Subgroups — C₇₉×#13440 #34125129

Order 20224.

398 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×SD₃₂×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₇₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#362×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#361×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₉×#1978	C₂	1	1	yes	maximal, characteristic
7	C₇₉×#1381	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
10	C₂×SD₃₂×C₇₉	—	2	2	no	—
11	C₂×SD₃₂×C₇₉	—	2	2	no	—
12	C₂×SD₃₂×C₇₉	—	2	2	no	—
13	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
15	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
16	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
17	#419×C₇₉	—	2	1	no	—
18	#418×C₇₉	—	2	1	no	—
19	C₂×#101×C₇₉	—	2	1	no	—
20	C₂×#100×C₇₉	—	2	1	no	—
21	#405×C₇₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₁₆×C₇₉	—	2	2	no	—
23	#362×C₇₉	—	2	2	no	—
24	#361×C₇₉	—	2	2	no	—
25	C₂²×Q₈×C₇₉	—	2	1	no	—
26	C₂²×D₄×C₇₉	—	2	1	no	—
27	C₂×Q₁₆×C₇₉	—	2	1	no	—
28	C₂×Q₁₆×C₇₉	—	2	1	no	—
29	C₂×Q₁₆×C₇₉	—	2	1	no	—
30	C₂×Q₁₆×C₇₉	—	2	1	no	—
31	C₂×Q₁₆×C₇₉	—	2	1	no	—
32	C₂×Q₁₆×C₇₉	D₄	1	2	yes	—
33	C₂×D₈×C₇₉	—	2	1	no	—
34	C₂×D₈×C₇₉	—	2	1	no	—
35	C₂×D₈×C₇₉	—	2	1	no	—
36	C₂×D₈×C₇₉	—	2	1	no	—
37	C₂×D₈×C₇₉	—	2	1	no	—
38	C₂×D₈×C₇₉	D₄	1	2	yes	—
39	C₂×M₁₆×C₇₉	—	2	1	no	—
40	C₂×M₁₆×C₇₉	—	2	1	no	—
41	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
43	#117×C₇₉	D₄	1	1	yes	characteristic
44	SD₃₂×C₇₉	—	2	8	no	—
45	M₃₂×C₇₉	—	2	1	no	—
46	M₃₂×C₇₉	—	2	1	no	—
47	C₂×C₁₆×C₇₉	—	2	1	no	—
48	C₂×C₁₆×C₇₉	—	2	1	no	—
49	C₂×C₁₆×C₇₉	—	2	1	no	—
50	C₂×C₁₆×C₇₉	D₄	1	2	yes	—
51	#108×C₇₉	—	2	1	no	—
52	#108×C₇₉	—	2	1	no	—
53	#108×C₇₉	—	2	1	no	—
54	#108×C₇₉	—	2	1	no	—
55	#103×C₇₉	—	4	2	no	—
56	#102×C₇₉	—	4	2	no	—
57	#101×C₇₉	—	4	2	no	—
58	#100×C₇₉	—	4	2	no	—
59	#97×C₇₉	—	2	1	no	—
60	C₄×C₈×C₇₉	—	2	1	no	—
61	C₂⁴×C₇₉	—	4	1	no	—
62	C₂×Q₈×C₇₉	—	2	2	no	—
63	C₂×Q₈×C₇₉	—	4	1	no	—
64	C₂×Q₈×C₇₉	—	4	1	no	—
65	C₂×Q₈×C₇₉	—	4	1	no	—
66	C₂×Q₈×C₇₉	—	4	1	no	—
67	C₂×Q₈×C₇₉	—	4	1	no	—
68	C₂×D₄×C₇₉	—	2	2	no	—
69	C₂×D₄×C₇₉	—	4	1	no	—
70	C₂×D₄×C₇₉	—	4	1	no	—
71	C₂×D₄×C₇₉	—	4	1	no	—
72	C₂×D₄×C₇₉	—	4	1	no	—
73	C₂×D₄×C₇₉	—	4	1	no	—
74	C₂²×C₄×C₇₉	—	4	1	no	—
75	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
76	Q₁₆×C₇₉	—	2	2	no	—
77	Q₁₆×C₇₉	—	2	2	no	—
78	Q₁₆×C₇₉	—	4	1	no	—
79	Q₁₆×C₇₉	—	4	1	no	—
80	D₈×C₇₉	—	2	2	no	—
81	D₈×C₇₉	—	2	2	no	—
82	D₈×C₇₉	—	4	1	no	—
83	D₈×C₇₉	—	4	1	no	—
84	M₁₆×C₇₉	—	2	1	no	—
85	M₁₆×C₇₉	—	2	1	no	—
86	M₁₆×C₇₉	—	4	1	no	—
87	M₁₆×C₇₉	—	4	1	no	—
88	M₁₆×C₇₉	—	4	1	no	—
89	M₁₆×C₇₉	—	4	1	no	—
90	C₂×C₈×C₇₉	—	2	1	no	—
91	C₂×C₈×C₇₉	—	4	1	no	—
92	C₂×C₈×C₇₉	—	4	1	no	—
93	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
97	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
99	C₄⋊C₄×C₇₉	D₈	1	2	yes	—
100	C₂²⋊C₄×C₇₉	—	2	1	no	—
101	C₄²×C₇₉	—	2	1	no	—
102	C₁₆×C₇₉	—	2	2	no	—
103	C₁₆×C₇₉	—	4	1	no	—
104	C₂³×C₇₉	—	4	2	no	—
105	C₂³×C₇₉	—	8	1	no	—
106	C₂³×C₇₉	—	8	1	no	—
107	C₂³×C₇₉	—	16	2	no	—
108	C₂³×C₇₉	#135	1	1	yes	characteristic
109	Q₈×C₇₉	—	4	2	no	—
110	Q₈×C₇₉	—	4	2	no	—
