Subgroups — C₄×Q₁₆×Dic₇₉ #34286746

Order 20224.

1110 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₇₉	C₂	1	2	yes	maximal
2	C₄×Q₈×Dic₇₉	C₂	1	2	yes	maximal
3	#107×Dic₇₉	C₂	1	2	yes	maximal
4	C₄×#14579095	C₂	1	1	yes	maximal, characteristic
5	#103×Dic₇₉	C₂	1	4	yes	maximal
6	C₄×#14579085	C₂	1	2	yes	maximal
7	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×Q₈×Dic₇₉	C₂²	1	4	yes	—
10	Q₁₆×Dic₇₉	C₄	1	16	yes	direct product factor
11	C₂×#14579095	C₂²	1	2	yes	—
12	C₂×#14579085	C₂²	1	4	yes	—
13	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
15	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
16	C₄⋊C₄×Dic₇₉	—	2	4	no	—
17	C₄⋊C₄×Dic₇₉	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₇₉	—	2	2	no	—
19	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
20	C₄²×Dic₇₉	—	2	2	no	—
21	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
22	#19098326	C₂²	1	2	yes	—
23	#19098325	C₂²	1	4	yes	—
24	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₇₉	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₇₉	C₄	1	8	yes	complemented
27	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
31	Q₈×Dic₇₉	—	2	8	no	—
32	Q₈×Dic₇₉	C₂×C₄	1	16	yes	—
33	C₂×C₄⋊Dic₇₉	—	2	4	no	—
34	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
35	C₂×C₄×Dic₇₉	—	2	4	no	—
36	C₂×C₄×Dic₇₉	—	2	2	no	—
37	C₂×C₄×Dic₇₉	—	2	4	no	—
38	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
40	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
42	#14579095	C₂×C₄	1	8	yes	—
43	#14579085	C₂×C₄	1	16	yes	—
44	C₄×C₇₉⋊C₈	—	2	2	no	—
45	C₈×Dic₇₉	—	2	4	no	—
46	C₈×Dic₇₉	C₂×C₄	1	8	yes	—
47	C₇₉⋊#95	—	2	2	no	—
48	C₇₉⋊#95	—	2	4	no	—
49	C₂²×Q₈×C₇₉	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₇₉	C₂×C₄	1	8	yes	—
51	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	C₄×Q₈×C₇₉	—	2	4	no	—
55	C₄×Q₈×C₇₉	C₂×C₄	1	8	yes	—
56	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
57	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
58	C₂×C₄²×C₇₉	—	2	2	no	—
59	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
60	#107×C₇₉	C₂×C₄	1	4	yes	—
61	#103×C₇₉	C₂×C₄	1	8	yes	—
62	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
63	C₂²×Dic₇₉	—	2	4	no	—
64	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
65	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
66	C₂×C₇₉⋊C₈	—	2	2	no	—
67	C₂×C₇₉⋊C₈	—	2	2	no	—
68	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
69	C₄⋊Dic₇₉	—	2	16	no	—
70	C₄⋊Dic₇₉	—	2	4	no	—
71	C₄⋊Dic₇₉	C₂²×C₄	1	8	yes	—
72	C₄×Dic₇₉	—	2	16	no	—
73	C₄×Dic₇₉	—	2	4	no	—
74	C₄×Dic₇₉	—	2	2	no	—
75	C₄×Dic₇₉	—	4	8	no	—
76	C₄×Dic₇₉	—	4	4	no	—
77	C₄×Dic₇₉	Q₁₆	1	4	yes	direct product factor
78	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
79	C₄×Dic₇₉	#41	1	4	yes	—
80	C₂×Q₈×C₇₉	—	2	2	no	—
81	C₂×Q₈×C₇₉	—	2	4	no	—
82	C₂×Q₈×C₇₉	C₂²×C₄	1	8	yes	—
83	C₂×Q₈×C₇₉	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₇₉	—	2	2	no	—
85	C₂²×C₄×C₇₉	—	2	2	no	—
86	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
89	Q₁₆×C₇₉	C₄²	1	16	yes	complemented
90	C₂×C₈×C₇₉	—	2	2	no	—
91	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
94	C₄⋊C₄×C₇₉	—	2	8	no	—
95	C₄⋊C₄×C₇₉	—	2	2	no	—
96	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
97	C₄²×C₇₉	—	2	8	no	—
98	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
99	C₄²×C₇₉	#41	1	2	yes	—
100	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
101	Q₈×C₇₉	—	2	8	no	—
102	Q₈×C₇₉	—	2	4	no	—
