Subgroups — #498×Dic₇₉ #34296998

Order 20224.

878 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×Dic₇₉	C₂	1	2	yes	maximal
2	#23764362	C₂	1	2	yes	maximal
3	#23764129	C₂	1	4	yes	maximal
4	#23763557	C₂	1	1	yes	maximal, characteristic
5	#106×Dic₇₉	C₂	1	4	yes	maximal
6	C₄×C₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#498×C₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₇₉	—	2	2	no	—
9	C₂×#14579093	C₂²	1	4	yes	—
10	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
12	C₂×C₈×Dic₇₉	C₂²	1	2	yes	—
13	#19098470	—	2	2	no	—
14	#19098462	—	2	2	no	—
15	#19098462	C₂²	1	4	yes	—
16	#19098433	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₇₉	—	2	2	no	—
18	C₄⋊C₄×Dic₇₉	C₂²	1	4	yes	—
19	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
20	#19098325	C₂²	1	8	yes	—
21	C₂×C₄⋊Q₈×C₇₉	C₂²	1	2	yes	—
22	#498×C₇₉	C₄	1	8	yes	complemented
23	C₂×#106×C₇₉	C₂²	1	4	yes	—
24	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
26	Q₈×Dic₇₉	—	4	8	no	—
27	C₂×C₄⋊Dic₇₉	—	2	2	no	—
28	C₂×C₄⋊Dic₇₉	—	2	2	no	—
29	C₂×C₄⋊Dic₇₉	—	2	4	no	—
30	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
31	C₂×C₄×Dic₇₉	—	2	4	no	—
32	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
33	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
34	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
35	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
36	#14579093	—	2	8	no	—
37	C₄×C₇₉⋊C₈	—	2	2	no	—
38	C₈×Dic₇₉	—	2	4	no	—
39	C₈×Dic₇₉	Q₈	1	8	yes	complemented
40	C₇₉⋊#95	—	2	4	no	—
41	C₇₉⋊#95	—	2	4	no	—
42	C₂²×Q₈×C₇₉	—	2	2	no	—
43	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₇₉	—	2	4	no	—
45	C₄⋊Q₈×C₇₉	C₂×C₄	1	8	yes	—
46	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
47	C₂×C₄⋊C₄×C₇₉	C₂³	1	4	yes	—
48	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
49	#106×C₇₉	C₂×C₄	1	16	yes	—
50	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
51	C₂²×Dic₇₉	—	2	4	no	—
52	C₂²×Dic₇₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×Dic₇₉	C₂×D₄	1	2	yes	—
54	C₂²×Dic₇₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₇₉⋊C₈	—	2	2	no	—
56	C₂×C₇₉⋊C₈	—	2	2	no	—
57	C₂×C₇₉⋊C₈	C₂×Q₈	1	4	yes	—
58	C₄⋊Dic₇₉	—	2	8	no	—
59	C₄⋊Dic₇₉	—	4	8	no	—
60	C₄⋊Dic₇₉	—	4	4	no	—
61	C₄⋊Dic₇₉	—	4	4	no	—
62	C₄×Dic₇₉	—	2	4	no	—
63	C₄×Dic₇₉	—	2	2	no	—
64	C₄×Dic₇₉	—	4	8	no	—
65	C₄×Dic₇₉	SD₁₆	1	8	yes	—
66	C₄×Dic₇₉	C₂×Q₈	1	4	yes	—
67	C₂×Q₈×C₇₉	—	2	8	no	—
68	C₂×Q₈×C₇₉	—	4	4	no	—
69	C₂×Q₈×C₇₉	—	4	4	no	—
70	C₂²×C₄×C₇₉	—	2	4	no	—
71	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
74	C₂×C₈×C₇₉	C₂×Q₈	1	4	yes	—
75	C₂×C₈×C₇₉	#41	1	4	yes	—
76	C₄⋊C₄×C₇₉	—	2	8	no	—
77	C₄⋊C₄×C₇₉	—	2	4	no	—
78	C₄⋊C₄×C₇₉	C₂²×C₄	1	8	yes	—
79	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
80	C₄²×C₇₉	#41	1	2	yes	—
81	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
