Subgroups — #34271419

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774523	C₂	1	1	yes	maximal, characteristic
2	#23774327	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098914	C₂	1	1	yes	maximal, characteristic, complemented
4	#23772113	C₂	1	1	yes	maximal, characteristic
5	#23762948	C₂	1	4	yes	maximal, complemented
6	#23762824	C₂	1	4	yes	maximal, complemented
7	#23762084	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762071	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579167	—	2	1	no	—
11	#19099113	C₂²	1	1	yes	characteristic
12	C₂×#14579161	C₂²	1	1	yes	characteristic
13	C₂×#14579161	C₂²	1	1	yes	characteristic
14	#19098968	—	2	1	no	—
15	#19098936	—	2	1	no	—
16	#19098914	—	2	1	no	—
17	#19098914	—	2	1	no	—
18	#19098914	—	2	1	no	—
19	#19098914	C₂²	1	2	yes	—
20	#19098914	C₂²	1	2	yes	—
21	#19098910	—	2	1	no	—
22	#19098877	—	2	2	no	—
23	#19098844	C₂²	1	4	yes	—
24	#19098769	C₂²	1	4	yes	—
25	#19098704	—	2	2	no	—
26	#19098642	—	2	1	no	—
27	#19098642	C₂²	1	2	yes	—
28	C₈×D₃₁₆	—	2	1	no	—
29	C₈×D₃₁₆	C₂²	1	2	yes	—
30	#19098587	C₂²	1	1	yes	characteristic
31	C₂×#14579086	—	2	1	no	—
32	C₂×#14579086	—	2	1	no	—
33	#19098529	—	2	1	no	—
34	#19098310	C₂²	1	4	yes	—
35	#19098308	C₂²	1	4	yes	—
36	#431×C₇₉	C₂²	1	1	yes	characteristic
37	#431×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
39	#335×C₇₉	C₂²	1	2	yes	—
40	#334×C₇₉	C₂²	1	2	yes	—
41	C₂×#11929624	—	2	1	no	—
42	C₂²×Dic₁₅₈	—	2	1	no	—
43	C₂²×Dic₁₅₈	—	2	1	no	—
44	C₂×#11929613	—	2	4	no	—
45	#14579167	—	4	4	no	—
46	C₂×#11929608	—	2	1	no	—
47	C₂×#11929607	—	2	2	no	—
48	C₂×C₈×D₇₉	—	2	2	no	—
49	#14579161	—	2	1	no	—
50	#14579161	—	2	1	no	—
51	#14579161	—	2	1	no	—
52	#14579161	—	2	1	no	—
53	#14579161	—	2	1	no	—
54	#14579161	—	2	1	no	—
55	#14579161	D₄	1	1	yes	characteristic
56	#14579161	C₂³	1	2	yes	—
57	#14579161	C₂³	1	2	yes	—
58	#14579158	D₄	1	1	yes	characteristic
59	#14579149	—	2	1	no	—
60	#14579135	—	2	1	no	—
61	#14579135	D₄	1	2	yes	—
62	C₄×D₃₁₆	—	2	1	no	—
63	C₄×D₃₁₆	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₇₉	—	2	1	no	—
65	C₂×C₄⋊Dic₇₉	—	2	1	no	—
66	C₂×C₄⋊Dic₇₉	—	2	1	no	—
67	C₂×C₄⋊Dic₇₉	—	2	1	no	—
68	#14579095	—	2	4	no	—
69	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
71	#14579086	—	2	2	no	—
72	#14579086	—	2	2	no	—
73	#14579086	—	2	2	no	—
74	#14579086	—	2	2	no	—
75	#14579086	—	2	4	no	—
76	#14579083	—	2	4	no	—
77	(C₂×D₇₉)⋊C₈	—	2	2	no	—
78	(C₂×D₇₉)⋊C₈	—	2	2	no	—
79	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
83	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₇₉	—	2	1	no	—
85	C₂²⋊C₈×C₇₉	—	2	1	no	—
86	C₄×C₈×C₇₉	—	2	1	no	—
87	C₄×C₈×C₇₉	—	2	1	no	—
88	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
89	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
90	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	#11929624	—	4	2	no	—
93	#11929624	—	4	2	no	—
94	C₂×Dic₁₅₈	—	2	1	no	—
95	C₂×Dic₁₅₈	—	2	2	no	—
