Subgroups — #34271371

Order 20224.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774530	C₂	1	1	yes	maximal, characteristic
2	#23774327	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098961	C₂	1	1	yes	maximal, characteristic, complemented
4	#23772113	C₂	1	1	yes	maximal, characteristic
5	#23762900	C₂	1	4	yes	maximal, complemented
6	#23762776	C₂	1	4	yes	maximal, complemented
7	#23762084	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762080	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	#19098961	—	2	1	no	—
15	#19098961	—	2	1	no	—
16	#19098961	—	2	1	no	—
17	#19098961	C₂²	1	2	yes	—
18	#19098961	C₂²	1	2	yes	—
19	#19098936	—	2	1	no	—
20	#19098914	—	2	1	no	—
21	#19098910	—	2	1	no	—
22	#19098882	—	2	2	no	—
23	#19098854	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	C₂×#14579093	—	2	1	no	—
31	C₂×#14579093	—	2	1	no	—
32	#19098587	C₂²	1	1	yes	characteristic
33	#19098529	—	2	1	no	—
34	#19098311	C₂²	1	4	yes	—
35	#19098310	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	#14579093	—	2	2	no	—
69	#14579093	—	2	2	no	—
70	#14579093	—	2	2	no	—
71	#14579093	—	2	2	no	—
72	#14579093	—	2	4	no	—
73	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
74	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
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	—	8	2	no	—
100	C₂×Dic₁₅₈	—	8	2	no	—
101	C₂×D₃₁₆	—	2	1	no	—
102	C₂×D₃₁₆	—	2	1	no	—
103	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
104	C₂×C₂²⋊D₇₉	—	2	2	no	—
105	C₂²×Dic₇₉	—	4	1	no	—
106	C₂²×Dic₇₉	—	4	1	no	—
107	C₂×C₄×D₇₉	—	2	2	no	—
108	#11929613	—	8	4	no	—
109	#11929608	—	4	2	no	—
110	#11929607	—	4	4	no	—
111	C₂×C₇₉⋊C₈	—	2	2	no	—
112	C₈×D₇₉	—	4	4	no	—
113	C₄⋊Dic₇₉	—	2	1	no	—
114	C₄⋊Dic₇₉	—	2	2	no	—
115	C₄⋊Dic₇₉	—	2	2	no	—
116	C₄⋊Dic₇₉	—	2	2	no	—
117	C₄⋊Dic₇₉	—	2	2	no	—
118	C₄⋊Dic₇₉	—	4	1	no	—
119	C₄⋊Dic₇₉	—	4	1	no	—
120	C₄⋊Dic₇₉	—	4	1	no	—
121	C₄⋊Dic₇₉	—	4	1	no	—
122	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
123	(C₂×D₇₉)⋊C₄	—	2	2	no	—
124	(C₂×D₇₉)⋊C₄	—	2	2	no	—
125	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
128	C₂×C₈×C₇₉	—	2	2	no	—
129	C₂×C₈×C₇₉	—	2	2	no	—
130	C₂×C₈×C₇₉	—	2	2	no	—
131	C₂×C₈×C₇₉	SD₁₆	1	4	yes	complemented
132	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
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	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₇₉	#41	1	1	yes	characteristic
139	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
140	C₂³×C₇₉	#121	1	1	yes	characteristic
141	C₂²×D₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	—	2	2	no	—
144	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₇₉	#137	1	2	yes	—
149	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₇₉	—	4	4	no	—
153	C₂×Dic₇₉	—	2	2	no	—
154	C₂×Dic₇₉	—	4	2	no	—
155	C₂×Dic₇₉	—	4	2	no	—
156	C₂×Dic₇₉	—	8	2	no	—
157	C₂×Dic₇₉	—	8	2	no	—
158	D₃₁₆	—	2	4	no	—
159	D₃₁₆	—	4	2	no	—
160	D₃₁₆	—	4	2	no	—
161	C₄×D₇₉	—	4	2	no	—
162	C₄×D₇₉	—	4	2	no	—
163	Dic₁₅₈	—	4	2	no	—
164	Dic₁₅₈	—	4	4	no	—
165	Dic₁₅₈	—	8	2	no	—
166	Dic₁₅₈	—	8	2	no	—
167	Dic₁₅₈	—	8	2	no	—
168	C₈×C₇₉	—	2	1	no	—
169	C₈×C₇₉	—	2	1	no	—
170	C₈×C₇₉	—	4	2	no	—
171	C₈×C₇₉	—	4	2	no	—
172	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
173	C₈×C₇₉	C₂×SD₁₆	1	2	yes	—
174	C₇₉⋊C₈	—	4	2	no	—
