Subgroups — #34443723

Order 20224.

858 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098998	C₂	1	2	yes	maximal, complemented
2	C₂²×D₁₂₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#23773887	C₂	1	2	yes	maximal, complemented
4	#23765665	C₂	1	4	yes	maximal, complemented
5	#23765662	C₂	1	2	yes	maximal, complemented
6	#23765648	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, complemented
9	C₂×#14579186	—	2	2	no	—
10	C₂²×D₆₃₂	C₂²	1	2	yes	—
11	#19099127	—	2	2	no	—
12	#19098998	—	2	8	no	—
13	C₂×#14579104	C₂²	1	4	yes	complemented
14	C₂×D₁₂₆₄	—	2	4	no	—
15	C₂×D₁₂₆₄	C₂²	1	2	yes	—
16	C₂×D₁₂₆₄	C₂²	1	2	yes	—
17	C₂×#14579102	C₂²	1	4	yes	complemented
18	C₂×#14579101	C₂²	1	1	yes	characteristic
19	#19098949	—	2	4	no	—
20	#19098948	C₂²	1	4	yes	complemented
21	#19098938	C₂²	1	4	yes	complemented
22	#19098518	C₂²	1	1	yes	characteristic
23	#19098338	C₂²	1	2	yes	complemented
24	C₁₆×Dic₇₉	C₂²	1	2	yes	complemented
25	C₂×#135×C₇₉	C₂²	1	2	yes	—
26	#508×C₇₉	—	2	4	no	—
27	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic, complemented
28	C₂×SD₃₂×C₇₉	C₂²	1	2	yes	complemented
29	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
30	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
31	C₂²×D₃₁₆	—	2	2	no	—
32	#14579186	—	4	8	no	—
33	C₂×#11929611	—	2	4	no	—
34	C₂×D₆₃₂	—	2	4	no	—
35	C₂×D₆₃₂	—	2	4	no	—
36	C₂×D₆₃₂	—	2	2	no	—
37	C₂×D₆₃₂	C₂³	1	4	yes	—
38	C₂×#11929609	—	2	4	no	—
39	C₂×#11929608	D₄	1	1	yes	characteristic
40	#14579157	—	2	4	no	—
41	#14579150	—	2	4	no	—
42	#14579135	—	4	4	no	—
43	#14579129	—	4	4	no	—
44	#14579120	D₄	1	1	yes	characteristic
45	#14579104	—	2	8	no	—
46	D₁₂₆₄	—	2	4	no	—
47	D₁₂₆₄	—	2	4	no	—
48	#14579102	—	2	8	no	—
49	#14579101	D₄	1	4	yes	complemented
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	complemented
64	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
66	C₂³×D₇₉	—	4	2	no	—
67	C₂×D₃₁₆	—	2	4	no	—
68	C₂×D₃₁₆	—	4	4	no	—
69	C₂×D₃₁₆	—	4	2	no	—
70	C₂×D₃₁₆	—	4	4	no	—
71	C₂×C₂²⋊D₇₉	—	4	8	no	—
72	#11929611	—	4	8	no	—
73	D₆₃₂	—	2	4	no	—
74	D₆₃₂	—	2	4	no	—
75	D₆₃₂	—	4	4	no	—
76	#11929609	—	4	8	no	—
77	#11929608	D₈	1	4	yes	—
78	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
79	C₄⋊Dic₇₉	D₈	1	2	yes	—
80	C₂²⋊Dic₇₉	D₈	1	2	yes	—
81	(C₂×D₇₉)⋊C₄	—	4	8	no	—
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	C₂²×D₇₉	—	4	4	no	—
105	C₂²×D₇₉	—	8	4	no	—
106	C₂²×D₇₉	—	16	4	no	—
107	Q₈×C₇₉	—	4	2	no	—
108	Q₈×C₇₉	—	4	2	no	—
109	D₄×C₇₉	—	4	2	no	—
110	D₄×C₇₉	—	4	2	no	—
111	D₄×C₇₉	—	8	4	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	8	4	no	—
114	C₂×C₄×C₇₉	#127	1	2	yes	—
115	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₇₉	—	16	8	no	—
119	C₂×Dic₇₉	C₂×D₈	1	4	yes	—
120	D₃₁₆	—	4	4	no	—
121	D₃₁₆	—	4	4	no	—
122	D₃₁₆	—	8	4	no	—
