Subgroups — #34258846

Order 20224.

614 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	#23773970	C₂	1	1	yes	maximal, characteristic
2	#23772375	C₂	1	2	yes	maximal, complemented
3	#23762315	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762310	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762304	C₂	1	2	yes	maximal, complemented
6	#23762225	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762220	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762214	C₂	1	2	yes	maximal, complemented
9	#23762163	C₂	1	1	yes	maximal, characteristic, complemented
10	#23761861	C₂	1	2	yes	maximal, complemented
11	C₂×#322×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099097	C₂²	1	2	yes	—
13	#19098965	C₄	1	1	yes	characteristic
14	#19098947	C₄	1	1	yes	characteristic
15	#19098918	C₄	1	2	yes	—
16	#19098804	C₄	1	1	yes	characteristic
17	#19098799	C₄	1	1	yes	characteristic
18	#19098792	C₄	1	2	yes	—
19	#19098693	C₂²	1	2	yes	—
20	C₈×Dic₁₅₈	C₂²	1	2	yes	—
21	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
22	C₄×#11929613	C₂²	1	2	yes	—
23	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
24	#19098642	C₂²	1	2	yes	—
25	#19098634	—	2	2	no	—
26	C₈×D₃₁₆	C₂²	1	2	yes	—
27	C₈×C₂²⋊D₇₉	—	2	2	no	—
28	#19098622	—	2	2	no	—
29	#19098620	C₂²	1	2	yes	—
30	#19098517	—	2	2	no	—
31	#19098316	C₂²	1	2	yes	—
32	#19098314	C₂²	1	1	yes	characteristic
33	#19098314	C₂²	1	1	yes	characteristic
34	#19098295	C₂²	1	2	yes	—
35	#19098295	C₂²	1	2	yes	—
36	#19098292	C₂²	1	2	yes	—
37	#19098292	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
41	#322×C₇₉	C₂²	1	1	yes	characteristic
42	#322×C₇₉	C₂²	1	1	yes	characteristic
43	#322×C₇₉	C₂²	1	1	yes	characteristic
44	#322×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11929613	C₂×C₄	1	2	yes	—
47	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
48	C₂×#11929607	—	2	2	no	—
49	C₂×C₈×D₇₉	—	2	2	no	—
50	#14579155	—	2	2	no	—
51	#14579153	C₂×C₄	1	2	yes	—
52	#14579149	C₂×C₄	1	2	yes	—
53	#14579144	—	2	2	no	—
54	#14579140	C₂×C₄	1	2	yes	—
55	#14579135	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₅₈	C₂³	1	2	yes	—
57	C₄×D₃₁₆	C₂³	1	2	yes	—
58	C₄×C₂²⋊D₇₉	—	2	2	no	—
59	#14579118	—	2	2	no	—
60	#14579095	C₂×C₄	1	1	yes	characteristic
61	#14579095	C₂×C₄	1	1	yes	characteristic
62	#14579093	C₂×C₄	1	2	yes	—
63	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
64	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
65	Dic₇₉⋊C₈	—	2	2	no	—
66	Dic₇₉⋊C₈	—	2	2	no	—
67	#14579086	C₂×C₄	1	2	yes	—
68	#14579086	C₂×C₄	1	2	yes	—
69	#14579083	C₂×C₄	1	2	yes	—
70	#14579083	C₂×C₄	1	2	yes	—
71	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
72	(C₂×D₇₉)⋊C₈	—	2	2	no	—
73	(C₂×D₇₉)⋊C₈	—	2	2	no	—
74	#14579072	—	2	2	no	—
75	#14579071	—	2	2	no	—
76	C₈×Dic₇₉	—	2	2	no	—
77	C₂²×C₈×C₇₉	—	2	1	no	—