103	Q₈×C₇₉	C₂×C₄²	1	8	yes	—
104	C₂×C₄×C₇₉	—	2	8	no	—
105	C₂×C₄×C₇₉	—	2	4	no	—
106	C₂×C₄×C₇₉	—	2	2	no	—
107	C₂×C₄×C₇₉	—	2	4	no	—
108	C₂×C₄×C₇₉	—	4	4	no	—
109	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
112	C₂×C₄×C₇₉	#135	1	2	yes	—
113	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
114	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
116	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₇₉	—	2	2	no	—
118	C₂×Dic₇₉	—	2	2	no	—
119	C₂×Dic₇₉	—	2	8	no	—
120	C₂×Dic₇₉	—	4	4	no	—
121	C₂×Dic₇₉	—	4	4	no	—
122	C₂×Dic₇₉	C₄×D₄	1	4	yes	—
123	C₂×Dic₇₉	C₄×D₄	1	4	yes	—
124	C₂×Dic₇₉	C₂×Q₁₆	1	4	yes	—
125	C₂×Dic₇₉	#135	1	4	yes	—
126	C₈×C₇₉	—	2	2	no	—
127	C₈×C₇₉	C₂×C₄²	1	4	yes	—
128	C₇₉⋊C₈	—	2	4	no	—
129	C₂²×C₇₉	C₂×C₄×D₄	1	2	yes	—
130	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
131	C₂²×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂²×C₇₉	C₂×#135	1	2	yes	—
133	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
134	C₄×C₇₉	—	2	2	no	—
135	C₄×C₇₉	—	2	8	no	—
136	C₄×C₇₉	—	4	4	no	—
137	C₄×C₇₉	C₄×Q₁₆	1	4	yes	complemented
138	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
139	C₄×C₇₉	C₄×#41	1	2	yes	—
140	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
141	Dic₇₉	—	2	4	no	—
142	Dic₇₉	—	4	8	no	—
143	Dic₇₉	C₄×Q₁₆	1	8	yes	direct product factor
144	C₄²×Q₁₆	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	C₄²×D₄	1	1	yes	characteristic
146	C₂×C₇₉	C₂×C₄×Q₁₆	1	2	yes	—
147	C₂×C₇₉	C₂×C₄×Q₁₆	1	1	yes	characteristic
148	C₂×C₇₉	C₄×#135	1	2	yes	—
149	C₂×C₇₉	C₄×#135	1	1	yes	characteristic
150	C₂×C₄×Q₁₆	—	79	2	no	—
151	C₂×C₄×Q₁₆	D₇₉	1	1	yes	characteristic
152	C₄²×Q₈	—	79	2	no	—
153	C₄×#107	—	79	2	no	—
154	C₄×#107	—	79	1	no	—
155	C₄×#103	—	79	4	no	—
156	C₄×#103	—	79	2	no	—
157	C₄²×C₈	—	79	1	no	—
158	C₇₉	C₄²×Q₁₆	1	1	yes	characteristic, complemented, Sylow
159	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
160	C₂×C₄×Q₈	—	79	4	no	—
161	C₂×C₄×Q₈	C₂×D₇₉	1	2	yes	—
162	C₄×Q₁₆	—	79	16	no	—
163	C₄×Q₁₆	Dic₇₉	1	8	yes	direct product factor
164	C₂×#107	—	79	2	no	—
165	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
166	C₂×#103	—	79	4	no	—
167	C₂×#103	C₂×D₇₉	1	2	yes	—
168	C₂×C₄×C₈	—	79	2	no	—
169	C₂×C₄×C₈	—	79	2	no	—
170	C₂×C₄×C₈	—	79	1	no	—
171	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
172	C₄×C₄⋊C₄	—	79	4	no	—
173	C₄×C₄⋊C₄	—	79	2	no	—
174	C₄×C₄⋊C₄	—	158	2	no	—
175	C₄×C₄⋊C₄	—	158	4	no	—
176	C₄³	—	79	1	no	—
177	C₄³	—	158	2	no	—
178	#340	—	79	2	no	—
179	#340	—	79	4	no	—
180	C₂²×Q₈	C₂²×D₇₉	1	2	yes	—
181	C₂×Q₁₆	C₄×D₇₉	1	8	yes	—
182	C₂×Q₁₆	C₂×Dic₇₉	1	4	yes	—
183	C₂²×C₈	—	79	2	no	—
184	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
186	C₄×Q₈	—	2	4	no	—
187	C₄×Q₈	—	79	16	no	—
188	C₄×Q₈	—	158	8	no	—
189	C₄×Q₈	C₂×Dic₇₉	1	8	yes	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	79	4	no	—
192	C₂×C₄⋊C₄	—	158	4	no	—
193	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	79	1	no	—
196	C₂×C₄²	—	79	2	no	—
197	C₂×C₄²	—	79	2	no	—
198	C₂×C₄²	—	79	1	no	—
199	C₂×C₄²	—	158	4	no	—
200	C₂×C₄²	—	158	2	no	—
201	C₂×C₄²	—	158	4	no	—
202	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
203	#107	—	79	8	no	—
204	#107	C₂×Dic₇₉	1	4	yes	—
205	#103	—	79	16	no	—
206	#103	C₂×Dic₇₉	1	8	yes	—
207	C₄×C₈	—	79	8	no	—
208	C₄×C₈	—	158	4	no	—
209	C₄×C₈	—	158	2	no	—
210	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
211	#95	—	158	4	no	—
212	#95	—	158	2	no	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	C₂×C₄×D₇₉	1	8	yes	—
