Subgroups — #34253993

Order 20224.

814 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772362	C₂	1	2	yes	maximal
2	C₄×#14579169	C₂	1	1	yes	maximal, characteristic
3	#23761702	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761698	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761697	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761686	C₂	1	2	yes	maximal, complemented
7	#23761685	C₂	1	1	yes	maximal, characteristic, complemented
8	#23761679	C₂	1	2	yes	maximal, complemented
9	M₁₆×C₇₉⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₈²×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1455	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579169	C₄	1	2	yes	—
13	C₄×#11929624	C₂²	1	1	yes	characteristic
14	#19098691	C₄	1	4	yes	—
15	#19098639	C₄	1	4	yes	—
16	#19098636	C₄	1	4	yes	—
17	#19098632	C₄	1	4	yes	—
18	#19098616	C₂²	1	2	yes	—
19	#19098614	C₂²	1	2	yes	—
20	#19098614	C₂²	1	2	yes	—
21	#19098521	C₄	1	2	yes	—
22	#19098520	C₄	1	2	yes	—
23	#19098520	C₄	1	2	yes	—
24	C₄×#11929607	C₂²	1	2	yes	—
25	C₄×#11929607	C₂²	1	1	yes	characteristic
26	C₄×#11929607	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₇₉	C₂²	1	2	yes	—
30	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
31	#19098285	C₂²	1	2	yes	—
32	#19098284	C₂²	1	2	yes	—
33	#19098284	C₂²	1	2	yes	—
34	#19098284	C₂²	1	1	yes	characteristic
35	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
36	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
37	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
38	C₄×M₁₆×C₇₉	C₂²	1	2	yes	—
39	C₄×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#322×C₇₉	C₂²	1	2	yes	—
41	#322×C₇₉	C₂²	1	1	yes	characteristic
42	C₈²×C₇₉	C₂²	1	1	yes	characteristic
43	C₂×#11929624	C₂×C₄	1	1	yes	characteristic
44	#14579169	—	2	8	no	—
45	C₂×#11929607	C₂×C₄	1	4	yes	—
46	C₂×#11929607	C₂×C₄	1	2	yes	—
47	C₂×#11929607	C₂×C₄	1	2	yes	—
48	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
49	C₂×C₈×D₇₉	C₂×C₄	1	4	yes	—
50	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
51	C₂×C₈×D₇₉	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
53	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
54	C₄×C₂²⋊D₇₉	C₂×C₄	1	2	yes	—
55	C₄×C₂²⋊D₇₉	C₂×C₄	1	2	yes	—
56	#14579120	C₂×C₄	1	1	yes	characteristic
57	#14579118	C₂×C₄	1	1	yes	characteristic
58	#14579118	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₇₉	C₂³	1	1	yes	characteristic
60	C₄²×D₇₉	C₂³	1	1	yes	characteristic
61	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
62	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
63	Dic₇₉⋊C₈	C₂×C₄	1	4	yes	—
64	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
65	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
66	(C₂×D₇₉)⋊C₈	C₂×C₄	1	4	yes	—
67	#14579071	C₂×C₄	1	4	yes	—
68	#14579071	C₂×C₄	1	2	yes	—
69	#14579071	C₂×C₄	1	2	yes	—
70	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
71	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
