Subgroups — #34258126

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772516	C₂	1	2	yes	maximal, complemented
2	C₄×#14579184	C₂	1	1	yes	maximal, characteristic
3	#23762315	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762313	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762309	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762290	C₂	1	2	yes	maximal, complemented
7	#23762287	C₂	1	2	yes	maximal, complemented
8	#334×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#23761786	C₂	1	2	yes	maximal, complemented
10	#23761685	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579184	C₄	1	1	yes	characteristic
13	C₄×#11929625	C₂²	1	2	yes	—
14	#19098694	C₂²	1	2	yes	—
15	Q₁₆×Dic₇₉	C₄	1	1	yes	characteristic
16	C₄×#11929611	C₂²	1	2	yes	—
17	#19098648	C₄	1	2	yes	—
18	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
19	#19098645	C₄	1	1	yes	characteristic
20	#19098642	C₂²	1	2	yes	—
21	#19098640	C₂²	1	2	yes	—
22	#19098636	—	2	2	no	—
23	#19098633	—	2	2	no	—
24	#19098618	—	2	2	no	—
25	#19098614	C₂²	1	1	yes	characteristic
26	#19098614	C₂²	1	1	yes	characteristic
27	#19098600	C₄	1	1	yes	characteristic
28	#19098585	—	2	2	no	—
29	C₄⋊C₈×D₇₉	C₂²	1	2	yes	—
30	#19098558	C₄	1	2	yes	—
31	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
32	#19098314	C₂²	1	1	yes	characteristic
33	#19098313	C₂²	1	2	yes	—
34	Q₈⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
35	#19098292	C₂²	1	2	yes	—
36	#19098290	C₂²	1	2	yes	—
37	#19098284	C₂²	1	1	yes	characteristic
38	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₇₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	#334×C₇₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₉	C₂²	1	2	yes	—
44	#322×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×#11929625	C₂×C₄	1	2	yes	—
46	#14579184	—	2	4	no	—
47	C₂×#11929611	C₂×C₄	1	2	yes	—
48	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
49	C₂×#11929607	—	2	2	no	—
50	C₂×#11929607	—	2	2	no	—
51	C₂×C₈×D₇₉	—	2	2	no	—
52	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₇₉	C₂×C₄	1	2	yes	—
55	C₄×D₃₁₆	—	2	2	no	—
56	C₄×D₃₁₆	C₂³	1	2	yes	—
57	#14579123	—	2	2	no	—
58	#14579123	C₂×C₄	1	2	yes	—
59	#14579119	—	2	2	no	—
60	#14579119	C₂×C₄	1	2	yes	—
61	C₄²×D₇₉	—	2	2	no	—
62	C₄²×D₇₉	C₂³	1	1	yes	characteristic
63	#14579095	C₂×C₄	1	1	yes	characteristic
64	#14579092	C₂×C₄	1	2	yes	—
65	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
66	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
67	Dic₇₉⋊C₈	—	2	2	no	—
68	Dic₇₉⋊C₈	—	2	1	no	—
69	Dic₇₉⋊C₈	D₄	1	2	yes	—
70	#14579085	C₂×C₄	1	2	yes	—
71	#14579083	C₂×C₄	1	2	yes	—
72	#14579081	C₂×C₄	1	2	yes	—
73	(C₂×D₇₉)⋊C₈	—	2	2	no	—
74	(C₂×D₇₉)⋊C₈	—	2	1	no	—
75	(C₂×D₇₉)⋊C₈	—	2	2	no	—
76	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
77	#14579072	—	2	2	no	—
78	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