82	Q₈×C₇₉	—	4	16	no	—
83	C₂×C₄×C₇₉	—	2	8	no	—
84	C₂×C₄×C₇₉	—	4	4	no	—
85	C₂×C₄×C₇₉	—	4	4	no	—
86	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₇₉	C₂×SD₁₆	1	4	yes	—
88	C₂×C₄×C₇₉	#135	1	4	yes	—
89	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
90	C₂×C₄×C₇₉	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
92	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
93	C₂×Dic₇₉	—	2	2	no	—
94	C₂×Dic₇₉	—	2	2	no	—
95	C₂×Dic₇₉	—	2	2	no	—
96	C₂×Dic₇₉	—	4	4	no	—
97	C₂×Dic₇₉	—	4	4	no	—
98	C₂×Dic₇₉	—	4	4	no	—
99	C₂×Dic₇₉	C₄⋊Q₈	1	2	yes	—
100	C₂×Dic₇₉	C₄⋊Q₈	1	2	yes	—
101	C₂×Dic₇₉	C₂×SD₁₆	1	4	yes	—
102	C₂×Dic₇₉	C₂×SD₁₆	1	4	yes	—
103	C₈×C₇₉	C₄×Q₈	1	8	yes	complemented
104	C₇₉⋊C₈	—	2	4	no	—
105	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
106	C₂²×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₇₉	#532	1	1	yes	characteristic
108	C₂²×C₇₉	C₂²×SD₁₆	1	2	yes	—
109	C₂²×C₇₉	C₂×#135	1	2	yes	—
110	C₄×C₇₉	—	4	8	no	—
111	C₄×C₇₉	C₄×SD₁₆	1	8	yes	—
112	C₄×C₇₉	C₂×C₄×Q₈	1	2	yes	—
113	C₄×C₇₉	#533	1	2	yes	—
114	Dic₇₉	—	2	2	no	—
115	Dic₇₉	—	2	4	no	—
116	Dic₇₉	—	8	4	no	—
117	Dic₇₉	#498	1	4	yes	direct product factor
118	C₄×#498	—	79	1	no	maximal, Sylow
119	C₂×C₇₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₇₉	C₂×C₄×SD₁₆	1	2	yes	—
121	C₂×C₇₉	C₂×#498	1	1	yes	characteristic
122	C₂×C₇₉	#3160	1	1	yes	characteristic
123	C₂×C₇₉	#3167	1	2	yes	—
124	C₂×#498	D₇₉	1	1	yes	characteristic
125	C₄×C₄⋊Q₈	—	79	2	no	—
126	#1990	—	79	2	no	—
127	#1941	—	79	4	no	—
128	#1844	—	79	1	no	—
129	C₄×#106	—	79	4	no	—
130	C₄²×C₈	—	79	1	no	—
131	C₇₉	C₄×#498	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄⋊Q₈	C₂×D₇₉	1	2	yes	—
133	C₂×C₄×Q₈	—	158	2	no	—
134	#498	Dic₇₉	1	8	yes	direct product factor
135	C₂×#106	—	79	4	no	—
136	C₂×#106	C₂×D₇₉	1	4	yes	—
137	C₂×C₄×C₈	—	79	2	no	—
138	C₂×C₄×C₈	—	79	2	no	—
139	C₂×C₄×C₈	—	79	1	no	—
140	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
141	#391	—	158	2	no	—
142	#389	—	79	4	no	—
143	#389	—	158	2	no	—
144	#384	—	79	2	no	—
145	C₄×C₄⋊C₄	—	79	4	no	—
146	C₄×C₄⋊C₄	—	158	2	no	—
147	C₄³	—	79	1	no	—
148	#340	—	79	8	no	—
149	C₂²×Q₈	—	2	2	no	—
150	C₂²×C₈	—	79	2	no	—
151	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	C₂×Dic₇₉	1	8	yes	—
154	C₄×Q₈	—	316	8	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	79	4	no	—
157	C₂×C₄⋊C₄	—	158	4	no	—
158	C₂×C₄⋊C₄	—	158	2	no	—
159	C₂×C₄⋊C₄	—	158	2	no	—
160	C₂×C₄⋊C₄	C₂²×D₇₉	1	4	yes	—
161	C₂×C₄²	—	79	2	no	—
162	C₂×C₄²	—	79	1	no	—
163	C₂×C₄²	—	79	2	no	—
164	C₂×C₄²	—	79	1	no	—
165	C₂×C₄²	—	158	4	no	—
166	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
167	#106	—	158	8	no	—
168	#106	C₂×Dic₇₉	1	16	yes	—
169	C₄×C₈	—	79	8	no	—
170	C₄×C₈	—	158	4	no	—
171	C₄×C₈	—	158	2	no	—
172	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
173	#95	—	158	4	no	—
174	#95	—	158	4	no	—
