Subgroups — #34178103

Order 20224.

522 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772030	C₂	1	1	yes	maximal, characteristic
2	#23772028	C₂	1	1	yes	maximal, characteristic
3	#432×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#23761263	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761262	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761261	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1287	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
9	#19098620	C₂²	1	1	yes	characteristic
10	#19098620	C₂²	1	1	yes	characteristic
11	#19098619	—	2	1	no	—
12	#19098619	—	2	1	no	—
13	#19098618	—	2	1	no	—
14	#19098618	C₄	1	2	yes	—
15	#19098617	—	2	1	no	—
16	#19098617	C₄	1	2	yes	—
17	#19098616	C₂²	1	1	yes	characteristic
18	#19098615	—	2	1	no	—
19	#19098614	C₄	1	2	yes	—
20	#19098613	—	2	1	no	—
21	#19098585	—	2	1	no	—
22	#19098585	—	2	1	no	—
23	C₄⋊C₈×D₇₉	—	2	1	no	—
24	#19098525	—	2	1	no	—
25	#19098525	—	2	1	no	—
26	C₂²⋊C₈×D₇₉	C₄	1	2	yes	—
27	#19098515	C₄	1	2	yes	—
28	C₄×#11929607	C₄	1	2	yes	—
29	#97×D₇₉	—	2	1	no	—
30	#432×C₇₉	C₂²	1	1	yes	characteristic
31	#432×C₇₉	C₂²	1	1	yes	characteristic
32	#431×C₇₉	C₂²	1	1	yes	characteristic
33	C₂²×C₄×D₇₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₇₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₇₉	Q₈	1	1	yes	characteristic
36	C₂×#11929607	—	2	2	no	—
37	C₂×#11929607	—	2	2	no	—
38	C₂×C₈×D₇₉	—	2	2	no	—
39	C₂×C₄×Dic₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₇₉	Q₈	1	1	yes	characteristic
42	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×D₇₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₇₉	D₄	1	1	yes	characteristic
47	C₄²×D₇₉	D₄	1	1	yes	characteristic
48	C₄²×D₇₉	D₄	1	1	yes	characteristic
49	C₄²×D₇₉	D₄	1	1	yes	characteristic
50	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
51	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
52	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
53	Dic₇₉⋊C₈	—	2	2	no	—
54	Dic₇₉⋊C₈	—	2	2	no	—
55	Dic₇₉⋊C₈	—	2	2	no	—
56	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
57	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
58	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	2	yes	—
59	(C₂×D₇₉)⋊C₈	—	2	2	no	—
60	(C₂×D₇₉)⋊C₈	—	2	2	no	—
61	(C₂×D₇₉)⋊C₈	—	2	2	no	—
62	#14579072	—	2	1	no	—
63	#14579072	C₂×C₄	1	2	yes	—
64	#14579071	—	2	2	no	—
65	#14579071	—	2	2	no	—
66	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
67	C₈×Dic₇₉	—	2	2	no	—
68	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₇₉	—	2	1	no	—
70	C₄⋊C₈×C₇₉	—	2	1	no	—
71	C₄⋊C₈×C₇₉	—	2	1	no	—
72	C₂²⋊C₈×C₇₉	—	2	1	no	—
73	C₂²⋊C₈×C₇₉	—	2	1	no	—
74	C₂²⋊C₈×C₇₉	C₂×C₄	1	2	yes	—
75	#97×C₇₉	—	2	1	no	—
76	#97×C₇₉	C₂×C₄	1	2	yes	—
77	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
78	C₂³×D₇₉	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×D₇₉	—	4	2	no	—
