉	—	2	2	no	—
94	C₂³×C₇₉	—	4	2	no	—
95	C₂³×C₇₉	—	8	1	no	—
96	C₂³×C₇₉	—	8	1	no	—
97	C₂³×C₇₉	—	8	2	no	—
98	C₂³×C₇₉	—	8	2	no	—
99	C₂³×C₇₉	#102	1	2	yes	—
100	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
101	D₄×C₇₉	—	4	2	no	—
102	D₄×C₇₉	—	4	2	no	—
103	D₄×C₇₉	—	8	8	no	—
104	D₄×C₇₉	—	8	1	no	—
105	D₄×C₇₉	—	8	1	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	2	no	—
109	C₂×C₄×C₇₉	—	2	2	no	—
110	C₂×C₄×C₇₉	—	2	2	no	—
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	8	2	no	—
114	C₂×C₄×C₇₉	#102	1	2	yes	—
115	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
118	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
121	C₂×Dic₇₉	—	4	4	no	—
122	C₂×Dic₇₉	—	8	2	no	—
123	C₂×Dic₇₉	—	8	2	no	—
124	C₈×C₇₉	—	2	2	no	—
125	C₈×C₇₉	C₄×D₄	1	2	yes	—
126	C₈×C₇₉	#121	1	2	yes	—
127	C₂²×C₇₉	—	2	2	no	—
128	C₂²×C₇₉	—	2	2	no	—
129	C₂²×C₇₉	—	2	2	no	—
130	C₂²×C₇₉	—	4	2	no	—
131	C₂²×C₇₉	—	4	2	no	—
132	C₂²×C₇₉	—	8	2	no	—
133	C₂²×C₇₉	—	8	2	no	—
134	C₂²×C₇₉	—	8	2	no	—
135	C₂²×C₇₉	—	16	2	no	—
136	C₂²×C₇₉	—	16	2	no	—
137	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
138	C₂²×C₇₉	#447	1	1	yes	characteristic
139	C₂²×C₇₉	#480	1	1	yes	characteristic
140	C₂²×C₇₉	#508	1	2	yes	—
141	C₂²×C₇₉	#509	1	2	yes	—
142	C₄×C₇₉	—	2	2	no	—
143	C₄×C₇₉	—	8	2	no	—
144	C₄×C₇₉	#386	1	1	yes	characteristic
145	C₄×C₇₉	#417	1	1	yes	characteristic
146	C₄×C₇₉	#448	1	1	yes	characteristic
147	C₄×C₇₉	#481	1	1	yes	characteristic
148	Dic₇₉	—	8	4	no	—
149	#12992	—	79	1	no	maximal, Sylow
150	C₂×C₇₉	—	4	2	no	—
151	C₂×C₇₉	—	16	2	no	—
152	C₂×C₇₉	#1874	1	1	yes	characteristic
153	C₂×C₇₉	#2138	1	1	yes	characteristic
154	C₂×C₇₉	#2140	1	1	yes	characteristic
155	C₂×C₇₉	#2186	1	2	yes	—
156	C₂×C₇₉	#2234	1	2	yes	—
157	C₂×#466	D₇₉	1	1	yes	characteristic
158	C₂×#427	—	79	1	no	—
159	C₂×#357	—	79	2	no	—
160	C₂×C₂²⋊C₁₆	—	79	1	no	—
161	#1380	—	79	2	no	—
162	C₇₉	#12992	1	1	yes	characteristic, complemented, Sylow
163	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
164	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
165	C₂×#121	—	2	1	no	—
166	C₂×#117	—	158	1	no	—
167	C₂²×C₁₆	—	79	2	no	—
168	#466	—	2	2	no	—
169	#466	Dic₇₉	1	4	yes	complemented
170	#427	—	158	4	no	—
171	C₂×#107	—	79	2	no	—
172	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
173	C₂×#106	—	158	1	no	—
174	C₂×#102	—	2	1	no	—
175	#357	—	158	4	no	—
176	C₂²⋊C₁₆	—	158	2	no	—
177	C₂²×D₄	—	2	1	no	—
178	C₂²×D₄	—	4	1	no	—
179	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
180	C₂×D₈	—	2	2	no	—
181	C₂×D₈	—	2	1	no	—
182	C₂×D₈	—	2	2	no	—
183	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
187	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
189	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
190	#121	—	2	4	no	—
191	#121	—	4	2	no	—
192	#117	—	316	2	no	—
193	#117	—	158	4	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	158	2	no	—
196	C₂×C₂²⋊C₄	—	4	1	no	—
197	C₂×C₂²⋊C₄	—	316	1	no	—
198	C₂×C₄²	—	316	1	no	—
199	C₂×C₁₆	—	158	2	no	—
200	C₂×C₁₆	—	158	2	no	—
201	C₂×C₁₆	—	158	2	no	—
202	#107	—	2	1	no	—
203	#107	—	79	4	no	—
204	#107	—	158	2	no	—
