Subgroups — #34272066

Order 20224.

690 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773986	C₂	1	1	yes	maximal, characteristic, complemented
2	#23773197	C₂	1	2	yes	maximal
3	#23762851	C₂	1	4	yes	maximal, complemented
4	#23762799	C₂	1	2	yes	maximal, complemented
5	#23762765	C₂	1	2	yes	maximal, complemented
6	#23762159	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762088	C₂	1	2	yes	maximal, complemented
8	C₂×C₈×D₄×C₇₉	C₂	1	1	yes	maximal, characteristic
9	#19099120	C₂²	1	2	yes	—
10	#19098968	C₂²	1	1	yes	characteristic, complemented
11	#19098943	C₂²	1	1	yes	characteristic, complemented
12	#19098914	C₂²	1	2	yes	complemented
13	#19098863	—	2	4	no	—
14	#19098853	C₂²	1	4	yes	—
15	#19098843	C₂²	1	4	yes	—
16	#19098804	—	2	1	no	—
17	#19098804	—	2	1	no	—
18	C₄×Dic₃₁₆	—	2	1	no	—
19	C₄×Dic₃₁₆	—	2	1	no	—
20	#19098603	—	2	2	no	—
21	#19098587	C₂²	1	2	yes	—
22	#19098568	—	2	2	no	—
23	#19098316	C₂²	1	2	yes	complemented
24	#19098316	C₂²	1	2	yes	complemented
25	#19098311	C₂²	1	4	yes	complemented
26	#19098310	C₂²	1	4	yes	complemented
27	C₂³×C₈×C₇₉	—	2	1	no	—
28	C₂×C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
29	C₈×D₄×C₇₉	—	2	1	no	—
30	C₈×D₄×C₇₉	—	2	1	no	—
31	C₈×D₄×C₇₉	C₂²	1	2	yes	—
32	C₈×D₄×C₇₉	C₂²	1	2	yes	—
33	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×C₂²⋊C₈×C₇₉	—	2	1	no	—
35	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×Dic₃₁₆	—	2	1	no	—
37	C₂×Dic₃₁₆	—	2	1	no	—
38	C₂×#11929608	—	2	2	no	—
39	#14579161	C₂³	1	2	yes	—
40	#14579161	C₂³	1	2	yes	—
41	#14579153	—	2	2	no	—
42	#14579140	—	2	2	no	—
43	#14579140	—	2	2	no	—
44	C₄×Dic₁₅₈	—	2	2	no	—
45	C₄×Dic₁₅₈	—	2	2	no	—
46	#14579120	—	2	2	no	—
47	#14579095	—	2	1	no	—
48	#14579095	—	2	1	no	—
49	#14579094	—	2	4	no	—
50	#14579092	—	2	4	no	—
51	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
53	#14579086	—	2	2	no	—
54	#14579086	—	2	2	no	—
55	#14579082	—	2	4	no	—
56	#14579082	—	2	4	no	—
57	C₂²×D₄×C₇₉	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₇₉	—	2	1	no	—
59	C₂²×C₈×C₇₉	—	2	1	no	—
60	C₂²×C₈×C₇₉	—	2	2	no	—
61	C₂²×C₈×C₇₉	—	2	2	no	—
62	C₂²×C₈×C₇₉	—	4	4	no	—
63	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
65	C₄×D₄×C₇₉	—	2	1	no	—
66	C₄×D₄×C₇₉	—	2	1	no	—
67	C₄×D₄×C₇₉	D₄	1	2	yes	—
68	C₄×D₄×C₇₉	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₇₉	—	2	1	no	—
71	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₇₉	—	2	1	no	—
73	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
74	C₂²⋊C₈×C₇₉	—	2	2	no	—
75	C₂²⋊C₈×C₇₉	—	2	2	no	—
76	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₅₈	—	2	2	no	—
81	C₂×Dic₁₅₈	—	2	2	no	—
82	Dic₃₁₆	—	4	2	no	—
83	Dic₃₁₆	—	4	2	no	—
84	#11929608	—	4	4	no	—
85	C₂×C₇₉⋊C₈	—	2	4	no	—
