Subgroups — C₂³×#14579190 #35202065

Order 20224.

11166 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11929623	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₈×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14579190	C₂	1	56	yes	maximal, direct product factor
4	C₂³×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11929614	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11929612	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#11929607	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×SD₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11929623	—	2	14	no	—
10	C₂²×#11929623	C₂²	1	28	yes	—
11	C₂³×Dic₁₅₈	—	2	1	no	—
12	C₂³×Dic₁₅₈	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₅₈	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×D₇₉	—	2	14	no	—
15	C₂²×Q₈×D₇₉	C₂²	1	28	yes	complemented
16	C₂³×C₂²⋊D₇₉	—	2	1	no	—
17	C₂⁴×Dic₇₉	—	2	1	no	—
18	C₂³×C₄×D₇₉	—	2	1	no	—
19	C₂³×C₄×D₇₉	C₂²	1	1	yes	characteristic
20	C₂×#14579190	C₂²	1	448	yes	direct product factor
21	C₂²×Dic₃₁₆	C₂²	1	28	yes	complemented
22	C₂²×#11929614	C₂²	1	28	yes	complemented
23	C₂²×#11929612	C₂²	1	28	yes	complemented
24	C₂²×#11929607	C₂²	1	28	yes	complemented
25	C₂³×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
26	C₂³×Q₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂³×D₄×C₇₉	C₂²	1	1	yes	characteristic
28	C₂²×SD₁₆×C₇₉	C₂²	1	28	yes	complemented
29	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂⁴×D₇₉	D₄	1	1	yes	characteristic
31	C₂×#11929623	—	2	168	no	—
32	C₂×#11929623	C₂³	1	112	yes	—
33	C₂²×Dic₁₅₈	—	2	7	no	—
34	C₂²×Dic₁₅₈	—	2	7	no	—
35	C₂²×Dic₁₅₈	—	2	28	no	—
36	C₂²×Dic₁₅₈	C₂³	1	14	yes	—
37	C₂²×Dic₁₅₈	C₂³	1	14	yes	—
38	C₂×Q₈×D₇₉	—	2	168	no	—
39	C₂×Q₈×D₇₉	C₂³	1	112	yes	complemented
40	C₂²×C₂²⋊D₇₉	—	2	28	no	—
41	C₂³×Dic₇₉	—	2	1	no	—
42	C₂³×Dic₇₉	—	2	1	no	—
43	C₂³×Dic₇₉	—	2	28	no	—
44	C₂³×Dic₇₉	D₄	1	1	yes	characteristic
45	C₂²×C₄×D₇₉	—	2	28	no	—
46	C₂²×C₄×D₇₉	D₄	1	14	yes	—
47	C₂²×C₄×D₇₉	C₂³	1	14	yes	—
48	#14579190	C₂³	1	512	yes	direct product factor
49	C₂×Dic₃₁₆	C₂³	1	112	yes	complemented
