Subgroups — #35157911

Order 20224.

1770 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776805	C₂	1	2	yes	maximal
2	#23773939	C₂	1	8	yes	maximal, complemented
3	#23773931	C₂	1	8	yes	maximal, complemented
4	#23773924	C₂	1	4	yes	maximal, complemented
5	#23773914	C₂	1	4	yes	maximal, complemented
6	#23771990	C₂	1	2	yes	maximal
7	C₂×#19098601	C₂	1	2	yes	maximal
8	C₂×#405×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#19099194	—	2	4	no	—
10	#19099191	C₂²	1	16	yes	—
11	#19099189	—	2	4	no	—
12	#19099120	C₂²	1	8	yes	—
13	#19099116	—	2	8	no	—
14	C₂×#14579120	C₂²	1	4	yes	—
15	#19098972	C₂²	1	32	yes	complemented
16	#19098968	C₂²	1	8	yes	complemented
17	#19098965	C₂²	1	8	yes	complemented
18	#19098965	C₂²	1	8	yes	complemented
19	#19098961	C₂²	1	8	yes	complemented
20	#19098605	C₂²	1	16	yes	—
21	#19098601	C₂²	1	4	yes	—
22	#19098601	C₂²	1	4	yes	—
23	#19098601	C₂²	1	4	yes	—
24	#19098601	C₂²	1	4	yes	—
25	#19098601	C₂²	1	8	yes	—
26	C₂×#14579095	C₂²	1	4	yes	—
27	C₂×#14579093	C₂²	1	4	yes	—
28	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
29	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
31	#405×C₇₉	C₂²	1	4	yes	—
32	#405×C₇₉	C₂²	1	4	yes	—
33	C₂×#97×C₇₉	C₂²	1	2	yes	—
34	C₂×C₄×C₈×C₇₉	C₂²	1	2	yes	—
35	#14579161	C₂³	1	16	yes	—
36	#14579160	—	2	16	no	—
37	#14579153	C₂³	1	16	yes	—
38	#14579152	—	2	16	no	—
39	#14579140	—	2	16	no	—
40	#14579138	—	2	32	no	—
41	C₄×Dic₁₅₈	—	2	16	no	—
42	#14579120	—	2	4	no	—
43	#14579120	—	2	4	no	—
44	#14579120	C₂³	1	8	yes	—
45	#14579120	C₂³	1	8	yes	—
46	C₂×C₄⋊Dic₇₉	C₂³	1	8	yes	—
47	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
48	C₂×C₄×Dic₇₉	—	2	4	no	—
49	#14579095	C₂³	1	8	yes	—
50	#14579095	C₂³	1	16	yes	—
51	#14579095	C₂³	1	8	yes	—
52	#14579093	C₂³	1	8	yes	—
53	#14579093	C₂³	1	16	yes	—
54	#14579093	C₂³	1	8	yes	—
55	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₇₉	Q₈	1	8	yes	complemented
57	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
58	C₂²×C₈×C₇₉	Q₈	1	8	yes	complemented
59	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
60	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
64	#117×C₇₉	D₄	1	4	yes	—
65	#117×C₇₉	C₂³	1	4	yes	—
66	C₂×C₄⋊C₄×C₇₉	D₄	1	2	yes	—
67	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
68	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
69	#97×C₇₉	C₂³	1	8	yes	—
70	C₄×C₈×C₇₉	C₂³	1	4	yes	—
71	C₄×C₈×C₇₉	C₂³	1	4	yes	—