86	C₄⋊Dic₇₉	—	2	2	no	—
87	C₄⋊Dic₇₉	—	2	2	no	—
88	C₄⋊Dic₇₉	—	2	2	no	—
89	C₄⋊Dic₇₉	—	2	2	no	—
90	Dic₇₉⋊C₄	—	4	2	no	—
91	Dic₇₉⋊C₄	—	4	2	no	—
92	C₂²⋊Dic₇₉	—	4	2	no	—
93	C₄×Dic₇₉	—	4	2	no	—
94	C₂⁴×C₇₉	—	2	1	no	—
95	C₂×D₄×C₇₉	—	2	1	no	—
96	C₂×D₄×C₇₉	—	2	2	no	—
97	C₂×D₄×C₇₉	Q₁₆	1	4	yes	complemented
98	C₂×D₄×C₇₉	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₇₉	—	2	1	no	—
100	C₂²×C₄×C₇₉	—	2	2	no	—
101	C₂²×C₄×C₇₉	—	2	2	no	—
102	C₂²×C₄×C₇₉	—	4	2	no	—
103	C₂²×C₄×C₇₉	—	4	2	no	—
104	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
107	C₂×C₈×C₇₉	—	2	2	no	—
108	C₂×C₈×C₇₉	—	2	2	no	—
109	C₂×C₈×C₇₉	—	4	1	no	—
110	C₂×C₈×C₇₉	—	4	2	no	—
111	C₂×C₈×C₇₉	—	4	4	no	—
112	C₂×C₈×C₇₉	—	4	4	no	—
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₇₉	—	2	1	no	—
118	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
119	C₂²⋊C₄×C₇₉	—	2	2	no	—
120	C₂²⋊C₄×C₇₉	—	2	2	no	—
121	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₉	#41	1	1	yes	characteristic
124	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
125	C₂³×C₇₉	—	2	2	no	—
126	C₂³×C₇₉	—	2	4	no	—
127	C₂³×C₇₉	—	4	2	no	—
128	C₂³×C₇₉	—	4	2	no	—
129	C₂³×C₇₉	#121	1	1	yes	characteristic
130	D₄×C₇₉	—	2	2	no	—
131	D₄×C₇₉	—	2	4	no	—
132	D₄×C₇₉	C₂×Q₁₆	1	4	yes	—
133	C₂×C₄×C₇₉	—	2	1	no	—
134	C₂×C₄×C₇₉	—	2	1	no	—
135	C₂×C₄×C₇₉	—	2	2	no	—
136	C₂×C₄×C₇₉	—	4	1	no	—
137	C₂×C₄×C₇₉	—	4	4	no	—
138	C₂×C₄×C₇₉	—	4	2	no	—
139	C₂×C₄×C₇₉	—	4	2	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
145	C₂×C₄×C₇₉	#135	1	2	yes	—
146	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₇₉	—	4	2	no	—
150	C₂×Dic₇₉	—	4	2	no	—
151	Dic₁₅₈	—	4	2	no	—
152	Dic₁₅₈	—	4	2	no	—
153	Dic₁₅₈	—	8	2	no	—
154	C₈×C₇₉	—	2	1	no	—
155	C₈×C₇₉	—	2	1	no	—
156	C₈×C₇₉	—	2	1	no	—
157	C₈×C₇₉	—	4	2	no	—
158	C₈×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
159	C₇₉⋊C₈	—	4	4	no	—
160	C₂²×C₇₉	—	2	1	no	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	—	2	4	no	—
163	C₂²×C₇₉	—	4	8	no	—
164	C₂²×C₇₉	—	4	2	no	—
165	C₂²×C₇₉	—	4	2	no	—
166	C₂²×C₇₉	#467	1	2	yes	—
167	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	1	no	—
171	C₄×C₇₉	—	4	2	no	—
172	C₄×C₇₉	#467	1	2	yes	—
173	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
174	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
175	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₄×C₇₉	#575	1	1	yes	characteristic
177	Dic₇₉	—	8	2	no	—
178	Dic₇₉	—	8	2	no	—
179	#19256	—	79	1	no	maximal, Sylow
180	C₂×C₇₉	—	2	2	no	—
181	C₂×C₇₉	—	4	4	no	—
182	C₂×C₇₉	#3036	1	1	yes	characteristic
183	C₂×C₇₉	C₂×#467	1	1	yes	characteristic
184	C₂×C₇₉	#3067	1	1	yes	characteristic
185	#3170	—	79	1	no	—
186	#3077	—	79	2	no	—
187	C₂×C₈×D₄	D₇₉	1	1	yes	characteristic