205	#107	C₂×Dic₇₉	1	2	yes	—
206	#106	—	158	4	no	—
207	#102	—	2	4	no	—
208	C₂⁴	—	4	1	no	—
209	C₂⁴	#11929609	1	1	yes	characteristic
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂×D₄	—	4	1	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	8	4	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	316	2	no	—
224	C₂²×C₄	C₂²⋊Dic₇₉	1	2	yes	—
225	C₂²×C₄	#11929609	1	1	yes	characteristic
226	C₂²×C₄	#11929612	1	2	yes	—
227	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
228	D₈	—	2	4	no	—
229	D₈	—	4	2	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
235	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
236	C₂×C₈	D₄×D₇₉	1	2	yes	—
237	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
238	C₂×C₈	#11929623	1	2	yes	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	316	2	no	—
242	C₄⋊C₄	—	158	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₂²⋊C₄	—	316	4	no	—
245	C₄²	—	316	4	no	—
246	C₁₆	—	316	2	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	1	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	8	1	no	—
253	C₂³	—	8	2	no	—
254	C₂³	—	8	2	no	—
255	C₂³	—	8	1	no	—
256	C₂³	#14579082	1	2	yes	—
257	C₂³	C₂×#11929609	1	1	yes	characteristic
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	8	no	—
261	D₄	—	8	1	no	—
262	D₄	—	8	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	8	2	no	—
271	C₂×C₄	—	316	4	no	—
272	C₂×C₄	—	632	2	no	—
273	C₂×C₄	—	632	2	no	—
274	C₂×C₄	#14579082	1	2	yes	—
275	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
276	C₂×C₄	#14579129	1	1	yes	characteristic
277	C₂×C₄	#14579142	1	1	yes	characteristic
278	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
279	C₂×C₄	#14579186	1	1	yes	characteristic
280	C₂×C₄	#14579191	1	1	yes	characteristic
281	C₈	—	2	2	no	—
282	C₈	D₄×Dic₇₉	1	2	yes	—
283	C₈	#14579133	1	2	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	2	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	—	16	2	no	—
294	C₂²	C₂×#14579082	1	1	yes	characteristic
295	C₂²	#19098696	1	1	yes	characteristic
296	C₂²	#19098874	1	1	yes	characteristic
297	C₂²	#19098993	1	2	yes	—
298	C₂²	#19098999	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	632	4	no	—
302	C₄	#19098443	1	1	yes	characteristic
303	C₄	#19098564	1	1	yes	characteristic
304	C₄	#19098701	1	1	yes	characteristic
305	C₄	#19098879	1	1	yes	characteristic
306	C₂	—	4	2	no	—
307	C₂	—	16	2	no	—
308	C₂	#23763718	1	1	yes	characteristic
309	C₂	#23765107	1	1	yes	characteristic
310	C₂	#23765116	1	1	yes	characteristic
311	C₂	#23765321	1	2	yes	—
312	C₂	#23765588	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#466×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#466	D₇₉
hypocenter	C₇₉	#12992
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×#19098601	C₂	semidirect
2. C₂×#19098364	C₂	semidirect
3. C₂×C₂²⋊(C₇₉⋊C₁₆)	C₂	semidirect
4. #23761491	C₂	semidirect
17. #466×C₇₉	C₄	semidirect
29. #14579095	D₄	semidirect
162. C₇₉	#12992	semidirect
169. #466	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098601	2
C₂×#19098364	2
C₂×C₂²⋊(C₇₉⋊C₁₆)	2
#23761491	2
C₂×#466×C₇₉	2
#12992	79