Subgroups — #1707×D₇₉ #34275914

Order 20224.

814 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×D₇₉	C₂	1	1	yes	maximal, characteristic
2	#498×D₇₉	C₂	1	2	yes	maximal
3	#494×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#487×D₇₉	C₂	1	2	yes	maximal
5	#23762998	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762997	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762996	C₂	1	2	yes	maximal, complemented
8	#23762995	C₂	1	2	yes	maximal, complemented
9	#23762994	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₇₉	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	—	2	2	no	—
13	C₄⋊Q₈×D₇₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₇₉	C₂²	1	2	yes	—
15	#19098968	C₂²	1	1	yes	characteristic
16	#19098967	—	2	2	no	—
17	#19098966	C₂²	1	2	yes	—
18	#19098961	C₂²	1	2	yes	—
19	#19098960	—	2	4	no	—
20	#19098959	C₂²	1	2	yes	—
21	#19098959	C₂²	1	2	yes	—
22	#19098943	C₂²	1	1	yes	characteristic
23	#19098942	C₂²	1	2	yes	—
24	#19098941	—	2	2	no	—
25	#19098915	C₂²	1	2	yes	—
26	#19098914	C₂²	1	2	yes	—
27	#19098913	C₂²	1	2	yes	—
28	#19098912	—	2	4	no	—
29	#107×D₇₉	—	2	2	no	—
30	#106×D₇₉	—	2	4	no	—
31	#103×D₇₉	—	2	4	no	—
32	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
35	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
36	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
38	#500×C₇₉	C₂²	1	1	yes	characteristic
39	#498×C₇₉	C₂²	1	2	yes	—
40	#494×C₇₉	C₂²	1	1	yes	characteristic
41	#487×C₇₉	C₂²	1	2	yes	—
42	C₈²×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₇₉	—	2	2	no	—
44	C₂×Q₈×D₇₉	—	2	2	no	—
45	C₂×Dic₃₁₆	—	2	2	no	—
46	C₂×#11929614	—	2	4	no	—
47	Q₁₆×D₇₉	—	4	8	no	—
48	C₂×C₈×D₇₉	D₄	1	2	yes	—
49	C₂×C₈×D₇₉	D₄	1	2	yes	—
50	C₂×C₈×D₇₉	D₄	1	2	yes	—
51	#14579161	C₂³	1	2	yes	—
52	#14579161	C₂³	1	2	yes	—
53	#14579160	—	2	2	no	—
54	#14579160	—	2	2	no	—
55	#14579160	—	2	2	no	—
56	#14579160	—	2	2	no	—
57	#14579159	—	2	2	no	—
58	#14579159	—	2	2	no	—
59	C₄⋊C₄×D₇₉	—	2	2	no	—
60	C₄⋊C₄×D₇₉	—	2	2	no	—
61	C₄⋊C₄×D₇₉	—	2	2	no	—
62	C₄⋊C₄×D₇₉	—	2	2	no	—
63	C₄²×D₇₉	C₂³	1	1	yes	characteristic
64	#14579095	—	2	2	no	—
65	#14579094	—	2	4	no	—
66	#14579093	—	2	4	no	—
67	#14579092	—	2	4	no	—
68	#14579092	—	2	4	no	—
69	#14579086	—	2	4	no	—
70	#14579085	—	2	4	no	—
71	#14579084	—	2	4	no	—
72	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×Dic₇₉	D₄	1	2	yes	—
76	C₈×Dic₇₉	D₄	1	2	yes	—
77	C₈×Dic₇₉	D₄	1	2	yes	—
78	C₂×Q₁₆×C₇₉	—	2	2	no	—
79	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
81	#107×C₇₉	—	2	2	no	—
