Subgroups — #34258800

Order 20224.

614 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773986	C₂	1	1	yes	maximal, characteristic
2	#23772330	C₂	1	2	yes	maximal
3	#23762296	C₂	1	4	yes	maximal, complemented
4	#23762206	C₂	1	4	yes	maximal, complemented
5	#23761854	C₂	1	2	yes	maximal
6	C₂×C₈×C₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1567	C₂	1	1	yes	maximal, characteristic
8	#19099120	C₂²	1	2	yes	—
9	#19098968	C₄	1	1	yes	characteristic
10	#19098943	C₄	1	1	yes	characteristic
11	#19098914	C₄	1	2	yes	—
12	#19098804	C₄	1	4	yes	—
13	#19098690	—	2	2	no	—
14	#19098690	C₂²	1	4	yes	—
15	#19098687	—	2	2	no	—
16	#19098687	C₂²	1	4	yes	—
17	C₄×Dic₃₁₆	C₂²	1	4	yes	—
18	#19098591	—	2	2	no	—
19	#19098521	—	2	2	no	—
20	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
22	#19098293	C₂²	1	8	yes	—
23	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
24	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
25	#423×C₇₉	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
27	#335×C₇₉	C₂²	1	4	yes	—
28	C₂×Dic₃₁₆	C₂×C₄	1	4	yes	—
29	C₂²×C₇₉⋊C₈	—	2	1	no	—
30	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
31	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
32	#14579161	C₂×C₄	1	2	yes	—
33	#14579161	C₂×C₄	1	2	yes	—
34	#14579153	—	2	2	no	—
35	#14579140	—	2	2	no	—
36	#14579140	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₅₈	—	2	2	no	—
38	C₄×Dic₁₅₈	C₂³	1	4	yes	—
39	#14579120	—	2	2	no	—
40	#14579095	C₂×C₄	1	4	yes	—
41	Dic₇₉⋊C₈	—	2	4	no	—
42	Dic₇₉⋊C₈	—	2	4	no	—
43	#14579086	C₂×C₄	1	8	yes	—
44	#14579071	—	2	4	no	—
45	C₄×C₇₉⋊C₈	—	2	1	no	—
46	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
47	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
48	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
49	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
50	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
51	C₈×Dic₇₉	—	2	4	no	—
52	C₂×M₁₆×C₇₉	—	2	2	no	—
53	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₉	—	2	2	no	—
56	C₄⋊C₈×C₇₉	C₂³	1	4	yes	—
57	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₇₉	C₂³	1	2	yes	—
60	C₂×Dic₁₅₈	—	2	2	no	—
61	C₂×Dic₁₅₈	C₂²×C₄	1	4	yes	—
62	Dic₃₁₆	—	2	8	no	—
63	C₂×C₇₉⋊C₈	—	2	4	no	—
64	C₂×C₇₉⋊C₈	—	2	2	no	—
65	C₂×C₇₉⋊C₈	—	2	2	no	—
66	C₂×C₇₉⋊C₈	Q₁₆	1	4	yes	complemented
67	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
70	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
71	C₄⋊Dic₇₉	—	2	2	no	—
72	C₄⋊Dic₇₉	—	2	2	no	—
73	C₄⋊Dic₇₉	—	2	2	no	—
74	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
75	Dic₇₉⋊C₄	—	2	4	no	—
76	Dic₇₉⋊C₄	—	2	4	no	—
77	C₂²⋊Dic₇₉	—	2	4	no	—
78	C₄×Dic₇₉	—	2	4	no	—
79	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
82	M₁₆×C₇₉	—	4	4	no	—
83	C₂×C₈×C₇₉	—	2	4	no	—
84	C₂×C₈×C₇₉	—	2	4	no	—
