Subgroups — #34181519

Order 20224.

362 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772334	C₂	1	1	yes	maximal, characteristic
2	#23763572	C₂	1	1	yes	maximal, characteristic
3	#23762088	C₂	1	2	yes	maximal
4	#23761275	C₂	1	2	yes	maximal
5	C₂×#334×C₇₉	C₂	1	1	yes	maximal, characteristic
6	#19099120	C₂²	1	1	yes	characteristic
7	#19098693	—	2	1	no	—
8	#19098693	C₄	1	2	yes	—
9	C₈×Dic₁₅₈	—	2	1	no	—
10	C₈×Dic₁₅₈	C₄	1	2	yes	—
11	C₂×#14579097	—	2	1	no	—
12	C₂×#14579097	—	2	1	no	—
13	#19098589	—	2	1	no	—
14	#19098587	C₂²	1	2	yes	—
15	#19098587	C₂²	1	1	yes	characteristic
16	#19098518	—	2	1	no	—
17	#19098311	—	2	2	no	—
18	#19098310	—	2	2	no	—
19	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
21	#334×C₇₉	—	2	1	no	—
22	#334×C₇₉	—	2	1	no	—
23	C₂×#11929608	—	2	1	no	—
24	C₂×#11929608	—	2	2	no	—
25	#14579161	D₄	1	1	yes	characteristic
26	#14579161	D₄	1	1	yes	characteristic
27	#14579153	—	2	1	no	—
28	#14579140	—	2	1	no	—
29	#14579140	Q₈	1	2	yes	—
30	C₄×Dic₁₅₈	—	2	1	no	—
31	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
32	#14579120	—	2	1	no	—
33	#14579097	—	4	2	no	—
34	#14579097	—	4	2	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
36	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
38	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
39	Dic₇₉⋊C₈	—	2	2	no	—
40	Dic₇₉⋊C₈	—	2	2	no	—
41	#14579071	—	2	2	no	—
42	C₈×Dic₇₉	—	2	2	no	—
43	C₂²×C₈×C₇₉	—	2	2	no	—
44	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₇₉	—	2	2	no	—
48	C₄⋊C₈×C₇₉	—	2	2	no	—
49	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
53	C₂×Dic₁₅₈	—	2	1	no	—
54	C₂×Dic₁₅₈	C₄⋊C₄	1	2	yes	—
55	#11929608	—	4	2	no	—
56	#11929608	—	4	1	no	—
57	#11929608	—	4	2	no	—
58	#11929608	—	4	1	no	—
59	C₂×C₇₉⋊C₈	—	2	2	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₄⋊Dic₇₉	—	2	1	no	—
62	C₄⋊Dic₇₉	—	2	1	no	—
63	C₄⋊Dic₇₉	—	2	1	no	—
64	C₄⋊Dic₇₉	C₄⋊C₄	1	2	yes	—
65	Dic₇₉⋊C₄	—	2	2	no	—
66	Dic₇₉⋊C₄	—	2	2	no	—
67	C₂²⋊Dic₇₉	—	2	2	no	—
68	C₄×Dic₇₉	—	2	2	no	—
69	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
72	C₂×C₈×C₇₉	—	2	1	no	—
73	C₂×C₈×C₇₉	—	2	1	no	—
74	C₂×C₈×C₇₉	—	4	2	no	—
75	C₂×C₈×C₇₉	—	8	2	no	—
76	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
81	C₂×C₈×C₇₉	Q₁₆	1	2	yes	—
82	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₇₉	#41	1	1	yes	characteristic
86	C₂³×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇₉	—	2	2	no	—
88	C₂×C₄×C₇₉	—	4	1	no	—
89	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#117	1	1	yes	characteristic
92	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
93	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
99	C₂×Dic₇₉	—	2	2	no	—
100	C₂×Dic₇₉	—	2	2	no	—
101	Dic₁₅₈	—	4	2	no	—
102	Dic₁₅₈	—	4	2	no	—
103	C₈×C₇₉	—	2	1	no	—
104	C₈×C₇₉	—	2	1	no	—
105	C₈×C₇₉	—	8	2	no	—
106	C₈×C₇₉	#103	1	2	yes	—
107	C₈×C₇₉	#103	1	2	yes	—
108	C₇₉⋊C₈	—	4	2	no	—
109	C₇₉⋊C₈	—	8	2	no	—
110	C₂²×C₇₉	—	2	1	no	—
111	C₂²×C₇₉	—	2	1	no	—
112	C₂²×C₇₉	#380	1	1	yes	characteristic
113	C₂²×C₇₉	#475	1	1	yes	characteristic
114	C₂²×C₇₉	#477	1	1	yes	characteristic
115	C₄×C₇₉	—	2	1	no	—
116	C₄×C₇₉	—	2	1	no	—
117	C₄×C₇₉	—	2	1	no	—
118	C₄×C₇₉	C₂×#103	1	1	yes	characteristic
119	C₄×C₇₉	#417	1	1	yes	characteristic
120	C₄×C₇₉	#460	1	1	yes	characteristic
121	C₄×C₇₉	#462	1	1	yes	characteristic
122	C₄×C₇₉	#471	1	2	yes	—
123	Dic₇₉	—	4	2	no	—
124	Dic₇₉	—	4	2	no	—
125	#8683	—	79	1	no	maximal, Sylow
126	C₂×C₇₉	—	2	1	no	—
127	C₂×C₇₉	—	2	1	no	—
128	C₂×C₇₉	#1803	1	1	yes	characteristic
129	C₂×C₇₉	#1813	1	1	yes	characteristic
130	C₂×C₇₉	#1815	1	1	yes	characteristic
131	#2960	—	79	1	no	—
132	#1848	—	79	1	no	—
133	C₂×#334	D₇₉	1	1	yes	characteristic
134	#1550	—	79	2	no	—
135	#1292	—	79	2	no	—
136	C₇₉	#8683	1	1	yes	characteristic, complemented, Sylow
137	#533	—	79	1	no	—
138	#446	—	79	2	no	—
139	#446	—	158	1	no	—
140	C₈×Q₈	—	79	2	no	—
141	C₈×Q₈	—	158	1	no	—
142	C₂×#108	—	158	1	no	—
143	C₂×#108	—	158	1	no	—
144	#424	—	158	1	no	—
145	#423	—	79	1	no	—
146	#423	—	79	2	no	—
147	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