82	#106×C₇₉	—	2	4	no	—
83	#103×C₇₉	—	2	4	no	—
84	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
87	C₂×Dic₁₅₈	—	2	2	no	—
88	C₂×Dic₁₅₈	—	2	2	no	—
89	C₂×Dic₁₅₈	—	4	2	no	—
90	C₂×Dic₁₅₈	—	4	2	no	—
91	Q₈×D₇₉	—	4	8	no	—
92	Q₈×D₇₉	—	4	8	no	—
93	C₂×C₄×D₇₉	—	4	2	no	—
94	C₂×C₄×D₇₉	—	4	2	no	—
95	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
98	Dic₃₁₆	—	4	4	no	—
99	#11929614	—	4	8	no	—
100	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
101	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
103	C₈×D₇₉	—	2	4	no	—
104	C₈×D₇₉	SD₁₆	1	8	yes	—
105	C₈×D₇₉	Q₁₆	1	8	yes	complemented
106	C₄⋊Dic₇₉	—	2	2	no	—
107	C₄⋊Dic₇₉	—	2	2	no	—
108	C₄⋊Dic₇₉	—	2	2	no	—
109	C₄⋊Dic₇₉	—	2	2	no	—
110	Dic₇₉⋊C₄	—	4	2	no	—
111	Dic₇₉⋊C₄	—	4	2	no	—
112	Dic₇₉⋊C₄	—	4	2	no	—
113	Dic₇₉⋊C₄	—	4	2	no	—
114	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Q₈×C₇₉	—	2	2	no	—
118	C₂×Q₈×C₇₉	—	2	2	no	—
119	Q₁₆×C₇₉	—	4	4	no	—
120	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₇₉	—	2	2	no	—
124	C₄⋊C₄×C₇₉	—	2	2	no	—
125	C₄⋊C₄×C₇₉	—	2	2	no	—
126	C₄⋊C₄×C₇₉	—	2	2	no	—
127	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
128	C₂²×D₇₉	#127	1	1	yes	characteristic
129	Q₈×C₇₉	—	4	4	no	—
130	Q₈×C₇₉	—	4	4	no	—
131	C₂×C₄×C₇₉	—	4	2	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	C₂×Dic₇₉	#127	1	1	yes	characteristic
139	C₂×Dic₇₉	#127	1	1	yes	characteristic
140	C₂×Dic₇₉	#127	1	1	yes	characteristic
141	C₄×D₇₉	—	8	4	no	—
142	C₄×D₇₉	—	8	4	no	—
143	C₄×D₇₉	C₂×SD₁₆	1	4	yes	—
144	C₄×D₇₉	C₂×Q₁₆	1	4	yes	—
145	C₄×D₇₉	#135	1	4	yes	—
146	Dic₁₅₈	—	4	4	no	—
147	Dic₁₅₈	—	4	4	no	—
148	Dic₁₅₈	—	8	4	no	—
149	Dic₁₅₈	—	8	4	no	—
150	C₈×C₇₉	—	2	2	no	—
151	C₈×C₇₉	C₂×SD₁₆	1	4	yes	—
152	C₈×C₇₉	C₂×Q₁₆	1	4	yes	complemented
153	C₇₉⋊C₈	—	2	2	no	—
154	C₇₉⋊C₈	C₂×SD₁₆	1	4	yes	—
155	C₇₉⋊C₈	C₂×Q₁₆	1	4	yes	complemented
156	C₂²×C₇₉	C₂×#127	1	1	yes	characteristic
157	C₂×D₇₉	#492	1	2	yes	—
158	C₂×D₇₉	#494	1	2	yes	—
159	C₂×D₇₉	#495	1	2	yes	—
160	C₄×C₇₉	—	8	2	no	—
161	C₄×C₇₉	—	8	2	no	—
162	C₄×C₇₉	C₂²×SD₁₆	1	2	yes	—
163	C₄×C₇₉	C₂²×Q₁₆	1	2	yes	—
164	C₄×C₇₉	C₂×#135	1	2	yes	—
165	Dic₇₉	—	8	2	no	—
166	Dic₇₉	—	8	2	no	—
167	Dic₇₉	#492	1	2	yes	—
168	Dic₇₉	#494	1	2	yes	—
169	Dic₇₉	#495	1	2	yes	—
170	C₂×#1707	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	C₂×#492	1	1	yes	characteristic
172	C₂×C₇₉	C₂×#494	1	1	yes	characteristic
173	C₂×C₇₉	C₂×#495	1	1	yes	characteristic
174	D₇₉	#1707	1	4	yes	direct product factor
175	C₂×#500	—	79	1	no	—
176	C₂×#498	—	79	2	no	—
177	C₂×#494	—	79	1	no	—
178	C₂×#487	—	79	2	no	—