72	C₂×Dic₁₅₈	—	2	16	no	—
73	C₂²×Dic₇₉	—	2	8	no	—
74	C₄⋊Dic₇₉	—	2	8	no	—
75	C₄⋊Dic₇₉	—	2	16	no	—
76	C₄⋊Dic₇₉	C₂⁴	1	16	yes	—
77	Dic₇₉⋊C₄	—	2	32	no	—
78	C₂²⋊Dic₇₉	—	2	8	no	—
79	C₂²⋊Dic₇₉	—	2	8	no	—
80	C₄×Dic₇₉	—	2	8	no	—
81	C₄×Dic₇₉	—	2	8	no	—
82	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂⁴	1	2	yes	—
90	M₁₆×C₇₉	C₂×Q₈	1	16	yes	—
91	C₂×C₈×C₇₉	C₂×Q₈	1	8	yes	—
92	C₂×C₈×C₇₉	C₂×Q₈	1	8	yes	—
93	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
94	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
95	C₂×C₈×C₇₉	C₂×Q₈	1	8	yes	—
96	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂⁴	1	4	yes	—
99	C₂×C₈×C₇₉	C₂⁴	1	2	yes	—
100	C₄⋊C₄×C₇₉	C₂×D₄	1	8	yes	—
101	C₂²⋊C₄×C₇₉	C₂×D₄	1	8	yes	—
102	C₄²×C₇₉	C₂×D₄	1	4	yes	—
103	C₄²×C₇₉	C₂⁴	1	4	yes	—
104	C₂³×C₇₉	—	4	2	no	—
105	C₂³×C₇₉	C₄⋊Q₈	1	4	yes	—
106	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
107	C₂³×C₇₉	C₂²×D₄	1	2	yes	—
108	C₂×C₄×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	C₄⋊Q₈	1	4	yes	—
110	C₂×C₄×C₇₉	C₄⋊Q₈	1	8	yes	—
111	C₂×C₄×C₇₉	C₂²×D₄	1	8	yes	—
112	C₂×C₄×C₇₉	C₂²×D₄	1	2	yes	—
113	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
115	C₂×C₄×C₇₉	C₂²×Q₈	1	4	yes	—
116	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
117	C₂×C₄×C₇₉	C₂²×Q₈	1	4	yes	—
118	C₂×C₄×C₇₉	C₂⁵	1	1	yes	characteristic
119	C₂×Dic₇₉	—	2	16	no	—
120	C₂×Dic₇₉	—	8	8	no	—
121	Dic₁₅₈	—	8	16	no	—
122	C₈×C₇₉	C₂²×Q₈	1	2	yes	—
123	C₈×C₇₉	C₂²×Q₈	1	2	yes	—
124	C₈×C₇₉	C₂²×Q₈	1	4	yes	—
125	C₈×C₇₉	C₂²×Q₈	1	8	yes	—
126	C₂²×C₇₉	—	2	2	no	—
127	C₂²×C₇₉	—	2	4	no	—
128	C₂²×C₇₉	—	4	4	no	—
129	C₂²×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
130	C₂²×C₇₉	C₂×C₄⋊Q₈	1	4	yes	—
131	C₂²×C₇₉	C₂³×D₄	1	1	yes	characteristic
132	C₄×C₇₉	—	2	8	no	—
133	C₄×C₇₉	C₂×C₄⋊Q₈	1	4	yes	—
134	C₄×C₇₉	C₂×C₄⋊Q₈	1	2	yes	—
135	C₄×C₇₉	C₂³×Q₈	1	1	yes	characteristic
136	C₄×C₇₉	C₂³×Q₈	1	1	yes	characteristic
137	Dic₇₉	—	4	16	no	—
138	#60660	—	79	1	no	maximal, Sylow
139	C₂×C₇₉	—	2	2	no	—
140	C₂×C₇₉	—	2	4	no	—
141	C₂×C₇₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
142	C₂×C₇₉	#3584	1	2	yes	—
143	#3458	—	79	2	no	—
144	#3164	—	79	8	no	—
145	#3163	—	79	8	no	—
146	#3162	—	79	4	no	—
147	#3160	—	79	4	no	—
148	#2911	—	79	2	no	—
149	C₂×#427	—	79	2	no	—
150	C₂×#405	D₇₉	1	1	yes	characteristic
151	C₇₉	#60660	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
153	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
154	#544	—	79	16	no	—
