Subgroups — #34292689

Order 20224.

730 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772201	C₂	1	1	yes	maximal, characteristic
2	#23772040	C₂	1	2	yes	maximal, complemented
3	#23764289	C₂	1	1	yes	maximal, characteristic
4	#23764153	C₂	1	1	yes	maximal, characteristic, complemented
5	#23764146	C₂	1	1	yes	maximal, characteristic, complemented
6	#23764114	C₂	1	2	yes	maximal, complemented
7	#23764101	C₂	1	2	yes	maximal, complemented
8	#23763308	C₂	1	1	yes	maximal, characteristic, complemented
9	#23763212	C₂	1	2	yes	maximal, complemented
10	#23763088	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1975	C₂	1	1	yes	maximal, characteristic
12	#41×Dic₇₉	C₂²	1	2	yes	—
13	#19098675	C₄	1	2	yes	complemented
14	#19098670	C₄	1	2	yes	complemented
15	#19098664	C₄	1	4	yes	complemented
16	#19098621	—	2	4	no	—
17	#19098613	—	2	4	no	—
18	C₂×#14579097	C₂²	1	1	yes	characteristic
19	#19098601	C₂²	1	1	yes	characteristic
20	#19098591	C₂²	1	2	yes	—
21	#19098589	C₂²	1	2	yes	—
22	C₂×#14579088	C₂²	1	2	yes	—
23	#19098559	C₂²	1	2	yes	—
24	#19098530	C₂²	1	2	yes	—
25	M₁₆×Dic₇₉	C₂²	1	2	yes	—
26	C₂×#14579071	C₂²	1	1	yes	characteristic
27	C₂×#14579071	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
31	#19098409	C₂²	1	1	yes	characteristic
32	#19098327	C₂²	1	2	yes	—
33	#19098324	C₂²	1	2	yes	—
34	#19098323	C₂²	1	2	yes	—
35	C₂×#135×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×#104×C₇₉	C₂²	1	2	yes	—
38	#408×C₇₉	C₂²	1	2	yes	—
39	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×#11929608	C₂³	1	2	yes	—
41	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
43	Q₈×Dic₇₉	C₂×C₄	1	4	yes	—
44	D₄×Dic₇₉	—	2	4	no	—
45	D₄×Dic₇₉	C₂×C₄	1	4	yes	—
46	#14579120	—	2	2	no	—
47	#14579120	C₂³	1	2	yes	—
48	C₂×C₄×Dic₇₉	—	2	2	no	—
49	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
51	#14579097	—	2	2	no	—
52	#14579095	C₂×C₄	1	2	yes	—
53	#14579093	C₂×C₄	1	2	yes	—
54	Dic₇₉⋊C₈	—	2	4	no	—
55	Dic₇₉⋊C₈	—	2	4	no	—
56	#14579088	—	2	4	no	—
57	#14579085	C₂×C₄	1	4	yes	—
58	#14579082	C₂×C₄	1	4	yes	—
59	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
60	#14579071	—	2	2	no	—
61	#14579071	—	2	2	no	—
62	#14579071	C₂×C₄	1	2	yes	—
63	#14579071	C₂×C₄	1	2	yes	—
64	C₄×C₇₉⋊C₈	—	2	2	no	—
65	C₈×Dic₇₉	—	2	2	no	—
66	C₈×Dic₇₉	—	2	2	no	—
67	C₈×Dic₇₉	D₄	1	4	yes	—
68	C₇₉⋊#95	—	2	2	no	—
69	C₂×#41×C₇₉	C₂³	1	2	yes	—
70	#135×C₇₉	—	2	2	no	—
71	#135×C₇₉	—	2	2	no	—
72	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₇₉	C₂×C₄	1	2	yes	—
74	C₂×D₈×C₇₉	C₂×C₄	1	1	yes	characteristic
75	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
76	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
79	#108×C₇₉	—	2	2	no	—
80	#104×C₇₉	—	2	4	no	—
81	C₂²⋊C₈×C₇₉	—	2	4	no	—
82	#97×C₇₉	—	2	2	no	—
83	C₂²×Dic₇₉	—	2	4	no	—