179	#1707	—	79	1	no	—
180	#1707	—	79	2	no	—
181	#1707	—	79	1	no	—
182	#1707	—	79	2	no	—
183	#1707	—	79	1	no	—
184	#1707	D₇₉	1	1	yes	characteristic, direct product factor
185	C₂×C₈²	—	79	1	no	—
186	C₇₉	C₂×#1707	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₁₆	—	158	2	no	—
188	C₂×C₄⋊Q₈	—	79	2	no	—
189	C₂×C₄⋊Q₈	—	79	2	no	—
190	#500	—	79	2	no	—
191	#500	—	79	1	no	—
192	#500	—	158	2	no	—
193	#500	C₂×D₇₉	1	1	yes	characteristic
194	#498	—	79	2	no	—
195	#498	—	79	2	no	—
196	#498	—	79	2	no	—
197	#498	—	158	4	no	—
198	#498	C₂×D₇₉	1	2	yes	—
199	#494	—	79	2	no	—
200	#494	—	79	1	no	—
201	#494	—	158	2	no	—
202	#494	C₂×D₇₉	1	1	yes	characteristic
203	#487	—	79	2	no	—
204	#487	—	79	2	no	—
205	#487	—	79	2	no	—
206	#487	—	158	4	no	—
207	#487	C₂×D₇₉	1	2	yes	—
208	C₂×#107	—	158	2	no	—
209	C₂×#106	—	158	4	no	—
210	C₂×#103	—	158	4	no	—
211	C₂×C₄×C₈	—	79	1	no	—
212	C₂×C₄×C₈	—	79	1	no	—
213	C₂×C₄×C₈	—	79	1	no	—
214	C₈²	—	79	1	no	—
215	C₈²	—	79	1	no	—
216	C₈²	—	79	1	no	—
217	C₈²	C₂×D₇₉	1	1	yes	characteristic
218	C₂²×Q₈	—	158	2	no	—
219	C₂²×Q₈	—	158	2	no	—
220	C₂×Q₁₆	—	2	2	no	—
221	C₂×Q₁₆	—	316	8	no	—
222	C₂×Q₁₆	—	158	4	no	—
223	C₂×Q₁₆	—	158	2	no	—
224	C₂²×C₈	—	79	2	no	—
225	C₂²×C₈	—	79	2	no	—
226	C₂²×C₈	—	79	2	no	—
227	C₄⋊Q₈	—	79	2	no	—
228	C₄⋊Q₈	—	79	2	no	—
229	C₄⋊Q₈	—	158	2	no	—
230	C₄⋊Q₈	—	158	2	no	—
231	C₄⋊Q₈	—	158	2	no	—
232	C₄⋊Q₈	—	158	2	no	—
233	C₄⋊Q₈	—	158	2	no	—
234	C₄⋊Q₈	—	158	2	no	—
235	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
236	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
237	C₂×C₄⋊C₄	—	158	2	no	—
238	C₂×C₄⋊C₄	—	158	2	no	—
239	C₂×C₄⋊C₄	—	158	2	no	—
240	C₂×C₄⋊C₄	—	158	2	no	—
241	C₂×C₄²	—	79	1	no	—
242	#107	—	2	2	no	—
243	#107	—	158	4	no	—
244	#107	—	158	2	no	—
245	#106	—	2	4	no	—
246	#106	—	158	4	no	—
247	#106	—	158	4	no	—
248	#106	—	158	4	no	—
249	#103	—	2	4	no	—
250	#103	—	158	4	no	—
251	#103	—	158	4	no	—
252	#103	—	158	4	no	—
253	C₄×C₈	—	79	2	no	—
254	C₄×C₈	—	79	2	no	—
255	C₄×C₈	—	79	2	no	—
256	C₄×C₈	—	79	1	no	—
257	C₄×C₈	—	79	1	no	—
258	C₄×C₈	—	79	1	no	—
259	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	316	2	no	—
265	C₂×Q₈	—	316	2	no	—
266	C₂×Q₈	—	316	8	no	—
267	C₂×Q₈	—	316	8	no	—
268	C₂×Q₈	—	158	2	no	—
269	C₂×Q₈	—	158	2	no	—
270	C₂²×C₄	—	316	2	no	—
271	C₂²×C₄	—	316	2	no	—
272	C₂²×C₄	—	79	1	no	—
273	C₂²×C₄	—	79	1	no	—
274	C₂²×C₄	—	79	1	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	—	316	8	no	—
277	Q₁₆	—	316	4	no	—
278	C₂×C₈	—	79	2	no	—
279	C₂×C₈	—	79	2	no	—
280	C₂×C₈	—	79	8	no	—