50	C₂×#11929614	C₂³	1	112	yes	complemented
51	C₂×#11929612	C₂³	1	112	yes	complemented
52	C₂×#11929607	C₂³	1	112	yes	complemented
53	C₂²×C₇₉⋊C₈	C₂³	1	14	yes	—
54	C₂⁵×C₇₉	—	2	1	no	—
55	C₂²×Q₈×C₇₉	—	2	7	no	—
56	C₂²×Q₈×C₇₉	C₂³	1	14	yes	—
57	C₂²×D₄×C₇₉	—	2	7	no	—
58	C₂²×D₄×C₇₉	C₂³	1	14	yes	—
59	C₂³×C₄×C₇₉	—	2	1	no	—
60	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
61	C₂×SD₁₆×C₇₉	C₂³	1	112	yes	complemented
62	C₂²×C₈×C₇₉	C₂³	1	14	yes	—
63	C₂³×D₇₉	—	2	8	no	—
64	C₂³×D₇₉	C₂×D₄	1	14	yes	—
65	#11929623	—	2	224	no	—
66	#11929623	C₂⁴	1	64	yes	—
67	C₂×Dic₁₅₈	—	2	112	no	—
68	C₂×Dic₁₅₈	—	2	42	no	—
69	C₂×Dic₁₅₈	—	2	42	no	—
70	C₂×Dic₁₅₈	C₂⁴	1	28	yes	—
71	C₂×Dic₁₅₈	C₂⁴	1	28	yes	—
72	Q₈×D₇₉	—	2	224	no	—
73	Q₈×D₇₉	C₂⁴	1	64	yes	complemented
74	C₂×C₂²⋊D₇₉	—	2	112	no	—
75	C₂²×Dic₇₉	—	2	112	no	—
76	C₂²×Dic₇₉	—	2	14	no	—
77	C₂²×Dic₇₉	—	2	14	no	—
78	C₂²×Dic₇₉	C₂×D₄	1	14	yes	—
79	C₂×C₄×D₇₉	—	2	112	no	—
80	C₂×C₄×D₇₉	C₂×D₄	1	84	yes	—
81	C₂×C₄×D₇₉	C₂⁴	1	28	yes	—
82	Dic₃₁₆	C₂⁴	1	64	yes	complemented
83	#11929614	C₂⁴	1	64	yes	complemented
84	#11929612	C₂⁴	1	64	yes	complemented
85	#11929607	C₂⁴	1	64	yes	complemented
86	C₂×C₇₉⋊C₈	C₂⁴	1	28	yes	—
87	C₂⁴×C₇₉	—	2	14	no	—
88	C₂⁴×C₇₉	—	4	8	no	—
89	C₂⁴×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂×Q₈×C₇₉	—	2	42	no	—
91	C₂×Q₈×C₇₉	C₂⁴	1	28	yes	—
92	C₂×D₄×C₇₉	—	2	42	no	—
93	C₂×D₄×C₇₉	C₂⁴	1	28	yes	—
94	C₂²×C₄×C₇₉	—	2	14	no	—
95	C₂²×C₄×C₇₉	C₂×D₄	1	7	yes	—
96	C₂²×C₄×C₇₉	C₂⁴	1	7	yes	—
97	SD₁₆×C₇₉	C₂⁴	1	64	yes	complemented
98	C₂×C₈×C₇₉	C₂⁴	1	28	yes	—
99	C₂³×C₇₉	—	2	28	no	—
100	C₂³×C₇₉	—	4	56	no	—
101	C₂³×C₇₉	#137	1	8	yes	complemented
102	C₂³×C₇₉	C₂²×D₄	1	7	yes	—
103	C₂²×D₇₉	—	2	56	no	—
104	C₂²×D₇₉	C₂²×D₄	1	28	yes	—
105	Q₈×C₇₉	—	2	28	no	—
106	Q₈×C₇₉	C₂⁵	1	8	yes	—
107	D₄×C₇₉	—	2	28	no	—
108	D₄×C₇₉	C₂⁵	1	8	yes	—
109	C₂×C₄×C₇₉	—	2	28	no	—
110	C₂×C₄×C₇₉	C₂²×D₄	1	21	yes	—
111	C₂×C₄×C₇₉	C₂⁵	1	7	yes	—
112	C₂²⋊D₇₉	—	2	64	no	—
113	C₂×Dic₇₉	—	2	28	no	—
114	C₂×Dic₇₉	—	2	28	no	—
115	C₂×Dic₇₉	—	2	64	no	—
116	C₂×Dic₇₉	C₂²×D₄	1	28	yes	—
117	C₄×D₇₉	—	2	64	no	—
118	C₄×D₇₉	C₂²×D₄	1	56	yes	—
119	C₄×D₇₉	C₂⁵	1	8	yes	—