83	C₂×C₄×D₇₉	—	4	2	no	—
84	C₂×C₄×D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	C₄²	1	1	yes	characteristic
86	C₂×C₄×D₇₉	C₄²	1	1	yes	characteristic
87	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×D₇₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×D₇₉	C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×D₇₉	C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×D₇₉	C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×D₇₉	C₄⋊C₄	1	1	yes	characteristic
97	#11929607	—	4	4	no	—
98	#11929607	—	4	4	no	—
99	C₂×C₇₉⋊C₈	—	2	2	no	—
100	C₂×C₇₉⋊C₈	—	2	2	no	—
101	C₂×C₇₉⋊C₈	—	2	2	no	—
102	C₈×D₇₉	—	4	4	no	—
103	C₄×Dic₇₉	C₄²	1	1	yes	characteristic
104	C₄×Dic₇₉	C₄²	1	1	yes	characteristic
105	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
106	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
110	C₄×Dic₇₉	C₂²⋊C₄	1	1	yes	characteristic
111	C₄×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
112	C₄×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
113	C₄×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
114	C₄×Dic₇₉	C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
118	C₂×C₈×C₇₉	—	2	2	no	—
119	C₂×C₈×C₇₉	—	2	2	no	—
120	C₂×C₈×C₇₉	—	2	2	no	—
121	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×D₇₉	—	2	2	no	—
127	C₂²×D₇₉	—	4	1	no	—
128	C₂²×D₇₉	—	4	1	no	—
129	C₂²×D₇₉	#95	1	1	yes	characteristic
130	C₂²×D₇₉	#95	1	1	yes	characteristic
131	C₂×C₄×C₇₉	—	2	2	no	—
132	C₂×C₄×C₇₉	—	4	1	no	—
133	C₂×C₄×C₇₉	—	4	1	no	—
134	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₇₉	—	2	2	no	—
141	C₂×Dic₇₉	—	4	1	no	—
142	C₂×Dic₇₉	—	4	1	no	—
143	C₂×Dic₇₉	#95	1	1	yes	characteristic
144	C₂×Dic₇₉	#95	1	1	yes	characteristic
145	C₂×Dic₇₉	#95	1	1	yes	characteristic
146	C₂×Dic₇₉	#95	1	1	yes	characteristic
147	C₂×Dic₇₉	#95	1	1	yes	characteristic
148	C₂×Dic₇₉	#95	1	1	yes	characteristic
149	C₄×D₇₉	—	2	2	no	—
150	C₄×D₇₉	—	2	2	no	—
151	C₄×D₇₉	—	2	2	no	—
152	C₄×D₇₉	—	2	2	no	—
153	C₄×D₇₉	—	2	2	no	—
154	C₄×D₇₉	—	2	2	no	—
155	C₄×D₇₉	—	4	1	no	—
156	C₄×D₇₉	—	4	1	no	—
157	C₄×D₇₉	—	4	1	no	—
158	C₄×D₇₉	—	4	1	no	—
159	C₄×D₇₉	—	4	1	no	—
160	C₄×D₇₉	—	4	1	no	—
161	C₈×C₇₉	—	4	2	no	—
162	C₈×C₇₉	—	4	2	no	—
163	C₈×C₇₉	—	4	2	no	—
164	C₇₉⋊C₈	—	4	2	no	—
165	C₇₉⋊C₈	—	4	2	no	—
166	C₇₉⋊C₈	—	4	2	no	—
167	C₂²×C₇₉	—	2	1	no	—
168	C₂²×C₇₉	—	2	1	no	—
169	C₂²×C₇₉	—	2	1	no	—
170	C₂²×C₇₉	C₂×#95	1	1	yes	characteristic
171	C₂×D₇₉	—	2	1	no	—
172	C₂×D₇₉	—	2	1	no	—
173	C₂×D₇₉	—	2	1	no	—
174	C₂×D₇₉	—	2	1	no	—
175	C₂×D₇₉	—	2	1	no	—
176	C₂×D₇₉	—	4	4	no	—
177	C₂×D₇₉	H(3,ℤ/4)	1	2	yes	—
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	2	1	no	—
180	C₄×C₇₉	—	2	1	no	—
181	C₄×C₇₉	—	2	1	no	—
182	C₄×C₇₉	—	2	1	no	—
183	C₄×C₇₉	#430	1	2	yes	—
184	Dic₇₉	—	2	1	no	—
185	Dic₇₉	—	2	1	no	—
186	Dic₇₉	—	2	1	no	—
187	Dic₇₉	—	2	1	no	—
188	Dic₇₉	—	2	1	no	—
189	Dic₇₉	#344	1	2	yes	—
190	#7970	—	79	1	no	maximal, Sylow