148	#405	—	158	1	no	—
149	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
150	#334	—	2	1	no	—
151	#334	—	2	1	no	—
152	#333	—	158	2	no	—
153	#332	—	158	2	no	—
154	C₂×M₁₆	—	158	2	no	—
155	C₂×M₁₆	—	158	1	no	—
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
158	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
159	C₄⋊Q₈	—	79	1	no	—
160	C₄⋊Q₈	—	79	1	no	—
161	#125	—	158	1	no	—
162	C₂²⋊Q₈	—	79	2	no	—
163	C₂²⋊Q₈	—	158	1	no	—
164	C₄×Q₈	—	79	2	no	—
165	C₄×Q₈	—	158	1	no	—
166	#117	—	158	1	no	—
167	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
168	#108	—	316	2	no	—
169	#108	—	316	2	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	C₄⋊C₈	—	79	2	no	—
173	C₄⋊C₈	—	158	2	no	—
174	C₄⋊C₈	—	158	2	no	—
175	C₄⋊C₈	—	158	2	no	—
176	C₄⋊C₈	—	158	1	no	—
177	C₄⋊C₈	—	158	2	no	—
178	#97	—	158	2	no	—
179	C₄×C₈	—	158	2	no	—
180	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
181	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
182	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
183	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
184	C₂×Q₈	—	79	2	no	—
185	C₂×Q₈	—	158	1	no	—
186	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
187	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
188	C₂²×C₄	#11929623	1	1	yes	characteristic
189	M₁₆	—	316	1	no	—
190	M₁₆	—	316	1	no	—
191	M₁₆	—	316	2	no	—
192	M₁₆	—	316	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	8	2	no	—
197	C₂×C₈	—	316	2	no	—
198	C₂×C₈	—	158	2	no	—
199	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
200	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
201	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
202	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
203	C₂×C₈	#11929613	1	2	yes	—
204	C₂×C₈	Dic₃₁₆	1	2	yes	—
205	C₄⋊C₄	—	79	2	no	—
206	C₄⋊C₄	—	158	2	no	—
207	C₄⋊C₄	—	158	1	no	—
208	C₄⋊C₄	—	158	1	no	—
209	C₄⋊C₄	—	158	1	no	—
210	C₄⋊C₄	—	158	2	no	—
211	C₂²⋊C₄	—	158	2	no	—
212	C₄²	—	158	2	no	—
213	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
214	C₄²	D₄×D₇₉	1	1	yes	characteristic
215	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
216	C₄²	#11929625	1	1	yes	characteristic
217	C₂³	#14579141	1	1	yes	characteristic
218	Q₈	—	316	2	no	—
219	Q₈	—	316	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	158	2	no	—
223	C₂×C₄	—	158	2	no	—
224	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
225	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
226	C₂×C₄	#14579119	1	1	yes	characteristic
227	C₂×C₄	#14579132	1	1	yes	characteristic
228	C₂×C₄	#14579136	1	1	yes	characteristic
229	C₂×C₄	#14579137	1	1	yes	characteristic
230	C₂×C₄	C₂×#11929613	1	1	yes	characteristic
231	C₂×C₄	C₂×Dic₃₁₆	1	1	yes	characteristic
232	C₂×C₄	#14579189	1	1	yes	characteristic
233	C₂×C₄	#14579194	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	2	1	no	—
236	C₈	—	8	2	no	—
237	C₈	—	316	2	no	—
238	C₈	—	632	2	no	—
239	C₈	#14579086	1	2	yes	—
240	C₈	#14579086	1	2	yes	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	#19098420	1	1	yes	characteristic
244	C₂²	#19098849	1	1	yes	characteristic
245	C₂²	#19098859	1	1	yes	characteristic
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	316	2	no	—
250	C₄	—	316	2	no	—
251	C₄	C₂×#14579086	1	1	yes	characteristic
252	C₄	#19098565	1	1	yes	characteristic
253	C₄	#19098764	1	1	yes	characteristic
254	C₄	#19098774	1	1	yes	characteristic
255	C₄	#19098823	1	2	yes	—
256	C₂	—	2	1	no	—
257	C₂	—	2	1	no	—
258	C₂	#23763406	1	1	yes	characteristic
259	C₂	#23763447	1	1	yes	characteristic
260	C₂	#23763460	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098420
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#334×C₇₉	C₂
socle	C₂²×C₇₉	#380
prime radical	C₂³×C₇₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	C₂×#334	D₇₉
hypocenter	C₇₉	#8683
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
136. C₇₉	#8683	semidirect

Maximal subgroups

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

subgroup	index
#23772334	2
#23763572	2
#23762088	2
#23761275	2
C₂×#334×C₇₉	2
#8683	79