Subgroups — #34290539

Order 20224.

790 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771964	C₂	1	1	yes	maximal, characteristic
2	#23771819	C₂	1	1	yes	maximal, characteristic
3	#23764001	C₂	1	8	yes	maximal, complemented
4	C₂×#19098328	C₂	1	1	yes	maximal, characteristic
5	#23763109	C₂	1	1	yes	maximal, characteristic
6	#23763109	C₂	1	1	yes	maximal, characteristic
7	#23763096	C₂	1	1	yes	maximal, characteristic
8	C₂×#427×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11929608	C₂²	1	1	yes	characteristic
10	C₂²×#11929608	C₂²	1	1	yes	characteristic
11	C₂²×#11929608	C₂²	1	1	yes	characteristic
12	C₂×#14579120	C₂²	1	1	yes	characteristic
13	#19098590	—	2	2	no	—
14	#19098590	C₂²	1	4	yes	—
15	#19098587	—	2	2	no	—
16	#19098519	—	2	2	no	—
17	#19098519	C₂²	1	4	yes	—
18	C₄×#11929608	—	2	2	no	—
19	#19098328	C₂²	1	4	yes	—
20	#19098322	C₂²	1	4	yes	—
21	#19098322	C₂²	1	4	yes	—
22	#19098317	C₂²	1	4	yes	—
23	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×#117×C₇₉	C₂²	1	1	yes	characteristic
26	#427×C₇₉	C₄	1	4	yes	complemented
27	#427×C₇₉	C₂²	1	4	yes	—
28	C₂×#107×C₇₉	C₄	1	2	yes	complemented
29	C₂×#106×C₇₉	C₄	1	2	yes	complemented
30	C₂×#11929608	—	2	1	no	—
31	C₂×#11929608	—	2	2	no	—
32	C₂×#11929608	—	2	2	no	—
33	C₂×#11929608	—	2	2	no	—
34	C₂×#11929608	—	2	1	no	—
35	C₂×#11929608	—	2	2	no	—
36	C₂×#11929608	—	2	2	no	—
37	C₂×#11929608	D₄	1	2	yes	—
38	C₂×#11929608	D₄	1	2	yes	—
39	C₂×#11929608	Q₈	1	4	yes	—
40	C₂×#11929608	C₂³	1	2	yes	—
41	C₂×#11929608	C₂³	1	2	yes	—
42	C₂²×C₇₉⋊C₈	—	2	1	no	—
43	C₂²×C₇₉⋊C₈	—	2	1	no	—
44	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
45	#14579120	—	2	2	no	—
46	#14579120	—	2	2	no	—
47	C₂×C₄⋊Dic₇₉	—	2	1	no	—
48	C₂×C₂²⋊Dic₇₉	—	2	1	no	—
49	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
50	C₄⋊(C₇₉⋊C₈)	—	2	8	no	—
51	#14579072	—	2	4	no	—
52	C₄×C₇₉⋊C₈	—	2	4	no	—
53	C₂³×C₄×C₇₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₉	—	2	2	no	—
55	C₂²×C₈×C₇₉	—	2	2	no	—
56	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₉	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
60	#117×C₇₉	—	2	1	no	—
61	#117×C₇₉	—	2	1	no	—
62	#117×C₇₉	—	2	1	no	—
63	#117×C₇₉	—	2	1	no	—
64	#117×C₇₉	C₂×C₄	1	2	yes	—
65	#117×C₇₉	C₂×C₄	1	2	yes	—
66	#117×C₇₉	C₂³	1	2	yes	—
67	#117×C₇₉	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₇₉	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
72	C₂×C₄²×C₇₉	—	2	1	no	—
73	C₂×C₄²×C₇₉	—	2	1	no	—
74	#107×C₇₉	C₂×C₄	1	8	yes	—
75	#106×C₇₉	C₂×C₄	1	8	yes	—
76	C₂²×Dic₇₉	—	2	2	no	—
77	#11929608	—	2	4	no	—
78	#11929608	—	2	4	no	—
79	#11929608	—	2	4	no	—
80	#11929608	—	2	2	no	—
81	#11929608	—	2	2	no	—
82	#11929608	—	2	2	no	—
83	#11929608	—	4	2	no	—
84	#11929608	C₂×Q₈	1	4	yes	—
85	C₂×C₇₉⋊C₈	—	2	4	no	—
86	C₂×C₇₉⋊C₈	—	2	2	no	—
87	C₂×C₇₉⋊C₈	—	2	2	no	—
88	C₂×C₇₉⋊C₈	—	2	2	no	—
89	C₂×C₇₉⋊C₈	—	2	2	no	—
90	C₂×C₇₉⋊C₈	—	4	2	no	—