96	C₂×Dic₁₅₈	—	2	2	no	—
97	C₂×Dic₁₅₈	—	4	1	no	—
98	C₂×Dic₁₅₈	—	4	1	no	—
99	C₂×Dic₁₅₈	—	4	2	no	—
100	C₂×Dic₁₅₈	—	4	2	no	—
101	C₂×Dic₁₅₈	—	8	2	no	—
102	C₂×D₃₁₆	—	2	1	no	—
103	C₂×D₃₁₆	—	2	1	no	—
104	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
105	C₂×C₂²⋊D₇₉	—	2	2	no	—
106	C₂²×Dic₇₉	—	4	1	no	—
107	C₂²×Dic₇₉	—	4	1	no	—
108	C₂×C₄×D₇₉	—	2	2	no	—
109	#11929613	—	8	4	no	—
110	#11929608	—	4	2	no	—
111	#11929607	—	4	4	no	—
112	C₂×C₇₉⋊C₈	—	2	2	no	—
113	C₈×D₇₉	—	4	4	no	—
114	C₄⋊Dic₇₉	—	2	1	no	—
115	C₄⋊Dic₇₉	—	2	2	no	—
116	C₄⋊Dic₇₉	—	2	2	no	—
117	C₄⋊Dic₇₉	—	2	2	no	—
118	C₄⋊Dic₇₉	—	2	2	no	—
119	C₄⋊Dic₇₉	—	4	1	no	—
120	C₄⋊Dic₇₉	—	4	1	no	—
121	C₄⋊Dic₇₉	—	4	1	no	—
122	C₄⋊Dic₇₉	—	4	1	no	—
123	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
124	(C₂×D₇₉)⋊C₄	—	2	2	no	—
125	(C₂×D₇₉)⋊C₄	—	2	2	no	—
126	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
129	C₂×C₈×C₇₉	—	2	2	no	—
130	C₂×C₈×C₇₉	—	2	2	no	—
131	C₂×C₈×C₇₉	—	2	2	no	—
132	C₂×C₈×C₇₉	—	2	2	no	—
133	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₇₉	#41	1	1	yes	characteristic
140	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
141	C₂³×C₇₉	#121	1	1	yes	characteristic
142	C₂²×D₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	—	2	2	no	—
144	C₂×C₄×C₇₉	—	2	2	no	—
145	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₇₉	#135	1	2	yes	—
149	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
150	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
153	C₂²⋊D₇₉	—	4	4	no	—
154	C₂×Dic₇₉	—	2	2	no	—
155	C₂×Dic₇₉	—	4	2	no	—
156	C₂×Dic₇₉	—	4	2	no	—
157	C₂×Dic₇₉	—	8	2	no	—
158	C₂×Dic₇₉	—	8	2	no	—
159	D₃₁₆	—	2	4	no	—
160	D₃₁₆	—	4	2	no	—
161	D₃₁₆	—	4	2	no	—
162	C₄×D₇₉	—	4	2	no	—
163	C₄×D₇₉	—	4	2	no	—
164	Dic₁₅₈	—	4	2	no	—
165	Dic₁₅₈	—	4	2	no	—
166	Dic₁₅₈	—	4	2	no	—
167	Dic₁₅₈	—	4	4	no	—
168	Dic₁₅₈	—	8	2	no	—
169	Dic₁₅₈	—	8	2	no	—
170	C₈×C₇₉	—	2	1	no	—
171	C₈×C₇₉	—	2	1	no	—
172	C₈×C₇₉	—	2	1	no	—
173	C₈×C₇₉	—	2	1	no	—
174	C₈×C₇₉	—	4	2	no	—
175	C₈×C₇₉	—	4	2	no	—
176	C₇₉⋊C₈	—	4	2	no	—
177	C₂²×C₇₉	—	2	1	no	—
178	C₂²×C₇₉	—	2	1	no	—
179	C₂²×C₇₉	#463	1	2	yes	—
180	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
181	C₂×D₇₉	—	4	2	no	—
182	C₂×D₇₉	—	4	2	no	—
183	C₂×D₇₉	—	4	2	no	—
184	C₄×C₇₉	—	2	1	no	—
185	C₄×C₇₉	—	2	1	no	—
186	C₄×C₇₉	#463	1	2	yes	—
187	C₄×C₇₉	#473	1	2	yes	—
188	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
189	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₄×C₇₉	#577	1	2	yes	—
191	Dic₇₉	—	4	2	no	—
192	Dic₇₉	—	8	2	no	—
193	Dic₇₉	—	8	2	no	—
194	#19197	—	79	1	no	maximal, Sylow
195	C₂×C₇₉	—	2	2	no	—
196	C₂×C₇₉	C₂×#463	1	1	yes	characteristic
197	C₂×C₇₉	#3057	1	1	yes	characteristic
198	C₂×C₇₉	C₂×#473	1	1	yes	characteristic
199	D₇₉	—	8	2	no	—
200	#3226	—	79	1	no	—
201	#3206	—	79	1	no	—
202	C₂×#487	—	79	1	no	—
203	#2931	—	79	1	no	—
204	#1699	—	79	4	no	—
205	#1679	—	79	4	no	—
206	#1564	D₇₉	1	1	yes	characteristic, complemented