85	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
87	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₇₉	#41	1	1	yes	characteristic
90	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
91	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₉	—	2	2	no	—
93	C₂×C₄×C₇₉	—	2	2	no	—
94	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₇₉	#135	1	2	yes	—
99	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₇₉	—	2	4	no	—
103	C₂×Dic₇₉	—	2	4	no	—
104	Dic₁₅₈	—	2	8	no	—
105	Dic₁₅₈	—	4	4	no	—
106	C₈×C₇₉	—	2	2	no	—
107	C₈×C₇₉	—	4	4	no	—
108	C₈×C₇₉	#131	1	4	yes	—
109	C₇₉⋊C₈	—	2	4	no	—
110	C₇₉⋊C₈	—	2	1	no	—
111	C₇₉⋊C₈	—	2	1	no	—
112	C₇₉⋊C₈	C₂×Q₁₆	1	2	yes	—
113	C₇₉⋊C₈	C₂×Q₁₆	1	2	yes	—
114	C₂²×C₇₉	—	2	1	no	—
115	C₂²×C₇₉	—	2	1	no	—
116	C₂²×C₇₉	C₄×Q₁₆	1	2	yes	—
117	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
118	C₄×C₇₉	—	2	1	no	—
119	C₄×C₇₉	—	2	1	no	—
120	C₄×C₇₉	C₄×Q₁₆	1	2	yes	—
121	C₄×C₇₉	#443	1	2	yes	—
122	C₄×C₇₉	C₂×#131	1	2	yes	—
123	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
124	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
125	Dic₇₉	—	4	4	no	—
126	Dic₇₉	—	4	4	no	—
127	#17958	—	79	1	no	maximal, Sylow
128	C₂×C₇₉	—	2	2	no	—
129	C₂×C₇₉	#2931	1	1	yes	characteristic
130	C₂×C₇₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
131	C₂×C₇₉	C₂×#443	1	1	yes	characteristic
132	#3170	—	79	1	no	—
133	#2960	—	79	2	no	—
134	#1591	—	79	4	no	—
135	C₈×Q₁₆	—	79	4	no	—
136	#1567	D₇₉	1	1	yes	characteristic
137	#1493	—	79	2	no	—
138	C₂×C₈²	—	79	1	no	—
139	C₇₉	#17958	1	1	yes	characteristic, complemented, Sylow
140	#533	—	79	2	no	—
141	#500	—	79	1	no	—
142	#494	—	79	1	no	—
143	#487	—	79	2	no	—
144	#467	—	79	4	no	—
145	#446	—	79	4	no	—
146	#446	—	158	2	no	—
147	C₈×Q₈	—	79	4	no	—
148	C₈×Q₈	—	158	2	no	—
149	C₄×Q₁₆	—	79	4	no	—
150	#424	—	158	2	no	—
151	#423	C₂×D₇₉	1	2	yes	—
152	#405	—	158	2	no	—
153	C₂×C₄×C₈	—	79	1	no	—
154	C₂×C₄×C₈	—	79	1	no	—
155	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
156	#335	C₂×D₇₉	1	4	yes	—
157	Q₈⋊C₈	—	79	8	no	—
158	C₈²	—	79	2	no	—
159	C₈²	—	79	2	no	—
160	C₂×Q₁₆	—	79	4	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂²×C₈	—	79	1	no	—
163	C₂²×C₈	—	79	1	no	—
164	C₂²×C₈	—	158	1	no	—
165	C₂²×C₈	C₄×D₇₉	1	2	yes	—
166	C₄⋊Q₈	—	79	2	no	—
167	C₄⋊Q₈	—	79	2	no	—
168	#125	—	158	2	no	—
169	C₂²⋊Q₈	—	79	4	no	—
170	C₂²⋊Q₈	—	158	2	no	—
171	C₄×Q₈	—	79	4	no	—
172	C₄×Q₈	—	158	2	no	—
173	#117	—	158	2	no	—
174	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
175	#107	—	79	4	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	158	4	no	—
178	C₄⋊C₈	—	158	4	no	—
179	C₄⋊C₈	C₂²×D₇₉	1	4	yes	—
180	#103	—	79	8	no	—
181	#97	—	158	4	no	—
182	C₄×C₈	—	79	1	no	—
183	C₄×C₈	—	79	1	no	—
184	C₄×C₈	—	79	1	no	—
185	C₄×C₈	—	79	1	no	—
186	C₄×C₈	—	79	2	no	—
187	C₄×C₈	—	158	4	no	—