91	C₂×C₇₉⋊C₈	C₂×D₄	1	4	yes	—
92	C₄⋊Dic₇₉	—	4	2	no	—
93	C₂²⋊Dic₇₉	—	4	2	no	—
94	C₄×Dic₇₉	—	2	2	no	—
95	C₄×Dic₇₉	—	2	2	no	—
96	C₂⁴×C₇₉	#41	1	1	yes	characteristic
97	C₂²×C₄×C₇₉	—	2	2	no	—
98	C₂²×C₄×C₇₉	—	2	2	no	—
99	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
104	C₂²×C₄×C₇₉	#41	1	2	yes	—
105	C₂²×C₄×C₇₉	#41	1	2	yes	—
106	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	#41	1	2	yes	—
108	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₇₉	—	2	2	no	—
110	C₂×C₈×C₇₉	—	2	2	no	—
111	C₂×C₈×C₇₉	—	2	4	no	—
112	C₂×C₈×C₇₉	—	2	2	no	—
113	C₂×C₈×C₇₉	—	2	2	no	—
114	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
116	C₄⋊C₄×C₇₉	—	2	2	no	—
117	C₄⋊C₄×C₇₉	—	2	2	no	—
118	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
119	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
120	C₂²⋊C₄×C₇₉	—	2	2	no	—
121	C₂²⋊C₄×C₇₉	—	2	2	no	—
122	C₂²⋊C₄×C₇₉	—	2	2	no	—
123	C₂²⋊C₄×C₇₉	—	2	2	no	—
124	C₄²×C₇₉	—	2	2	no	—
125	C₄²×C₇₉	—	2	2	no	—
126	C₄²×C₇₉	—	2	2	no	—
127	C₄²×C₇₉	—	2	2	no	—
128	C₂³×C₇₉	—	4	2	no	—
129	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
130	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
131	C₂³×C₇₉	#125	1	2	yes	—
132	C₂³×C₇₉	#126	1	2	yes	—
133	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₇₉	—	2	4	no	—
135	C₂×C₄×C₇₉	—	2	4	no	—
136	C₂×C₄×C₇₉	—	8	2	no	—
137	C₂×C₄×C₇₉	—	8	2	no	—
138	C₂×C₄×C₇₉	#117	1	2	yes	—
139	C₂×C₄×C₇₉	#117	1	2	yes	—
140	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
144	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₇₉	C₂²⋊Q₈	1	2	yes	—
146	C₂×C₄×C₇₉	#123	1	2	yes	—
147	C₂×C₄×C₇₉	#123	1	2	yes	—
148	C₂×C₄×C₇₉	#125	1	2	yes	—
149	C₂×C₄×C₇₉	#126	1	2	yes	—
150	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
156	C₂×Dic₇₉	—	4	4	no	—
157	C₂×Dic₇₉	—	4	2	no	—
158	C₈×C₇₉	—	2	2	no	—
159	C₈×C₇₉	—	2	2	no	—
160	C₇₉⋊C₈	—	2	4	no	—
161	C₇₉⋊C₈	—	2	4	no	—
162	C₇₉⋊C₈	—	4	2	no	—
163	C₂²×C₇₉	—	2	2	no	—
164	C₂²×C₇₉	—	2	2	no	—
165	C₂²×C₇₉	—	2	2	no	—
166	C₂²×C₇₉	—	4	2	no	—
167	C₂²×C₇₉	—	4	2	no	—
168	C₂²×C₇₉	#387	1	2	yes	—
169	C₂²×C₇₉	#387	1	2	yes	—
170	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂²×C₇₉	C₂×#125	1	1	yes	characteristic
172	C₂²×C₇₉	C₂×#126	1	1	yes	characteristic
173	C₄×C₇₉	—	4	4	no	—
174	C₄×C₇₉	—	4	4	no	—
175	C₄×C₇₉	#387	1	2	yes	—
176	C₄×C₇₉	#387	1	2	yes	—
177	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
178	C₄×C₇₉	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₄×C₇₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
180	C₄×C₇₉	C₂×#123	1	1	yes	characteristic
181	Dic₇₉	—	8	2	no	—
182	#21043	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	2	1	no	—
184	C₂×C₇₉	—	2	1	no	—
185	C₂×C₇₉	—	2	2	no	—
186	C₂×C₇₉	—	2	2	no	—
187	C₂×C₇₉	C₂×#387	1	1	yes	characteristic
188	C₂×C₇₉	#2957	1	2	yes	—
189	C₂×#427	D₇₉	1	1	yes	characteristic
190	#2905	—	79	1	no	—