83	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
84	C₄×Q₈×C₇₉	C₂³	1	2	yes	—
85	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
86	C₄⋊C₈×C₇₉	—	2	2	no	—
87	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
88	#103×C₇₉	C₂×C₄	1	2	yes	—
89	#97×C₇₉	—	2	2	no	—
90	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
93	#11929625	—	2	4	no	—
94	#11929625	—	2	4	no	—
95	C₂×D₃₁₆	—	2	2	no	—
96	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₇₉	—	2	2	no	—
98	C₂×C₄×D₇₉	—	2	2	no	—
99	C₂×C₄×D₇₉	—	2	2	no	—
100	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
103	#11929611	—	2	4	no	—
104	D₆₃₂	—	2	2	no	—
105	#11929607	—	8	2	no	—
106	#11929607	—	8	2	no	—
107	C₂×C₇₉⋊C₈	—	2	2	no	—
108	C₂×C₇₉⋊C₈	—	2	1	no	—
109	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
112	C₈×D₇₉	—	2	2	no	—
113	C₈×D₇₉	—	8	2	no	—
114	C₈×D₇₉	M₁₆	1	4	yes	complemented
115	C₄⋊Dic₇₉	—	2	2	no	—
116	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
117	(C₂×D₇₉)⋊C₄	—	2	2	no	—
118	(C₂×D₇₉)⋊C₄	—	2	2	no	—
119	(C₂×D₇₉)⋊C₄	—	2	2	no	—
120	C₄×Dic₇₉	—	2	2	no	—
121	C₄×Dic₇₉	—	2	2	no	—
122	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₇₉	#41	1	1	yes	characteristic
125	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
126	Q₁₆×C₇₉	—	2	2	no	—
127	C₂×C₈×C₇₉	—	2	2	no	—
128	C₂×C₈×C₇₉	—	2	1	no	—
129	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
132	C₄⋊C₄×C₇₉	—	2	2	no	—
133	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
134	C₄²×C₇₉	—	2	2	no	—
135	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
136	C₂²×D₇₉	—	2	2	no	—
137	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₇₉	—	2	2	no	—
139	Q₈×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
145	C₂×Dic₇₉	—	2	2	no	—
146	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
147	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
149	D₃₁₆	—	2	2	no	—
150	D₃₁₆	—	2	2	no	—
151	D₃₁₆	—	4	4	no	—
152	C₄×D₇₉	—	2	2	no	—
153	C₄×D₇₉	—	4	1	no	—
154	C₄×D₇₉	—	4	2	no	—
155	C₄×D₇₉	—	4	2	no	—
156	C₄×D₇₉	—	4	4	no	—
157	C₄×D₇₉	—	8	2	no	—
158	C₄×D₇₉	C₂×M₁₆	1	2	yes	—
159	C₄×D₇₉	#131	1	2	yes	—
160	C₈×C₇₉	—	2	1	no	—
161	C₈×C₇₉	—	2	2	no	—
162	C₈×C₇₉	—	4	1	no	—
163	C₈×C₇₉	—	4	2	no	—
164	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
165	C₇₉⋊C₈	—	2	1	no	—
166	C₇₉⋊C₈	—	2	2	no	—
167	C₇₉⋊C₈	—	4	1	no	—
168	C₇₉⋊C₈	—	4	2	no	—
169	C₇₉⋊C₈	C₂×M₁₆	1	2	yes	—
170	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₇₉	—	2	1	no	—
172	C₂×D₇₉	—	2	1	no	—
173	C₂×D₇₉	—	4	2	no	—
174	C₂×D₇₉	—	4	2	no	—
175	C₂×D₇₉	—	4	2	no	—
176	C₂×D₇₉	#436	1	2	yes	—
177	C₄×C₇₉	—	2	1	no	—
178	C₄×C₇₉	—	4	2	no	—
179	C₄×C₇₉	—	4	2	no	—
180	C₄×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
182	C₄×C₇₉	#577	1	2	yes	—
183	Dic₇₉	—	2	1	no	—
184	Dic₇₉	—	2	1	no	—
185	Dic₇₉	—	4	2	no	—
186	Dic₇₉	#436	1	2	yes	—
187	#17899	—	79	1	no	maximal, Sylow
188	C₂×C₇₉	C₂×#436	1	1	yes	characteristic
