Subgroups — C₇₉×#13009 #34124698

Order 20224.

496 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#470×C₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#430×C₇₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#362×C₇₉	C₂	1	2	yes	maximal, complemented
4	C₂×#349×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₇₉×#1383	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₇₉	—	2	1	no	—
8	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊Q₈×C₇₉	—	2	1	no	—
10	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
11	#470×C₇₉	—	2	2	no	—
12	#470×C₇₉	C₄	1	4	yes	—
13	#430×C₇₉	—	2	4	no	—
14	C₂×#108×C₇₉	—	2	1	no	—
15	C₂×#108×C₇₉	C₂²	1	2	yes	—
16	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
17	C₂×#103×C₇₉	—	2	1	no	—
18	#362×C₇₉	—	2	4	no	—
19	#349×C₇₉	—	2	2	no	—
20	C₂²×Q₈×C₇₉	—	2	1	no	—
21	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₇₉	—	2	2	no	—
23	C₂×Q₁₆×C₇₉	—	2	1	no	—
24	C₂×Q₁₆×C₇₉	—	2	2	no	—
25	C₂×Q₁₆×C₇₉	C₂×C₄	1	2	yes	—
26	C₂×M₁₆×C₇₉	—	2	2	no	—
27	C₂×M₁₆×C₇₉	—	2	2	no	—
28	C₂×M₁₆×C₇₉	—	2	4	no	—
29	C₂×M₁₆×C₇₉	—	2	2	no	—
30	C₂×M₁₆×C₇₉	—	4	1	no	—
31	C₂×M₁₆×C₇₉	—	4	2	no	—
32	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
33	C₂×M₁₆×C₇₉	D₄	1	2	yes	—
34	C₂²×C₈×C₇₉	—	4	1	no	—
35	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
37	C₂²⋊Q₈×C₇₉	—	2	4	no	—
38	C₂²⋊Q₈×C₇₉	—	4	2	no	—
39	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
41	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
42	M₃₂×C₇₉	—	2	2	no	—
43	M₃₂×C₇₉	—	2	2	no	—
44	C₂×C₁₆×C₇₉	—	2	2	no	—
45	#108×C₇₉	—	2	4	no	—
46	#108×C₇₉	—	2	2	no	—
47	#108×C₇₉	D₄	1	4	yes	—
48	#106×C₇₉	—	2	1	no	—
49	#106×C₇₉	C₂×C₄	1	2	yes	—
50	#103×C₇₉	—	2	4	no	—
51	C₂⁴×C₇₉	SD₁₆	1	1	yes	characteristic
52	C₂×Q₈×C₇₉	—	2	2	no	—
53	C₂×Q₈×C₇₉	—	4	1	no	—
54	C₂×Q₈×C₇₉	—	4	1	no	—
55	C₂×Q₈×C₇₉	—	4	1	no	—
56	C₂×Q₈×C₇₉	—	4	1	no	—
57	C₂×Q₈×C₇₉	—	4	1	no	—
58	C₂²×C₄×C₇₉	—	2	1	no	—
59	C₂²×C₄×C₇₉	—	2	1	no	—
60	C₂²×C₄×C₇₉	—	2	1	no	—
61	C₂²×C₄×C₇₉	—	2	1	no	—
62	C₂²×C₄×C₇₉	—	4	1	no	—
63	C₂²×C₄×C₇₉	—	4	1	no	—
64	C₂²×C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₇₉	D₈	1	2	yes	—
66	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
68	Q₁₆×C₇₉	—	2	4	no	—
69	Q₁₆×C₇₉	—	4	2	no	—
70	M₁₆×C₇₉	—	2	4	no	—
71	M₁₆×C₇₉	—	2	4	no	—
72	M₁₆×C₇₉	—	2	4	no	—
73	M₁₆×C₇₉	—	4	4	no	—
74	M₁₆×C₇₉	—	4	2	no	—
75	C₂×C₈×C₇₉	—	2	1	no	—
76	C₂×C₈×C₇₉	—	2	1	no	—
77	C₂×C₈×C₇₉	—	4	4	no	—
78	C₂×C₈×C₇₉	—	4	2	no	—
79	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₇₉	#41	1	2	yes	—
84	C₄⋊C₄×C₇₉	—	2	2	no	—
85	C₄⋊C₄×C₇₉	—	4	4	no	—
86	C₄⋊C₄×C₇₉	—	4	1	no	—
87	C₂²⋊C₄×C₇₉	—	4	4	no	—
88	C₁₆×C₇₉	—	4	2	no	—
89	C₂³×C₇₉	—	2	1	no	—
90	C₂³×C₇₉	—	2	1	no	—
91	C₂³×C₇₉	—	4	2	no	—
92	C₂³×C₇₉	#102	1	2	yes	—
