Subgroups — #34179113

Order 20224.

462 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098694	C₂	1	1	yes	maximal, characteristic
2	#23763270	C₂	1	1	yes	maximal, characteristic
3	C₄×#14579085	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098285	C₂	1	1	yes	maximal, characteristic
5	#23761240	C₂	1	1	yes	maximal, characteristic
6	#23761234	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₈⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19098694	—	2	2	no	—
9	#19098694	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#14579085	C₄	1	2	yes	—
13	C₂×#14579072	—	2	1	no	—
14	C₂×#14579071	—	2	1	no	—
15	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
18	#19098325	C₄	1	2	yes	—
19	#19098317	—	2	1	no	—
20	#19098285	—	2	2	no	—
21	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	Q₈⋊C₈×C₇₉	C₄	1	4	yes	complemented
25	C₂²×C₇₉⋊C₈	—	2	1	no	—
26	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
28	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
30	C₂×C₄×Dic₇₉	—	2	1	no	—
31	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
33	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
34	#14579085	—	2	4	no	—
35	#14579072	—	2	4	no	—
36	#14579071	—	4	2	no	—
37	C₄×C₇₉⋊C₈	—	2	1	no	—
38	C₄×C₇₉⋊C₈	—	2	1	no	—
39	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
40	C₄×C₇₉⋊C₈	D₄	1	2	yes	—
41	C₇₉⋊#95	—	2	1	no	—
42	C₂²×Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	—	2	1	no	—
44	C₂²×C₈×C₇₉	—	2	1	no	—
45	C₄×Q₈×C₇₉	—	2	1	no	—
46	C₄×Q₈×C₇₉	—	2	1	no	—
47	C₄×Q₈×C₇₉	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₇₉	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₉	—	2	1	no	—
52	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₇₉	—	2	1	no	—
54	C₄⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₇₉	—	2	1	no	—
56	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
57	C₂²×Dic₇₉	—	2	2	no	—
58	C₂×C₇₉⋊C₈	—	2	2	no	—
59	C₂×C₇₉⋊C₈	—	2	2	no	—
60	C₂×C₇₉⋊C₈	—	2	2	no	—
61	C₂×C₇₉⋊C₈	—	2	1	no	—
62	C₂×C₇₉⋊C₈	—	2	1	no	—
63	C₂×C₇₉⋊C₈	—	2	1	no	—
64	C₂×C₇₉⋊C₈	—	2	1	no	—
65	C₂×C₇₉⋊C₈	—	4	1	no	—
66	C₂×C₇₉⋊C₈	—	4	1	no	—
67	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
68	C₂×C₇₉⋊C₈	C₂²⋊C₄	1	2	yes	—
69	C₂×C₇₉⋊C₈	M₁₆	1	4	yes	—
70	C₄⋊Dic₇₉	—	2	2	no	—
71	C₄⋊Dic₇₉	—	2	2	no	—
72	C₄×Dic₇₉	—	4	1	no	—
73	C₄×Dic₇₉	—	4	1	no	—
74	C₂×Q₈×C₇₉	—	2	1	no	—
75	C₂×Q₈×C₇₉	—	2	2	no	—
76	C₂×Q₈×C₇₉	—	2	2	no	—
77	C₂×Q₈×C₇₉	C₄²	1	2	yes	—
78	C₂²×C₄×C₇₉	—	2	1	no	—
79	C₂²×C₄×C₇₉	—	2	1	no	—
80	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₇₉	—	2	2	no	—
84	C₂×C₈×C₇₉	—	2	2	no	—
85	C₂×C₈×C₇₉	—	4	2	no	—
86	C₂×C₈×C₇₉	—	4	1	no	—
87	C₂×C₈×C₇₉	—	4	1	no	—
88	C₄⋊C₄×C₇₉	—	2	2	no	—
89	C₄⋊C₄×C₇₉	—	2	2	no	—
90	C₄⋊C₄×C₇₉	—	2	1	no	—
91	C₄⋊C₄×C₇₉	C₄²	1	2	yes	—
92	C₄²×C₇₉	—	2	2	no	—
93	C₄²×C₇₉	—	2	2	no	—
94	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇₉	#41	1	1	yes	characteristic
98	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
99	Q₈×C₇₉	—	2	2	no	—
100	Q₈×C₇₉	—	2	2	no	—