216	C₂×Q₈	C₂²×Dic₇₉	1	4	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	79	2	no	—
220	C₂²×C₄	—	79	2	no	—
221	C₂²×C₄	—	158	4	no	—
222	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
223	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
224	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
225	Q₁₆	C₄×Dic₇₉	1	16	yes	direct product factor
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	79	4	no	—
228	C₂×C₈	—	158	2	no	—
229	C₂×C₈	—	158	2	no	—
230	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
231	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
232	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	79	8	no	—
236	C₄⋊C₄	—	158	16	no	—
237	C₄⋊C₄	—	158	4	no	—
238	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
239	C₄²	—	2	8	no	—
240	C₄²	—	316	8	no	—
241	C₄²	—	316	4	no	—
242	C₄²	—	79	4	no	—
243	C₄²	—	79	4	no	—
244	C₄²	—	79	4	no	—
245	C₄²	—	158	4	no	—
246	C₄²	—	158	16	no	—
247	C₄²	—	158	2	no	—
248	C₄²	C₂²×Dic₇₉	1	2	yes	—
249	C₄²	#11929623	1	2	yes	—
250	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
251	Q₈	—	2	8	no	—
252	Q₈	—	2	4	no	—
253	Q₈	C₂×C₄×Dic₇₉	1	8	yes	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	316	4	no	—
260	C₂×C₄	—	316	4	no	—
261	C₂×C₄	—	79	4	no	—
262	C₂×C₄	—	79	4	no	—
263	C₂×C₄	—	79	4	no	—
264	C₂×C₄	—	79	4	no	—
265	C₂×C₄	—	158	8	no	—
266	C₂×C₄	—	158	2	no	—
267	C₂×C₄	—	158	2	no	—
268	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
269	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
270	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
271	C₂×C₄	#14579184	1	2	yes	—
272	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
273	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
274	C₂×C₄	#41×D₇₉	1	2	yes	—
275	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	—	158	4	no	—
278	C₈	C₂×C₄×Dic₇₉	1	4	yes	—
279	C₂²	C₄×D₄×D₇₉	1	2	yes	—
280	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
281	C₂²	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
282	C₂²	#135×D₇₉	1	2	yes	—
283	C₂²	C₂×#14579184	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	8	no	—
286	C₄	—	4	4	no	—
287	C₄	—	316	8	no	—
288	C₄	—	79	8	no	—
289	C₄	—	158	4	no	—
290	C₄	Q₁₆×Dic₇₉	1	4	yes	direct product factor
291	C₄	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
292	C₄	C₄×#11929623	1	2	yes	—
293	C₄	#41×Dic₇₉	1	1	yes	characteristic
294	C₂	C₄×D₄×Dic₇₉	1	1	yes	characteristic
295	C₂	C₄×Q₁₆×D₇₉	1	2	yes	—
296	C₂	C₂×Q₁₆×Dic₇₉	1	1	yes	characteristic
297	C₂	C₄×#14579184	1	2	yes	—
298	C₂	#135×Dic₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₄×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₄×Q₁₆×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇₉
hypocenter	C₇₉	C₄²×Q₁₆
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
10. Q₁₆×Dic₇₉	C₄	direct (N × H)
26. C₄×Q₁₆×C₇₉	C₄	semidirect
77. C₄×Dic₇₉	Q₁₆	direct (N × H)
89. Q₁₆×C₇₉	C₄²	semidirect
137. C₄×C₇₉	C₄×Q₁₆	semidirect
143. Dic₇₉	C₄×Q₁₆	direct (N × H)
158. C₇₉	C₄²×Q₁₆	semidirect
163. C₄×Q₁₆	Dic₇₉	direct (N × H)
225. Q₁₆	C₄×Dic₇₉	direct (N × H)
290. C₄	Q₁₆×Dic₇₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×Dic₇₉	2
C₄×Q₈×Dic₇₉	2
#107×Dic₇₉	2
C₄×#14579095	2
#103×Dic₇₉	2
C₄×#14579085	2
C₄×C₈×Dic₇₉	2
C₂×C₄×Q₁₆×C₇₉	2
C₄²×Q₁₆	79

Subgroups — C4×Q16×Dic79 #34286746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₁₆×Dic₇₉ #34286746