84	C₂²×Dic₇₉	—	2	1	no	—
85	C₂²×Dic₇₉	#41	1	2	yes	—
86	#11929608	—	2	2	no	—
87	#11929608	—	2	2	no	—
88	C₂×C₇₉⋊C₈	—	2	2	no	—
89	C₂×C₇₉⋊C₈	—	2	1	no	—
90	C₂×C₇₉⋊C₈	—	2	1	no	—
91	C₂×C₇₉⋊C₈	—	2	1	no	—
92	C₂×C₇₉⋊C₈	—	2	1	no	—
93	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
95	C₄⋊Dic₇₉	—	2	4	no	—
96	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
97	C₂²⋊Dic₇₉	—	2	4	no	—
98	C₂²⋊Dic₇₉	—	2	2	no	—
99	C₄×Dic₇₉	—	2	4	no	—
100	C₄×Dic₇₉	—	2	2	no	—
101	C₄×Dic₇₉	—	2	1	no	—
102	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
103	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
104	C₄×Dic₇₉	#41	1	2	yes	—
105	#41×C₇₉	—	2	2	no	—
106	#41×C₇₉	—	2	2	no	—
107	#41×C₇₉	—	2	2	no	—
108	#41×C₇₉	—	2	2	no	—
109	C₂×Q₈×C₇₉	C₂²×C₄	1	2	yes	—
110	C₂×D₄×C₇₉	—	2	2	no	—
111	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₇₉	—	2	1	no	—
113	C₂²×C₄×C₇₉	—	2	2	no	—
114	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
115	Q₁₆×C₇₉	—	2	2	no	—
116	SD₁₆×C₇₉	—	2	4	no	—
117	D₈×C₇₉	—	2	2	no	—
118	M₁₆×C₇₉	—	2	2	no	—
119	M₁₆×C₇₉	—	2	2	no	—
120	C₂×C₈×C₇₉	—	2	2	no	—
121	C₂×C₈×C₇₉	—	2	1	no	—
122	C₂×C₈×C₇₉	—	2	1	no	—
123	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₇₉	#41	1	2	yes	—
127	C₂×C₈×C₇₉	#41	1	2	yes	—
128	C₄²×C₇₉	—	2	1	no	—
129	C₄²×C₇₉	#41	1	2	yes	—
130	C₂³×C₇₉	—	2	2	no	—
131	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
132	Q₈×C₇₉	—	2	2	no	—
133	Q₈×C₇₉	—	2	2	no	—
134	D₄×C₇₉	—	2	2	no	—
135	D₄×C₇₉	—	2	2	no	—
136	D₄×C₇₉	—	4	2	no	—
137	D₄×C₇₉	—	4	2	no	—
138	C₂×C₄×C₇₉	—	2	2	no	—
139	C₂×C₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	—	4	2	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₇₉	—	2	4	no	—
150	C₂×Dic₇₉	—	2	2	no	—
151	C₂×Dic₇₉	—	4	2	no	—
152	C₂×Dic₇₉	—	4	8	no	—
153	C₂×Dic₇₉	—	4	2	no	—
154	C₂×Dic₇₉	C₄×D₄	1	4	yes	—
155	C₈×C₇₉	—	2	2	no	—
156	C₈×C₇₉	—	4	2	no	—
157	C₈×C₇₉	#117	1	2	yes	—
158	C₈×C₇₉	#117	1	2	yes	—
159	C₇₉⋊C₈	—	2	2	no	—
160	C₇₉⋊C₈	—	2	2	no	—
161	C₇₉⋊C₈	—	4	2	no	—
162	C₂²×C₇₉	—	2	2	no	—
163	C₂²×C₇₉	—	4	4	no	—
164	C₂²×C₇₉	—	4	2	no	—
165	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
166	C₂²×C₇₉	#529	1	1	yes	characteristic
167	C₂²×C₇₉	#529	1	1	yes	characteristic
168	C₄×C₇₉	—	4	2	no	—
169	C₄×C₇₉	—	4	2	no	—
170	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
171	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
172	C₄×C₇₉	#532	1	1	yes	characteristic
173	C₄×C₇₉	#532	1	1	yes	characteristic
174	Dic₇₉	—	2	4	no	—
175	Dic₇₉	—	4	2	no	—
176	Dic₇₉	—	4	4	no	—
177	#21296	—	79	1	no	maximal, Sylow
178	C₂×C₇₉	—	2	1	no	—
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	—	8	2	no	—
181	C₂×C₇₉	#2293	1	1	yes	characteristic
182	C₂×C₇₉	#2955	1	2	yes	—
183	#2943	—	79	1	no	—
184	#2922	—	79	2	no	—
185	#1975	D₇₉	1	1	yes	characteristic
