Subgroups — #34188271

Order 20224.

512 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579161	C₂	1	1	yes	maximal, characteristic
2	#23763253	C₂	1	1	yes	maximal, characteristic
3	#23761240	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₈×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579161	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₅₈	—	2	1	no	—
7	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×#14579072	—	2	1	no	—
9	#19098469	—	2	1	no	—
10	#19098461	—	2	1	no	—
11	#19098461	C₄	1	2	yes	—
12	#19098433	C₄	1	1	yes	characteristic
13	C₄×C₄⋊Dic₇₉	—	2	1	no	—
14	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
15	#19098318	—	2	1	no	—
16	C₂×C₄⋊C₈×C₇₉	—	2	1	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
18	C₄³×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	1	no	—
20	C₂²×Dic₁₅₈	—	2	1	no	—
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	#14579161	—	2	4	no	—
23	C₄×Dic₁₅₈	—	4	2	no	—
24	C₄×Dic₁₅₈	—	4	2	no	—
25	C₂×C₄⋊Dic₇₉	—	2	1	no	—
26	C₂×C₄⋊Dic₇₉	—	2	1	no	—
27	C₂×C₄⋊Dic₇₉	—	2	1	no	—
28	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
29	C₂×C₄⋊Dic₇₉	C₂×C₄	1	2	yes	—
30	C₂×Dic₇₉⋊C₄	—	2	2	no	—
31	C₂×C₄×Dic₇₉	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₇₉⋊C₈)	Q₈	1	4	yes	complemented
34	#14579072	—	4	2	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₇₉⋊#95	—	2	2	no	—
37	C₂²×C₈×C₇₉	—	2	2	no	—
38	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₇₉	—	2	4	no	—
46	C₄×C₈×C₇₉	—	2	4	no	—
47	C₂×Dic₁₅₈	—	2	4	no	—
48	C₂×Dic₁₅₈	—	4	2	no	—
49	C₂×Dic₁₅₈	—	4	2	no	—
50	C₂²×Dic₇₉	—	2	2	no	—
51	C₂²×Dic₇₉	—	2	2	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	4	2	no	—
54	C₂×C₇₉⋊C₈	—	4	2	no	—
55	C₄⋊Dic₇₉	—	2	4	no	—
56	C₄⋊Dic₇₉	—	2	2	no	—
57	C₄⋊Dic₇₉	—	2	4	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	C₄⋊Dic₇₉	—	4	2	no	—
60	C₄⋊Dic₇₉	M₁₆	1	4	yes	—
61	Dic₇₉⋊C₄	—	4	4	no	—
62	C₄×Dic₇₉	—	4	2	no	—
63	C₄×Dic₇₉	—	4	2	no	—
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₇₉	—	2	4	no	—
72	C₂×C₈×C₇₉	—	4	2	no	—
73	C₂×C₈×C₇₉	—	4	2	no	—
74	C₄²×C₇₉	—	2	2	no	—
75	C₄²×C₇₉	—	2	2	no	—
76	C₄²×C₇₉	—	2	2	no	—
77	C₄²×C₇₉	—	2	2	no	—
78	C₄²×C₇₉	—	4	1	no	—
79	C₄²×C₇₉	D₈	1	2	yes	—
80	C₄²×C₇₉	SD₁₆	1	2	yes	—
81	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
82	C₄²×C₇₉	#41	1	2	yes	—
83	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₉	—	2	1	no	—
85	C₂×C₄×C₇₉	—	2	1	no	—
86	C₂×C₄×C₇₉	—	2	1	no	—
87	C₂×C₄×C₇₉	—	2	1	no	—
88	C₂×C₄×C₇₉	—	2	1	no	—
89	C₂×C₄×C₇₉	—	2	1	no	—
90	C₂×C₄×C₇₉	—	2	1	no	—
91	C₂×C₄×C₇₉	—	2	1	no	—
92	C₂×C₄×C₇₉	—	2	1	no	—
93	C₂×C₄×C₇₉	—	2	1	no	—
94	C₂×C₄×C₇₉	—	2	1	no	—
95	C₂×C₄×C₇₉	#102	1	2	yes	—
96	C₂×C₄×C₇₉	#102	1	2	yes	—