120	Dic₁₅₈	—	2	28	no	—
121	Dic₁₅₈	—	2	28	no	—
122	Dic₁₅₈	—	2	64	no	—
123	Dic₁₅₈	C₂⁵	1	8	yes	—
124	Dic₁₅₈	C₂⁵	1	8	yes	—
125	C₈×C₇₉	C₂⁵	1	8	yes	—
126	C₇₉⋊C₈	C₂⁵	1	8	yes	—
127	C₂²×C₇₉	—	2	8	no	—
128	C₂²×C₇₉	—	4	56	no	—
129	C₂²×C₇₉	C₂×#137	1	28	yes	complemented
130	C₂²×C₇₉	C₂³×D₄	1	7	yes	—
131	C₂×D₇₉	—	2	56	no	—
132	C₂×D₇₉	C₂³×D₄	1	8	yes	—
133	C₄×C₇₉	—	2	8	no	—
134	C₄×C₇₉	C₂³×D₄	1	7	yes	—
135	C₄×C₇₉	C₂⁶	1	1	yes	characteristic
136	Dic₇₉	—	2	8	no	—
137	Dic₇₉	—	2	8	no	—
138	Dic₇₉	C₂³×D₄	1	8	yes	—
139	C₂³×#137	—	79	1	no	maximal, Sylow
140	C₂×C₇₉	—	4	8	no	—
141	C₂×C₇₉	C₂²×#137	1	14	yes	complemented
142	C₂×C₇₉	C₂⁴×D₄	1	1	yes	characteristic
143	D₇₉	—	2	8	no	—
144	C₂³×#41	—	79	1	no	—
145	C₂⁴×Q₈	—	79	1	no	—
146	C₂²×#137	—	79	56	no	—
147	C₂³×Q₁₆	—	79	1	no	—
148	C₂³×Q₁₆	—	79	1	no	—
149	C₂³×SD₁₆	—	79	1	no	—
150	C₂³×SD₁₆	D₇₉	1	1	yes	characteristic, complemented
151	C₂³×M₁₆	—	79	1	no	—
152	C₇₉	C₂³×#137	1	1	yes	characteristic, complemented, Sylow
153	C₂²×#41	—	79	28	no	—
154	C₂²×#41	—	158	14	no	—
155	C₂³×Q₈	—	79	28	no	—
156	C₂³×Q₈	—	79	1	no	—
157	C₂³×Q₈	—	79	1	no	—
158	C₂³×Q₈	—	158	1	no	—
159	C₂³×Q₈	—	158	14	no	—
160	C₂³×Q₈	C₂×D₇₉	1	1	yes	characteristic
161	C₂³×D₄	—	158	1	no	—
162	C₂³×D₄	C₂×D₇₉	1	1	yes	characteristic
163	C₂⁴×C₄	—	79	1	no	—
164	C₂⁴×C₄	—	158	1	no	—
165	C₂⁴×C₄	—	158	1	no	—
166	C₂×#137	—	79	448	no	—
167	C₂²×Q₁₆	—	79	28	no	—
168	C₂²×Q₁₆	—	79	28	no	—
169	C₂²×SD₁₆	—	79	28	no	—
170	C₂²×SD₁₆	C₂×D₇₉	1	28	yes	complemented
171	C₂²×M₁₆	—	79	28	no	—
172	C₂³×C₈	—	79	1	no	—
173	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
174	C₂⁵	—	2	1	no	—
175	C₂⁵	—	79	1	no	—
176	C₂×#41	—	79	112	no	—
177	C₂×#41	—	158	168	no	—
178	C₂²×Q₈	—	2	7	no	—
179	C₂²×Q₈	—	79	112	no	—
180	C₂²×Q₈	—	79	14	no	—
181	C₂²×Q₈	—	79	14	no	—
182	C₂²×Q₈	—	158	28	no	—
183	C₂²×Q₈	—	158	168	no	—
184	C₂²×Q₈	—	158	7	no	—
185	C₂²×Q₈	—	158	7	no	—
186	C₂²×Q₈	C₂²×D₇₉	1	14	yes	—
187	C₂²×D₄	—	2	7	no	—
188	C₂²×D₄	—	158	28	no	—
189	C₂²×D₄	C₂²×D₇₉	1	14	yes	—
190	C₂³×C₄	—	2	1	no	—
191	C₂³×C₄	—	79	1	no	—
192	C₂³×C₄	—	79	14	no	—
193	C₂³×C₄	—	79	14	no	—
194	C₂³×C₄	—	158	28	no	—