78	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
79	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
80	C₂²×C₈×C₇₉	D₄	1	2	yes	—
81	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₇₉	—	2	1	no	—
83	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₇₉	D₄	1	2	yes	—
86	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
93	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
94	C₂×C₂²⋊D₇₉	—	2	2	no	—
95	C₂×C₄×D₇₉	—	2	2	no	—
96	Dic₃₁₆	—	2	2	no	—
97	#11929613	—	2	4	no	—
98	D₆₃₂	—	2	2	no	—
99	#11929607	—	4	4	no	—
100	C₂×C₇₉⋊C₈	—	2	2	no	—
101	C₂×C₇₉⋊C₈	—	2	2	no	—
102	C₈×D₇₉	—	4	4	no	—
103	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
104	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
105	Dic₇₉⋊C₄	—	2	2	no	—
106	Dic₇₉⋊C₄	—	2	2	no	—
107	Dic₇₉⋊C₄	—	2	2	no	—
108	Dic₇₉⋊C₄	—	2	2	no	—
109	C₂²⋊Dic₇₉	—	2	2	no	—
110	(C₂×D₇₉)⋊C₄	—	2	2	no	—
111	(C₂×D₇₉)⋊C₄	—	2	2	no	—
112	(C₂×D₇₉)⋊C₄	—	2	2	no	—
113	(C₂×D₇₉)⋊C₄	—	2	2	no	—
114	C₄×Dic₇₉	—	2	2	no	—
115	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₉	—	2	2	no	—
119	C₂×C₈×C₇₉	—	2	2	no	—
120	C₂×C₈×C₇₉	—	2	1	no	—
121	C₂×C₈×C₇₉	—	2	1	no	—
122	C₂×C₈×C₇₉	—	4	2	no	—
123	C₂×C₈×C₇₉	—	4	2	no	—
124	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
130	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₇₉	#41	1	1	yes	characteristic
133	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
134	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
135	C₂²×D₇₉	—	2	2	no	—
136	C₂×C₄×C₇₉	—	2	2	no	—
137	C₂×C₄×C₇₉	—	2	1	no	—
138	C₂×C₄×C₇₉	—	2	1	no	—
139	C₂×C₄×C₇₉	—	2	2	no	—
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×D₄	1	1	yes	characteristic
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	C₂²⋊D₇₉	—	4	4	no	—
147	C₂×Dic₇₉	—	2	2	no	—
148	C₂×Dic₇₉	—	2	2	no	—
149	C₂×Dic₇₉	—	2	2	no	—
150	D₃₁₆	—	2	2	no	—
151	D₃₁₆	—	2	2	no	—
152	C₄×D₇₉	—	4	2	no	—
153	C₄×D₇₉	—	4	2	no	—
154	Dic₁₅₈	—	2	2	no	—
155	Dic₁₅₈	—	2	2	no	—
156	C₈×C₇₉	—	2	1	no	—
157	C₈×C₇₉	—	2	1	no	—
158	C₈×C₇₉	—	2	2	no	—
159	C₈×C₇₉	—	2	2	no	—
160	C₈×C₇₉	—	4	1	no	—
161	C₈×C₇₉	—	4	1	no	—
162	C₈×C₇₉	#131	1	2	yes	—
163	C₈×C₇₉	#131	1	2	yes	—
164	C₇₉⋊C₈	—	4	2	no	—
165	C₇₉⋊C₈	—	4	2	no	—
166	C₂²×C₇₉	—	2	1	no	—
167	C₂²×C₇₉	—	2	1	no	—
168	C₂²×C₇₉	#444	1	2	yes	—
169	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×D₇₉	—	4	2	no	—
171	C₂×D₇₉	—	4	2	no	—
172	C₂×D₇₉	—	4	2	no	—
173	C₄×C₇₉	—	2	1	no	—
174	C₄×C₇₉	—	2	1	no	—
175	C₄×C₇₉	—	2	1	no	—
176	C₄×C₇₉	—	2	1	no	—
177	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
178	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
179	C₄×C₇₉	#575	1	2	yes	—
180	Dic₇₉	—	4	2	no	—
181	Dic₇₉	—	4	2	no	—
182	Dic₇₉	—	4	2	no	—
183	#17963	—	79	1	no	maximal, Sylow
184	C₂×C₇₉	—	2	2	no	—
185	C₂×C₇₉	#2931	1	1	yes	characteristic
186	C₂×C₇₉	#2943	1	1	yes	characteristic
