Subgroups — #34187277

Order 20224.

500 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771763	C₂	1	1	yes	maximal, characteristic
2	#23763253	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098304	C₂	1	4	yes	maximal
4	C₄×#97×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14579153	C₂²	1	2	yes	—
6	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
7	C₂×#14579072	C₂²	1	2	yes	—
8	#19098461	—	2	1	no	—
9	#19098461	—	2	1	no	—
10	#19098461	—	2	1	no	—
11	#19098461	—	2	1	no	—
12	#19098433	C₄	1	1	yes	characteristic
13	#19098433	C₄	1	1	yes	characteristic
14	#19098429	C₄	1	2	yes	—
15	#19098318	—	2	1	no	—
16	#19098304	—	2	8	no	—
17	C₂×#97×C₇₉	—	2	1	no	—
18	C₂×#97×C₇₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	—	2	1	no	—
20	C₄³×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	#14579153	—	2	4	no	—
23	#14579153	—	2	4	no	—
24	C₂×C₄⋊Dic₇₉	—	2	1	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×Dic₇₉⋊C₄	—	2	2	no	—
32	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
33	#14579072	—	2	4	no	—
34	C₇₉⋊#95	—	2	2	no	—
35	C₇₉⋊#95	—	2	2	no	—
36	C₂²×C₈×C₇₉	—	2	2	no	—
37	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₇₉	D₄	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	C₂³	1	1	yes	characteristic
44	#97×C₇₉	—	2	4	no	—
45	#97×C₇₉	—	2	2	no	—
46	#97×C₇₉	D₄	1	4	yes	—
47	C₄×C₈×C₇₉	—	2	4	no	—
48	C₂²×Dic₇₉	—	2	2	no	—
49	C₂²×Dic₇₉	—	2	2	no	—
50	C₂×C₇₉⋊C₈	—	2	4	no	—
51	C₂×C₇₉⋊C₈	—	4	2	no	—
52	C₂×C₇₉⋊C₈	—	4	2	no	—
53	C₄⋊Dic₇₉	—	2	4	no	—
54	C₄⋊Dic₇₉	—	2	4	no	—
55	C₄⋊Dic₇₉	—	4	2	no	—
56	C₄⋊Dic₇₉	—	4	2	no	—
57	C₄⋊Dic₇₉	—	4	2	no	—
58	C₄⋊Dic₇₉	—	4	2	no	—
59	Dic₇₉⋊C₄	—	4	4	no	—
60	Dic₇₉⋊C₄	—	4	2	no	—
61	Dic₇₉⋊C₄	—	4	2	no	—
62	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₉	—	2	4	no	—
70	C₂×C₈×C₇₉	—	2	4	no	—
71	C₂×C₈×C₇₉	—	2	4	no	—
72	C₄²×C₇₉	—	2	2	no	—
73	C₄²×C₇₉	—	2	2	no	—
74	C₄²×C₇₉	—	2	2	no	—
75	C₄²×C₇₉	—	2	2	no	—
76	C₄²×C₇₉	—	2	2	no	—
77	C₄²×C₇₉	—	4	1	no	—
78	C₄²×C₇₉	C₂×D₄	1	2	yes	—
79	C₄²×C₇₉	#41	1	2	yes	—
80	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₇₉	—	2	1	no	—
82	C₂×C₄×C₇₉	—	2	1	no	—
83	C₂×C₄×C₇₉	—	2	1	no	—
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	#101	1	2	yes	—
95	C₂×C₄×C₇₉	#101	1	2	yes	—
96	C₂×C₄×C₇₉	#104	1	2	yes	—
97	C₂×C₄×C₇₉	#104	1	2	yes	—
98	C₂×C₄×C₇₉	#104	1	2	yes	—
99	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₇₉	#121	1	2	yes	—
101	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
102	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
103	C₂×Dic₇₉	—	4	2	no	—
104	C₂×Dic₇₉	—	4	2	no	—
105	C₂×Dic₇₉	—	4	2	no	—
106	C₂×Dic₇₉	—	4	2	no	—
107	C₂×Dic₇₉	—	4	2	no	—
108	C₂×Dic₇₉	—	4	2	no	—
109	C₈×C₇₉	—	2	8	no	—
110	C₇₉⋊C₈	—	4	4	no	—
111	C₂²×C₇₉	#390	1	1	yes	characteristic
112	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
113	C₂²×C₇₉	C₂×#101	1	1	yes	characteristic
114	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
115	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
116	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
117	C₂²×C₇₉	#421	1	1	yes	characteristic
118	C₄×C₇₉	—	2	2	no	—
119	C₄×C₇₉	—	2	2	no	—
120	C₄×C₇₉	—	2	1	no	—
121	C₄×C₇₉	—	2	1	no	—
122	C₄×C₇₉	—	2	2	no	—
123	C₄×C₇₉	—	2	2	no	—
124	C₄×C₇₉	—	4	1	no	—
125	C₄×C₇₉	#330	1	4	yes	complemented
126	Dic₇₉	—	8	2	no	—
127	Dic₇₉	—	8	2	no	—
128	#9686	—	79	1	no	maximal, Sylow
129	C₂×C₇₉	C₂×#330	1	1	yes	characteristic
130	C₂×C₇₉	#1526	1	1	yes	characteristic
131	C₂×C₇₉	#1528	1	1	yes	characteristic
132	C₂×C₇₉	#1531	1	1	yes	characteristic
133	C₂×C₇₉	#1962	1	1	yes	characteristic
134	C₂×C₇₉	#1966	1	1	yes	characteristic
135	C₂×C₇₉	#1980	1	1	yes	characteristic
136	#2859	—	79	1	no	—
137	#1767	—	79	1	no	—
138	C₄×#97	D₇₉	1	1	yes	characteristic
139	C₂×#330	—	79	4	no	—
140	C₇₉	#9686	1	1	yes	characteristic, complemented, Sylow