175	C₂²×C₇₉	—	2	1	no	—
176	C₂²×C₇₉	—	2	1	no	—
177	C₂²×C₇₉	#461	1	2	yes	—
178	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
179	C₂×D₇₉	—	4	2	no	—
180	C₂×D₇₉	—	4	2	no	—
181	C₂×D₇₉	—	4	2	no	—
182	C₄×C₇₉	—	2	1	no	—
183	C₄×C₇₉	—	2	1	no	—
184	C₄×C₇₉	#461	1	2	yes	—
185	C₄×C₇₉	#471	1	2	yes	—
186	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
187	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
188	C₄×C₇₉	#577	1	2	yes	—
189	Dic₇₉	—	4	2	no	—
190	Dic₇₉	—	8	2	no	—
191	Dic₇₉	—	8	2	no	—
192	#19191	—	79	1	no	maximal, Sylow
193	C₂×C₇₉	—	2	2	no	—
194	C₂×C₇₉	C₂×#461	1	1	yes	characteristic
195	C₂×C₇₉	#3057	1	1	yes	characteristic
196	C₂×C₇₉	C₂×#471	1	1	yes	characteristic
197	D₇₉	—	8	2	no	—
198	#3227	—	79	1	no	—
199	#3206	—	79	1	no	—
200	C₂×#498	—	79	1	no	—
201	#2931	—	79	1	no	—
202	#1691	—	79	4	no	—
203	#1671	—	79	4	no	—
204	#1564	D₇₉	1	1	yes	characteristic, complemented
205	#1549	—	79	1	no	—
206	#1548	—	79	1	no	—
207	C₇₉	#19191	1	1	yes	characteristic, complemented, Sylow
208	C₂×#131	—	158	1	no	—
209	#532	—	79	1	no	—
210	C₂×C₄⋊Q₈	—	79	1	no	—
211	C₂×C₄⋊Q₈	—	79	1	no	—
212	#498	—	79	2	no	—
213	#498	—	79	2	no	—
214	#498	—	158	1	no	—
215	#498	—	158	1	no	—
216	#498	—	158	1	no	—
217	#492	—	158	1	no	—
218	#487	—	158	1	no	—
219	#486	—	158	1	no	—
220	#481	—	158	2	no	—
221	#476	—	79	4	no	—
222	#461	—	79	4	no	—
223	#448	—	158	2	no	—
224	#436	—	79	2	no	—
225	#436	—	158	1	no	—
226	C₈×D₄	—	79	2	no	—
227	C₈×D₄	—	158	1	no	—
228	#431	C₂×D₇₉	1	1	yes	characteristic
229	#431	C₂×D₇₉	1	1	yes	characteristic
230	C₂×#106	—	158	1	no	—
231	C₂×#106	—	158	1	no	—
232	#423	—	79	1	no	—
233	#408	—	158	1	no	—
234	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
235	#335	C₂×D₇₉	1	2	yes	—
236	#334	C₂×D₇₉	1	2	yes	—
237	#333	—	79	4	no	—
238	#332	—	79	4	no	—
239	C₂×#41	—	158	1	no	—
240	C₂²×Q₈	—	158	1	no	—
241	C₂²×Q₈	—	158	1	no	—
242	C₂×SD₁₆	—	158	4	no	—
243	#131	—	316	4	no	—
244	C₂×M₁₆	—	158	1	no	—
245	C₂×M₁₆	—	158	2	no	—
246	C₂²×C₈	—	158	2	no	—
247	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
248	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
249	C₄⋊Q₈	—	79	1	no	—
250	C₄⋊Q₈	—	79	2	no	—
251	C₄⋊Q₈	—	79	2	no	—
252	C₄⋊Q₈	—	158	1	no	—
253	C₄⋊Q₈	—	158	1	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	#127	—	79	1	no	—
259	#124	—	158	1	no	—
260	#121	—	79	2	no	—
261	#121	—	158	1	no	—
262	C₄×D₄	—	79	2	no	—
263	C₄×D₄	—	158	1	no	—
264	C₂×C₄⋊C₄	—	158	1	no	—
265	C₂×C₄⋊C₄	—	158	1	no	—
266	C₂×C₄⋊C₄	—	158	1	no	—
267	C₂×C₄⋊C₄	—	158	1	no	—
268	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
269	#106	—	158	4	no	—
270	#106	—	158	2	no	—
271	#106	—	158	2	no	—
272	#106	—	158	2	no	—
273	#106	—	158	2	no	—
274	C₄⋊C₈	—	79	2	no	—
275	C₄⋊C₈	—	158	1	no	—
276	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
277	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
278	#103	—	158	4	no	—
279	#102	—	158	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	158	2	no	—
283	C₂²⋊C₈	—	158	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	D₃₁₆	1	1	yes	characteristic
287	C₄×C₈	D₃₁₆	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
290	#41	—	316	2	no	—
291	#41	—	316	2	no	—
292	C₂×Q₈	—	316	1	no	—
293	C₂×Q₈	—	316	1	no	—
294	C₂×Q₈	—	632	2	no	—
295	C₂×Q₈	—	632	2	no	—