189	C₂×C₇₉	#2952	1	1	yes	characteristic
190	C₂×C₇₉	#2957	1	1	yes	characteristic
191	D₇₉	—	2	2	no	—
192	D₇₉	—	8	2	no	—
193	#2989	—	79	2	no	—
194	C₄×#135	—	79	1	no	—
195	#1594	—	79	1	no	—
196	#1594	—	79	1	no	—
197	#1593	—	79	1	no	—
198	#1593	D₇₉	1	1	yes	characteristic, complemented
199	#1590	—	79	2	no	—
200	#1590	—	79	2	no	—
201	C₂×#334	—	79	1	no	—
202	#1477	—	79	2	no	—
203	#1450	—	79	1	no	—
204	C₇₉	#17899	1	1	yes	characteristic, complemented, Sylow
205	C₂×#135	—	79	1	no	—
206	C₄×#41	—	79	2	no	—
207	#446	—	79	2	no	—
208	#446	C₂×D₇₉	1	2	yes	—
209	C₄×Q₁₆	—	79	1	no	—
210	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
211	C₄×SD₁₆	—	79	2	no	—
212	C₄×SD₁₆	—	79	2	no	—
213	C₄×D₈	—	79	1	no	—
214	C₄×D₈	—	79	1	no	—
215	#436	—	79	2	no	—
216	#436	—	79	2	no	—
217	#435	—	158	2	no	—
218	#435	—	158	2	no	—
219	#432	—	158	2	no	—
220	#431	—	79	1	no	—
221	#431	—	79	1	no	—
222	#427	—	79	1	no	—
223	#423	—	158	2	no	—
224	C₂×C₄⋊C₈	—	79	2	no	—
225	#416	—	79	2	no	—
226	C₂×C₄×C₈	—	79	1	no	—
227	#334	—	79	2	no	—
228	#334	—	79	1	no	—
229	#334	C₂×D₇₉	1	1	yes	characteristic
230	Q₈⋊C₈	—	79	2	no	—
231	Q₈⋊C₈	C₂×D₇₉	1	2	yes	—
232	D₄⋊C₈	—	79	2	no	—
233	D₄⋊C₈	—	79	2	no	—
234	#322	—	79	1	no	—
235	#322	C₂×D₇₉	1	1	yes	characteristic
236	C₈²	—	79	1	no	—
237	C₈²	—	79	1	no	—
238	C₂×#41	—	79	2	no	—
239	#135	—	158	4	no	—
240	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
241	C₂×SD₁₆	—	79	2	no	—
242	C₂×D₈	—	79	1	no	—
243	C₂×M₁₆	—	158	2	no	—
244	C₂×M₁₆	—	158	2	no	—
245	C₂²×C₈	—	79	1	no	—
246	C₂²×C₈	—	79	1	no	—
247	C₂²×C₈	—	158	2	no	—
248	C₄×Q₈	—	79	2	no	—
249	C₄×Q₈	C₂²×D₇₉	1	2	yes	—
250	C₄×D₄	—	79	2	no	—
251	C₄×D₄	—	79	2	no	—
252	C₄×D₄	—	158	2	no	—
253	C₄×D₄	—	158	2	no	—
254	#117	—	79	2	no	—
255	#117	—	158	2	no	—
256	C₂×C₄²	—	79	1	no	—
257	C₂×C₄²	—	158	2	no	—
258	#107	—	79	1	no	—
259	#107	C₄×D₇₉	1	1	yes	characteristic
260	#106	—	79	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	79	2	no	—
263	C₄⋊C₈	—	79	2	no	—
264	C₄⋊C₈	—	158	2	no	—
265	C₄⋊C₈	—	158	2	no	—
266	C₄⋊C₈	—	158	1	no	—
267	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
268	#103	—	79	2	no	—
269	#103	C₄×D₇₉	1	2	yes	—
270	#102	—	79	2	no	—
271	#102	—	79	2	no	—
272	C₂²⋊C₈	—	79	2	no	—
273	C₂²⋊C₈	—	158	2	no	—
274	C₂²⋊C₈	—	158	1	no	—
275	C₂²⋊C₈	—	158	2	no	—
276	#97	—	2	2	no	—
277	#97	—	158	2	no	—
278	C₄×C₈	—	79	1	no	—
279	C₄×C₈	—	79	1	no	—
280	C₄×C₈	—	79	1	no	—
281	C₄×C₈	—	79	1	no	—
282	C₄×C₈	—	79	1	no	—
283	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
286	#41	—	158	4	no	—
287	#41	—	158	4	no	—
288	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
289	C₂×D₄	—	79	2	no	—
290	C₂×D₄	—	158	2	no	—
291	C₂²×C₄	—	79	1	no	—
292	C₂²×C₄	—	79	1	no	—
293	C₂²×C₄	—	79	1	no	—
294	C₂²×C₄	—	158	2	no	—
295	C₂²×C₄	—	158	2	no	—
296	C₂²×C₄	—	158	2	no	—
297	Q₁₆	—	2	2	no	—
298	SD₁₆	—	158	4	no	—
299	D₈	—	158	2	no	—
300	M₁₆	—	632	2	no	—