93	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
94	Q₈×C₇₉	—	4	2	no	—
95	Q₈×C₇₉	—	4	2	no	—
96	Q₈×C₇₉	—	8	1	no	—
97	Q₈×C₇₉	—	8	1	no	—
98	C₂×C₄×C₇₉	—	2	1	no	—
99	C₂×C₄×C₇₉	—	2	1	no	—
100	C₂×C₄×C₇₉	—	2	2	no	—
101	C₂×C₄×C₇₉	—	2	2	no	—
102	C₂×C₄×C₇₉	—	2	2	no	—
103	C₂×C₄×C₇₉	—	2	2	no	—
104	C₂×C₄×C₇₉	—	4	2	no	—
105	C₂×C₄×C₇₉	—	4	2	no	—
106	C₂×C₄×C₇₉	—	8	2	no	—
107	C₂×C₄×C₇₉	—	8	1	no	—
108	C₂×C₄×C₇₉	—	8	1	no	—
109	C₂×C₄×C₇₉	#102	1	2	yes	—
110	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
112	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
116	C₈×C₇₉	—	2	2	no	—
117	C₈×C₇₉	—	4	4	no	—
118	C₈×C₇₉	C₄×D₄	1	2	yes	—
119	C₈×C₇₉	#121	1	2	yes	—
120	C₂²×C₇₉	—	2	2	no	—
121	C₂²×C₇₉	—	2	2	no	—
122	C₂²×C₇₉	—	4	2	no	—
123	C₂²×C₇₉	—	4	2	no	—
124	C₂²×C₇₉	—	8	1	no	—
125	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
126	C₂²×C₇₉	#448	1	1	yes	characteristic
127	C₂²×C₇₉	#481	1	1	yes	characteristic
128	C₄×C₇₉	—	2	2	no	—
129	C₄×C₇₉	—	8	2	no	—
130	C₄×C₇₉	—	8	2	no	—
131	C₄×C₇₉	#386	1	1	yes	characteristic
132	C₄×C₇₉	#417	1	1	yes	characteristic
133	C₄×C₇₉	#447	1	1	yes	characteristic
134	C₄×C₇₉	#480	1	1	yes	characteristic
135	#13009	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₇₉	—	2	1	no	—
137	C₂×C₇₉	—	2	1	no	—
138	C₂×C₇₉	—	4	2	no	—
139	C₂×C₇₉	#1874	1	1	yes	characteristic
140	C₂×C₇₉	#2239	1	2	yes	—
141	C₂×#470	C₂×C₇₉	1	1	yes	characteristic
142	C₂×#430	C₂×C₇₉	1	1	yes	characteristic
143	C₂×#362	C₂×C₇₉	1	2	yes	complemented
144	C₂×#349	C₂×C₇₉	1	1	yes	characteristic
145	#1383	C₂×C₇₉	1	2	yes	complemented
146	C₇₉	#13009	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×Q₁₆	C₂²×C₇₉	1	1	yes	characteristic
148	C₂²×M₁₆	—	2	1	no	—
149	C₂²×M₁₆	C₂²×C₇₉	1	1	yes	characteristic
150	C₂×C₂²⋊Q₈	—	2	1	no	—
151	C₂×M₃₂	C₂²×C₇₉	1	2	yes	—
152	#470	—	2	2	no	—
153	#470	C₄×C₇₉	1	4	yes	—
154	#430	—	2	4	no	—
155	C₂×#108	—	2	1	no	—
156	C₂×#108	C₂²×C₇₉	1	2	yes	—
157	C₂×#106	C₂²×C₇₉	1	1	yes	characteristic
158	C₂×#103	—	2	1	no	—
159	#362	—	2	4	no	—
160	#349	—	2	2	no	—
161	C₂²×Q₈	—	2	1	no	—
162	C₂³×C₄	D₄×C₇₉	1	1	yes	characteristic
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	2	1	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	C₂×C₄×C₇₉	1	2	yes	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	4	1	no	—
172	C₂×M₁₆	—	4	2	no	—
173	C₂×M₁₆	C₂×C₄×C₇₉	1	2	yes	—
174	C₂×M₁₆	D₄×C₇₉	1	2	yes	—
175	C₂²×C₈	—	4	1	no	—
176	C₂²×C₈	D₄×C₇₉	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₇₉	1	1	yes	characteristic
178	C₂²⋊Q₈	—	2	4	no	—
179	C₂²⋊Q₈	—	4	2	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	4	1	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	#108	—	2	4	no	—
187	#108	—	2	2	no	—
188	#108	D₄×C₇₉	1	4	yes	—
189	#106	—	2	1	no	—
190	#106	C₂×C₄×C₇₉	1	2	yes	—
191	#103	—	2	4	no	—