191	#2873	—	79	1	no	—
192	#1922	—	79	8	no	—
193	C₂×#341	—	79	1	no	—
194	#1733	—	79	1	no	—
195	#1733	—	79	1	no	—
196	#1729	—	79	1	no	—
197	C₇₉	#21043	1	1	yes	characteristic, complemented, Sylow
198	C₂²×M₁₆	—	79	1	no	—
199	C₂²×M₁₆	—	79	1	no	—
200	C₂²×M₁₆	—	79	1	no	—
201	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
202	C₂×#117	—	79	1	no	—
203	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
204	C₂×#117	C₂×D₇₉	1	1	yes	characteristic
205	#427	Dic₇₉	1	4	yes	complemented
206	#427	C₂×D₇₉	1	4	yes	—
207	C₂×#107	Dic₇₉	1	2	yes	complemented
208	C₂×#106	Dic₇₉	1	2	yes	complemented
209	#424	—	79	4	no	—
210	#424	—	158	2	no	—
211	#423	—	158	2	no	—
212	#405	—	79	4	no	—
213	#405	—	158	2	no	—
214	C₄×M₁₆	—	158	2	no	—
215	#341	—	79	4	no	—
216	#339	—	79	4	no	—
217	#339	—	79	4	no	—
218	#336	—	79	4	no	—
219	C₂³×C₄	C₂²×D₇₉	1	1	yes	characteristic
220	C₂×M₁₆	—	79	2	no	—
221	C₂×M₁₆	—	79	2	no	—
222	C₂×M₁₆	—	79	4	no	—
223	C₂×M₁₆	—	79	2	no	—
224	C₂×M₁₆	—	79	2	no	—
225	C₂×M₁₆	—	158	2	no	—
226	C₂×M₁₆	—	158	2	no	—
227	C₂×M₁₆	—	158	1	no	—
228	C₂×M₁₆	—	158	1	no	—
229	C₂×M₁₆	—	158	2	no	—
230	C₂×M₁₆	—	158	2	no	—
231	C₂×M₁₆	—	158	2	no	—
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	—	2	2	no	—
234	C₂²×C₈	—	79	2	no	—
235	C₂²×C₈	—	158	1	no	—
236	C₂²×C₈	—	158	1	no	—
237	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
238	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
239	C₂²×C₈	C₂×Dic₇₉	1	2	yes	—
240	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
241	#117	—	2	1	no	—
242	#117	—	2	1	no	—
243	#117	—	2	1	no	—
244	#117	—	2	1	no	—
245	#117	—	158	2	no	—
246	#117	—	158	2	no	—
247	#117	C₂×Dic₇₉	1	2	yes	—
248	#117	C₂×Dic₇₉	1	2	yes	—
249	#117	C₂²×D₇₉	1	2	yes	—
250	#117	C₂²×D₇₉	1	2	yes	—
251	C₂×C₄⋊C₄	—	158	1	no	—
252	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
253	C₂×C₄⋊C₄	C₂×Dic₇₉	1	2	yes	—
254	C₂×C₂²⋊C₄	—	2	1	no	—
255	C₂×C₂²⋊C₄	—	2	1	no	—
256	C₂×C₂²⋊C₄	—	158	1	no	—
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	—	2	1	no	—
259	C₂×C₄²	—	79	2	no	—
260	#107	C₂×Dic₇₉	1	8	yes	—
261	#106	C₂×Dic₇₉	1	8	yes	—
262	C₄⋊C₈	—	158	8	no	—
263	#97	—	158	4	no	—
264	C₄×C₈	—	158	4	no	—
265	C₂⁴	#11929623	1	1	yes	characteristic
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	158	2	no	—
269	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
273	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
274	C₂²×C₄	#11929623	1	1	yes	characteristic
275	C₂²×C₄	#11929624	1	2	yes	—
276	C₂²×C₄	#11929624	1	2	yes	—
277	C₂²×C₄	#11929625	1	2	yes	—
278	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
279	M₁₆	—	316	2	no	—
280	M₁₆	—	79	4	no	—
281	M₁₆	—	158	2	no	—
282	M₁₆	—	158	2	no	—
283	M₁₆	—	158	2	no	—
284	M₁₆	—	158	4	no	—
285	M₁₆	—	158	4	no	—
286	M₁₆	—	158	4	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	4	no	—
292	C₂×C₈	—	316	2	no	—
293	C₂×C₈	—	79	4	no	—
