Subgroups — #34443731

Order 20224.

794 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099003	C₂	1	2	yes	maximal, complemented
2	C₂²×Dic₆₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#23773895	C₂	1	2	yes	maximal, complemented
4	#23765670	C₂	1	4	yes	maximal, complemented
5	#23765658	C₂	1	2	yes	maximal, complemented
6	#23765651	C₂	1	2	yes	maximal, complemented
7	#23764996	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#508×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579191	—	2	2	no	—
10	C₂²×Dic₃₁₆	C₂²	1	2	yes	—
11	#19099134	—	2	2	no	—
12	#19099003	—	2	8	no	—
13	C₂×Dic₆₃₂	—	2	4	no	—
14	C₂×Dic₆₃₂	C₂²	1	2	yes	—
15	C₂×Dic₆₃₂	C₂²	1	2	yes	—
16	C₂×#14579107	C₂²	1	4	yes	—
17	C₂×#14579105	C₂²	1	4	yes	—
18	C₂×#14579101	C₂²	1	1	yes	characteristic
19	#19098954	—	2	4	no	—
20	#19098946	C₂²	1	4	yes	—
21	#19098941	C₂²	1	4	yes	—
22	#19098518	C₂²	1	1	yes	characteristic
23	#19098338	C₂²	1	2	yes	—
24	C₁₆×Dic₇₉	C₂²	1	2	yes	—
25	C₂×#135×C₇₉	C₂²	1	2	yes	—
26	#508×C₇₉	—	2	4	no	—
27	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×SD₃₂×C₇₉	C₂²	1	2	yes	—
29	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
30	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²×Dic₁₅₈	—	2	2	no	—
32	#14579191	—	4	8	no	—
33	C₂×Dic₃₁₆	—	2	4	no	—
34	C₂×Dic₃₁₆	—	2	4	no	—
35	C₂×Dic₃₁₆	—	2	2	no	—
36	C₂×Dic₃₁₆	C₂³	1	4	yes	—
37	C₂×#11929614	—	2	4	no	—
38	C₂×#11929612	—	2	4	no	—
39	C₂×#11929608	D₄	1	1	yes	characteristic
40	#14579159	—	2	4	no	—
41	#14579148	—	2	4	no	—
42	#14579142	—	4	4	no	—
43	#14579140	—	4	4	no	—
44	#14579120	D₄	1	1	yes	characteristic
45	Dic₆₃₂	—	2	4	no	—
46	Dic₆₃₂	—	2	4	no	—
47	#14579107	—	2	8	no	—
48	#14579105	—	2	8	no	—
49	#14579101	D₄	1	4	yes	—
50	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
51	#14579071	D₄	1	2	yes	—
52	C₈×Dic₇₉	C₂³	1	2	yes	—
53	C₂×#41×C₇₉	—	2	2	no	—
54	#135×C₇₉	—	2	4	no	—
55	#135×C₇₉	—	2	4	no	—
56	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
57	C₂×SD₁₆×C₇₉	—	2	2	no	—
58	C₂×D₈×C₇₉	C₂³	1	2	yes	—
59	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
60	Q₃₂×C₇₉	—	2	2	no	—
61	SD₃₂×C₇₉	—	2	4	no	—
62	D₁₆×C₇₉	—	2	2	no	—
63	C₂×C₁₆×C₇₉	D₄	1	4	yes	—
64	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₅₈	—	2	4	no	—
67	C₂×Dic₁₅₈	—	4	4	no	—
68	C₂×Dic₁₅₈	—	4	4	no	—
69	C₂×Dic₁₅₈	—	4	2	no	—
70	C₂²×Dic₇₉	—	4	2	no	—
71	Dic₃₁₆	—	2	4	no	—
72	Dic₃₁₆	—	2	4	no	—
73	Dic₃₁₆	—	4	4	no	—
74	#11929614	—	4	8	no	—
75	#11929612	—	4	8	no	—
76	#11929608	D₈	1	4	yes	—
77	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
78	C₄⋊Dic₇₉	D₈	1	2	yes	—
79	Dic₇₉⋊C₄	—	4	8	no	—
80	C₂²⋊Dic₇₉	—	4	8	no	—
81	C₂²⋊Dic₇₉	D₈	1	2	yes	—
82	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
83	C₇₉⋊C₁₆	C₂×D₄	1	4	yes	—
84	#41×C₇₉	—	4	4	no	—
85	#41×C₇₉	—	4	4	no	—
86	C₂×Q₈×C₇₉	—	2	2	no	—
87	C₂×D₄×C₇₉	—	2	2	no	—
88	C₂×D₄×C₇₉	—	4	2	no	—
89	C₂²×C₄×C₇₉	—	4	2	no	—