111	Q₈×C₇₉	—	8	1	no	—
112	Q₈×C₇₉	—	8	1	no	—
113	D₄×C₇₉	—	4	2	no	—
114	D₄×C₇₉	—	4	2	no	—
115	D₄×C₇₉	—	8	1	no	—
116	D₄×C₇₉	—	8	1	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	—	8	1	no	—
120	C₂×C₄×C₇₉	—	8	1	no	—
121	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
122	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₈×C₇₉	—	4	1	no	—
128	C₈×C₇₉	—	8	1	no	—
129	C₈×C₇₉	—	8	1	no	—
130	C₈×C₇₉	#120	1	1	yes	characteristic
131	C₈×C₇₉	#123	1	1	yes	characteristic
132	C₈×C₇₉	#123	1	1	yes	characteristic
133	C₈×C₇₉	#124	1	1	yes	characteristic
134	C₂²×C₇₉	—	2	2	no	—
135	C₂²×C₇₉	—	8	2	no	—
136	C₂²×C₇₉	—	16	2	no	—
137	C₂²×C₇₉	—	16	2	no	—
138	C₂²×C₇₉	—	16	2	no	—
139	C₂²×C₇₉	#463	1	1	yes	characteristic
140	C₂²×C₇₉	#463	1	1	yes	characteristic
141	C₂²×C₇₉	#466	1	1	yes	characteristic
142	C₄×C₇₉	—	4	2	no	—
143	C₄×C₇₉	—	8	2	no	—
144	C₄×C₇₉	#394	1	1	yes	characteristic
145	C₄×C₇₉	#459	1	1	yes	characteristic
146	C₄×C₇₉	#459	1	1	yes	characteristic
147	C₄×C₇₉	#469	1	1	yes	characteristic
148	#13440	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₇₉	—	2	1	no	—
150	C₂×C₇₉	—	2	1	no	—
151	C₂×C₇₉	—	16	2	no	—
152	C₂×C₇₉	#2054	1	1	yes	characteristic
153	C₂×C₇₉	#2223	1	2	yes	—
154	C₂²×SD₃₂	C₂×C₇₉	1	1	yes	characteristic
155	#2151	C₂×C₇₉	1	1	yes	characteristic, complemented
156	C₂×#362	C₂×C₇₉	1	1	yes	characteristic, complemented
157	C₂×#361	C₂×C₇₉	1	1	yes	characteristic
158	#1979	C₂×C₇₉	1	1	yes	characteristic, complemented
159	#1978	C₂×C₇₉	1	1	yes	characteristic
160	#1381	C₂×C₇₉	1	1	yes	characteristic, complemented
161	C₇₉	#13440	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic
163	C₂²×D₈	C₂²×C₇₉	1	1	yes	characteristic
164	C₂×SD₃₂	—	2	2	no	—
165	C₂×SD₃₂	—	2	2	no	—
166	C₂×SD₃₂	—	2	2	no	—
167	C₂×M₃₂	C₂²×C₇₉	1	1	yes	characteristic
168	C₂²×C₁₆	C₂²×C₇₉	1	1	yes	characteristic
169	C₂×#108	C₂²×C₇₉	1	1	yes	characteristic
170	C₂×#108	C₂²×C₇₉	1	1	yes	characteristic
171	#419	—	2	1	no	—
172	#418	—	2	1	no	—
173	C₂×#101	—	2	1	no	—
174	C₂×#100	—	2	1	no	—
175	#405	C₂²×C₇₉	1	1	yes	characteristic
176	C₄⋊C₁₆	—	2	2	no	—
177	#362	—	2	2	no	—
178	#361	—	2	2	no	—
179	C₂²×Q₈	—	2	1	no	—
180	C₂²×D₄	—	2	1	no	—
181	C₂×Q₁₆	—	2	1	no	—
182	C₂×Q₁₆	—	2	1	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	—	2	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	D₄×C₇₉	1	2	yes	—
187	C₂×D₈	—	2	1	no	—
188	C₂×D₈	—	2	1	no	—
189	C₂×D₈	—	2	1	no	—
190	C₂×D₈	—	2	1	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	D₄×C₇₉	1	2	yes	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	D₄×C₇₉	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
197	#117	D₄×C₇₉	1	1	yes	characteristic
198	SD₃₂	—	2	8	no	—
199	M₃₂	—	2	1	no	—
200	M₃₂	—	2	1	no	—
201	C₂×C₁₆	—	2	1	no	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	D₄×C₇₉	1	2	yes	—
205	#108	—	2	1	no	—
206	#108	—	2	1	no	—
207	#108	—	2	1	no	—
208	#108	—	2	1	no	—
209	#103	—	4	2	no	—
210	#102	—	4	2	no	—
211	#101	—	4	2	no	—
212	#100	—	4	2	no	—
213	#97	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₂⁴	—	4	1	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	—	4	1	no	—