97	C₂×C₄×C₇₉	#104	1	2	yes	—
98	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
99	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₇₉	#131	1	2	yes	—
104	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₇₉	#137	1	2	yes	—
107	C₂×Dic₇₉	—	2	4	no	—
108	C₂×Dic₇₉	—	4	2	no	—
109	C₂×Dic₇₉	—	4	2	no	—
110	C₂×Dic₇₉	—	4	2	no	—
111	C₂×Dic₇₉	—	4	2	no	—
112	C₂×Dic₇₉	—	4	2	no	—
113	Dic₁₅₈	—	4	8	no	—
114	C₈×C₇₉	—	4	4	no	—
115	C₇₉⋊C₈	—	4	4	no	—
116	C₂²×C₇₉	#391	1	1	yes	characteristic
117	C₂²×C₇₉	#407	1	1	yes	characteristic
118	C₂²×C₇₉	#408	1	1	yes	characteristic
119	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
120	C₂²×C₇₉	#419	1	1	yes	characteristic
121	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
122	C₂²×C₇₉	#421	1	1	yes	characteristic
123	C₄×C₇₉	—	2	1	no	—
124	C₄×C₇₉	—	2	1	no	—
125	C₄×C₇₉	—	2	2	no	—
126	C₄×C₇₉	—	2	2	no	—
127	C₄×C₇₉	—	2	2	no	—
128	C₄×C₇₉	—	4	1	no	—
129	C₄×C₇₉	#446	1	4	yes	—
130	C₄×C₇₉	#474	1	2	yes	—
131	C₄×C₇₉	#475	1	2	yes	—
132	Dic₇₉	—	4	4	no	—
133	Dic₇₉	—	8	2	no	—
134	#9869	—	79	1	no	maximal, Sylow
135	C₂×C₇₉	#1486	1	1	yes	characteristic
136	C₂×C₇₉	#1491	1	1	yes	characteristic
137	C₂×C₇₉	#1492	1	1	yes	characteristic
138	C₂×C₇₉	#1497	1	1	yes	characteristic
139	C₂×C₇₉	#1985	1	1	yes	characteristic
140	C₂×C₇₉	#1986	1	1	yes	characteristic
141	C₂×C₇₉	#1988	1	1	yes	characteristic
142	C₄×C₄⋊Q₈	—	79	1	no	—
143	#1767	—	79	1	no	—
144	C₄×C₄⋊C₈	D₇₉	1	1	yes	characteristic
145	#1279	—	79	4	no	—
146	C₇₉	#9869	1	1	yes	characteristic, complemented, Sylow
147	C₂×C₄⋊Q₈	—	79	1	no	—
148	C₂×C₄×Q₈	—	158	1	no	—
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	79	2	no	—
151	C₂×#97	—	158	1	no	—
152	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
153	#391	—	158	1	no	—
154	#389	—	79	2	no	—
155	#389	—	158	1	no	—
156	#384	—	79	1	no	—
157	C₄×C₄⋊C₄	—	79	2	no	—
158	C₄×C₄⋊C₄	—	158	1	no	—
159	C₄³	C₂×D₇₉	1	1	yes	characteristic
160	#336	—	2	1	no	—
161	#336	—	158	1	no	—
162	C₂²×Q₈	—	158	1	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	158	2	no	—
165	C₄⋊Q₈	—	158	4	no	—
166	C₄×Q₈	—	316	2	no	—
167	C₄×Q₈	—	316	2	no	—
168	C₂×C₄⋊C₄	—	79	2	no	—
169	C₂×C₄⋊C₄	—	79	2	no	—
170	C₂×C₄⋊C₄	—	158	1	no	—
171	C₂×C₄⋊C₄	—	158	2	no	—
172	C₂×C₄⋊C₄	—	158	1	no	—
173	C₂×C₄⋊C₄	—	158	1	no	—
174	C₂×C₄²	—	158	2	no	—
175	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
176	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
177	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
178	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
180	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
182	C₄⋊C₈	—	2	4	no	—
183	C₄⋊C₈	—	79	4	no	—
184	C₄⋊C₈	—	158	2	no	—
185	#97	—	316	2	no	—
186	C₄×C₈	—	2	4	no	—