155	#544	—	158	4	no	—
156	#543	—	158	4	no	—
157	#533	—	79	8	no	—
158	#533	—	158	8	no	—
159	C₂×#117	—	79	4	no	—
160	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
161	#501	—	79	32	no	—
162	#500	—	79	8	no	—
163	#499	—	79	8	no	—
164	#499	—	79	8	no	—
165	#498	—	79	8	no	—
166	#428	—	79	16	no	—
167	#427	—	79	4	no	—
168	#427	—	79	8	no	—
169	#427	—	79	4	no	—
170	#427	—	79	4	no	—
171	#427	—	79	4	no	—
172	C₂×#107	—	79	4	no	—
173	C₂×#106	—	79	4	no	—
174	#405	C₂×D₇₉	1	4	yes	—
175	#405	C₂×D₇₉	1	4	yes	—
176	C₂×#97	C₂×D₇₉	1	2	yes	—
177	C₂×C₄×C₈	C₂×D₇₉	1	2	yes	—
178	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
179	C₂×M₁₆	Dic₁₅₈	1	8	yes	complemented
180	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
181	C₂²×C₈	Dic₁₅₈	1	8	yes	complemented
182	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
185	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
186	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
187	C₄⋊Q₈	—	79	16	no	—
188	C₄⋊Q₈	—	158	16	no	—
189	#125	—	79	16	no	—
190	#125	—	158	16	no	—
191	C₂²⋊Q₈	—	158	16	no	—
192	C₂²⋊Q₈	—	158	32	no	—
193	C₄×Q₈	—	158	16	no	—
194	#117	—	79	8	no	—
195	#117	—	79	8	no	—
196	#117	—	158	4	no	—
197	#117	—	158	4	no	—
198	#117	D₃₁₆	1	4	yes	—
199	#117	C₂²×D₇₉	1	4	yes	—
200	C₂×C₄⋊C₄	—	79	8	no	—
201	C₂×C₄⋊C₄	D₃₁₆	1	2	yes	—
202	C₂×C₂²⋊C₄	—	158	4	no	—
203	C₂×C₂²⋊C₄	D₃₁₆	1	2	yes	—
204	C₂×C₄²	—	158	4	no	—
205	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
206	#107	—	79	8	no	—
207	#107	—	79	8	no	—
208	#107	—	79	16	no	—
209	#106	—	79	8	no	—
210	#106	—	79	8	no	—
211	#106	—	79	16	no	—
212	#97	C₂²×D₇₉	1	8	yes	—
213	C₄×C₈	C₂²×D₇₉	1	4	yes	—
214	C₄×C₈	C₂²×D₇₉	1	4	yes	—
215	C₂⁴	C₂×D₃₁₆	1	1	yes	characteristic
216	C₂×Q₈	—	158	16	no	—
217	C₂²×C₄	—	158	8	no	—
218	C₂²×C₄	C₂×D₃₁₆	1	4	yes	—
219	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
221	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
222	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
223	C₂²×C₄	C₂³×D₇₉	1	2	yes	—
224	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
225	M₁₆	C₂×Dic₁₅₈	1	16	yes	—
226	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
227	C₂×C₈	C₂×Dic₁₅₈	1	4	yes	—
228	C₂×C₈	C₂×Dic₁₅₈	1	8	yes	—
229	C₂×C₈	C₂×Dic₁₅₈	1	8	yes	—
230	C₂×C₈	C₂×Dic₁₅₈	1	8	yes	—
231	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
232	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
233	C₂×C₈	C₂³×D₇₉	1	4	yes	—