207	#1549	—	79	1	no	—
208	#1546	—	79	1	no	—
209	C₇₉	#19197	1	1	yes	characteristic, complemented, Sylow
210	C₂×#131	—	158	1	no	—
211	#532	—	79	1	no	—
212	C₂×C₄⋊Q₈	—	79	1	no	—
213	C₂×C₄⋊Q₈	—	79	1	no	—
214	#500	—	158	1	no	—
215	#492	—	158	1	no	—
216	#487	—	79	2	no	—
217	#487	—	79	2	no	—
218	#487	—	158	1	no	—
219	#487	—	158	1	no	—
220	#487	—	158	1	no	—
221	#486	—	158	1	no	—
222	#480	—	158	2	no	—
223	#474	—	79	4	no	—
224	#461	—	79	4	no	—
225	#448	—	158	2	no	—
226	#436	—	79	2	no	—
227	#436	—	158	1	no	—
228	C₈×D₄	—	79	2	no	—
229	C₈×D₄	—	158	1	no	—
230	#431	C₂×D₇₉	1	1	yes	characteristic
231	#431	C₂×D₇₉	1	1	yes	characteristic
232	#423	—	79	1	no	—
233	C₂×#103	—	158	1	no	—
234	C₂×#103	—	158	1	no	—
235	#408	—	158	1	no	—
236	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
237	#335	C₂×D₇₉	1	2	yes	—
238	#334	C₂×D₇₉	1	2	yes	—
239	#332	—	79	4	no	—
240	#332	—	79	4	no	—
241	C₂×#41	—	158	1	no	—
242	C₂²×Q₈	—	158	1	no	—
243	C₂²×Q₈	—	158	1	no	—
244	C₂×SD₁₆	—	158	4	no	—
245	#131	—	316	4	no	—
246	C₂×M₁₆	—	158	1	no	—
247	C₂×M₁₆	—	158	2	no	—
248	C₂²×C₈	—	158	2	no	—
249	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
250	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
251	C₄⋊Q₈	—	79	1	no	—
252	C₄⋊Q₈	—	79	2	no	—
253	C₄⋊Q₈	—	79	2	no	—
254	C₄⋊Q₈	—	158	1	no	—
255	C₄⋊Q₈	—	158	1	no	—
256	C₄⋊Q₈	—	158	1	no	—
257	C₄⋊Q₈	—	158	1	no	—
258	C₄⋊Q₈	—	158	1	no	—
259	C₄⋊Q₈	—	158	1	no	—
260	#127	—	79	1	no	—
261	#124	—	158	1	no	—
262	#121	—	79	2	no	—
263	#121	—	158	1	no	—
264	C₄×D₄	—	79	2	no	—
265	C₄×D₄	—	158	1	no	—
266	C₂×C₄⋊C₄	—	158	1	no	—
267	C₂×C₄⋊C₄	—	158	1	no	—
268	C₂×C₄⋊C₄	—	158	1	no	—
269	C₂×C₄⋊C₄	—	158	1	no	—
270	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
271	#107	—	158	4	no	—
272	C₄⋊C₈	—	79	2	no	—
273	C₄⋊C₈	—	158	1	no	—
274	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
275	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
276	#103	—	158	4	no	—
277	#103	—	158	2	no	—
278	#103	—	158	2	no	—
279	#103	—	158	2	no	—
280	#103	—	158	2	no	—
281	#102	—	158	4	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	158	2	no	—
285	C₂²⋊C₈	—	158	2	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	—	2	1	no	—
288	C₄×C₈	D₃₁₆	1	1	yes	characteristic
289	C₄×C₈	D₃₁₆	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
292	#41	—	316	2	no	—
293	#41	—	316	2	no	—
294	C₂×Q₈	—	316	1	no	—
295	C₂×Q₈	—	316	1	no	—
296	C₂×Q₈	—	316	2	no	—
297	C₂×Q₈	—	316	2	no	—
298	C₂×Q₈	—	632	2	no	—
299	C₂×Q₈	—	158	2	no	—
300	C₂×Q₈	—	158	2	no	—
301	C₂×Q₈	—	158	1	no	—
302	C₂×D₄	—	79	2	no	—
303	C₂×D₄	—	158	2	no	—
304	C₂×D₄	—	158	1	no	—
305	C₂×D₄	—	158	1	no	—
306	C₂²×C₄	—	316	1	no	—
307	C₂²×C₄	—	316	1	no	—
308	C₂²×C₄	—	158	2	no	—
309	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
311	C₂²×C₄	#11929625	1	1	yes	characteristic
312	SD₁₆	—	632	4	no	—
313	M₁₆	—	316	4	no	—