188	C₄×C₈	—	158	1	no	—
189	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
190	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
191	C₄×C₈	C₂²×D₇₉	1	2	yes	—
192	C₂×Q₈	—	79	4	no	—
193	C₂×Q₈	—	158	2	no	—
194	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
195	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
196	C₂²×C₄	#11929625	1	1	yes	characteristic
197	Q₁₆	—	158	8	no	—
198	M₁₆	—	4	4	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	79	4	no	—
202	C₂×C₈	—	79	1	no	—
203	C₂×C₈	—	79	1	no	—
204	C₂×C₈	—	79	1	no	—
205	C₂×C₈	—	79	1	no	—
206	C₂×C₈	—	158	4	no	—
207	C₂×C₈	—	158	2	no	—
208	C₂×C₈	—	158	2	no	—
209	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
211	C₄⋊C₄	—	79	4	no	—
212	C₄⋊C₄	—	158	4	no	—
213	C₄⋊C₄	—	158	4	no	—
214	C₄⋊C₄	—	158	2	no	—
215	C₄⋊C₄	—	158	2	no	—
216	C₄⋊C₄	—	158	2	no	—
217	C₂²⋊C₄	—	158	4	no	—
218	C₄²	—	158	4	no	—
219	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
220	C₄²	D₄×D₇₉	1	1	yes	characteristic
221	C₄²	#11929625	1	1	yes	characteristic
222	C₄²	C₂³×D₇₉	1	1	yes	characteristic
223	C₂³	#14579123	1	1	yes	characteristic
224	Q₈	—	316	4	no	—
225	Q₈	—	158	8	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	158	4	no	—
229	C₂×C₄	—	158	4	no	—
230	C₂×C₄	#14579123	1	1	yes	characteristic
231	C₂×C₄	#14579123	1	1	yes	characteristic
232	C₂×C₄	#14579123	1	1	yes	characteristic
233	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
234	C₂×C₄	#14579180	1	2	yes	—
235	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
237	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	4	4	no	—
240	C₈	—	79	2	no	—
241	C₈	—	79	2	no	—
242	C₈	—	158	4	no	—
243	C₈	—	158	1	no	—
244	C₈	—	158	1	no	—
245	C₈	#14579169	1	4	yes	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	#19098657	1	2	yes	—
249	C₂²	C₂×#14579123	1	1	yes	characteristic
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	316	4	no	—
253	C₄	—	316	4	no	—
254	C₄	#19098657	1	2	yes	—
255	C₄	#19098679	1	2	yes	—
256	C₄	C₂×#14579169	1	2	yes	—
257	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
258	C₄	C₂×#14579180	1	1	yes	characteristic
259	C₂	—	2	2	no	—
260	C₂	#23772111	1	1	yes	characteristic
261	C₂	C₂×#19098657	1	1	yes	characteristic
262	C₂	C₂×#19098679	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1567	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂³×C₇₉	C₄×D₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1567	D₇₉
hypocenter	C₇₉	#17958
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #23762296	C₂	semidirect
4. #23762206	C₂	semidirect
66. C₂×C₇₉⋊C₈	Q₁₆	semidirect
139. C₇₉	#17958	semidirect

Maximal subgroups

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

subgroup	index
#23773986	2
#23772330	2
#23762296	2
#23762206	2
#23761854	2
C₂×C₈×C₇₉⋊C₈	2
C₇₉×#1567	2
#17958	79