175	C₂×Q₈	—	2	8	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	—	79	1	no	—
180	C₂²×C₄	—	79	1	no	—
181	C₂²×C₄	—	79	2	no	—
182	C₂²×C₄	—	158	4	no	—
183	C₂²×C₄	D₄×D₇₉	1	2	yes	—
184	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
185	C₂×C₈	—	79	4	no	—
186	C₂×C₈	—	158	2	no	—
187	C₂×C₈	—	158	2	no	—
188	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
189	C₂×C₈	Q₈×D₇₉	1	4	yes	—
190	C₂×C₈	#11929625	1	4	yes	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	316	8	no	—
194	C₄⋊C₄	—	316	4	no	—
195	C₄⋊C₄	—	316	4	no	—
196	C₄⋊C₄	—	158	8	no	—
197	C₄⋊C₄	C₂²×Dic₇₉	1	8	yes	—
198	C₄²	—	316	8	no	—
199	C₄²	—	79	4	no	—
200	C₄²	—	79	8	no	—
201	C₄²	—	158	2	no	—
202	C₄²	—	158	4	no	—
203	C₄²	C₂²×Dic₇₉	1	2	yes	—
204	C₄²	#11929623	1	2	yes	—
205	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
206	Q₈	—	4	16	no	—
207	C₂×C₄	—	2	8	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	316	4	no	—
211	C₂×C₄	—	316	4	no	—
212	C₂×C₄	—	316	4	no	—
213	C₂×C₄	—	79	4	no	—
214	C₂×C₄	—	79	2	no	—
215	C₂×C₄	—	79	4	no	—
216	C₂×C₄	—	79	2	no	—
217	C₂×C₄	—	158	2	no	—
218	C₂×C₄	—	158	2	no	—
219	C₂×C₄	—	158	2	no	—
220	C₂×C₄	D₄×Dic₇₉	1	4	yes	—
221	C₂×C₄	SD₁₆×D₇₉	1	4	yes	—
222	C₂×C₄	#14579182	1	4	yes	—
223	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
224	C₂×C₄	C₂×Q₈×D₇₉	1	2	yes	—
225	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
226	C₂×C₄	C₂×#11929625	1	2	yes	—
227	C₈	—	158	4	no	—
228	C₈	Q₈×Dic₇₉	1	8	yes	complemented
229	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
230	C₂²	C₄⋊Q₈×D₇₉	1	1	yes	characteristic
231	C₂²	#19099114	1	1	yes	characteristic
232	C₂²	C₂×SD₁₆×D₇₉	1	2	yes	—
233	C₂²	C₂×#14579182	1	2	yes	—
234	C₄	—	4	8	no	—
235	C₄	—	79	4	no	—
236	C₄	—	632	4	no	—
237	C₄	—	158	4	no	—
238	C₄	—	158	2	no	—
239	C₄	SD₁₆×Dic₇₉	1	8	yes	—
240	C₄	C₂×Q₈×Dic₇₉	1	2	yes	—
241	C₄	#19099118	1	2	yes	—
242	C₂	C₄⋊Q₈×Dic₇₉	1	1	yes	characteristic
243	C₂	C₂×SD₁₆×Dic₇₉	1	2	yes	—
244	C₂	#498×D₇₉	1	1	yes	characteristic
245	C₂	#23773909	1	1	yes	characteristic
246	C₂	#23773961	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#498×C₇₉	C₂
socle, prime radical	C₂³×C₇₉	C₂²×D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#498	D₇₉
hypocenter	C₇₉	C₄×#498
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#498

Complements

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

N	H ≅ G/N	type
22. #498×C₇₉	C₄	semidirect
39. C₈×Dic₇₉	Q₈	semidirect
103. C₈×C₇₉	C₄×Q₈	semidirect
117. Dic₇₉	#498	direct (N × H)
131. C₇₉	C₄×#498	semidirect
134. #498	Dic₇₉	direct (N × H)
228. C₈	Q₈×Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×Dic₇₉	2
#23764362	2
#23764129	2
#23763557	2
#106×Dic₇₉	2
C₄×C₈×Dic₇₉	2
C₂×#498×C₇₉	2
C₄×#498	79

Subgroups — #498×Dic79 #34296998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #498×Dic₇₉ #34296998