Subgroups — #34443704

Order 20224.

814 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099004	C₂	1	4	yes	maximal, complemented
2	#23773828	C₂	1	1	yes	maximal, characteristic
3	#23765669	C₂	1	4	yes	maximal, complemented
4	#23765657	C₂	1	2	yes	maximal, complemented
5	#23765652	C₂	1	2	yes	maximal, complemented
6	C₄×#14579101	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#494×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579183	—	2	1	no	—
9	C₂×#14579183	C₂²	1	2	yes	—
10	#19099113	—	2	1	no	—
11	#19099004	—	2	16	no	—
12	C₂×#14579107	C₂²	1	8	yes	—
13	C₂×#14579104	C₂²	1	8	yes	—
14	C₂×#14579101	C₂²	1	2	yes	—
15	#19098947	—	2	1	no	—
16	#19098947	C₂²	1	2	yes	—
17	#19098943	C₂²	1	1	yes	characteristic
18	#19098940	C₂²	1	1	yes	characteristic
19	#19098936	—	2	1	no	—
20	#19098339	C₂²	1	2	yes	—
21	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
22	C₂²×Q₁₆×C₇₉	—	2	1	no	—
23	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
24	C₂×C₄⋊Q₈×C₇₉	—	2	1	no	—
25	#494×C₇₉	—	2	1	no	—
26	#494×C₇₉	—	2	1	no	—
27	#494×C₇₉	C₂²	1	2	yes	—
28	#494×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#11929624	—	2	2	no	—
31	#14579183	—	2	8	no	—
32	#14579183	—	4	4	no	—
33	C₂×Dic₃₁₆	—	2	1	no	—
34	C₂×Dic₃₁₆	C₂³	1	2	yes	—
35	C₂×#11929613	—	2	2	no	—
36	C₂×#11929613	—	2	2	no	—
37	C₂×D₆₃₂	—	2	1	no	—
38	C₂×D₆₃₂	C₂³	1	2	yes	—
39	#14579161	—	2	1	no	—
40	#14579158	—	2	1	no	—
41	#14579149	—	2	2	no	—
42	#14579135	—	4	2	no	—
43	C₄×D₃₁₆	—	4	2	no	—
44	#14579107	—	2	16	no	—
45	#14579104	—	2	16	no	—
46	#14579101	D₄	1	8	yes	complemented
47	C₂×C₇₉⋊C₁₆	C₂³	1	4	yes	—
48	C₂²×Q₈×C₇₉	—	2	2	no	—
49	C₂×Q₁₆×C₇₉	—	2	8	no	—
50	C₂×Q₁₆×C₇₉	—	2	2	no	—
51	C₂×Q₁₆×C₇₉	—	2	2	no	—
52	C₂×Q₁₆×C₇₉	—	2	2	no	—
53	C₂×Q₁₆×C₇₉	—	4	4	no	—
54	C₂×Q₁₆×C₇₉	C₂³	1	4	yes	—
55	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₇₉	—	2	2	no	—
58	C₄⋊Q₈×C₇₉	—	2	2	no	—
59	C₄⋊Q₈×C₇₉	—	4	2	no	—
60	C₂×C₄⋊C₄×C₇₉	—	4	2	no	—
61	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
63	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
64	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	#11929624	—	4	4	no	—
67	#11929624	—	4	4	no	—
68	C₂×Dic₁₅₈	—	2	2	no	—
69	C₂×D₃₁₆	—	2	2	no	—
70	C₂×D₃₁₆	—	4	2	no	—
71	C₂×C₂²⋊D₇₉	—	4	2	no	—
72	C₂×C₄×D₇₉	—	4	2	no	—
73	Dic₃₁₆	—	2	4	no	—
74	Dic₃₁₆	—	4	2	no	—
75	#11929613	—	4	4	no	—
76	#11929613	—	4	4	no	—
77	D₆₃₂	—	2	4	no	—
78	D₆₃₂	—	4	2	no	—
79	C₄⋊Dic₇₉	—	4	2	no	—
80	(C₂×D₇₉)⋊C₄	—	4	4	no	—
81	C₇₉⋊C₁₆	C₂×D₄	1	8	yes	—
82	C₂×Q₈×C₇₉	—	2	4	no	—
83	C₂×Q₈×C₇₉	—	4	2	no	—
84	C₂×Q₈×C₇₉	—	4	4	no	—
85	C₂×Q₈×C₇₉	—	4	4	no	—
86	C₂²×C₄×C₇₉	—	4	2	no	—
87	C₂²×C₄×C₇₉	D₈	1	2	yes	—
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	Q₁₆×C₇₉	—	2	8	no	—
90	Q₁₆×C₇₉	—	4	4	no	—
91	Q₁₆×C₇₉	—	4	4	no	—
92	Q₁₆×C₇₉	—	4	4	no	—
93	C₂×C₈×C₇₉	D₈	1	4	yes	—
94	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₇₉	—	4	4	no	—
100	C₄⋊C₄×C₇₉	—	4	4	no	—
101	C₄²×C₇₉	D₈	1	2	yes	—
102	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