123	C₈×C₇₉	#127	1	2	yes	—
124	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
126	C₇₉⋊C₈	C₂×D₈	1	4	yes	—
127	C₂²×C₇₉	—	2	2	no	—
128	C₂²×C₇₉	—	8	2	no	—
129	C₂²×C₇₉	—	8	4	no	—
130	C₂²×C₇₉	#493	1	2	yes	—
131	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
132	C₂×D₇₉	—	8	4	no	—
133	C₂×D₇₉	—	8	8	no	—
134	C₂×D₇₉	—	16	8	no	—
135	C₄×C₇₉	—	8	2	no	—
136	C₄×C₇₉	#493	1	2	yes	—
137	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
138	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
139	Dic₇₉	—	2	4	no	—
140	#33720	—	79	1	no	maximal, Sylow
141	C₂×C₇₉	—	2	2	no	—
142	C₂×C₇₉	—	16	2	no	—
143	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
144	C₂×C₇₉	#3415	1	2	yes	—
145	D₇₉	—	16	4	no	—
146	C₂×#509	—	79	2	no	—
147	C₂×#508	D₇₉	1	1	yes	characteristic, complemented
148	C₂²×D₁₆	—	79	1	no	—
149	#3155	—	79	2	no	—
150	#2250	—	79	4	no	—
151	#2249	—	79	2	no	—
152	#2246	—	79	2	no	—
153	#2108	—	79	1	no	—
154	C₇₉	#33720	1	1	yes	characteristic, complemented, Sylow
155	C₂×AGL(1,ℤ/8)	—	158	2	no	—
156	C₂×#135	C₂×D₇₉	1	2	yes	—
157	C₂²×D₈	—	79	2	no	—
158	#535	—	158	2	no	—
159	#509	—	158	8	no	—
160	#508	—	2	4	no	—
161	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic, complemented
162	C₂×SD₃₂	—	79	4	no	—
163	C₂×SD₃₂	C₂×D₇₉	1	2	yes	complemented
164	C₂×D₁₆	—	79	4	no	—
165	C₂×D₁₆	—	79	2	no	—
166	C₂×D₁₆	—	79	2	no	—
167	C₂×D₁₆	—	158	4	no	—
168	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
169	C₂×M₃₂	—	79	1	no	—
170	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
171	#496	—	158	4	no	—
172	#495	—	79	4	no	—
173	#493	—	79	4	no	—
174	#405	—	79	1	no	—
175	#346	—	79	2	no	—
176	C₄×C₁₆	—	79	2	no	—
177	C₂×#41	—	2	2	no	—
178	C₂²×D₄	—	158	2	no	—
179	AGL(1,ℤ/8)	—	316	8	no	—
180	#135	—	2	4	no	—
181	#135	—	2	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	—	79	4	no	—
186	C₂×D₈	—	158	4	no	—
187	C₂×D₈	—	158	2	no	—
188	C₂×D₈	—	158	4	no	—
189	C₂×D₈	—	158	4	no	—
190	C₂×D₈	C₂²×D₇₉	1	2	yes	—
191	C₂×M₁₆	—	79	1	no	—
192	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
193	#127	—	158	4	no	—
194	#124	—	158	4	no	—
195	#121	—	316	4	no	—
196	#120	—	316	4	no	—
197	#117	—	79	1	no	—
198	Q₃₂	—	2	2	no	—
199	SD₃₂	—	2	4	no	—
200	SD₃₂	—	158	8	no	—
201	D₁₆	—	2	2	no	—
202	D₁₆	—	158	8	no	—
203	D₁₆	—	158	4	no	—
204	D₁₆	—	158	4	no	—
205	M₃₂	—	79	4	no	—
206	C₂×C₁₆	—	79	2	no	—
207	C₂×C₁₆	C₂²⋊D₇₉	1	4	yes	complemented
208	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
209	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
210	#97	—	79	2	no	—
211	C₄×C₈	—	79	2	no	—
212	C₂⁴	—	316	2	no	—
213	#41	—	4	4	no	—
214	#41	—	4	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	316	8	no	—
219	C₂×D₄	—	316	4	no	—
220	C₂×D₄	—	316	4	no	—
221	C₂×D₄	—	316	2	no	—