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₇₉	—	2	2	no	—
92	Q₁₆×C₇₉	—	2	2	no	—
93	SD₁₆×C₇₉	—	4	4	no	—
94	D₈×C₇₉	—	2	2	no	—
95	D₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
100	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
101	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
102	C₂³×C₇₉	—	4	2	no	—
103	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
104	Q₈×C₇₉	—	4	2	no	—
105	Q₈×C₇₉	—	4	2	no	—
106	D₄×C₇₉	—	4	2	no	—
107	D₄×C₇₉	—	4	2	no	—
108	D₄×C₇₉	—	8	4	no	—
109	C₂×C₄×C₇₉	—	4	2	no	—
110	C₂×C₄×C₇₉	—	8	4	no	—
111	C₂×C₄×C₇₉	#127	1	2	yes	—
112	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₇₉	—	4	4	no	—
116	C₂×Dic₇₉	—	8	4	no	—
117	C₂×Dic₇₉	C₂×D₈	1	4	yes	—
118	Dic₁₅₈	—	4	4	no	—
119	Dic₁₅₈	—	4	4	no	—
120	Dic₁₅₈	—	8	4	no	—
121	C₈×C₇₉	#127	1	2	yes	—
122	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
124	C₇₉⋊C₈	C₂×D₈	1	4	yes	—
125	C₂²×C₇₉	—	2	2	no	—
126	C₂²×C₇₉	—	8	2	no	—
127	C₂²×C₇₉	—	8	4	no	—
128	C₂²×C₇₉	#493	1	2	yes	—
129	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₇₉	—	8	2	no	—
131	C₄×C₇₉	#493	1	2	yes	—
132	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
133	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
134	Dic₇₉	—	2	4	no	—
135	Dic₇₉	—	8	4	no	—
136	#33721	—	79	1	no	maximal, Sylow
137	C₂×C₇₉	—	2	2	no	—
138	C₂×C₇₉	—	16	2	no	—
139	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
140	C₂×C₇₉	#3417	1	2	yes	—
141	C₂×#510	—	79	2	no	—
142	C₂×#508	D₇₉	1	1	yes	characteristic
143	C₂²×Q₃₂	—	79	1	no	—
144	#3156	—	79	2	no	—
145	#2251	—	79	4	no	—
146	#2249	—	79	2	no	—
147	#2247	—	79	2	no	—
148	#2108	—	79	1	no	—
149	C₇₉	#33721	1	1	yes	characteristic, complemented, Sylow
150	C₂×#137	—	158	2	no	—
151	C₂×#135	C₂×D₇₉	1	2	yes	—
152	C₂²×Q₁₆	—	79	2	no	—
153	#536	—	158	2	no	—
154	#510	—	158	8	no	—
155	#508	—	2	4	no	—
156	C₂×Q₃₂	—	79	4	no	—
157	C₂×Q₃₂	—	79	2	no	—
158	C₂×Q₃₂	—	79	2	no	—
159	C₂×Q₃₂	—	158	4	no	—
160	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
161	C₂×SD₃₂	—	79	4	no	—
162	C₂×SD₃₂	C₂×D₇₉	1	2	yes	—
163	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
164	C₂×M₃₂	—	79	1	no	—
165	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
166	#497	—	158	4	no	—
167	#495	—	79	4	no	—
168	#494	—	79	4	no	—
169	#405	—	79	1	no	—
170	#346	—	79	2	no	—
171	C₄×C₁₆	—	79	2	no	—
172	C₂×#41	—	2	2	no	—
173	C₂²×Q₈	—	158	2	no	—
174	#137	—	316	8	no	—
175	#135	—	2	4	no	—
176	#135	—	2	4	no	—
177	C₂×Q₁₆	—	79	4	no	—
178	C₂×Q₁₆	—	158	4	no	—
179	C₂×Q₁₆	—	158	2	no	—
180	C₂×Q₁₆	—	158	4	no	—
181	C₂×Q₁₆	—	158	4	no	—
182	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×SD₁₆	—	158	4	no	—
185	C₂×D₈	C₂²×D₇₉	1	2	yes	—
186	C₂×M₁₆	—	79	1	no	—
187	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	158	4	no	—
189	#124	—	158	4	no	—
190	#123	—	316	4	no	—
191	C₂²⋊Q₈	—	316	4	no	—
192	#117	—	79	1	no	—
193	Q₃₂	—	2	2	no	—
194	Q₃₂	—	158	8	no	—