281	C₂×C₈	—	79	8	no	—
282	C₂×C₈	—	79	2	no	—
283	C₂×C₈	—	158	4	no	—
284	C₂×C₈	D₄×D₇₉	1	2	yes	—
285	C₂×C₈	D₄×D₇₉	1	2	yes	—
286	C₂×C₈	D₄×D₇₉	1	2	yes	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	316	2	no	—
292	C₄⋊C₄	—	316	2	no	—
293	C₄⋊C₄	—	316	2	no	—
294	C₄⋊C₄	—	316	2	no	—
295	C₄⋊C₄	—	158	2	no	—
296	C₄⋊C₄	—	158	2	no	—
297	C₄⋊C₄	—	158	2	no	—
298	C₄⋊C₄	—	158	2	no	—
299	C₄²	—	79	1	no	—
300	C₄²	—	79	1	no	—
301	C₄²	—	79	1	no	—
302	C₄²	C₂³×D₇₉	1	1	yes	characteristic
303	C₂³	—	79	1	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	4	no	—
306	Q₈	—	316	4	no	—
307	Q₈	—	316	4	no	—
308	Q₈	—	632	4	no	—
309	Q₈	—	632	4	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	316	2	no	—
313	C₂×C₄	—	316	2	no	—
314	C₂×C₄	—	79	1	no	—
315	C₂×C₄	—	79	1	no	—
316	C₂×C₄	—	79	1	no	—
317	C₂×C₄	—	79	4	no	—
318	C₂×C₄	—	79	4	no	—
319	C₂×C₄	—	79	4	no	—
320	C₂×C₄	—	632	4	no	—
321	C₂×C₄	—	632	4	no	—
322	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
323	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	79	4	no	—
327	C₈	—	79	4	no	—
328	C₈	—	158	2	no	—
329	C₈	SD₁₆×D₇₉	1	4	yes	—
330	C₈	Q₁₆×D₇₉	1	4	yes	complemented
331	C₂²	—	79	2	no	—
332	C₂²	—	79	2	no	—
333	C₂²	—	79	2	no	—
334	C₂²	#127×D₇₉	1	1	yes	characteristic
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	79	2	no	—
338	C₄	—	79	2	no	—
339	C₄	—	79	2	no	—
340	C₄	—	632	2	no	—
341	C₄	—	632	2	no	—
342	C₄	C₂×SD₁₆×D₇₉	1	2	yes	—
343	C₄	C₂×Q₁₆×D₇₉	1	2	yes	—
344	C₄	#135×D₇₉	1	2	yes	—
345	C₂	—	79	4	no	—
346	C₂	#492×D₇₉	1	1	yes	characteristic
347	C₂	#494×D₇₉	1	1	yes	characteristic
348	C₂	#495×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₇₉
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1707	C₂
socle	C₂²×C₇₉	C₂×#127
prime radical	C₂²×D₇₉	#127
hypercenter	#1707	D₇₉
hypocenter	C₇₉	C₂×#1707
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #23762998	C₂	semidirect
6. #23762997	C₂	semidirect
7. #23762996	C₂	semidirect
8. #23762995	C₂	semidirect
9. #23762994	C₂	semidirect
11. C₇₉×#1707	C₂	semidirect
105. C₈×D₇₉	Q₁₆	semidirect
152. C₈×C₇₉	C₂×Q₁₆	semidirect
155. C₇₉⋊C₈	C₂×Q₁₆	semidirect
174. D₇₉	#1707	direct (N × H)
184. #1707	D₇₉	direct (N × H)
186. C₇₉	C₂×#1707	semidirect
330. C₈	Q₁₆×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#500×D₇₉	2
#498×D₇₉	2
#494×D₇₉	2
#487×D₇₉	2
#23762998	2
#23762997	2
#23762996	2
#23762995	2
#23762994	2
C₈²×D₇₉	2
C₇₉×#1707	2
C₂×#1707	79

Subgroups — #1707×D79 #34275914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1707×D₇₉ #34275914