186	#1974	—	79	1	no	—
187	#1946	—	79	1	no	—
188	#1945	—	79	1	no	—
189	#1938	—	79	2	no	—
190	#1937	—	79	2	no	—
191	#1778	—	79	1	no	—
192	#1759	—	79	2	no	—
193	#1726	—	79	1	no	—
194	C₇₉	#21296	1	1	yes	characteristic, complemented, Sylow
195	C₂×#135	C₂×D₇₉	1	1	yes	characteristic
196	C₄×#41	—	79	2	no	—
197	#442	—	79	2	no	—
198	#441	—	79	2	no	—
199	#440	—	79	4	no	—
200	#433	—	158	4	no	—
201	#431	—	158	4	no	—
202	C₂×#108	—	79	1	no	—
203	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
204	#427	—	79	1	no	—
205	#424	—	79	2	no	—
206	#424	—	79	2	no	—
207	C₂×#104	—	79	2	no	—
208	C₂×#104	C₂×D₇₉	1	2	yes	—
209	#416	—	79	2	no	—
210	#408	—	79	2	no	—
211	#408	C₂×D₇₉	1	2	yes	—
212	C₄×M₁₆	—	79	2	no	—
213	C₂×#97	—	79	1	no	—
214	C₂×#97	—	79	1	no	—
215	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
216	C₂×C₄×C₈	—	79	1	no	—
217	C₂×C₄×C₈	—	79	1	no	—
218	C₂×C₄×C₈	—	79	1	no	—
219	#376	—	79	1	no	—
220	#341	—	79	2	no	—
221	#339	—	79	2	no	—
222	#339	—	79	2	no	—
223	C₂×#41	C₂²×D₇₉	1	2	yes	—
224	#135	—	2	2	no	—
225	#135	—	2	2	no	—
226	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
227	C₂×SD₁₆	C₄×D₇₉	1	2	yes	—
228	C₂×D₈	C₄×D₇₉	1	1	yes	characteristic
229	C₂×M₁₆	—	79	2	no	—
230	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
231	C₂²×C₈	—	79	1	no	—
232	C₂²×C₈	—	79	1	no	—
233	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
234	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
235	C₄×Q₈	—	79	4	no	—
236	C₄×D₄	—	79	4	no	—
237	C₄×D₄	—	158	4	no	—
238	#117	—	79	2	no	—
239	#117	—	158	2	no	—
240	C₂×C₄²	—	79	1	no	—
241	C₂×C₄²	—	79	1	no	—
242	C₂×C₄²	—	158	2	no	—
243	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
244	#108	—	2	2	no	—
245	#108	—	158	2	no	—
246	#107	—	79	2	no	—
247	#106	—	79	2	no	—
248	C₄⋊C₈	—	158	4	no	—
249	C₄⋊C₈	—	158	4	no	—
250	#104	—	2	4	no	—
251	#104	—	158	4	no	—
252	#103	—	79	4	no	—
253	#102	—	79	4	no	—
254	C₂²⋊C₈	—	2	4	no	—
255	C₂²⋊C₈	—	158	4	no	—
256	#97	—	2	2	no	—
257	#97	—	79	2	no	—
258	#97	—	79	2	no	—
259	#97	—	158	2	no	—
260	#97	—	158	2	no	—
261	C₄×C₈	—	79	4	no	—
262	C₄×C₈	—	158	2	no	—
263	C₄×C₈	—	158	2	no	—
264	C₄×C₈	—	158	2	no	—
265	#95	—	158	2	no	—
266	#41	—	2	2	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	C₂×Q₈	C₂×C₄×D₇₉	1	2	yes	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	C₂×C₄×D₇₉	1	2	yes	—
273	C₂²×C₄	—	2	1	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	79	2	no	—
276	C₂²×C₄	—	158	4	no	—
277	C₂²×C₄	—	158	1	no	—
278	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
279	Q₁₆	—	2	2	no	—
280	SD₁₆	—	2	4	no	—
281	D₈	—	2	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	158	2	no	—
285	M₁₆	—	158	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	79	2	no	—
290	C₂×C₈	—	79	2	no	—