234	C₂×C₈	C₂³×D₇₉	1	2	yes	—
235	C₄⋊C₄	—	79	16	no	—
236	C₄⋊C₄	—	158	32	no	—
237	C₄⋊C₄	—	158	8	no	—
238	C₄⋊C₄	—	158	16	no	—
239	C₄⋊C₄	C₂×D₃₁₆	1	8	yes	—
240	C₂²⋊C₄	—	158	8	no	—
241	C₂²⋊C₄	—	158	8	no	—
242	C₂²⋊C₄	C₂×D₃₁₆	1	8	yes	—
243	C₄²	—	158	8	no	—
244	C₄²	—	158	8	no	—
245	C₄²	C₂×D₃₁₆	1	4	yes	—
246	C₄²	C₂³×D₇₉	1	4	yes	—
247	C₂³	—	4	2	no	—
248	C₂³	#14579161	1	4	yes	—
249	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
250	C₂³	C₂²×D₃₁₆	1	2	yes	—
251	Q₈	—	632	16	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	632	8	no	—
254	C₂×C₄	—	158	16	no	—
255	C₂×C₄	#14579161	1	4	yes	—
256	C₂×C₄	#14579161	1	8	yes	—
257	C₂×C₄	C₂²×D₃₁₆	1	8	yes	—
258	C₂×C₄	C₂²×D₃₁₆	1	2	yes	—
259	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
260	C₂×C₄	C₂²×Dic₁₅₈	1	4	yes	—
261	C₂×C₄	C₂²×Dic₁₅₈	1	2	yes	—
262	C₂×C₄	C₂²×Dic₁₅₈	1	2	yes	—
263	C₂×C₄	C₂²×Dic₁₅₈	1	4	yes	—
264	C₂×C₄	C₂⁴×D₇₉	1	1	yes	characteristic
265	C₈	C₂²×Dic₁₅₈	1	2	yes	—
266	C₈	C₂²×Dic₁₅₈	1	2	yes	—
267	C₈	C₂²×Dic₁₅₈	1	4	yes	—
268	C₈	C₂²×Dic₁₅₈	1	8	yes	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	4	4	no	—
272	C₂²	C₂×#14579161	1	2	yes	—
273	C₂²	C₂×#14579161	1	4	yes	—
274	C₂²	C₂³×D₃₁₆	1	1	yes	characteristic
275	C₄	—	2	8	no	—
276	C₄	—	316	16	no	—
277	C₄	C₂×#14579161	1	4	yes	—
278	C₄	C₂×#14579161	1	2	yes	—
279	C₄	C₂³×Dic₁₅₈	1	1	yes	characteristic
280	C₄	C₂³×Dic₁₅₈	1	1	yes	characteristic
281	C₂	—	2	2	no	—
282	C₂	—	2	4	no	—
283	C₂	C₂²×#14579161	1	1	yes	characteristic
284	C₂	#23778174	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator, cosocle	C₂×C₄×C₇₉	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₉
Fitting	C₂×#405×C₇₉	C₂
socle	C₂²×C₇₉	C₂³×D₄
prime radical	C₂⁴×C₇₉	C₂×D₄
hypercenter	C₂×#405	D₇₉
hypocenter	C₇₉	#60660
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #23773939	C₂	semidirect
3. #23773931	C₂	semidirect
4. #23773924	C₂	semidirect
5. #23773914	C₂	semidirect
15. #19098972	C₂²	semidirect
16. #19098968	C₂²	semidirect
17. #19098965	C₂²	semidirect
18. #19098965	C₂²	semidirect
19. #19098961	C₂²	semidirect
56. C₂×M₁₆×C₇₉	Q₈	semidirect
58. C₂²×C₈×C₇₉	Q₈	semidirect
151. C₇₉	#60660	semidirect
179. C₂×M₁₆	Dic₁₅₈	semidirect
181. C₂²×C₈	Dic₁₅₈	semidirect

Maximal subgroups

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

subgroup	index
#23776805	2
#23773939	2
#23773931	2
#23773924	2
#23773914	2
#23771990	2
C₂×#19098601	2
C₂×#405×C₇₉	2
#60660	79