72	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
75	C₈×Dic₇₉	C₂×C₄	1	4	yes	—
76	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
77	C₈×Dic₇₉	C₂×C₄	1	2	yes	—
78	C₂×M₁₆×C₇₉	C₂×C₄	1	4	yes	—
79	C₂×M₁₆×C₇₉	C₂×C₄	1	2	yes	—
80	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
81	#97×C₇₉	C₂×C₄	1	4	yes	—
82	#97×C₇₉	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₇₉	C₂³	1	2	yes	—
85	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₉	C₂³	1	2	yes	—
87	#11929624	—	2	4	no	—
88	C₂×Dic₁₅₈	C₄²	1	1	yes	characteristic
89	C₂×D₃₁₆	C₄²	1	1	yes	characteristic
90	C₂×C₂²⋊D₇₉	C₄²	1	1	yes	characteristic
91	C₂×C₂²⋊D₇₉	C₄²	1	1	yes	characteristic
92	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
96	#11929607	—	2	4	no	—
97	#11929607	—	2	4	no	—
98	#11929607	—	2	8	no	—
99	C₂×C₇₉⋊C₈	—	2	4	no	—
100	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
102	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
103	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
104	C₈×D₇₉	—	2	8	no	—
105	C₈×D₇₉	—	2	4	no	—
106	C₈×D₇₉	M₁₆	1	8	yes	complemented
107	C₄⋊Dic₇₉	C₄²	1	2	yes	—
108	Dic₇₉⋊C₄	C₄²	1	2	yes	—
109	Dic₇₉⋊C₄	C₄²	1	2	yes	—
110	C₂²⋊Dic₇₉	C₄²	1	2	yes	—
111	(C₂×D₇₉)⋊C₄	C₄²	1	2	yes	—
112	(C₂×D₇₉)⋊C₄	C₄²	1	2	yes	—
113	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
114	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
115	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
116	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
119	M₁₆×C₇₉	—	2	4	no	—
120	M₁₆×C₇₉	—	2	8	no	—
121	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
124	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
125	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
127	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
128	C₂³×C₇₉	C₂×C₄²	1	1	yes	characteristic
129	C₂²×D₇₉	C₂×C₄²	1	1	yes	characteristic
130	C₂²×D₇₉	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₇₉	—	2	2	no	—
132	C₂×C₄×C₇₉	—	2	4	no	—
133	C₂×C₄×C₇₉	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₇₉	C₂×C₄²	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
137	C₂²⋊D₇₉	—	2	2	no	—
138	C₂²⋊D₇₉	—	2	2	no	—
139	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
140	C₂×Dic₇₉	C₂×C₄²	1	2	yes	—
141	C₂×Dic₇₉	C₂×C₄²	1	1	yes	characteristic
142	C₂×Dic₇₉	C₂×C₄²	1	1	yes	characteristic
143	D₃₁₆	—	2	2	no	—
144	C₄×D₇₉	—	2	2	no	—
145	C₄×D₇₉	—	2	4	no	—
146	C₄×D₇₉	—	2	4	no	—
147	C₄×D₇₉	C₂×M₁₆	1	4	yes	—
148	Dic₁₅₈	—	2	2	no	—
149	C₈×C₇₉	—	2	2	no	—
150	C₈×C₇₉	—	2	4	no	—
151	C₈×C₇₉	C₂×M₁₆	1	4	yes	—
152	C₈×C₇₉	#131	1	8	yes	complemented
153	C₇₉⋊C₈	—	2	2	no	—
154	C₇₉⋊C₈	—	2	4	no	—
155	C₇₉⋊C₈	—	2	4	no	—
156	C₇₉⋊C₈	C₂×M₁₆	1	4	yes	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	—	2	2	no	—
159	C₂²×C₇₉	C₂²×C₄²	1	1	yes	characteristic
160	C₂×D₇₉	—	2	2	no	—
161	C₂×D₇₉	—	2	1	no	—
162	C₂×D₇₉	—	2	2	no	—
163	C₂×D₇₉	C₄×M₁₆	1	2	yes	—
164	C₄×C₇₉	—	2	1	no	—
165	C₄×C₇₉	—	2	2	no	—