191	C₂×C₇₉	—	4	1	no	—
192	C₂×C₇₉	C₂×H(3,ℤ/4)	1	1	yes	characteristic
193	C₂×C₇₉	#1738	1	1	yes	characteristic
194	C₂×C₇₉	C₂×#344	1	1	yes	characteristic
195	D₇₉	—	2	2	no	—
196	D₇₉	—	4	1	no	—
197	#2920	—	79	1	no	—
198	#2920	—	79	1	no	—
199	C₂×#432	—	79	1	no	—
200	#1287	—	79	1	no	—
201	#1287	—	79	1	no	—
202	#1287	—	79	1	no	—
203	#1287	D₇₉	1	1	yes	characteristic, complemented
204	C₇₉	#7970	1	1	yes	characteristic, complemented, Sylow
205	C₂²×C₄²	—	79	1	no	—
206	#432	—	79	1	no	—
207	#432	—	79	2	no	—
208	#432	—	79	1	no	—
209	#432	—	79	2	no	—
210	#432	—	158	1	no	—
211	#432	—	158	1	no	—
212	#432	—	158	1	no	—
213	#432	—	158	1	no	—
214	#432	C₂×D₇₉	1	1	yes	characteristic
215	#432	C₂×D₇₉	1	1	yes	characteristic
216	#431	—	79	1	no	—
217	#431	—	79	2	no	—
218	#431	—	158	1	no	—
219	#431	—	158	1	no	—
220	#431	C₂×D₇₉	1	1	yes	characteristic
221	#423	—	158	1	no	—
222	#423	—	158	1	no	—
223	C₂×C₄⋊C₈	—	158	1	no	—
224	#407	—	158	1	no	—
225	#407	—	158	1	no	—
226	C₂×C₂²⋊C₈	—	79	2	no	—
227	C₄×M₁₆	—	79	2	no	—
228	C₄×M₁₆	—	79	2	no	—
229	C₂×#97	—	158	1	no	—
230	C₂³×C₄	—	79	1	no	—
231	C₂³×C₄	—	79	1	no	—
232	C₂³×C₄	—	79	1	no	—
233	C₂×M₁₆	—	158	2	no	—
234	C₂×M₁₆	—	158	2	no	—
235	C₂²×C₈	—	158	2	no	—
236	C₂×C₄²	—	79	1	no	—
237	C₂×C₄²	—	79	1	no	—
238	C₂×C₄²	—	79	1	no	—
239	C₂×C₄²	—	79	1	no	—
240	C₂×C₄²	—	79	1	no	—
241	C₂×C₄²	—	79	1	no	—
242	C₂×C₄²	—	79	1	no	—
243	C₂×C₄²	—	79	1	no	—
244	C₂×C₄²	—	79	1	no	—
245	C₂×C₄²	—	79	1	no	—
246	C₂×C₄²	—	79	1	no	—
247	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	158	2	no	—
252	C₄⋊C₈	—	158	2	no	—
253	C₄⋊C₈	—	158	2	no	—
254	C₄⋊C₈	—	158	1	no	—
255	C₄⋊C₈	—	158	1	no	—
256	C₄⋊C₈	—	158	1	no	—
257	C₂²⋊C₈	—	2	1	no	—
258	C₂²⋊C₈	—	2	1	no	—
259	C₂²⋊C₈	—	79	2	no	—
260	C₂²⋊C₈	—	158	2	no	—
261	C₂²⋊C₈	—	158	2	no	—
262	C₂²⋊C₈	—	158	2	no	—
263	C₂²⋊C₈	—	158	1	no	—
264	C₂²⋊C₈	—	158	1	no	—
265	C₂²⋊C₈	C₄×D₇₉	1	2	yes	—
266	#97	—	2	1	no	—
267	#97	—	79	2	no	—
268	#97	—	158	2	no	—
269	#97	—	158	2	no	—
270	#97	—	158	1	no	—
271	#97	C₄×D₇₉	1	2	yes	—
272	C₄×C₈	—	79	2	no	—
273	C₄×C₈	—	158	2	no	—
274	C₄×C₈	C₄×D₇₉	1	2	yes	—
275	C₂⁴	—	79	1	no	—
276	C₂²×C₄	—	316	2	no	—
277	C₂²×C₄	—	316	2	no	—
278	C₂²×C₄	—	316	2	no	—
279	C₂²×C₄	—	79	1	no	—
280	C₂²×C₄	—	79	1	no	—
281	C₂²×C₄	—	79	1	no	—
282	C₂²×C₄	—	79	1	no	—
283	C₂²×C₄	—	79	1	no	—
284	C₂²×C₄	—	79	1	no	—
285	C₂²×C₄	—	79	1	no	—
286	C₂²×C₄	—	79	1	no	—
287	C₂²×C₄	—	79	1	no	—
288	C₂²×C₄	—	79	1	no	—
289	C₂²×C₄	—	79	1	no	—
290	C₂²×C₄	—	79	1	no	—
291	C₂²×C₄	—	79	1	no	—
292	C₂²×C₄	—	79	1	no	—
293	C₂²×C₄	—	79	1	no	—
294	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
296	C₂²×C₄	Q₈×D₇₉	1	1	yes	characteristic
297	M₁₆	—	316	4	no	—
298	M₁₆	—	316	4	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	316	4	no	—
303	C₂×C₈	—	158	2	no	—
304	C₂×C₈	—	158	2	no	—
305	C₂×C₈	—	158	2	no	—
306	C₄²	—	79	1	no	—
307	C₄²	—	79	1	no	—
308	C₄²	—	79	1	no	—