195	C₂³×C₄	—	158	28	no	—
196	C₂³×C₄	—	158	1	no	—
197	C₂³×C₄	—	158	1	no	—
198	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
199	#137	—	79	512	no	—
200	C₂×Q₁₆	—	79	112	no	—
201	C₂×Q₁₆	—	79	112	no	—
202	C₂×SD₁₆	—	79	112	no	—
203	C₂×SD₁₆	C₂²×D₇₉	1	112	yes	complemented
204	C₂×M₁₆	—	79	112	no	—
205	C₂²×C₈	—	79	14	no	—
206	C₂²×C₈	C₂²×D₇₉	1	14	yes	—
207	C₂⁴	—	2	14	no	—
208	C₂⁴	—	4	8	no	—
209	C₂⁴	—	79	14	no	—
210	C₂⁴	—	158	8	no	—
211	C₂⁴	D₄×D₇₉	1	1	yes	characteristic
212	#41	—	79	64	no	—
213	#41	—	158	224	no	—
214	C₂×Q₈	—	2	42	no	—
215	C₂×Q₈	—	79	64	no	—
216	C₂×Q₈	—	79	28	no	—
217	C₂×Q₈	—	79	28	no	—
218	C₂×Q₈	—	158	112	no	—
219	C₂×Q₈	—	158	224	no	—
220	C₂×Q₈	—	158	42	no	—
221	C₂×Q₈	—	158	42	no	—
222	C₂×Q₈	C₂³×D₇₉	1	28	yes	—
223	C₂×D₄	—	2	42	no	—
224	C₂×D₄	—	158	112	no	—
225	C₂×D₄	C₂³×D₇₉	1	28	yes	—
226	C₂²×C₄	—	2	14	no	—
227	C₂²×C₄	—	79	14	no	—
228	C₂²×C₄	—	79	28	no	—
229	C₂²×C₄	—	79	84	no	—
230	C₂²×C₄	—	158	112	no	—
231	C₂²×C₄	—	158	112	no	—
232	C₂²×C₄	—	158	14	no	—
233	C₂²×C₄	—	158	14	no	—
234	C₂²×C₄	D₄×D₇₉	1	7	yes	—
235	C₂²×C₄	C₂³×D₇₉	1	7	yes	—
236	Q₁₆	—	79	64	no	—
237	Q₁₆	—	79	64	no	—
238	SD₁₆	—	79	64	no	—
239	SD₁₆	C₂³×D₇₉	1	64	yes	complemented
240	M₁₆	—	79	64	no	—
241	C₂×C₈	—	79	28	no	—
242	C₂×C₈	C₂³×D₇₉	1	28	yes	—
243	C₂³	—	2	28	no	—
244	C₂³	—	4	56	no	—
245	C₂³	—	79	28	no	—
246	C₂³	—	158	56	no	—
247	C₂³	#14579190	1	8	yes	direct product factor
248	C₂³	C₂×D₄×D₇₉	1	7	yes	—
249	Q₈	—	2	28	no	—
250	Q₈	—	79	8	no	—
251	Q₈	—	79	8	no	—
252	Q₈	—	158	64	no	—
253	Q₈	—	158	28	no	—
254	Q₈	—	158	28	no	—
255	Q₈	C₂⁴×D₇₉	1	8	yes	—
256	D₄	—	2	28	no	—
257	D₄	—	158	64	no	—
258	D₄	C₂⁴×D₇₉	1	8	yes	—
259	C₂×C₄	—	2	28	no	—
260	C₂×C₄	—	79	28	no	—
261	C₂×C₄	—	79	8	no	—
262	C₂×C₄	—	79	56	no	—
263	C₂×C₄	—	158	64	no	—
264	C₂×C₄	—	158	64	no	—
265	C₂×C₄	—	158	28	no	—
266	C₂×C₄	—	158	28	no	—
267	C₂×C₄	C₂×D₄×D₇₉	1	21	yes	—
268	C₂×C₄	C₂⁴×D₇₉	1	7	yes	—
269	C₈	—	79	8	no	—
270	C₈	C₂⁴×D₇₉	1	8	yes	—
271	C₂²	—	2	8	no	—
272	C₂²	—	4	56	no	—
273	C₂²	—	79	8	no	—
274	C₂²	—	158	56	no	—
275	C₂²	C₂×#14579190	1	28	yes	direct product factor