187	#95	—	158	2	no	—
188	#95	—	158	2	no	—
189	C₂×Q₈	—	316	2	no	—
190	C₂×Q₈	—	316	2	no	—
191	C₂×Q₈	—	158	4	no	—
192	C₂²×C₄	—	158	2	no	—
193	C₂²×C₄	—	158	2	no	—
194	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
195	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
196	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
197	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	316	2	no	—
205	C₂×C₈	—	316	2	no	—
206	C₂×C₈	—	158	4	no	—
207	C₄⋊C₄	—	316	4	no	—
208	C₄⋊C₄	—	316	2	no	—
209	C₄⋊C₄	—	316	2	no	—
210	C₄⋊C₄	—	79	4	no	—
211	C₄⋊C₄	—	158	4	no	—
212	C₄⋊C₄	—	158	4	no	—
213	C₄⋊C₄	—	158	2	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	1	no	—
219	C₄²	—	316	2	no	—
220	C₄²	—	316	2	no	—
221	C₄²	D₆₃₂	1	2	yes	—
222	C₄²	#11929613	1	2	yes	—
223	C₄²	Q₈×D₇₉	1	2	yes	—
224	C₄²	#11929623	1	2	yes	—
225	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
226	Q₈	—	316	8	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	316	2	no	—
239	C₂×C₄	—	316	2	no	—
240	C₂×C₄	—	316	2	no	—
241	C₂×C₄	—	316	2	no	—
242	C₂×C₄	—	316	2	no	—
243	C₂×C₄	—	158	4	no	—
244	C₂×C₄	#14579083	1	2	yes	—
245	C₂×C₄	#14579083	1	2	yes	—
246	C₂×C₄	#14579089	1	2	yes	—
247	C₂×C₄	#14579126	1	2	yes	—
248	C₂×C₄	#14579141	1	2	yes	—
249	C₂×C₄	#14579148	1	1	yes	characteristic
250	C₂×C₄	#14579159	1	1	yes	characteristic
251	C₂×C₄	M₁₆×D₇₉	1	2	yes	—
252	C₂×C₄	#14579167	1	2	yes	—
253	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
254	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
255	C₂×C₄	#14579194	1	2	yes	—
256	C₈	—	4	4	no	—
257	C₈	—	316	4	no	—
258	C₂²	#19098468	1	1	yes	characteristic
259	C₂²	#19098526	1	1	yes	characteristic
260	C₂²	#19098529	1	1	yes	characteristic
261	C₂²	C₂×#14579083	1	1	yes	characteristic
262	C₂²	#19098573	1	1	yes	characteristic
263	C₂²	C₂×#14579089	1	1	yes	characteristic
264	C₂²	#19098581	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	316	4	no	—
272	C₄	—	632	2	no	—
273	C₄	#19098692	1	4	yes	—
274	C₄	#19098844	1	2	yes	—
275	C₄	#19098849	1	2	yes	—
276	C₂	#23761826	1	1	yes	characteristic
277	C₂	#23761848	1	1	yes	characteristic
278	C₂	#23761852	1	1	yes	characteristic
279	C₂	#23761874	1	1	yes	characteristic
280	C₂	#23764339	1	1	yes	characteristic
281	C₂	#23764343	1	1	yes	characteristic
282	C₂	#23764351	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₁₆
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×C₄⋊C₈×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₇₉
hypocenter	C₇₉	#9869
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
33. C₄⋊(C₇₉⋊C₈)	Q₈	semidirect
146. C₇₉	#9869	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579161	2
#23763253	2
#23761240	2
C₄×C₄⋊C₈×C₇₉	2
#9869	79