187	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
188	D₇₉	—	8	2	no	—
189	#3168	—	79	1	no	—
190	#2966	—	79	2	no	—
191	#1594	—	79	1	no	—
192	#1593	—	79	1	no	—
193	#1592	—	79	2	no	—
194	#1580	—	79	1	no	—
195	#1579	—	79	1	no	—
196	SD₁₆⋊C₈	—	79	2	no	—
197	#1568	—	79	1	no	—
198	#1494	—	79	2	no	—
199	C₂×#322	D₇₉	1	1	yes	characteristic, complemented
200	C₇₉	#17963	1	1	yes	characteristic, complemented, Sylow
201	#529	—	79	2	no	—
202	#499	—	79	1	no	—
203	#495	—	79	1	no	—
204	#488	—	79	2	no	—
205	#467	—	79	1	no	—
206	#466	—	79	1	no	—
207	#465	—	79	2	no	—
208	#446	—	79	2	no	—
209	C₈×Q₈	—	79	2	no	—
210	C₄×Q₁₆	—	79	1	no	—
211	C₄×SD₁₆	—	79	2	no	—
212	C₄×D₈	—	79	1	no	—
213	#436	—	79	2	no	—
214	#435	—	158	2	no	—
215	C₈×D₄	—	79	2	no	—
216	C₈×D₄	—	158	2	no	—
217	#433	—	158	2	no	—
218	#432	—	79	2	no	—
219	#405	—	158	2	no	—
220	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
221	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
222	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
223	#335	—	79	2	no	—
224	#334	—	79	1	no	—
225	#334	—	79	1	no	—
226	Q₈⋊C₈	—	79	2	no	—
227	Q₈⋊C₈	—	79	2	no	—
228	D₄⋊C₈	—	79	2	no	—
229	D₄⋊C₈	—	79	2	no	—
230	#322	C₂×D₇₉	1	1	yes	characteristic
231	#322	C₂×D₇₉	1	1	yes	characteristic
232	#322	C₂×D₇₉	1	1	yes	characteristic
233	#322	C₂×D₇₉	1	1	yes	characteristic
234	C₂×Q₁₆	—	79	1	no	—
235	C₂×SD₁₆	—	79	2	no	—
236	C₂×D₈	—	79	1	no	—
237	C₂×M₁₆	—	158	2	no	—
238	C₂²×C₈	—	2	1	no	—
239	C₂²×C₈	—	158	2	no	—
240	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
241	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
242	C₂²×C₈	D₃₁₆	1	2	yes	—
243	#126	—	158	2	no	—
244	#125	—	79	2	no	—
245	#124	—	79	2	no	—
246	#123	—	158	2	no	—
247	C₂²⋊Q₈	—	79	2	no	—
248	#121	—	79	2	no	—
249	C₄×Q₈	—	79	2	no	—
250	C₄×D₄	—	79	2	no	—
251	C₄×D₄	—	158	2	no	—
252	#117	—	158	2	no	—
253	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
254	#107	—	79	1	no	—
255	#107	—	79	1	no	—
256	#106	—	79	2	no	—
257	C₄⋊C₈	—	79	2	no	—
258	C₄⋊C₈	—	79	2	no	—
259	C₄⋊C₈	—	158	2	no	—
260	C₄⋊C₈	—	158	2	no	—
261	#103	—	79	2	no	—
262	#103	—	79	2	no	—
263	#102	—	79	2	no	—
264	#102	—	79	2	no	—
265	C₂²⋊C₈	—	158	2	no	—
266	C₂²⋊C₈	—	158	2	no	—
267	C₂²⋊C₈	—	158	2	no	—
268	#97	—	158	2	no	—
269	#97	—	158	2	no	—
270	C₄×C₈	—	2	1	no	—
271	C₄×C₈	—	158	2	no	—
272	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
273	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
274	C₄×C₈	D₃₁₆	1	2	yes	—
275	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
281	C₂×Q₈	—	79	2	no	—
282	C₂×D₄	—	79	2	no	—
283	C₂×D₄	—	158	2	no	—
284	C₂²×C₄	—	158	2	no	—
285	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
286	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
287	C₂²×C₄	#11929624	1	1	yes	characteristic
288	Q₁₆	—	158	2	no	—
289	SD₁₆	—	158	4	no	—
290	D₈	—	158	2	no	—