309	C₄²	—	79	1	no	—
310	C₄²	—	79	1	no	—
311	C₄²	—	79	1	no	—
312	C₄²	—	79	1	no	—
313	C₄²	—	79	1	no	—
314	C₄²	—	79	1	no	—
315	C₄²	—	79	1	no	—
316	C₄²	—	79	1	no	—
317	C₄²	—	79	1	no	—
318	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
319	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
320	C₄²	D₄×D₇₉	1	1	yes	characteristic
321	C₄²	D₄×D₇₉	1	1	yes	characteristic
322	C₂³	—	316	1	no	—
323	C₂³	—	316	1	no	—
324	C₂³	—	79	1	no	—
325	C₂³	—	79	1	no	—
326	C₂³	—	158	2	no	—
327	C₂³	C₄⋊C₄×D₇₉	1	1	yes	characteristic
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	316	1	no	—
332	C₂×C₄	—	316	1	no	—
333	C₂×C₄	—	316	1	no	—
334	C₂×C₄	—	316	1	no	—
335	C₂×C₄	—	316	1	no	—
336	C₂×C₄	—	316	1	no	—
337	C₂×C₄	—	316	1	no	—
338	C₂×C₄	—	316	1	no	—
339	C₂×C₄	—	79	1	no	—
340	C₂×C₄	—	79	1	no	—
341	C₂×C₄	—	79	1	no	—
342	C₂×C₄	—	79	1	no	—
343	C₂×C₄	—	79	1	no	—
344	C₂×C₄	—	79	1	no	—
345	C₂×C₄	—	158	2	no	—
346	C₂×C₄	—	158	2	no	—
347	C₂×C₄	—	158	2	no	—
348	C₂×C₄	—	158	2	no	—
349	C₂×C₄	—	158	2	no	—
350	C₂×C₄	—	158	2	no	—
351	C₂×C₄	—	158	2	no	—
352	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
353	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
354	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
355	C₂×C₄	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
356	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
357	C₂×C₄	C₄⋊C₄×D₇₉	1	1	yes	characteristic
358	C₈	—	4	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	4	2	no	—
361	C₈	—	316	2	no	—
362	C₈	—	316	2	no	—
363	C₈	—	316	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	316	4	no	—
368	C₂²	—	79	2	no	—
369	C₂²	—	158	1	no	—
370	C₂²	—	158	1	no	—
371	C₂²	—	158	1	no	—
372	C₂²	—	158	1	no	—
373	C₂²	—	158	1	no	—
374	C₂²	#95×D₇₉	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	79	2	no	—
381	C₄	—	158	1	no	—
382	C₄	—	158	1	no	—
383	C₄	—	158	1	no	—
384	C₄	—	158	1	no	—
385	C₄	—	158	1	no	—
386	C₄	#19098610	1	2	yes	—
387	C₂	—	4	1	no	—
388	C₂	—	316	1	no	—
389	C₂	—	158	2	no	—
390	C₂	H(3,ℤ/4)×D₇₉	1	1	yes	characteristic
391	C₂	#23763125	1	1	yes	characteristic
392	C₂	#344×D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#95×D₇₉
commutator	C₂×C₄×C₇₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#1287	C₂
socle	C₂²×C₇₉	C₂×#95
prime radical	C₂³×D₇₉	C₄⋊C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#1287	D₇₉
hypocenter	C₇₉	#7970
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1287

Complements

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

N	H ≅ G/N	type
4. #23761263	C₂	semidirect
5. #23761262	C₂	semidirect
6. #23761261	C₂	semidirect
7. C₇₉×#1287	C₂	semidirect
203. #1287	D₇₉	semidirect
204. C₇₉	#7970	semidirect

Maximal subgroups

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

subgroup	index
#23772030	2
#23772028	2
#432×D₇₉	2
#23761263	2
#23761262	2
#23761261	2
C₇₉×#1287	2
#7970	79