105	C₂²×D₇₉	—	4	2	no	—
106	Q₈×C₇₉	—	4	4	no	—
107	Q₈×C₇₉	—	4	4	no	—
108	Q₈×C₇₉	—	8	4	no	—
109	C₂×C₄×C₇₉	—	2	4	no	—
110	C₂×C₄×C₇₉	—	4	4	no	—
111	C₂×C₄×C₇₉	—	8	4	no	—
112	C₂×C₄×C₇₉	#127	1	2	yes	—
113	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
115	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₇₉	—	8	4	no	—
119	C₂×Dic₇₉	—	4	2	no	—
120	D₃₁₆	—	4	2	no	—
121	D₃₁₆	—	4	2	no	—
122	D₃₁₆	—	8	4	no	—
123	C₄×D₇₉	—	8	4	no	—
124	Dic₁₅₈	—	4	2	no	—
125	Dic₁₅₈	—	4	2	no	—
126	C₈×C₇₉	#127	1	2	yes	—
127	C₈×C₇₉	C₂×D₈	1	2	yes	—
128	C₈×C₇₉	C₂×D₈	1	2	yes	—
129	C₈×C₇₉	C₂²×D₄	1	2	yes	—
130	C₂²×C₇₉	—	2	2	no	—
131	C₂²×C₇₉	#493	1	2	yes	—
132	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
133	C₂×D₇₉	—	8	2	no	—
134	C₂×D₇₉	—	8	4	no	—
135	C₄×C₇₉	—	2	2	no	—
136	C₄×C₇₉	—	8	4	no	—
137	C₄×C₇₉	#493	1	2	yes	—
138	C₄×C₇₉	#510	1	4	yes	complemented
139	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
140	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
141	Dic₇₉	—	8	2	no	—
142	#33718	—	79	1	no	maximal, Sylow
143	C₂×C₇₉	—	2	2	no	—
144	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
145	C₂×C₇₉	C₂×#510	1	2	yes	—
146	D₇₉	—	16	2	no	—
147	C₂×#510	—	79	4	no	—
148	#3147	—	79	1	no	—
149	C₂×#494	D₇₉	1	1	yes	characteristic, complemented
150	#2251	—	79	4	no	—
151	#2248	—	79	2	no	—
152	#2247	—	79	2	no	—
153	C₄×M₃₂	—	79	1	no	—
154	C₇₉	#33718	1	1	yes	characteristic, complemented, Sylow
155	C₂×#135	—	79	2	no	—
156	C₂×#135	—	158	1	no	—
157	C₂²×Q₁₆	—	2	1	no	—
158	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
159	#532	—	158	1	no	—
160	C₂×C₄⋊Q₈	—	2	1	no	—
161	#510	—	158	16	no	—
162	C₂×Q₃₂	—	79	8	no	—
163	C₂×SD₃₂	—	79	8	no	—
164	C₂×M₃₂	—	79	2	no	—
165	#495	—	79	2	no	—
166	#495	—	158	1	no	—
167	#494	—	2	1	no	—
168	#494	—	2	1	no	—
169	#494	—	79	1	no	—
170	#494	C₂×D₇₉	1	2	yes	—
171	#494	C₂×D₇₉	1	2	yes	—
172	#493	—	79	1	no	—
173	#492	—	158	1	no	—
174	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
175	#346	—	79	2	no	—
176	C₄×C₁₆	—	79	2	no	—
177	C₂×#41	—	158	2	no	—
178	C₂²×Q₈	—	2	2	no	—
179	#135	—	316	4	no	—
180	#135	—	158	8	no	—
181	C₂×Q₁₆	—	2	8	no	—
182	C₂×Q₁₆	—	2	2	no	—
183	C₂×Q₁₆	—	2	2	no	—
184	C₂×Q₁₆	—	2	2	no	—
185	C₂×Q₁₆	—	4	4	no	—
186	C₂×Q₁₆	—	79	2	no	—
187	C₂×Q₁₆	—	158	1	no	—
188	C₂×Q₁₆	C₂²×D₇₉	1	4	yes	—
189	C₂×SD₁₆	—	158	2	no	—
190	C₂×SD₁₆	—	158	2	no	—
191	C₂×D₈	—	79	2	no	—
192	C₂×D₈	—	158	1	no	—
193	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
194	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
195	C₄⋊Q₈	—	2	2	no	—
196	C₄⋊Q₈	—	2	2	no	—
197	C₄⋊Q₈	—	4	2	no	—
198	C₄⋊Q₈	—	158	1	no	—
199	#127	—	158	1	no	—
200	#124	—	158	2	no	—
201	#121	—	316	2	no	—
202	C₄×D₄	—	316	2	no	—
203	C₂×C₄⋊C₄	—	4	2	no	—
204	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
205	Q₃₂	—	158	16	no	—
206	SD₃₂	—	158	16	no	—
207	M₃₂	—	79	8	no	—
208	C₂×C₁₆	—	79	4	no	—
209	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
210	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