166	C₄×C₇₉	C₂²×M₁₆	1	2	yes	—
167	C₄×C₇₉	C₂×#131	1	4	yes	—
168	Dic₇₉	—	2	1	no	—
169	Dic₇₉	—	2	2	no	—
170	Dic₇₉	—	2	2	no	—
171	Dic₇₉	C₄×M₁₆	1	2	yes	—
172	#17335	—	79	1	no	maximal, Sylow
173	C₂×C₇₉	—	4	1	no	—
174	C₂×C₇₉	C₂×C₄×M₁₆	1	1	yes	characteristic
175	C₂×C₇₉	C₄×#131	1	2	yes	—
176	D₇₉	—	2	2	no	—
177	D₇₉	—	4	1	no	—
178	#2965	—	79	2	no	—
179	C₄×#131	—	79	1	no	—
180	#1455	—	79	1	no	—
181	#1455	D₇₉	1	1	yes	characteristic, complemented
182	#1453	—	79	1	no	—
183	#1453	—	79	1	no	—
184	#1451	—	79	2	no	—
185	#1450	—	79	1	no	—
186	C₈×M₁₆	—	79	2	no	—
187	C₈×M₁₆	—	79	2	no	—
188	C₂×C₈²	—	79	1	no	—
189	C₇₉	#17335	1	1	yes	characteristic, complemented, Sylow
190	C₂×#131	—	79	2	no	—
191	C₄×#41	—	79	1	no	—
192	#446	—	79	4	no	—
193	#436	—	79	4	no	—
194	#435	—	79	4	no	—
195	#435	—	79	4	no	—
196	#431	—	79	2	no	—
197	#431	—	79	2	no	—
198	#431	—	79	2	no	—
199	#405	—	79	2	no	—
200	#405	—	79	2	no	—
201	#405	—	79	2	no	—
202	C₄×M₁₆	—	79	1	no	—
203	C₄×M₁₆	—	79	1	no	—
204	C₄×M₁₆	—	79	2	no	—
205	C₄×M₁₆	C₂×D₇₉	1	1	yes	characteristic
206	C₄×M₁₆	C₂×D₇₉	1	2	yes	—
207	C₂×C₄×C₈	—	79	1	no	—
208	C₂×C₄×C₈	—	79	1	no	—
209	C₂×C₄×C₈	—	79	1	no	—
210	C₂×C₄×C₈	—	79	2	no	—
211	#322	—	79	2	no	—
212	#322	—	79	2	no	—
213	#322	—	79	2	no	—
214	#322	—	79	1	no	—
215	#322	C₂×D₇₉	1	2	yes	—
216	#322	C₂×D₇₉	1	1	yes	characteristic
217	C₈²	—	79	2	no	—
218	C₈²	—	79	1	no	—
219	C₈²	—	79	2	no	—
220	C₈²	C₂×D₇₉	1	1	yes	characteristic
221	C₂×#41	—	79	1	no	—
222	#131	—	158	8	no	—
223	C₂×M₁₆	—	79	2	no	—
224	C₂×M₁₆	—	79	2	no	—
225	C₂×M₁₆	—	79	4	no	—
226	C₂×M₁₆	C₄×D₇₉	1	2	yes	—
227	C₂×M₁₆	C₄×D₇₉	1	4	yes	—
228	C₂²×C₈	—	79	2	no	—
229	C₂²×C₈	—	79	2	no	—
230	C₂²×C₈	—	79	2	no	—
231	C₂²×C₈	—	79	4	no	—
232	C₄×Q₈	—	79	2	no	—
233	C₄×D₄	—	79	2	no	—
234	C₄×D₄	—	79	2	no	—
235	C₄×D₄	—	79	2	no	—
236	#117	—	79	1	no	—
237	#117	—	79	1	no	—
238	#117	—	79	1	no	—
239	C₂×C₄²	—	79	1	no	—
240	C₂×C₄²	—	79	1	no	—
241	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
242	C₄⋊C₈	—	79	4	no	—
243	C₄⋊C₈	—	79	4	no	—
244	C₄⋊C₈	—	79	4	no	—
245	C₂²⋊C₈	—	79	4	no	—
246	C₂²⋊C₈	—	79	4	no	—
247	C₂²⋊C₈	—	79	4	no	—
248	#97	—	79	2	no	—
249	#97	—	79	2	no	—
250	#97	—	79	4	no	—
251	#97	C₄×D₇₉	1	2	yes	—
252	#97	C₄×D₇₉	1	4	yes	—
253	C₄×C₈	—	79	2	no	—
254	C₄×C₈	—	79	2	no	—
255	C₄×C₈	—	79	4	no	—
256	C₄×C₈	—	79	2	no	—
257	C₄×C₈	—	79	1	no	—
258	C₄×C₈	—	79	2	no	—
259	C₄×C₈	—	79	2	no	—
260	C₄×C₈	—	79	1	no	—
261	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₇₉	1	2	yes	—
263	C₄×C₈	C₂²×D₇₉	1	2	yes	—
264	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
265	#41	—	158	4	no	—
266	C₂×Q₈	—	79	1	no	—
267	C₂×D₄	—	79	1	no	—
268	C₂×D₄	—	79	1	no	—
269	C₂×D₄	—	79	1	no	—
270	C₂²×C₄	—	79	1	no	—
271	C₂²×C₄	—	79	2	no	—
272	C₂²×C₄	—	79	2	no	—
273	C₂²×C₄	—	79	1	no	—
274	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
275	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	2	8	no	—
278	M₁₆	—	158	4	no	—
279	M₁₆	—	158	4	no	—
280	M₁₆	—	158	8	no	—
281	C₂×C₈	—	79	2	no	—