276	C₂²	C₂²×D₄×D₇₉	1	7	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	79	8	no	—
279	C₄	—	158	8	no	—
280	C₄	—	158	8	no	—
281	C₄	C₂²×D₄×D₇₉	1	7	yes	—
282	C₄	C₂⁵×D₇₉	1	1	yes	characteristic
283	C₂	—	4	8	no	—
284	C₂	—	158	8	no	—
285	C₂	C₂²×#14579190	1	14	yes	direct product factor
286	C₂	C₂³×D₄×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₉
commutator, cosocle	C₄×C₇₉	C₂⁶
Frattini	C₄	C₂⁵×D₇₉
Fitting	C₂³×SD₁₆×C₇₉	C₂
socle	C₂⁴×C₇₉	C₂×D₄
prime radical	C₂³×#11929623	C₂
hypercenter	C₂³×SD₁₆	D₇₉
hypocenter	C₇₉	C₂³×#137
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×SD₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₈×D₇₉	C₂	semidirect
3. C₂²×#14579190	C₂	direct (N × H)
4. C₂³×Dic₃₁₆	C₂	semidirect
5. C₂³×#11929614	C₂	semidirect
6. C₂³×#11929612	C₂	semidirect
7. C₂³×#11929607	C₂	semidirect
8. C₂³×SD₁₆×C₇₉	C₂	semidirect
15. C₂²×Q₈×D₇₉	C₂²	semidirect
20. C₂×#14579190	C₂²	direct (N × H)
21. C₂²×Dic₃₁₆	C₂²	semidirect
22. C₂²×#11929614	C₂²	semidirect
23. C₂²×#11929612	C₂²	semidirect
24. C₂²×#11929607	C₂²	semidirect
28. C₂²×SD₁₆×C₇₉	C₂²	semidirect
39. C₂×Q₈×D₇₉	C₂³	semidirect
48. #14579190	C₂³	direct (N × H)
49. C₂×Dic₃₁₆	C₂³	semidirect
50. C₂×#11929614	C₂³	semidirect
51. C₂×#11929612	C₂³	semidirect
52. C₂×#11929607	C₂³	semidirect
61. C₂×SD₁₆×C₇₉	C₂³	semidirect
73. Q₈×D₇₉	C₂⁴	semidirect
82. Dic₃₁₆	C₂⁴	semidirect
83. #11929614	C₂⁴	semidirect
84. #11929612	C₂⁴	semidirect
85. #11929607	C₂⁴	semidirect
97. SD₁₆×C₇₉	C₂⁴	semidirect
101. C₂³×C₇₉	#137	semidirect
129. C₂²×C₇₉	C₂×#137	semidirect
141. C₂×C₇₉	C₂²×#137	semidirect
150. C₂³×SD₁₆	D₇₉	semidirect
152. C₇₉	C₂³×#137	semidirect
170. C₂²×SD₁₆	C₂×D₇₉	semidirect
203. C₂×SD₁₆	C₂²×D₇₉	semidirect
239. SD₁₆	C₂³×D₇₉	semidirect
247. C₂³	#14579190	direct (N × H)
275. C₂²	C₂×#14579190	direct (N × H)
285. C₂	C₂²×#14579190	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11929623	2
C₂³×Q₈×D₇₉	2
C₂²×#14579190	2
C₂³×Dic₃₁₆	2
C₂³×#11929614	2
C₂³×#11929612	2
C₂³×#11929607	2
C₂³×SD₁₆×C₇₉	2
C₂³×#137	79

Subgroups — C23×#14579190 #35202065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14579190 #35202065