213	#41	—	316	4	no	—
214	#41	—	316	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂×Q₈	—	4	4	no	—
218	C₂×Q₈	—	4	4	no	—
219	C₂×Q₈	—	158	2	no	—
220	C₂×D₄	—	316	2	no	—
221	C₂×D₄	—	316	2	no	—
222	C₂×D₄	—	158	2	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	316	2	no	—
225	C₂²×C₄	#11929609	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
227	Q₁₆	—	2	8	no	—
228	Q₁₆	—	4	4	no	—
229	Q₁₆	—	4	4	no	—
230	Q₁₆	—	4	4	no	—
231	Q₁₆	—	316	2	no	—
232	Q₁₆	—	158	4	no	—
233	SD₁₆	—	316	4	no	—
234	SD₁₆	—	316	4	no	—
235	D₈	—	316	2	no	—
236	D₈	—	158	4	no	—
237	C₂×C₈	#11929609	1	4	yes	—
238	C₂×C₈	D₄×D₇₉	1	4	yes	—
239	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
240	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
241	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
242	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	316	2	no	—
246	C₂²⋊C₄	—	316	4	no	—
247	C₄²	#11929609	1	2	yes	—
248	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
249	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
250	C₁₆	—	79	8	no	—
251	C₂³	—	316	2	no	—
252	C₂³	C₂×#11929609	1	1	yes	characteristic
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	Q₈	—	8	4	no	—
256	Q₈	—	316	2	no	—
257	Q₈	—	316	2	no	—
258	D₄	—	316	2	no	—
259	D₄	—	316	2	no	—
260	D₄	—	632	4	no	—
261	D₄	—	632	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	—	316	2	no	—
266	C₂×C₄	—	632	4	no	—
267	C₂×C₄	#14579157	1	2	yes	—
268	C₂×C₄	C₂×#11929609	1	2	yes	—
269	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
270	C₂×C₄	C₂×#11929609	1	2	yes	—
271	C₂×C₄	C₂×#11929609	1	2	yes	—
272	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
273	C₈	#14579157	1	2	yes	—
274	C₈	C₂×#11929609	1	2	yes	—
275	C₈	C₂×#11929609	1	2	yes	—
276	C₈	C₂×D₄×D₇₉	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	632	4	no	—
279	C₂²	—	632	2	no	—
280	C₂²	#19098939	1	2	yes	—
281	C₂²	C₂²×#11929609	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	4	no	—
284	C₄	—	632	2	no	—
285	C₄	#19098939	1	2	yes	—
286	C₄	#19099004	1	4	yes	complemented
287	C₄	C₂×#14579157	1	1	yes	characteristic
288	C₄	C₂²×#11929609	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	1264	2	no	—
291	C₂	C₂×#19098939	1	1	yes	characteristic
292	C₂	C₂×#19099004	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#494×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23773828	C₂
hypercenter	C₂×#494	D₇₉
hypocenter	C₇₉	#33718
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
1. C₂×#19099004	C₂	semidirect
3. #23765669	C₂	semidirect
4. #23765657	C₂	semidirect
5. #23765652	C₂	semidirect
6. C₄×#14579101	C₂	semidirect
7. C₂×#494×C₇₉	C₂	semidirect
46. #14579101	D₄	semidirect
138. C₄×C₇₉	#510	semidirect
149. C₂×#494	D₇₉	semidirect
154. C₇₉	#33718	semidirect
286. C₄	#19099004	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099004	2
#23773828	2
#23765669	2
#23765657	2
#23765652	2
C₄×#14579101	2
C₂×#494×C₇₉	2
#33718	79