Subgroups — #34187565

Order 20224.

506 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	C₂×#19098943	C₂	1	1	yes	maximal, characteristic
2	#23764352	C₂	1	2	yes	maximal
3	C₂×#19098314	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098310	C₂	1	2	yes	maximal
5	C₇₉×#1770	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₆	—	2	1	no	—
7	C₂²×Dic₃₁₆	—	2	1	no	—
8	C₂×#14579161	C₂²	1	2	yes	—
9	#19098943	—	2	2	no	—
10	#19098943	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
12	#19098325	—	2	4	no	—
13	#19098314	—	2	2	no	—
14	#19098310	—	2	4	no	—
15	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#97×C₇₉	—	2	1	no	—
17	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
19	#336×C₇₉	—	2	1	no	—
20	C₂²×Dic₁₅₈	—	2	2	no	—
21	C₂×Dic₃₁₆	—	2	4	no	—
22	C₂×Dic₃₁₆	—	2	4	no	—
23	C₂×Dic₃₁₆	—	4	2	no	—
24	C₂×Dic₃₁₆	—	4	2	no	—
25	C₂×Dic₃₁₆	—	4	2	no	—
26	C₂×Dic₃₁₆	—	4	2	no	—
27	C₂²×C₇₉⋊C₈	—	2	2	no	—
28	#14579161	—	2	2	no	—
29	#14579161	—	2	2	no	—
30	#14579161	—	2	2	no	—
31	#14579161	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄⋊Dic₇₉	—	2	2	no	—
34	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
36	C₂²×C₈×C₇₉	—	2	1	no	—
37	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
40	C₂×C₄²×C₇₉	—	2	1	no	—
41	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₇₉	—	2	1	no	—
43	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
44	#97×C₇₉	—	4	2	no	—
45	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₇₉	D₄	1	2	yes	—
47	#95×C₇₉	—	2	1	no	—
48	C₂×Dic₁₅₈	—	2	4	no	—
49	C₂×Dic₁₅₈	—	4	2	no	—
50	C₂×Dic₁₅₈	—	4	2	no	—
51	C₂×Dic₁₅₈	—	4	2	no	—
52	C₂×Dic₁₅₈	—	4	2	no	—
53	C₂×Dic₁₅₈	—	4	2	no	—
54	C₂²×Dic₇₉	—	4	2	no	—
55	Dic₃₁₆	—	4	8	no	—
56	Dic₃₁₆	—	4	8	no	—
57	C₂×C₇₉⋊C₈	—	4	2	no	—
58	C₂×C₇₉⋊C₈	—	4	4	no	—
59	C₄⋊Dic₇₉	—	2	4	no	—
60	C₄⋊Dic₇₉	—	2	4	no	—
61	C₄⋊Dic₇₉	—	4	2	no	—
62	C₄⋊Dic₇₉	—	4	2	no	—
63	C₂²×C₄×C₇₉	—	2	2	no	—
64	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
67	C₂×C₈×C₇₉	—	2	2	no	—
68	C₂×C₈×C₇₉	—	2	1	no	—
69	C₂×C₈×C₇₉	—	2	1	no	—
70	C₂×C₈×C₇₉	—	4	1	no	—
71	C₂×C₈×C₇₉	—	4	1	no	—
72	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₇₉	D₈	1	2	yes	—
75	C₂×C₈×C₇₉	SD₁₆	1	2	yes	—
76	C₄⋊C₄×C₇₉	—	2	4	no	—
77	C₄²×C₇₉	—	4	1	no	—
78	C₄²×C₇₉	—	4	1	no	—
79	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₇₉	#41	1	1	yes	characteristic
83	C₂³×C₇₉	#121	1	1	yes	characteristic
84	C₂×C₄×C₇₉	—	2	2	no	—
85	C₂×C₄×C₇₉	—	4	2	no	—
86	C₂×C₄×C₇₉	—	4	2	no	—
87	C₂×C₄×C₇₉	—	4	2	no	—
88	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
92	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
93	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
94	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₄×C₇₉	#137	1	4	yes	—
99	C₂×Dic₇₉	—	4	4	no	—
100	C₂×Dic₇₉	—	8	2	no	—
101	C₂×Dic₇₉	—	8	2	no	—
102	Dic₁₅₈	—	4	4	no	—
103	Dic₁₅₈	—	4	4	no	—
104	Dic₁₅₈	—	8	2	no	—
105	Dic₁₅₈	—	8	2	no	—
106	C₈×C₇₉	—	2	2	no	—
107	C₈×C₇₉	—	4	2	no	—
108	C₈×C₇₉	#102	1	4	yes	—
109	C₇₉⋊C₈	—	8	2	no	—
110	C₂²×C₇₉	#390	1	1	yes	characteristic
111	C₂²×C₇₉	#459	1	1	yes	characteristic
112	C₂²×C₇₉	#460	1	1	yes	characteristic
113	C₂²×C₇₉	#470	1	2	yes	—
114	C₂²×C₇₉	#473	1	2	yes	—
115	C₄×C₇₉	—	2	1	no	—
116	C₄×C₇₉	—	2	1	no	—
117	C₄×C₇₉	—	8	2	no	—
118	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
119	C₄×C₇₉	#418	1	1	yes	characteristic
120	C₄×C₇₉	#441	1	2	yes	—
121	C₄×C₇₉	#459	1	1	yes	characteristic
122	C₄×C₇₉	#460	1	1	yes	characteristic
123	C₄×C₇₉	#497	1	2	yes	—
124	Dic₇₉	—	8	4	no	—
125	#9742	—	79	1	no	maximal, Sylow
126	C₂×C₇₉	#1689	1	2	yes	—
127	C₂×C₇₉	#1690	1	2	yes	—
128	C₂×C₇₉	#1971	1	1	yes	characteristic
129	C₂×C₇₉	#1979	1	1	yes	characteristic
130	C₂×C₇₉	#1982	1	1	yes	characteristic
131	C₂×#494	—	79	1	no	—
132	#1988	—	79	2	no	—
133	#1770	D₇₉	1	1	yes	characteristic
134	C₂×#334	—	79	1	no	—
135	C₂×#332	—	79	2	no	—
136	C₇₉	#9742	1	1	yes	characteristic, complemented, Sylow
137	C₂²×Q₁₆	—	158	1	no	—
138	C₂²×Q₁₆	—	158	1	no	—
139	C₂×C₄⋊Q₈	—	79	2	no	—