101	Q₈×C₇₉	—	4	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	2	2	no	—
105	C₂×C₄×C₇₉	—	4	2	no	—
106	C₂×C₄×C₇₉	—	4	2	no	—
107	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₇₉	#131	1	2	yes	—
115	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
117	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
118	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
119	C₂×Dic₇₉	—	4	2	no	—
120	C₂×Dic₇₉	—	4	2	no	—
121	C₂×Dic₇₉	—	4	2	no	—
122	C₈×C₇₉	—	4	2	no	—
123	C₈×C₇₉	—	4	2	no	—
124	C₇₉⋊C₈	—	2	2	no	—
125	C₇₉⋊C₈	—	2	2	no	—
126	C₇₉⋊C₈	—	2	4	no	—
127	C₇₉⋊C₈	—	4	2	no	—
128	C₂²×C₇₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₇₉	#436	1	2	yes	—
130	C₂²×C₇₉	#440	1	1	yes	characteristic
131	C₂²×C₇₉	#441	1	1	yes	characteristic
132	C₂²×C₇₉	#444	1	2	yes	—
133	C₄×C₇₉	—	2	1	no	—
134	C₄×C₇₉	—	2	1	no	—
135	C₄×C₇₉	—	4	2	no	—
136	C₄×C₇₉	—	4	2	no	—
137	C₄×C₇₉	C₄×M₁₆	1	1	yes	characteristic
138	C₄×C₇₉	#405	1	1	yes	characteristic
139	C₄×C₇₉	#417	1	1	yes	characteristic
140	C₄×C₇₉	#419	1	1	yes	characteristic
141	C₄×C₇₉	#435	1	2	yes	—
142	C₄×C₇₉	#440	1	1	yes	characteristic
143	C₄×C₇₉	#441	1	1	yes	characteristic
144	Dic₇₉	—	8	2	no	—
145	#8209	—	79	1	no	maximal, Sylow
146	C₂×C₇₉	#1592	1	2	yes	—
147	C₂×C₇₉	#1593	1	2	yes	—
148	C₂×C₇₉	#1753	1	1	yes	characteristic
149	C₂×C₇₉	#1763	1	1	yes	characteristic
150	C₂×C₇₉	#1765	1	1	yes	characteristic
151	C₂×#446	—	79	1	no	—
152	#1770	—	79	1	no	—
153	C₄×#103	—	79	1	no	—
154	C₂×Q₈⋊C₈	D₇₉	1	1	yes	characteristic
155	C₂×#322	—	79	1	no	—
156	#1279	—	79	1	no	—
157	#1277	—	79	1	no	—
158	C₇₉	#8209	1	1	yes	characteristic, complemented, Sylow
159	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
160	#446	—	79	4	no	—
161	#446	—	158	2	no	—
162	C₂×C₄⋊C₈	—	79	1	no	—
163	C₂×C₄⋊C₈	—	158	1	no	—
164	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
165	C₂×#103	—	79	2	no	—
166	C₂×#97	—	158	1	no	—
167	C₂×#97	—	158	1	no	—
168	C₂×C₄×C₈	—	79	1	no	—
169	C₂×C₄×C₈	—	79	1	no	—
170	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
171	C₄×C₄⋊C₄	—	79	1	no	—
172	#340	—	79	2	no	—
173	#336	—	158	1	no	—
174	Q₈⋊C₈	Dic₇₉	1	4	yes	complemented
175	#322	—	158	2	no	—
176	C₂²×Q₈	C₄×D₇₉	1	1	yes	characteristic
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	79	2	no	—
180	C₂²×C₈	—	79	1	no	—
181	C₂²×C₈	—	79	1	no	—
182	C₂²×C₈	—	158	1	no	—
183	C₄×Q₈	—	2	1	no	—
184	C₄×Q₈	—	2	1	no	—
185	C₄×Q₈	C₄×D₇₉	1	2	yes	—
186	C₄×Q₈	C₂×Dic₇₉	1	2	yes	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	79	2	no	—
189	C₂×C₄⋊C₄	C₄×D₇₉	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	158	1	no	—
192	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	79	2	no	—
195	C₄⋊C₈	—	158	4	no	—
196	C₄⋊C₈	—	158	1	no	—
197	C₄⋊C₈	C₂×Dic₇₉	1	2	yes	—
198	#103	—	158	4	no	—
199	#97	—	316	2	no	—
200	#97	—	158	4	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	79	2	no	—
203	C₄×C₈	—	79	2	no	—
204	C₄×C₈	—	158	1	no	—
205	C₄×C₈	—	158	1	no	—
206	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
207	#95	—	158	1	no	—