141	C₂×#125	—	79	2	no	—
142	C₂×C₄⋊C₈	—	158	1	no	—
143	C₂×#97	—	2	1	no	—
144	C₂×#97	—	79	2	no	—
145	C₂×#97	C₂×D₇₉	1	2	yes	—
146	C₂×C₄×C₈	—	2	1	no	—
147	#389	—	158	1	no	—
148	#389	—	158	1	no	—
149	#389	—	158	1	no	—
150	#389	—	158	1	no	—
151	#384	—	79	1	no	—
152	#384	—	79	1	no	—
153	#382	—	79	2	no	—
154	C₄³	C₂×D₇₉	1	1	yes	characteristic
155	#336	—	158	1	no	—
156	#330	—	158	8	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	158	2	no	—
159	#125	—	158	4	no	—
160	#125	—	158	4	no	—
161	C₂×C₄⋊C₄	—	79	2	no	—
162	C₂×C₄⋊C₄	—	79	2	no	—
163	C₂×C₄⋊C₄	—	158	1	no	—
164	C₂×C₄⋊C₄	—	158	1	no	—
165	C₂×C₄⋊C₄	—	158	1	no	—
166	C₂×C₄⋊C₄	—	158	1	no	—
167	C₂×C₄⋊C₄	—	158	2	no	—
168	C₂×C₄⋊C₄	—	158	2	no	—
169	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
170	C₂×C₄²	C₄×D₇₉	1	1	yes	characteristic
171	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
172	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
173	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
175	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
176	C₄⋊C₈	—	158	4	no	—
177	#97	—	2	4	no	—
178	#97	—	2	2	no	—
179	#97	—	158	4	no	—
180	#97	D₃₁₆	1	4	yes	—
181	C₄×C₈	—	2	4	no	—
182	#95	—	158	2	no	—
183	#95	—	158	2	no	—
184	C₂²×C₄	—	158	2	no	—
185	C₂²×C₄	—	158	2	no	—
186	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
187	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
188	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
189	C₂²×C₄	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	316	2	no	—
197	C₂×C₈	—	316	2	no	—
198	C₂×C₈	—	158	4	no	—
199	C₄⋊C₄	—	316	2	no	—
200	C₄⋊C₄	—	316	2	no	—
201	C₄⋊C₄	—	316	2	no	—
202	C₄⋊C₄	—	316	2	no	—
203	C₄⋊C₄	—	316	4	no	—
204	C₄⋊C₄	—	316	2	no	—
205	C₄⋊C₄	—	316	2	no	—
206	C₄⋊C₄	—	158	4	no	—
207	C₄⋊C₄	—	158	4	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	2	2	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₂×D₃₁₆	1	2	yes	—
215	C₄²	#11929624	1	2	yes	—
216	C₂³	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	316	2	no	—
231	C₂×C₄	—	316	2	no	—
232	C₂×C₄	—	316	2	no	—
233	C₂×C₄	—	316	2	no	—
234	C₂×C₄	—	316	2	no	—
235	C₂×C₄	—	316	2	no	—
236	C₂×C₄	#14579079	1	2	yes	—
237	C₂×C₄	#14579079	1	2	yes	—
238	C₂×C₄	#14579087	1	2	yes	—
239	C₂×C₄	#14579089	1	2	yes	—
240	C₂×C₄	#14579089	1	2	yes	—
241	C₂×C₄	C₄×D₃₁₆	1	2	yes	—
242	C₂×C₄	#14579135	1	2	yes	—
243	C₂×C₄	#14579149	1	1	yes	characteristic
244	C₂×C₄	#14579158	1	1	yes	characteristic
245	C₈	—	2	8	no	—
246	C₈	—	316	4	no	—
247	C₂²	#19098465	1	1	yes	characteristic
248	C₂²	C₂×#14579079	1	1	yes	characteristic
249	C₂²	C₂×#14579079	1	1	yes	characteristic
250	C₂²	C₂×#14579087	1	1	yes	characteristic
251	C₂²	C₂×#14579089	1	1	yes	characteristic
252	C₂²	C₂×#14579089	1	1	yes	characteristic
253	C₂²	#19098579	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	1	no	—
261	C₄	—	632	2	no	—
262	C₄	—	632	2	no	—
263	C₄	#19098304	1	4	yes	complemented
264	C₂	C₂×#19098304	1	1	yes	characteristic
265	C₂	#23761994	1	1	yes	characteristic
266	C₂	#23762001	1	1	yes	characteristic
267	C₂	#23762014	1	1	yes	characteristic
268	C₂	#23764228	1	1	yes	characteristic
269	C₂	#23764244	1	1	yes	characteristic
270	C₂	#23764319	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₇₉)⋊C₄
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₄×#97×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₄×#97	D₇₉
hypocenter	C₇₉	#9686
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#97

Complements

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

N	H ≅ G/N	type
125. C₄×C₇₉	#330	semidirect
140. C₇₉	#9686	semidirect
263. C₄	#19098304	semidirect

Maximal subgroups

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

subgroup	index
#23771763	2
#23763253	2
C₂×#19098304	2
C₄×#97×C₇₉	2
#9686	79