282	C₂×C₈	—	79	8	no	—
283	C₂×C₈	—	79	2	no	—
284	C₂×C₈	—	79	4	no	—
285	C₂×C₈	—	79	4	no	—
286	C₂×C₈	—	158	4	no	—
287	C₂×C₈	—	158	4	no	—
288	C₂×C₈	—	158	8	no	—
289	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
290	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
291	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
292	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
293	C₄⋊C₄	—	79	2	no	—
294	C₄⋊C₄	—	79	2	no	—
295	C₄⋊C₄	—	79	2	no	—
296	C₂²⋊C₄	—	79	2	no	—
297	C₂²⋊C₄	—	79	2	no	—
298	C₂²⋊C₄	—	79	2	no	—
299	C₄²	—	79	1	no	—
300	C₄²	—	79	1	no	—
301	C₄²	—	79	2	no	—
302	C₄²	—	79	2	no	—
303	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
304	C₄²	C₂×C₄×D₇₉	1	2	yes	—
305	C₄²	C₂³×D₇₉	1	1	yes	characteristic
306	C₂³	—	79	1	no	—
307	C₂³	—	79	1	no	—
308	C₂³	C₄²×D₇₉	1	1	yes	characteristic
309	Q₈	—	158	2	no	—
310	D₄	—	158	2	no	—
311	D₄	—	158	2	no	—
312	D₄	—	158	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	79	1	no	—
316	C₂×C₄	—	79	4	no	—
317	C₂×C₄	—	79	2	no	—
318	C₂×C₄	—	79	2	no	—
319	C₂×C₄	—	79	1	no	—
320	C₂×C₄	—	158	2	no	—
321	C₂×C₄	—	158	4	no	—
322	C₂×C₄	—	158	4	no	—
323	C₂×C₄	C₄²×D₇₉	1	2	yes	—
324	C₂×C₄	C₄²×D₇₉	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
326	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
327	C₈	—	2	2	no	—
328	C₈	—	2	4	no	—
329	C₈	—	79	4	no	—
330	C₈	—	158	2	no	—
331	C₈	—	158	4	no	—
332	C₈	—	158	4	no	—
333	C₈	M₁₆×D₇₉	1	4	yes	—
334	C₈	#14579169	1	8	yes	complemented
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	79	2	no	—
338	C₂²	—	158	1	no	—
339	C₂²	—	158	2	no	—
340	C₂²	—	158	2	no	—
341	C₂²	C₂×C₄²×D₇₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	79	2	no	—
345	C₄	—	158	1	no	—
346	C₄	—	158	2	no	—
347	C₄	—	158	2	no	—
348	C₄	C₂×M₁₆×D₇₉	1	2	yes	—
349	C₄	C₂×#14579169	1	4	yes	—
350	C₂	—	4	1	no	—
351	C₂	—	316	1	no	—
352	C₂	—	158	2	no	—
353	C₂	C₄×M₁₆×D₇₉	1	1	yes	characteristic
354	C₂	C₄×#14579169	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇₉
commutator, socle	C₂²×C₇₉	C₂²×C₄²
Fitting	C₇₉×#1455	C₂
prime radical	C₂×#11929624	C₂×C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1455	D₇₉
hypocenter	C₇₉	#17335
LCS	G ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ #1455

Complements

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

N	H ≅ G/N	type
3. #23761702	C₂	semidirect
4. #23761698	C₂	semidirect
5. #23761697	C₂	semidirect
6. #23761686	C₂	semidirect
7. #23761685	C₂	semidirect
8. #23761679	C₂	semidirect
9. M₁₆×C₇₉⋊C₈	C₂	semidirect
10. C₈²×D₇₉	C₂	semidirect
11. C₇₉×#1455	C₂	semidirect
106. C₈×D₇₉	M₁₆	semidirect
152. C₈×C₇₉	#131	semidirect
181. #1455	D₇₉	semidirect
189. C₇₉	#17335	semidirect
334. C₈	#14579169	semidirect

Maximal subgroups

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

subgroup	index
#23772362	2
C₄×#14579169	2
#23761702	2
#23761698	2
#23761697	2
#23761686	2
#23761685	2
#23761679	2
M₁₆×C₇₉⋊C₈	2
C₈²×D₇₉	2
C₇₉×#1455	2
#17335	79