Subgroups — #519×Dic₇₉ #34446521

Order 20224.

3402 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099041	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765765	C₂	1	8	yes	maximal, complemented
4	#23765756	C₂	1	6	yes	maximal
5	C₄×Q₈×Dic₇₉	C₂	1	8	yes	maximal, complemented
6	#117×Dic₇₉	C₂	1	6	yes	maximal
7	C₂×#519×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19099041	C₄	1	16	yes	complemented
9	C₂×Q₈×Dic₇₉	C₄	1	16	yes	complemented
10	C₂×Q₈×Dic₇₉	C₂²	1	4	yes	—
11	C₂×Q₈×Dic₇₉	C₂²	1	4	yes	—
12	C₂×Q₈×Dic₇₉	C₂²	1	8	yes	—
13	C₂×#14579120	C₂²	1	6	yes	—
14	C₂²×C₄⋊Dic₇₉	C₂²	1	3	yes	—
15	C₂²×C₄×Dic₇₉	C₂²	1	3	yes	—
16	C₄⋊C₄×Dic₇₉	C₂²	1	12	yes	—
17	C₄⋊C₄×Dic₇₉	C₂²	1	24	yes	—
18	C₄⋊C₄×Dic₇₉	C₂²	1	12	yes	—
19	C₄×C₄⋊Dic₇₉	C₂²	1	24	yes	—
20	C₂²⋊C₄×Dic₇₉	C₂²	1	4	yes	—
21	C₂²⋊C₄×Dic₇₉	C₂²	1	12	yes	—
22	#19098409	C₂²	1	12	yes	—
23	C₄²×Dic₇₉	C₂²	1	12	yes	—
24	C₂³×Q₈×C₇₉	C₂²	1	1	yes	characteristic
25	#519×C₇₉	C₄	1	16	yes	complemented
26	C₂×C₄×Q₈×C₇₉	C₂²	1	8	yes	—
27	C₂×#117×C₇₉	C₂²	1	6	yes	—
28	C₂³×Dic₇₉	C₂³	1	1	yes	characteristic
29	C₂³×Dic₇₉	C₂³	1	3	yes	—
30	Q₈×Dic₇₉	C₂×C₄	1	32	yes	—
31	Q₈×Dic₇₉	C₂×C₄	1	32	yes	—
32	Q₈×Dic₇₉	C₂×C₄	1	64	yes	complemented
33	#14579120	C₂×C₄	1	48	yes	—
34	C₂×C₄⋊Dic₇₉	C₂×C₄	1	24	yes	—
35	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
36	C₂×C₄⋊Dic₇₉	C₂³	1	12	yes	—
37	C₂×C₂²⋊Dic₇₉	C₂³	1	4	yes	—
38	C₂×C₄×Dic₇₉	C₂×C₄	1	24	yes	—
39	C₂×C₄×Dic₇₉	C₂³	1	8	yes	—
40	C₂×C₄×Dic₇₉	C₂³	1	24	yes	—
41	C₂×C₄×Dic₇₉	C₂³	1	6	yes	—
42	C₂×C₄×Dic₇₉	C₂³	1	12	yes	—
43	C₂×C₄×Dic₇₉	C₂³	1	6	yes	—
44	C₇₉⋊#95	C₂³	1	8	yes	—
45	C₇₉⋊#95	C₂³	1	24	yes	—
46	C₂²×Q₈×C₇₉	C₂×C₄	1	8	yes	—
47	C₂²×Q₈×C₇₉	C₂×C₄	1	8	yes	—
48	C₂²×Q₈×C₇₉	C₂×C₄	1	8	yes	—
49	C₂²×Q₈×C₇₉	C₂³	1	4	yes	—
50	C₂³×C₄×C₇₉	C₂³	1	3	yes	—
51	C₄×Q₈×C₇₉	C₂×C₄	1	64	yes	complemented
52	#117×C₇₉	C₂×C₄	1	48	yes	—
53	C₂×C₄⋊C₄×C₇₉	C₂³	1	12	yes	—
54	C₂×C₂²⋊C₄×C₇₉	C₂³	1	4	yes	—
55	C₂×C₄²×C₇₉	C₂³	1	12	yes	—
56	C₂²×Dic₇₉	—	2	4	no	—
57	C₂²×Dic₇₉	—	4	6	no	—
58	C₂²×Dic₇₉	C₂²×C₄	1	4	yes	—
59	C₂²×Dic₇₉	C₂²×C₄	1	12	yes	—
60	C₂²×Dic₇₉	C₂⁴	1	1	yes	characteristic
61	C₂²×Dic₇₉	C₂⁴	1	8	yes	—
62	C₂²×Dic₇₉	C₂⁴	1	6	yes	—
63	C₂²×Dic₇₉	C₂⁴	1	1	yes	characteristic
64	C₄⋊Dic₇₉	C₂²×C₄	1	48	yes	—
65	C₄⋊Dic₇₉	C₂²×C₄	1	48	yes	—
66	C₂²⋊Dic₇₉	C₂²×C₄	1	16	yes	—