192	C₂⁴	SD₁₆×C₇₉	1	1	yes	characteristic
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×Q₈	—	4	1	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	4	1	no	—
205	C₂²×C₄	C₂²⋊C₄×C₇₉	1	2	yes	—
206	C₂²×C₄	D₈×C₇₉	1	2	yes	—
207	C₂²×C₄	SD₁₆×C₇₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₇₉	1	1	yes	characteristic
209	Q₁₆	—	2	4	no	—
210	Q₁₆	—	4	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	4	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	4	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	C₂²⋊C₄×C₇₉	1	2	yes	—
221	C₂×C₈	C₂²×C₄×C₇₉	1	1	yes	characteristic
222	C₂×C₈	C₂×D₄×C₇₉	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₇₉	1	1	yes	characteristic
224	C₂×C₈	#41×C₇₉	1	2	yes	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	4	4	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₂²⋊C₄	—	4	4	no	—
229	C₁₆	—	4	2	no	—
230	C₂³	—	2	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	—	4	2	no	—
233	C₂³	#102×C₇₉	1	2	yes	—
234	C₂³	C₂×SD₁₆×C₇₉	1	1	yes	characteristic
235	Q₈	—	4	2	no	—
236	Q₈	—	4	2	no	—
237	Q₈	—	8	1	no	—
238	Q₈	—	8	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	8	2	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	#102×C₇₉	1	2	yes	—
251	C₂×C₄	C₂×C₂²⋊C₄×C₇₉	1	1	yes	characteristic
252	C₂×C₄	#120×C₇₉	1	1	yes	characteristic
253	C₂×C₄	#123×C₇₉	1	1	yes	characteristic
254	C₂×C₄	C₂×D₈×C₇₉	1	1	yes	characteristic
255	C₂×C₄	AGL(1,ℤ/8)×C₇₉	1	1	yes	characteristic
256	C₂×C₄	#137×C₇₉	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	4	no	—
259	C₈	C₄×D₄×C₇₉	1	2	yes	—
260	C₈	#121×C₇₉	1	2	yes	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	8	1	no	—
266	C₂²	C₂×#102×C₇₉	1	1	yes	characteristic
267	C₂²	#448×C₇₉	1	1	yes	characteristic
268	C₂²	#481×C₇₉	1	1	yes	characteristic
269	C₄	—	2	2	no	—
270	C₄	—	8	2	no	—
271	C₄	—	8	2	no	—
272	C₄	#386×C₇₉	1	1	yes	characteristic
273	C₄	#417×C₇₉	1	1	yes	characteristic
274	C₄	#447×C₇₉	1	1	yes	characteristic
275	C₄	#480×C₇₉	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	C₇₉×#1874	1	1	yes	characteristic
280	C₂	C₇₉×#2239	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₉	#481
commutator	C₂×C₈	C₂²×C₄×C₇₉
Frattini, cosocle	C₂²×C₈	C₂³×C₇₉
prime radical	C₂⁴×C₇₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₉ ◁ C₂²×C₄×C₇₉ ◁ C₂²×M₁₆×C₇₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#362×C₇₉	C₂	semidirect
5. C₇₉×#1383	C₂	semidirect
135. #13009	C₇₉	direct (N × H)
143. C₂×#362	C₂×C₇₉	semidirect
145. #1383	C₂×C₇₉	semidirect
146. C₇₉	#13009	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×#470×C₇₉	2
C₂×#430×C₇₉	2
C₂×#362×C₇₉	2
C₂×#349×C₇₉	2
C₇₉×#1383	2
#13009	79

Subgroups — C79×#13009 #34124698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₉×#13009 #34124698