188	#1683	—	79	4	no	—
189	#1675	—	79	2	no	—
190	#1669	—	79	2	no	—
191	#1567	—	79	1	no	—
192	#1550	—	79	2	no	—
193	C₇₉	#19256	1	1	yes	characteristic, complemented, Sylow
194	C₂³×C₈	—	2	1	no	—
195	#533	—	79	2	no	—
196	C₂×C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
197	#500	—	79	1	no	—
198	#494	—	79	1	no	—
199	#487	—	79	2	no	—
200	#478	—	158	4	no	—
201	#476	—	79	4	no	—
202	#474	—	79	4	no	—
203	#467	—	158	1	no	—
204	#467	—	158	1	no	—
205	C₄×Q₁₆	—	158	1	no	—
206	C₄×Q₁₆	—	158	1	no	—
207	C₈×D₄	—	2	1	no	—
208	C₈×D₄	—	2	1	no	—
209	C₈×D₄	C₂×D₇₉	1	2	yes	—
210	C₈×D₄	C₂×D₇₉	1	2	yes	—
211	#428	—	158	2	no	—
212	#423	—	79	2	no	—
213	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
214	#418	—	158	2	no	—
215	C₂×C₂²⋊C₈	—	2	1	no	—
216	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
217	#335	—	79	2	no	—
218	#335	—	79	2	no	—
219	#333	—	79	4	no	—
220	#332	—	79	4	no	—
221	C₂²×D₄	D₃₁₆	1	1	yes	characteristic
222	C₂³×C₄	—	2	1	no	—
223	C₂×Q₁₆	—	158	1	no	—
224	C₂×Q₁₆	—	158	1	no	—
225	C₂×M₁₆	—	158	2	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	2	2	no	—
228	C₂²×C₈	—	2	2	no	—
229	C₂²×C₈	—	4	4	no	—
230	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
232	C₄⋊Q₈	—	79	2	no	—
233	C₄⋊Q₈	—	79	2	no	—
234	#125	—	158	2	no	—
235	C₂²⋊Q₈	—	158	2	no	—
236	C₂²⋊Q₈	—	158	2	no	—
237	C₄×Q₈	—	158	2	no	—
238	C₄×Q₈	—	158	2	no	—
239	C₄×D₄	—	2	1	no	—
240	C₄×D₄	—	2	1	no	—
241	C₄×D₄	D₃₁₆	1	2	yes	—
242	C₄×D₄	C₂²×D₇₉	1	2	yes	—
243	#117	—	158	2	no	—
244	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
245	C₂×C₂²⋊C₄	—	2	1	no	—
246	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
247	#107	—	158	1	no	—
248	#107	—	158	1	no	—
249	#107	—	158	4	no	—
250	#106	—	158	4	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	79	4	no	—
253	C₄⋊C₈	—	158	2	no	—
254	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
255	#103	—	158	2	no	—
256	#103	—	158	2	no	—
257	#102	—	158	4	no	—
258	#102	—	158	4	no	—
259	C₂²⋊C₈	—	2	2	no	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
262	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
265	C₂⁴	—	2	1	no	—
266	C₂×Q₈	—	158	2	no	—
267	C₂×Q₈	—	158	2	no	—
268	C₂×D₄	—	2	1	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	Dic₃₁₆	1	4	yes	complemented
271	C₂×D₄	C₂×D₃₁₆	1	2	yes	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
279	C₂²×C₄	#11929624	1	1	yes	characteristic
280	Q₁₆	—	316	2	no	—
281	Q₁₆	—	316	2	no	—
282	M₁₆	—	316	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	4	no	—
287	C₂×C₈	—	4	2	no	—
288	C₂×C₈	—	4	4	no	—
289	C₂×C₈	—	158	4	no	—
290	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	316	2	no	—
296	C₄⋊C₄	—	316	2	no	—
297	C₄⋊C₄	—	158	2	no	—
298	C₄⋊C₄	—	158	2	no	—