67	C₄×Dic₇₉	—	2	16	no	—
68	C₄×Dic₇₉	—	4	24	no	—
69	C₄×Dic₇₉	C₂²×C₄	1	24	yes	—
70	C₄×Dic₇₉	C₂²×C₄	1	24	yes	—
71	C₄×Dic₇₉	C₂²×C₄	1	48	yes	—
72	C₂⁴×C₇₉	C₂⁴	1	1	yes	characteristic
73	C₂×Q₈×C₇₉	C₄²	1	64	yes	complemented
74	C₂×Q₈×C₇₉	C₂²×C₄	1	16	yes	—
75	C₂×Q₈×C₇₉	C₂²×C₄	1	16	yes	—
76	C₂×Q₈×C₇₉	C₂²×C₄	1	16	yes	—
77	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
78	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
79	C₂²×C₄×C₇₉	C₂²×C₄	1	12	yes	—
80	C₂²×C₄×C₇₉	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₇₉	C₂⁴	1	6	yes	—
82	C₄⋊C₄×C₇₉	C₂²×C₄	1	48	yes	—
83	C₂²⋊C₄×C₇₉	C₂²×C₄	1	16	yes	—
84	C₄²×C₇₉	C₂²×C₄	1	48	yes	—
85	C₂³×C₇₉	—	2	2	no	—
86	C₂³×C₇₉	—	2	2	no	—
87	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
88	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
89	C₂³×C₇₉	C₂³×C₄	1	2	yes	—
90	C₂³×C₇₉	C₂⁵	1	1	yes	characteristic
91	Q₈×C₇₉	C₂×C₄²	1	64	yes	—
92	C₂×C₄×C₇₉	—	2	8	no	—
93	C₂×C₄×C₇₉	—	2	8	no	—
94	C₂×C₄×C₇₉	C₂×C₄²	1	48	yes	—
95	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
96	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
97	C₂×C₄×C₇₉	C₂³×C₄	1	16	yes	—
98	C₂×C₄×C₇₉	C₂³×C₄	1	12	yes	—
99	C₂×Dic₇₉	—	2	8	no	—
100	C₂×Dic₇₉	—	2	2	no	—
101	C₂×Dic₇₉	—	2	8	no	—
102	C₂×Dic₇₉	—	2	12	no	—
103	C₂×Dic₇₉	—	2	8	no	—
104	C₂×Dic₇₉	—	4	12	no	—
105	C₂×Dic₇₉	C₂³×C₄	1	2	yes	—
106	C₂×Dic₇₉	C₂³×C₄	1	16	yes	—
107	C₂×Dic₇₉	C₂³×C₄	1	12	yes	—
108	C₂×Dic₇₉	C₂³×C₄	1	2	yes	—
109	C₂×Dic₇₉	2₋¹⁺⁴	1	4	yes	—
110	C₂²×C₇₉	—	2	2	no	—
111	C₂²×C₇₉	—	2	2	no	—
112	C₂²×C₇₉	—	2	4	no	—
113	C₂²×C₇₉	—	2	4	no	—
114	C₂²×C₇₉	C₂²×C₄²	1	4	yes	—
115	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
116	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
117	C₂²×C₇₉	C₂⁴×C₄	1	1	yes	characteristic
118	C₂²×C₇₉	C₂×2₋¹⁺⁴	1	2	yes	—
119	C₂²×C₇₉	#585	1	2	yes	—
120	C₄×C₇₉	—	2	16	no	—
121	C₄×C₇₉	C₂²×C₄²	1	24	yes	—
122	Dic₇₉	—	2	2	no	—
123	Dic₇₉	—	2	12	no	—
124	Dic₇₉	—	2	16	no	—
125	Dic₇₉	#519	1	4	yes	direct product factor
126	C₄×#519	—	79	1	no	maximal, Sylow
127	C₂×C₇₉	—	2	4	no	—
128	C₂×C₇₉	C₂³×C₄²	1	1	yes	characteristic
129	C₂×C₇₉	C₂×#519	1	1	yes	characteristic
130	C₂×C₇₉	#3428	1	1	yes	characteristic
131	C₂×C₇₉	C₄×2₋¹⁺⁴	1	2	yes	—
132	C₂×C₇₉	C₄×2₋¹⁺⁴	1	2	yes	—
133	C₂×#519	—	79	1	no	—
134	C₂×#519	D₇₉	1	1	yes	characteristic