301	M₁₆	—	632	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	79	1	no	—
305	C₂×C₈	—	79	4	no	—
306	C₂×C₈	—	79	1	no	—
307	C₂×C₈	—	79	2	no	—
308	C₂×C₈	—	632	2	no	—
309	C₂×C₈	—	158	2	no	—
310	C₂×C₈	—	158	1	no	—
311	C₂×C₈	—	158	2	no	—
312	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
314	C₂×C₈	D₄×D₇₉	1	2	yes	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	79	2	no	—
317	C₄⋊C₄	—	158	2	no	—
318	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
319	C₂²⋊C₄	—	158	2	no	—
320	C₂²⋊C₄	—	158	2	no	—
321	C₂²⋊C₄	—	158	2	no	—
322	C₄²	—	2	2	no	—
323	C₄²	—	79	1	no	—
324	C₄²	—	79	1	no	—
325	C₄²	—	79	1	no	—
326	C₄²	—	158	2	no	—
327	C₄²	—	158	2	no	—
328	C₄²	C₂³×D₇₉	1	1	yes	characteristic
329	C₂³	—	79	1	no	—
330	C₂³	—	158	2	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	D₄	—	316	4	no	—
334	D₄	—	158	2	no	—
335	D₄	—	158	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	316	1	no	—
339	C₂×C₄	—	316	4	no	—
340	C₂×C₄	—	316	2	no	—
341	C₂×C₄	—	316	2	no	—
342	C₂×C₄	—	79	1	no	—
343	C₂×C₄	—	79	2	no	—
344	C₂×C₄	—	79	2	no	—
345	C₂×C₄	—	79	1	no	—
346	C₂×C₄	—	79	1	no	—
347	C₂×C₄	—	632	2	no	—
348	C₂×C₄	—	158	2	no	—
349	C₂×C₄	—	158	2	no	—
350	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
352	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
353	C₈	—	2	1	no	—
354	C₈	—	2	2	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	316	1	no	—
358	C₈	—	316	2	no	—
359	C₈	—	79	2	no	—
360	C₈	—	158	1	no	—
361	C₈	—	158	2	no	—
362	C₈	M₁₆×D₇₉	1	2	yes	—
363	C₂²	—	316	2	no	—
364	C₂²	—	316	2	no	—
365	C₂²	—	316	2	no	—
366	C₂²	—	79	2	no	—
367	C₂²	—	158	1	no	—
368	C₂²	—	158	1	no	—
369	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	4	2	no	—
372	C₄	—	4	2	no	—
373	C₄	—	316	2	no	—
374	C₄	—	79	2	no	—
375	C₄	—	158	1	no	—
376	C₄	—	158	1	no	—
377	C₄	C₂×M₁₆×D₇₉	1	1	yes	characteristic
378	C₄	#131×D₇₉	1	1	yes	characteristic
379	C₄	#19099372	1	2	yes	—
380	C₂	—	632	2	no	—
381	C₂	—	158	2	no	—
382	C₂	#436×D₇₉	1	1	yes	characteristic
383	C₂	#23772280	1	1	yes	characteristic
384	C₂	#23772314	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1593	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂×#14579184	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1593	D₇₉
hypocenter	C₇₉	#17899
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
1. #23772516	C₂	semidirect
3. #23762315	C₂	semidirect
4. #23762313	C₂	semidirect
5. #23762309	C₂	semidirect
6. #23762290	C₂	semidirect
7. #23762287	C₂	semidirect
8. #334×D₇₉	C₂	semidirect
9. #23761786	C₂	semidirect
10. #23761685	C₂	semidirect
11. C₇₉×#1593	C₂	semidirect
114. C₈×D₇₉	M₁₆	semidirect
198. #1593	D₇₉	semidirect
204. C₇₉	#17899	semidirect

Maximal subgroups

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

subgroup	index
#23772516	2
C₄×#14579184	2
#23762315	2
#23762313	2
#23762309	2
#23762290	2
#23762287	2
#334×D₇₉	2
#23761786	2
#23761685	2
C₇₉×#1593	2
#17899	79