296	C₂×Q₈	—	158	2	no	—
297	C₂×Q₈	—	158	2	no	—
298	C₂×Q₈	—	158	1	no	—
299	C₂×D₄	—	79	2	no	—
300	C₂×D₄	—	158	2	no	—
301	C₂×D₄	—	158	1	no	—
302	C₂×D₄	—	158	1	no	—
303	C₂²×C₄	—	316	1	no	—
304	C₂²×C₄	—	316	1	no	—
305	C₂²×C₄	—	158	2	no	—
306	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
308	C₂²×C₄	#11929625	1	1	yes	characteristic
309	SD₁₆	—	632	4	no	—
310	M₁₆	—	316	2	no	—
311	M₁₆	—	316	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	316	4	no	—
316	C₂×C₈	—	158	2	no	—
317	C₂×C₈	#11929613	1	4	yes	complemented
318	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
319	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
320	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
321	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
322	C₄⋊C₄	—	316	1	no	—
323	C₄⋊C₄	—	316	1	no	—
324	C₄⋊C₄	—	316	1	no	—
325	C₄⋊C₄	—	316	1	no	—
326	C₄⋊C₄	—	79	2	no	—
327	C₄⋊C₄	—	158	2	no	—
328	C₄⋊C₄	—	158	2	no	—
329	C₄⋊C₄	—	158	2	no	—
330	C₄⋊C₄	—	158	2	no	—
331	C₄⋊C₄	—	158	1	no	—
332	C₂²⋊C₄	—	158	2	no	—
333	C₂²⋊C₄	—	158	2	no	—
334	C₄²	D₄×D₇₉	1	1	yes	characteristic
335	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
336	C₄²	#11929625	1	1	yes	characteristic
337	C₄²	C₂³×D₇₉	1	1	yes	characteristic
338	C₂³	—	158	2	no	—
339	C₂³	#14579136	1	1	yes	characteristic
340	Q₈	—	316	2	no	—
341	Q₈	—	316	4	no	—
342	Q₈	—	632	2	no	—
343	Q₈	—	632	2	no	—
344	Q₈	—	632	2	no	—
345	D₄	—	316	4	no	—
346	D₄	—	316	2	no	—
347	D₄	—	316	2	no	—
348	D₄	—	158	4	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	316	2	no	—
352	C₂×C₄	—	316	2	no	—
353	C₂×C₄	—	316	2	no	—
354	C₂×C₄	—	316	2	no	—
355	C₂×C₄	—	632	2	no	—
356	C₂×C₄	—	632	2	no	—
357	C₂×C₄	—	158	2	no	—
358	C₂×C₄	#14579136	1	1	yes	characteristic
359	C₂×C₄	#14579136	1	1	yes	characteristic
360	C₂×C₄	#14579136	1	1	yes	characteristic
361	C₂×C₄	C₂×#11929613	1	2	yes	—
362	C₂×C₄	#14579194	1	2	yes	—
363	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
365	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	316	2	no	—
371	C₈	C₂×#11929613	1	2	yes	—
372	C₈	C₂×#11929613	1	2	yes	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	316	2	no	—
376	C₂²	—	316	2	no	—
377	C₂²	—	316	2	no	—
378	C₂²	#19098769	1	2	yes	—
379	C₂²	C₂×#14579136	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	316	2	no	—
383	C₄	—	632	2	no	—
384	C₄	—	632	2	no	—
385	C₄	#19098769	1	2	yes	—
386	C₄	#19098827	1	2	yes	—
387	C₄	C₂²×#11929613	1	1	yes	characteristic
388	C₄	C₂×#14579194	1	1	yes	characteristic
389	C₄	#19099370	1	2	yes	—
390	C₂	—	2	2	no	—
391	C₂	—	632	2	no	—
392	C₂	C₂×#19098769	1	1	yes	characteristic
393	C₂	#23773040	1	1	yes	characteristic
394	C₂	C₂×#19098827	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	#19191
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×#19098961	C₂	semidirect
5. #23762900	C₂	semidirect
6. #23762776	C₂	semidirect
7. #23762084	C₂	semidirect
8. #23762080	C₂	semidirect
9. C₇₉×#1564	C₂	semidirect
131. C₂×C₈×C₇₉	SD₁₆	semidirect
204. #1564	D₇₉	semidirect
207. C₇₉	#19191	semidirect
317. C₂×C₈	#11929613	semidirect

Maximal subgroups

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

subgroup	index
#23774530	2
#23774327	2
C₂×#19098961	2
#23772113	2
#23762900	2
#23762776	2
#23762084	2
#23762080	2
C₇₉×#1564	2
#19191	79