135	C₂²×C₄×Q₈	—	79	1	no	—
136	#2276	—	79	8	no	—
137	#2274	—	79	6	no	—
138	C₄²×Q₈	—	79	8	no	—
139	C₄×#117	—	79	6	no	—
140	C₇₉	C₄×#519	1	1	yes	characteristic, complemented, Sylow
141	C₂³×Q₈	C₂×D₇₉	1	1	yes	characteristic
142	#519	—	79	16	no	—
143	#519	Dic₇₉	1	16	yes	direct product factor
144	C₂×C₄×Q₈	—	79	4	no	—
145	C₂×C₄×Q₈	—	79	4	no	—
146	C₂×C₄×Q₈	—	79	8	no	—
147	C₂×C₄×Q₈	—	79	16	no	—
148	C₂×C₄×Q₈	C₂×D₇₉	1	8	yes	—
149	C₂×#117	—	79	6	no	—
150	C₂×#117	C₂×D₇₉	1	6	yes	—
151	C₂²×C₄⋊C₄	—	79	3	no	—
152	C₂²×C₄²	—	79	3	no	—
153	C₄×C₄⋊C₄	—	79	12	no	—
154	C₄×C₄⋊C₄	—	79	12	no	—
155	C₄×C₄⋊C₄	—	79	24	no	—
156	C₄×C₄⋊C₄	—	79	24	no	—
157	C₄×C₂²⋊C₄	—	79	4	no	—
158	C₄×C₂²⋊C₄	—	79	12	no	—
159	#376	—	79	12	no	—
160	C₄³	—	79	12	no	—
161	C₂²×Q₈	C₄×D₇₉	1	8	yes	—
162	C₂²×Q₈	C₄×D₇₉	1	8	yes	—
163	C₂²×Q₈	C₂×Dic₇₉	1	8	yes	—
164	C₂²×Q₈	C₂²×D₇₉	1	4	yes	—
165	C₂³×C₄	—	79	3	no	—
166	C₂³×C₄	—	79	1	no	—
167	C₂³×C₄	C₂²×D₇₉	1	3	yes	—
168	C₄×Q₈	—	79	32	no	—
169	C₄×Q₈	—	79	32	no	—
170	C₄×Q₈	—	79	64	no	—
171	C₄×Q₈	C₂×Dic₇₉	1	64	yes	complemented
172	#117	—	79	48	no	—
173	#117	C₂×Dic₇₉	1	48	yes	—
174	C₂×C₄⋊C₄	—	79	24	no	—
175	C₂×C₄⋊C₄	—	79	12	no	—
176	C₂×C₄⋊C₄	—	79	12	no	—
177	C₂×C₄⋊C₄	C₂²×D₇₉	1	12	yes	—
178	C₂×C₂²⋊C₄	—	79	4	no	—
179	C₂×C₂²⋊C₄	C₂²×D₇₉	1	4	yes	—
180	C₂×C₄²	—	79	24	no	—
181	C₂×C₄²	—	79	24	no	—
182	C₂×C₄²	—	79	12	no	—
183	C₂×C₄²	—	79	6	no	—
184	C₂×C₄²	—	79	6	no	—
185	C₂×C₄²	—	79	8	no	—
186	C₂×C₄²	C₂²×D₇₉	1	12	yes	—
187	#95	—	79	8	no	—
188	#95	—	79	24	no	—
189	C₂⁴	C₂³×D₇₉	1	1	yes	characteristic
190	C₂×Q₈	C₄×Dic₇₉	1	64	yes	complemented
191	C₂×Q₈	C₂×C₄×D₇₉	1	16	yes	—
192	C₂×Q₈	C₂×C₄×D₇₉	1	16	yes	—
193	C₂×Q₈	C₂²×Dic₇₉	1	16	yes	—
194	C₂²×C₄	—	316	6	no	—
195	C₂²×C₄	—	79	12	no	—
196	C₂²×C₄	—	79	1	no	—
197	C₂²×C₄	—	79	1	no	—
198	C₂²×C₄	—	79	8	no	—
199	C₂²×C₄	—	79	6	no	—
200	C₂²×C₄	—	79	4	no	—
201	C₂²×C₄	—	158	4	no	—
202	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
203	C₂²×C₄	C₂×C₄×D₇₉	1	12	yes	—
204	C₂²×C₄	C₂²×Dic₇₉	1	12	yes	—
205	C₂²×C₄	C₂³×D₇₉	1	8	yes	—
206	C₂²×C₄	C₂³×D₇₉	1	6	yes	—
207	C₄⋊C₄	—	79	48	no	—
208	C₄⋊C₄	—	79	48	no	—
209	C₄⋊C₄	C₂²×Dic₇₉	1	48	yes	—
210	C₂²⋊C₄	—	79	16	no	—
211	C₂²⋊C₄	C₂²×Dic₇₉	1	16	yes	—
212	C₄²	—	316	24	no	—
213	C₄²	—	79	24	no	—
214	C₄²	—	79	24	no	—
215	C₄²	—	79	48	no	—
216	C₄²	—	158	16	no	—