222	C₂×D₄	—	158	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	SD₁₆	—	316	8	no	—
229	D₈	—	2	2	no	—
230	D₈	—	2	2	no	—
231	D₈	—	316	8	no	—
232	D₈	—	316	4	no	—
233	D₈	—	158	4	no	—
234	D₈	—	158	4	no	—
235	M₁₆	—	79	4	no	—
236	C₂×C₈	—	79	2	no	—
237	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
238	C₂×C₈	D₄×D₇₉	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
240	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
241	C₄⋊C₄	—	79	2	no	—
242	C₂²⋊C₄	—	316	8	no	—
243	C₂²⋊C₄	—	79	2	no	—
244	C₄²	—	79	2	no	—
245	C₁₆	—	79	4	no	—
246	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
247	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
248	C₂³	—	4	2	no	—
249	C₂³	—	316	4	no	—
250	C₂³	—	632	4	no	—
251	C₂³	—	1264	4	no	—
252	C₂³	D₈×D₇₉	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	316	4	no	—
259	D₄	—	316	4	no	—
260	D₄	—	632	4	no	—
261	D₄	—	1264	8	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	—	79	4	no	—
265	C₂×C₄	#14579157	1	2	yes	—
266	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
267	C₂×C₄	D₈×D₇₉	1	2	yes	—
268	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
269	C₈	—	79	4	no	—
270	C₈	#14579157	1	2	yes	—
271	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
272	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	632	8	no	—
277	C₂²	—	632	4	no	—
278	C₂²	—	1264	8	no	—
279	C₂²	#19098938	1	2	yes	—
280	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
281	C₄	—	8	2	no	—
282	C₄	—	158	4	no	—
283	C₄	#19098938	1	2	yes	—
284	C₄	C₂×#14579157	1	1	yes	characteristic
285	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	16	2	no	—
288	C₂	—	1264	4	no	—
289	C₂	C₂×#19098938	1	1	yes	characteristic
290	C₂	#23776538	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#508×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	C₂×#508	D₇₉
hypocenter	C₇₉	#33720
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×#19098998	C₂	semidirect
2. C₂²×D₁₂₆₄	C₂	semidirect
3. #23773887	C₂	semidirect
4. #23765665	C₂	semidirect
5. #23765662	C₂	semidirect
6. #23765648	C₂	semidirect
7. #23764996	C₂	semidirect
8. C₂×#508×C₇₉	C₂	semidirect
13. C₂×#14579104	C₂²	semidirect
17. C₂×#14579102	C₂²	semidirect
20. #19098948	C₂²	semidirect
21. #19098938	C₂²	semidirect
23. #19098338	C₂²	semidirect
24. C₁₆×Dic₇₉	C₂²	semidirect
27. C₂×Q₃₂×C₇₉	C₂²	semidirect
28. C₂×SD₃₂×C₇₉	C₂²	semidirect
29. C₂×D₁₆×C₇₉	C₂²	semidirect
49. #14579101	D₄	semidirect
63. C₂×C₁₆×C₇₉	D₄	semidirect
147. C₂×#508	D₇₉	semidirect
154. C₇₉	#33720	semidirect
161. C₂×Q₃₂	C₂×D₇₉	semidirect
163. C₂×SD₃₂	C₂×D₇₉	semidirect
168. C₂×D₁₆	C₂×D₇₉	semidirect
207. C₂×C₁₆	C₂²⋊D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098998	2
C₂²×D₁₂₆₄	2
#23773887	2
#23765665	2
#23765662	2
#23765648	2
#23764996	2
C₂×#508×C₇₉	2
#33720	79