140	#494	—	79	4	no	—
141	#494	—	158	2	no	—
142	C₂×C₄⋊C₈	—	79	2	no	—
143	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
144	C₂×#97	—	2	1	no	—
145	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
146	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
147	#340	—	158	4	no	—
148	#336	—	2	1	no	—
149	#334	—	158	2	no	—
150	#332	—	158	4	no	—
151	C₂²×Q₈	—	158	2	no	—
152	C₂×Q₁₆	—	316	2	no	—
153	C₂×Q₁₆	—	316	2	no	—
154	C₂×Q₁₆	—	316	2	no	—
155	C₂×Q₁₆	—	316	2	no	—
156	C₂×Q₁₆	—	158	4	no	—
157	C₂×Q₁₆	—	158	4	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	158	2	no	—
160	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
161	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
162	C₄⋊Q₈	—	79	4	no	—
163	C₄⋊Q₈	—	158	2	no	—
164	C₄⋊Q₈	—	158	2	no	—
165	C₄⋊Q₈	—	158	2	no	—
166	C₂×C₄⋊C₄	—	158	2	no	—
167	C₂×C₄⋊C₄	—	158	2	no	—
168	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
169	C₂×C₄²	—	2	1	no	—
170	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	—	158	2	no	—
173	C₄⋊C₈	—	158	2	no	—
174	C₄⋊C₈	D₃₁₆	1	2	yes	—
175	#97	—	4	2	no	—
176	C₄×C₈	C₄×D₇₉	1	2	yes	—
177	C₄×C₈	D₃₁₆	1	2	yes	—
178	#95	—	2	1	no	—
179	C₂×Q₈	—	316	2	no	—
180	C₂×Q₈	—	316	2	no	—
181	C₂×Q₈	—	316	2	no	—
182	C₂×Q₈	—	316	2	no	—
183	C₂×Q₈	—	316	2	no	—
184	C₂×Q₈	—	158	4	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	316	2	no	—
187	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
189	C₂²×C₄	#11929624	1	1	yes	characteristic
190	Q₁₆	—	316	8	no	—
191	Q₁₆	—	316	8	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	316	2	no	—
198	C₂×C₈	—	316	4	no	—
199	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
200	C₂×C₈	(C₂×D₇₉)⋊C₄	1	2	yes	—
201	C₂×C₈	#11929609	1	2	yes	—
202	C₂×C₈	#11929611	1	2	yes	—
203	C₄⋊C₄	—	2	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	—	4	1	no	—
209	C₄²	—	4	1	no	—
210	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
211	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
212	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
213	C₄²	#11929624	1	1	yes	characteristic
214	C₂³	#14579135	1	1	yes	characteristic
215	Q₈	—	316	4	no	—
216	Q₈	—	316	4	no	—
217	Q₈	—	632	2	no	—
218	Q₈	—	632	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	316	4	no	—
224	C₂×C₄	—	632	2	no	—
225	C₂×C₄	—	632	2	no	—
226	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
228	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
229	C₂×C₄	#14579135	1	1	yes	characteristic
230	C₂×C₄	#14579149	1	1	yes	characteristic
231	C₂×C₄	#14579158	1	1	yes	characteristic
232	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
233	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
234	C₂×C₄	#14579186	1	1	yes	characteristic
235	C₂×C₄	#14579186	1	1	yes	characteristic
236	C₂×C₄	#14579194	1	4	yes	—
237	C₈	—	2	2	no	—
238	C₈	—	4	2	no	—
239	C₈	—	632	2	no	—
240	C₈	#14579081	1	4	yes	—
241	C₂²	#19098465	1	1	yes	characteristic
242	C₂²	#19098758	1	1	yes	characteristic
243	C₂²	#19098763	1	1	yes	characteristic
244	C₂²	#19098821	1	2	yes	—
245	C₂²	#19098838	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	8	2	no	—
249	C₄	—	632	4	no	—
250	C₄	C₂×#14579081	1	1	yes	characteristic
251	C₄	#19098567	1	1	yes	characteristic
252	C₄	#19098671	1	2	yes	—
253	C₄	#19098758	1	1	yes	characteristic
254	C₄	#19098763	1	1	yes	characteristic
255	C₄	#19098957	1	2	yes	—
256	C₂	#23762889	1	2	yes	—
257	C₂	#23762894	1	2	yes	—
258	C₂	#23764271	1	1	yes	characteristic
259	C₂	#23764315	1	1	yes	characteristic
260	C₂	#23764327	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579135
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1770	C₂
socle, prime radical	C₂³×C₇₉	#121
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1770	D₇₉
hypocenter	C₇₉	#9742
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
10. #19098943	C₄	semidirect
136. C₇₉	#9742	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098943	2
#23764352	2
C₂×#19098314	2
C₂×#19098310	2
C₇₉×#1770	2
#9742	79