294	C₂×C₈	—	158	2	no	—
295	C₂×C₈	—	158	2	no	—
296	C₂×C₈	—	158	2	no	—
297	C₂×C₈	—	158	4	no	—
298	C₂×C₈	—	158	2	no	—
299	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
300	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	316	2	no	—
304	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
305	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₂²⋊C₄	—	2	2	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	316	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	158	2	no	—
316	C₄²	—	158	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
319	C₂³	D₄×Dic₇₉	1	1	yes	characteristic
320	C₂³	#14579151	1	2	yes	—
321	C₂³	#14579156	1	2	yes	—
322	C₂³	C₂×#11929623	1	1	yes	characteristic
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	—	316	4	no	—
328	C₂×C₄	—	316	2	no	—
329	C₂×C₄	#14579120	1	2	yes	—
330	C₂×C₄	#14579120	1	2	yes	—
331	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
332	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
333	C₂×C₄	Q₈×Dic₇₉	1	2	yes	—
334	C₂×C₄	Q₈×Dic₇₉	1	2	yes	—
335	C₂×C₄	#14579139	1	2	yes	—
336	C₂×C₄	#14579139	1	2	yes	—
337	C₂×C₄	#14579146	1	2	yes	—
338	C₂×C₄	#14579146	1	2	yes	—
339	C₂×C₄	#14579151	1	2	yes	—
340	C₂×C₄	#14579156	1	2	yes	—
341	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	C₂×Q₈×D₇₉	1	1	yes	characteristic
344	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
345	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
346	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	316	2	no	—
350	C₈	—	158	4	no	—
351	C₈	—	158	4	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	#19098449	1	2	yes	—
358	C₂²	#19098449	1	2	yes	—
359	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
360	C₂²	C₂×#14579151	1	1	yes	characteristic
361	C₂²	C₂×#14579156	1	1	yes	characteristic
362	C₄	—	4	4	no	—
363	C₄	—	4	4	no	—
364	C₄	—	632	2	no	—
365	C₄	#19098449	1	2	yes	—
366	C₄	#19098449	1	2	yes	—
367	C₄	C₂×#14579120	1	1	yes	characteristic
368	C₄	C₂×Q₈×Dic₇₉	1	1	yes	characteristic
369	C₄	C₂×#14579139	1	1	yes	characteristic
370	C₄	C₂×#14579146	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	2	no	—
373	C₂	—	2	2	no	—
374	C₂	—	2	1	no	—
375	C₂	C₂×#19098449	1	1	yes	characteristic
376	C₂	#23772317	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×#427×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂⁴×C₇₉	#41
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×#427	D₇₉
hypocenter	C₇₉	#21043
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
3. #23764001	C₂	semidirect
26. #427×C₇₉	C₄	semidirect
28. C₂×#107×C₇₉	C₄	semidirect
29. C₂×#106×C₇₉	C₄	semidirect
197. C₇₉	#21043	semidirect
205. #427	Dic₇₉	semidirect
207. C₂×#107	Dic₇₉	semidirect
208. C₂×#106	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23771964	2
#23771819	2
#23764001	2
C₂×#19098328	2
#23763109	2
#23763096	2
C₂×#427×C₇₉	2
#21043	79