291	M₁₆	—	316	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	4	2	no	—
297	C₂×C₈	—	4	2	no	—
298	C₂×C₈	—	316	4	no	—
299	C₂×C₈	—	158	2	no	—
300	C₂×C₈	—	158	2	no	—
301	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
305	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
306	C₂×C₈	C₂×D₃₁₆	1	2	yes	—
307	C₄⋊C₄	—	79	2	no	—
308	C₄⋊C₄	—	79	2	no	—
309	C₄⋊C₄	—	158	2	no	—
310	C₄⋊C₄	—	158	2	no	—
311	C₄⋊C₄	—	158	2	no	—
312	C₄⋊C₄	—	158	2	no	—
313	C₂²⋊C₄	—	158	2	no	—
314	C₂²⋊C₄	—	158	2	no	—
315	C₂²⋊C₄	—	158	2	no	—
316	C₂²⋊C₄	—	158	2	no	—
317	C₂²⋊C₄	—	158	2	no	—
318	C₄²	—	158	2	no	—
319	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
320	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
321	C₄²	#11929624	1	1	yes	characteristic
322	C₄²	C₂³×D₇₉	1	1	yes	characteristic
323	C₂³	—	158	2	no	—
324	C₂³	C₄×D₃₁₆	1	1	yes	characteristic
325	Q₈	—	158	2	no	—
326	Q₈	—	158	2	no	—
327	D₄	—	316	4	no	—
328	D₄	—	158	2	no	—
329	D₄	—	158	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	316	2	no	—
335	C₂×C₄	—	316	2	no	—
336	C₂×C₄	—	158	2	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	—	158	2	no	—
339	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
340	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
341	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
344	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	4	1	no	—
350	C₈	—	4	1	no	—
351	C₈	—	316	2	no	—
352	C₈	—	316	2	no	—
353	C₈	#14579167	1	2	yes	—
354	C₈	#14579167	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	316	2	no	—
358	C₂²	—	316	2	no	—
359	C₂²	—	316	2	no	—
360	C₂²	#19098686	1	2	yes	—
361	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	316	2	no	—
367	C₄	—	316	2	no	—
368	C₄	—	316	2	no	—
369	C₄	C₂×#14579167	1	1	yes	characteristic
370	C₄	C₂×#14579167	1	1	yes	characteristic
371	C₄	#19099363	1	2	yes	—
372	C₂	—	2	2	no	—
373	C₂	—	632	2	no	—
374	C₂	#23772113	1	1	yes	characteristic
375	C₂	#23772202	1	1	yes	characteristic
376	C₂	C₂×#19098686	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#322×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#19098799	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₂×#322	D₇₉
hypocenter	C₇₉	#17963
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
2. #23772375	C₂	semidirect
3. #23762315	C₂	semidirect
4. #23762310	C₂	semidirect
5. #23762304	C₂	semidirect
6. #23762225	C₂	semidirect
7. #23762220	C₂	semidirect
8. #23762214	C₂	semidirect
9. #23762163	C₂	semidirect
10. #23761861	C₂	semidirect
11. C₂×#322×C₇₉	C₂	semidirect
199. C₂×#322	D₇₉	semidirect
200. C₇₉	#17963	semidirect

Maximal subgroups

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

subgroup	index
#23773970	2
#23772375	2
#23762315	2
#23762310	2
#23762304	2
#23762225	2
#23762220	2
#23762214	2
#23762163	2
#23761861	2
C₂×#322×C₇₉	2
#17963	79