217	C₄²	C₂²×Dic₇₉	1	48	yes	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	C₂²×C₄×D₇₉	1	2	yes	—
221	C₂³	C₂²×C₄×D₇₉	1	2	yes	—
222	C₂³	C₂³×Dic₇₉	1	2	yes	—
223	C₂³	C₂⁴×D₇₉	1	1	yes	characteristic
224	Q₈	C₂×C₄×Dic₇₉	1	64	yes	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	316	12	no	—
228	C₂×C₄	—	79	4	no	—
229	C₂×C₄	—	79	2	no	—
230	C₂×C₄	—	79	2	no	—
231	C₂×C₄	—	79	16	no	—
232	C₂×C₄	—	79	12	no	—
233	C₂×C₄	—	158	8	no	—
234	C₂×C₄	—	158	8	no	—
235	C₂×C₄	—	158	2	no	—
236	C₂×C₄	—	158	8	no	—
237	C₂×C₄	—	158	12	no	—
238	C₂×C₄	C₂×C₄×Dic₇₉	1	48	yes	—
239	C₂×C₄	C₂²×C₄×D₇₉	1	12	yes	—
240	C₂×C₄	C₂²×C₄×D₇₉	1	12	yes	—
241	C₂×C₄	C₂³×Dic₇₉	1	16	yes	—
242	C₂×C₄	C₂³×Dic₇₉	1	12	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	C₂²×C₄×Dic₇₉	1	4	yes	—
248	C₂²	C₂³×C₄×D₇₉	1	1	yes	characteristic
249	C₂²	C₂³×C₄×D₇₉	1	1	yes	characteristic
250	C₂²	C₂⁴×Dic₇₉	1	1	yes	characteristic
251	C₂²	2₋¹⁺⁴×D₇₉	1	2	yes	—
252	C₂²	#19099400	1	2	yes	—
253	C₄	—	2	16	no	—
254	C₄	—	79	4	no	—
255	C₄	—	158	2	no	—
256	C₄	—	158	12	no	—
257	C₄	—	158	16	no	—
258	C₄	C₂²×C₄×Dic₇₉	1	24	yes	—
259	C₂	—	2	4	no	—
260	C₂	C₂³×C₄×Dic₇₉	1	1	yes	characteristic
261	C₂	#519×D₇₉	1	1	yes	characteristic
262	C₂	#23776602	1	1	yes	characteristic
263	C₂	#23776609	1	2	yes	—
264	C₂	2₋¹⁺⁴×Dic₇₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇₉
commutator	C₂×C₇₉	C₂³×C₄²
Fitting	C₂×#519×C₇₉	C₂
socle, cosocle	C₂³×C₇₉	C₂⁵
prime radical	C₂⁴×C₇₉	C₂⁴
hypercenter	C₂×#519	D₇₉
hypocenter	C₇₉	C₄×#519
LCS	G ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×#519

Complements

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

N	H ≅ G/N	type
3. #23765765	C₂	semidirect
5. C₄×Q₈×Dic₇₉	C₂	semidirect
8. #19099041	C₄	semidirect
9. C₂×Q₈×Dic₇₉	C₄	semidirect
25. #519×C₇₉	C₄	semidirect
32. Q₈×Dic₇₉	C₂×C₄	semidirect
51. C₄×Q₈×C₇₉	C₂×C₄	semidirect
73. C₂×Q₈×C₇₉	C₄²	semidirect
125. Dic₇₉	#519	direct (N × H)
140. C₇₉	C₄×#519	semidirect
143. #519	Dic₇₉	direct (N × H)
171. C₄×Q₈	C₂×Dic₇₉	semidirect
190. C₂×Q₈	C₄×Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099041	2
C₂²×Q₈×Dic₇₉	2
#23765765	2
#23765756	2
C₄×Q₈×Dic₇₉	2
#117×Dic₇₉	2
C₂×#519×C₇₉	2
C₄×#519	79

Subgroups — #519×Dic79 #34446521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #519×Dic₇₉ #34446521