299	C₄⋊C₄	—	158	2	no	—
300	C₄⋊C₄	—	158	2	no	—
301	C₄⋊C₄	C₂×D₃₁₆	1	2	yes	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	316	2	no	—
305	C₄²	—	316	2	no	—
306	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
307	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
308	C₄²	#11929624	1	1	yes	characteristic
309	C₄²	C₂³×D₇₉	1	1	yes	characteristic
310	C₂³	—	2	2	no	—
311	C₂³	—	2	4	no	—
312	C₂³	—	4	2	no	—
313	C₂³	—	4	2	no	—
314	C₂³	#14579135	1	1	yes	characteristic
315	Q₈	—	316	2	no	—
316	Q₈	—	316	2	no	—
317	Q₈	—	632	2	no	—
318	D₄	—	2	4	no	—
319	D₄	—	2	2	no	—
320	D₄	C₂×Dic₃₁₆	1	4	yes	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	316	2	no	—
330	C₂×C₄	—	316	2	no	—
331	C₂×C₄	#14579135	1	1	yes	characteristic
332	C₂×C₄	#14579135	1	1	yes	characteristic
333	C₂×C₄	#14579135	1	1	yes	characteristic
334	C₂×C₄	C₂×Dic₃₁₆	1	2	yes	—
335	C₂×C₄	#14579183	1	2	yes	—
336	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
337	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
338	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	4	2	no	—
343	C₈	—	316	4	no	—
344	C₈	#14579187	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	4	no	—
348	C₂²	—	4	8	no	—
349	C₂²	—	4	2	no	—
350	C₂²	—	4	2	no	—
351	C₂²	#19098804	1	2	yes	—
352	C₂²	C₂×#14579135	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	632	2	no	—
358	C₄	—	632	2	no	—
359	C₄	#19098804	1	2	yes	—
360	C₄	C₂²×Dic₃₁₆	1	1	yes	characteristic
361	C₄	C₂×#14579183	1	1	yes	characteristic
362	C₄	C₂×#14579187	1	1	yes	characteristic
363	C₄	#19099361	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	4	4	no	—
366	C₂	#23772864	1	1	yes	characteristic
367	C₂	C₂×#19098804	1	1	yes	characteristic
368	C₂	#23773120	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579135
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×C₈×D₄×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#121
prime radical	C₂²×D₄×C₇₉	D₄
hypercenter	C₂×C₈×D₄	D₇₉
hypocenter	C₇₉	#19256
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #23773986	C₂	semidirect
3. #23762851	C₂	semidirect
4. #23762799	C₂	semidirect
5. #23762765	C₂	semidirect
6. #23762159	C₂	semidirect
7. #23762088	C₂	semidirect
10. #19098968	C₂²	semidirect
11. #19098943	C₂²	semidirect
12. #19098914	C₂²	semidirect
23. #19098316	C₂²	semidirect
24. #19098316	C₂²	semidirect
25. #19098311	C₂²	semidirect
26. #19098310	C₂²	semidirect
97. C₂×D₄×C₇₉	Q₁₆	semidirect
193. C₇₉	#19256	semidirect
270. C₂×D₄	Dic₃₁₆	semidirect

Maximal subgroups

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

subgroup	index
#23773986	2
#23773197	2
#23762851	2
#23762799	2
#23762765	2
#23762159	2
#23762088	2
C₂×C₈×D₄×C₇₉	2
#19256	79