314	M₁₆	—	316	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	316	4	no	—
320	C₂×C₈	—	158	2	no	—
321	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
322	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
323	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
324	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
325	C₄⋊C₄	—	316	1	no	—
326	C₄⋊C₄	—	316	1	no	—
327	C₄⋊C₄	—	316	1	no	—
328	C₄⋊C₄	—	316	1	no	—
329	C₄⋊C₄	—	79	2	no	—
330	C₄⋊C₄	—	158	2	no	—
331	C₄⋊C₄	—	158	2	no	—
332	C₄⋊C₄	—	158	2	no	—
333	C₄⋊C₄	—	158	2	no	—
334	C₄⋊C₄	—	158	1	no	—
335	C₂²⋊C₄	—	158	2	no	—
336	C₂²⋊C₄	—	158	2	no	—
337	C₄²	D₄×D₇₉	1	1	yes	characteristic
338	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
339	C₄²	#11929625	1	1	yes	characteristic
340	C₄²	C₂³×D₇₉	1	1	yes	characteristic
341	C₂³	—	158	2	no	—
342	C₂³	#14579136	1	1	yes	characteristic
343	Q₈	—	316	2	no	—
344	Q₈	—	316	2	no	—
345	Q₈	—	316	2	no	—
346	Q₈	—	316	4	no	—
347	Q₈	—	632	2	no	—
348	Q₈	—	632	2	no	—
349	D₄	—	316	4	no	—
350	D₄	—	316	2	no	—
351	D₄	—	316	2	no	—
352	D₄	—	158	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	316	2	no	—
356	C₂×C₄	—	316	2	no	—
357	C₂×C₄	—	316	2	no	—
358	C₂×C₄	—	316	2	no	—
359	C₂×C₄	—	632	2	no	—
360	C₂×C₄	—	632	2	no	—
361	C₂×C₄	—	158	2	no	—
362	C₂×C₄	#14579136	1	1	yes	characteristic
363	C₂×C₄	#14579136	1	1	yes	characteristic
364	C₂×C₄	#14579136	1	1	yes	characteristic
365	C₂×C₄	#14579183	1	2	yes	—
366	C₂×C₄	#14579189	1	2	yes	—
367	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
369	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	4	2	no	—
375	C₈	—	4	2	no	—
376	C₈	—	316	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	316	2	no	—
380	C₂²	—	316	2	no	—
381	C₂²	—	316	2	no	—
382	C₂²	#19098779	1	2	yes	—
383	C₂²	C₂×#14579136	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	316	2	no	—
387	C₄	—	632	2	no	—
388	C₄	—	632	2	no	—
389	C₄	#19098779	1	2	yes	—
390	C₄	#19098839	1	2	yes	—
391	C₄	C₂×#14579183	1	1	yes	characteristic
392	C₄	C₂×#14579189	1	1	yes	characteristic
393	C₄	#19099370	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	632	2	no	—
396	C₂	C₂×#19098779	1	1	yes	characteristic
397	C₂	#23773040	1	1	yes	characteristic
398	C₂	C₂×#19098839	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579136
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1564	C₂
socle	C₂²×C₇₉	C₂×#121
prime radical	#19099113	C₂²
hypercenter	#1564	D₇₉
hypocenter	C₇₉	#19197
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₂×#19098914	C₂	semidirect
5. #23762948	C₂	semidirect
6. #23762824	C₂	semidirect
7. #23762084	C₂	semidirect
8. #23762071	C₂	semidirect
9. C₇₉×#1564	C₂	semidirect
206. #1564	D₇₉	semidirect
209. C₇₉	#19197	semidirect

Maximal subgroups

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

subgroup	index
#23774523	2
#23774327	2
C₂×#19098914	2
#23772113	2
#23762948	2
#23762824	2
#23762084	2
#23762071	2
C₇₉×#1564	2
#19197	79