218	C₂×Q₈	—	4	1	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	—	4	1	no	—
226	C₂×D₄	—	4	1	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂²×C₄	—	4	1	no	—
229	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
230	Q₁₆	—	2	2	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	4	1	no	—
233	Q₁₆	—	4	1	no	—
234	D₈	—	2	2	no	—
235	D₈	—	2	2	no	—
236	D₈	—	4	1	no	—
237	D₈	—	4	1	no	—
238	M₁₆	—	2	1	no	—
239	M₁₆	—	2	1	no	—
240	M₁₆	—	4	1	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
249	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
250	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
251	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
252	C₂×C₈	#41×C₇₉	1	1	yes	characteristic
253	C₄⋊C₄	D₈×C₇₉	1	2	yes	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	—	2	1	no	—
256	C₁₆	—	2	2	no	—
257	C₁₆	—	4	1	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	8	1	no	—
260	C₂³	—	8	1	no	—
261	C₂³	—	16	2	no	—
262	C₂³	#135×C₇₉	1	1	yes	characteristic
263	Q₈	—	4	2	no	—
264	Q₈	—	4	2	no	—
265	Q₈	—	8	1	no	—
266	Q₈	—	8	1	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	2	no	—
269	D₄	—	8	1	no	—
270	D₄	—	8	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	#121×C₇₉	1	1	yes	characteristic
276	C₂×C₄	#121×C₇₉	1	1	yes	characteristic
277	C₂×C₄	#121×C₇₉	1	1	yes	characteristic
278	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
279	C₂×C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
280	C₂×C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
281	C₈	—	4	1	no	—
282	C₈	—	8	1	no	—
283	C₈	—	8	1	no	—
284	C₈	#120×C₇₉	1	1	yes	characteristic
285	C₈	#123×C₇₉	1	1	yes	characteristic
286	C₈	#123×C₇₉	1	1	yes	characteristic
287	C₈	#124×C₇₉	1	1	yes	characteristic
288	C₂²	—	2	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	16	2	no	—
291	C₂²	—	16	2	no	—
292	C₂²	—	16	2	no	—
293	C₂²	#463×C₇₉	1	1	yes	characteristic
294	C₂²	#463×C₇₉	1	1	yes	characteristic
295	C₂²	#466×C₇₉	1	1	yes	characteristic
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	#394×C₇₉	1	1	yes	characteristic
299	C₄	#459×C₇₉	1	1	yes	characteristic
300	C₄	#459×C₇₉	1	1	yes	characteristic
301	C₄	#469×C₇₉	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₇₉×#2054	1	1	yes	characteristic
306	C₂	C₇₉×#2223	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₉	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
prime radical	C₂²×D₈×C₇₉	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ #405×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₉×#2151	C₂	semidirect
3. C₂×#362×C₇₉	C₂	semidirect
5. C₇₉×#1979	C₂	semidirect
7. C₇₉×#1381	C₂	semidirect
148. #13440	C₇₉	direct (N × H)
155. #2151	C₂×C₇₉	semidirect
156. C₂×#362	C₂×C₇₉	semidirect
158. #1979	C₂×C₇₉	semidirect
160. #1381	C₂×C₇₉	semidirect
161. C₇₉	#13440	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×SD₃₂×C₇₉	2
C₇₉×#2151	2
C₂×#362×C₇₉	2
C₂×#361×C₇₉	2
C₇₉×#1979	2
C₇₉×#1978	2
C₇₉×#1381	2
#13440	79

Subgroups — C79×#13440 #34125129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#13440 #34125129