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	C₄×Dic₇₉	1	2	yes	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	—	158	2	no	—
215	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
217	C₂²×C₄	#11929623	1	1	yes	characteristic
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	316	1	no	—
224	C₂×C₈	—	316	1	no	—
225	C₂×C₈	—	79	2	no	—
226	C₂×C₈	—	79	2	no	—
227	C₂×C₈	—	79	4	no	—
228	C₂×C₈	—	158	2	no	—
229	C₂×C₈	—	158	2	no	—
230	C₂×C₈	—	158	2	no	—
231	C₂×C₈	—	158	1	no	—
232	C₂×C₈	—	158	1	no	—
233	C₂×C₈	—	158	1	no	—
234	C₂×C₈	—	158	1	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	158	2	no	—
239	C₄⋊C₄	—	158	2	no	—
240	C₄⋊C₄	C₄×Dic₇₉	1	2	yes	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	316	1	no	—
244	C₄²	—	316	1	no	—
245	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
246	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
247	C₄²	D₄×D₇₉	1	1	yes	characteristic
248	C₄²	#11929623	1	1	yes	characteristic
249	C₂³	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
250	Q₈	—	2	2	no	—
251	Q₈	—	2	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	316	2	no	—
259	C₂×C₄	—	316	2	no	—
260	C₂×C₄	—	316	2	no	—
261	C₂×C₄	C₂×C₄×Dic₇₉	1	1	yes	characteristic
262	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
263	C₂×C₄	#14579117	1	1	yes	characteristic
264	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
266	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
267	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
268	C₂×C₄	#14579167	1	2	yes	—
269	C₂×C₄	#14579188	1	1	yes	characteristic
270	C₂×C₄	#14579190	1	1	yes	characteristic
271	C₂×C₄	#14579193	1	1	yes	characteristic
272	C₂×C₄	#14579193	1	1	yes	characteristic
273	C₈	—	4	2	no	—
274	C₈	—	4	2	no	—
275	C₈	—	316	2	no	—
276	C₈	—	158	4	no	—
277	C₈	—	158	2	no	—
278	C₈	—	158	2	no	—
279	C₂²	C₂²⋊C₄×Dic₇₉	1	1	yes	characteristic
280	C₂²	#19098638	1	2	yes	—
281	C₂²	#19098666	1	1	yes	characteristic
282	C₂²	#19098667	1	1	yes	characteristic
283	C₂²	#19098682	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	4	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	632	2	no	—
289	C₄	M₁₆×Dic₇₉	1	1	yes	characteristic
290	C₄	#19098518	1	1	yes	characteristic
291	C₄	#19098563	1	1	yes	characteristic
292	C₄	#19098570	1	1	yes	characteristic
293	C₄	#19098637	1	2	yes	—
294	C₄	#19098664	1	1	yes	characteristic
295	C₄	#19098670	1	1	yes	characteristic
296	C₂	#23762305	1	2	yes	—
297	C₂	#23762306	1	2	yes	—
298	C₂	#23763186	1	1	yes	characteristic
299	C₂	#23763232	1	1	yes	characteristic
300	C₂	#23763245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×Q₈⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×Q₈⋊C₈	D₇₉
hypocenter	C₇₉	#8209
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
9. #19098694	C₄	semidirect
24. Q₈⋊C₈×C₇₉	C₄	semidirect
158. C₇₉	#8209	semidirect
174. Q₈⋊C₈	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098694	2
#23763270	2
C₄×#14579085	2
C₂×#19098285	2
#23761240	2
#23761234	2
C₂×Q₈⋊C₈×C₇₉	2
#8209	79