291	C₂×C₈	—	158	2	no	—
292	C₂×C₈	—	158	1	no	—
293	C₂×C₈	—	158	1	no	—
294	C₂×C₈	—	158	1	no	—
295	C₂×C₈	—	158	1	no	—
296	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
298	C₂×C₈	D₄×D₇₉	1	2	yes	—
299	C₂×C₈	#11929623	1	2	yes	—
300	C₂×C₈	#11929623	1	2	yes	—
301	C₄⋊C₄	—	79	4	no	—
302	C₄⋊C₄	—	158	4	no	—
303	C₂²⋊C₄	—	158	4	no	—
304	C₂²⋊C₄	—	158	2	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	79	2	no	—
307	C₄²	—	79	2	no	—
308	C₄²	—	79	2	no	—
309	C₄²	—	158	4	no	—
310	C₄²	—	158	1	no	—
311	C₄²	—	158	2	no	—
312	C₄²	#11929623	1	2	yes	—
313	C₂³	—	2	2	no	—
314	C₂³	#41×D₇₉	1	1	yes	characteristic
315	Q₈	—	2	2	no	—
316	Q₈	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	316	2	no	—
327	C₂×C₄	—	316	8	no	—
328	C₂×C₄	—	316	2	no	—
329	C₂×C₄	—	79	4	no	—
330	C₂×C₄	—	158	4	no	—
331	C₂×C₄	—	158	2	no	—
332	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
334	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
335	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
336	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
337	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
338	C₈	—	2	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	316	2	no	—
341	C₈	—	158	2	no	—
342	C₈	—	158	2	no	—
343	C₈	#14579117	1	2	yes	—
344	C₈	#14579117	1	2	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
349	C₂²	#19099092	1	1	yes	characteristic
350	C₂²	#19099094	1	1	yes	characteristic
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	316	2	no	—
354	C₄	—	316	4	no	—
355	C₄	—	158	4	no	—
356	C₄	C₂×#14579117	1	1	yes	characteristic
357	C₄	C₄×#11929623	1	1	yes	characteristic
358	C₄	#19099108	1	1	yes	characteristic
359	C₄	#19099111	1	1	yes	characteristic
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	8	2	no	—
363	C₂	#23765862	1	1	yes	characteristic
364	C₂	#23772300	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1975	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂×#135×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1975	D₇₉
hypocenter	C₇₉	#21296
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1975

Complements

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

N	H ≅ G/N	type
2. #23772040	C₂	semidirect
4. #23764153	C₂	semidirect
5. #23764146	C₂	semidirect
6. #23764114	C₂	semidirect
7. #23764101	C₂	semidirect
8. #23763308	C₂	semidirect
9. #23763212	C₂	semidirect
10. #23763088	C₂	semidirect
13. #19098675	C₄	semidirect
14. #19098670	C₄	semidirect
15. #19098664	C₄	semidirect
194. C₇₉	#21296	semidirect

Maximal subgroups

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

subgroup	index
#23772201	2
#23772040	2
#23764289	2
#23764153	2
#23764146	2
#23764114	2
#23764101	2
#23763308	2
#23763212	2
#23763088	2
C₇₉×#1975	2
#21296	79