195	Q₃₂	—	158	4	no	—
196	Q₃₂	—	158	4	no	—
197	SD₃₂	—	2	4	no	—
198	SD₃₂	—	158	8	no	—
199	D₁₆	—	2	2	no	—
200	M₃₂	—	79	4	no	—
201	C₂×C₁₆	—	79	2	no	—
202	C₂×C₁₆	C₂²⋊D₇₉	1	4	yes	—
203	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
204	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
205	#97	—	79	2	no	—
206	C₄×C₈	—	79	2	no	—
207	#41	—	4	4	no	—
208	#41	—	4	4	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	316	2	no	—
211	C₂×Q₈	—	316	4	no	—
212	C₂×Q₈	—	316	4	no	—
213	C₂×Q₈	—	158	4	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	316	2	no	—
218	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
219	Q₁₆	—	2	2	no	—
220	Q₁₆	—	2	2	no	—
221	Q₁₆	—	316	8	no	—
222	Q₁₆	—	316	4	no	—
223	Q₁₆	—	158	4	no	—
224	Q₁₆	—	158	4	no	—
225	SD₁₆	—	4	4	no	—
226	SD₁₆	—	316	8	no	—
227	D₈	—	2	2	no	—
228	D₈	—	2	2	no	—
229	M₁₆	—	79	4	no	—
230	C₂×C₈	—	79	2	no	—
231	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
232	C₂×C₈	D₄×D₇₉	1	2	yes	—
233	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
234	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
235	C₄⋊C₄	—	316	8	no	—
236	C₄⋊C₄	—	79	2	no	—
237	C₂²⋊C₄	—	316	8	no	—
238	C₂²⋊C₄	—	79	2	no	—
239	C₄²	—	79	2	no	—
240	C₁₆	—	79	4	no	—
241	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
242	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
243	C₂³	—	4	2	no	—
244	C₂³	D₈×D₇₉	1	1	yes	characteristic
245	Q₈	—	4	2	no	—
246	Q₈	—	4	2	no	—
247	Q₈	—	316	4	no	—
248	Q₈	—	316	4	no	—
249	Q₈	—	632	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	316	4	no	—
256	C₂×C₄	—	79	4	no	—
257	C₂×C₄	—	632	4	no	—
258	C₂×C₄	#14579157	1	2	yes	—
259	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
260	C₂×C₄	D₈×D₇₉	1	2	yes	—
261	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
262	C₈	—	79	4	no	—
263	C₈	#14579157	1	2	yes	—
264	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
265	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	8	2	no	—
269	C₂²	#19098938	1	2	yes	—
270	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
271	C₄	—	8	2	no	—
272	C₄	—	632	4	no	—
273	C₄	—	158	4	no	—
274	C₄	#19098938	1	2	yes	—
275	C₄	C₂×#14579157	1	1	yes	characteristic
276	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×#19098938	1	1	yes	characteristic
280	C₂	#23776550	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting, prime radical	C₂×#508×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	C₂×#508	D₇₉
hypocenter	C₇₉	#33721
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#19099003	C₂	semidirect
2. C₂²×Dic₆₃₂	C₂	semidirect
3. #23773895	C₂	semidirect
4. #23765670	C₂	semidirect
5. #23765658	C₂	semidirect
6. #23765651	C₂	semidirect
7. #23764996	C₂	semidirect
149. C₇₉	#33721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099003	2
C₂²×Dic₆₃₂	2
#23773895	2
#23765670	2
#23765658	2
#23765651	2
#23764996	2
C₂×#508×C₇₉	2
#33721	79