Subgroups — #34271378

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774327	C₂	1	1	yes	maximal, characteristic
2	#23774325	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098966	C₂	1	1	yes	maximal, characteristic, complemented
4	#23772111	C₂	1	1	yes	maximal, characteristic
5	#23762912	C₂	1	4	yes	maximal, complemented
6	#23762788	C₂	1	4	yes	maximal, complemented
7	#23762146	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762086	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#1549	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579169	—	2	1	no	—
11	#19099113	C₂²	1	1	yes	characteristic
12	C₂×#14579161	C₂²	1	1	yes	characteristic
13	#19098966	—	2	1	no	—
14	#19098966	—	2	1	no	—
15	#19098966	C₂²	1	2	yes	—
16	#19098966	C₂²	1	2	yes	—
17	#19098937	—	2	1	no	—
18	#19098936	—	2	1	no	—
19	#19098914	—	2	1	no	—
20	#19098913	—	2	1	no	—
21	#19098769	C₂²	1	4	yes	—
22	#19098765	C₂²	1	4	yes	—
23	#19098704	—	2	2	no	—
24	#19098702	—	2	2	no	—
25	#19098640	—	2	1	no	—
26	#19098640	C₂²	1	2	yes	—
27	#19098628	—	2	1	no	—
28	#19098628	C₂²	1	2	yes	—
29	#19098616	C₂²	1	1	yes	characteristic
30	C₂×#14579094	—	2	1	no	—
31	C₂×#14579094	—	2	1	no	—
32	#19098531	—	2	1	no	—
33	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
34	#19098315	C₂²	1	2	yes	—
35	#19098313	C₂²	1	2	yes	—
36	#19098308	C₂²	1	4	yes	—
37	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
38	#431×C₇₉	C₂²	1	1	yes	characteristic
39	#423×C₇₉	C₂²	1	1	yes	characteristic
40	#332×C₇₉	C₂²	1	4	yes	—
41	C₂×#11929624	—	2	1	no	—
42	C₂²×Dic₁₅₈	—	2	1	no	—
43	C₂×#11929613	—	2	4	no	—
44	C₂×#11929611	—	2	4	no	—
45	#14579169	—	4	4	no	—
46	C₂×#11929607	—	2	2	no	—
47	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
48	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
49	C₂×C₈×D₇₉	—	2	2	no	—
50	#14579161	—	2	1	no	—
51	#14579161	—	2	1	no	—
52	#14579161	—	2	1	no	—
53	#14579161	D₄	1	1	yes	characteristic
54	#14579161	C₂³	1	2	yes	—
55	#14579158	D₄	1	1	yes	characteristic
56	#14579149	—	2	1	no	—
57	#14579135	—	2	1	no	—
58	#14579135	D₄	1	2	yes	—
59	C₄×D₃₁₆	—	2	1	no	—
60	C₄×D₃₁₆	C₂³	1	2	yes	—
61	C₂×C₄⋊Dic₇₉	—	2	1	no	—
62	C₂×C₄⋊Dic₇₉	—	2	1	no	—
63	#14579094	—	2	2	no	—
64	#14579094	—	2	2	no	—
65	#14579094	—	2	2	no	—
66	#14579094	—	2	2	no	—
67	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
68	#14579086	—	2	4	no	—
69	#14579085	—	2	4	no	—
70	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
71	(C₂×D₇₉)⋊C₈	—	2	2	no	—
72	(C₂×D₇₉)⋊C₈	—	2	2	no	—
73	C₄×C₇₉⋊C₈	—	2	1	no	—
74	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
76	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₇₉	—	2	1	no	—
79	C₂×M₁₆×C₇₉	—	2	1	no	—
80	C₄⋊Q₈×C₇₉	—	2	1	no	—
81	C₄⋊Q₈×C₇₉	—	2	1	no	—
82	C₄⋊Q₈×C₇₉	—	2	1	no	—
83	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
86	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₇₉	—	2	1	no	—
88	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
89	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
90	C₂²⋊C₈×C₇₉	—	2	1	no	—
91	C₄×C₈×C₇₉	—	2	1	no	—
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	—	4	1	no	—
97	C₂×Dic₁₅₈	—	8	2	no	—
98	C₂×D₃₁₆	—	2	1	no	—
99	C₂×D₃₁₆	—	2	1	no	—
100	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
101	C₂×C₂²⋊D₇₉	—	2	2	no	—
102	C₂²×Dic₇₉	—	4	1	no	—
103	C₂×C₄×D₇₉	—	2	2	no	—
104	#11929613	—	8	4	no	—
105	#11929611	—	8	4	no	—
106	#11929607	—	4	4	no	—
107	C₂×C₇₉⋊C₈	—	2	2	no	—
108	C₂×C₇₉⋊C₈	—	2	2	no	—
109	C₂×C₇₉⋊C₈	SD₁₆	1	4	yes	complemented
110	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
114	C₈×D₇₉	—	4	4	no	—
115	C₄⋊Dic₇₉	—	2	1	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	C₂×D₄	1	2	yes	—
121	(C₂×D₇₉)⋊C₄	—	2	2	no	—
122	(C₂×D₇₉)⋊C₄	—	2	2	no	—
123	C₂×Q₈×C₇₉	—	2	2	no	—
124	C₂×Q₈×C₇₉	—	4	1	no	—
125	C₂×Q₈×C₇₉	—	8	2	no	—
126	C₂²×C₄×C₇₉	—	4	1	no	—
127	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
130	M₁₆×C₇₉	—	4	2	no	—
131	C₂×C₈×C₇₉	—	2	2	no	—
132	C₂×C₈×C₇₉	—	2	2	no	—
133	C₄⋊C₄×C₇₉	—	2	2	no	—
134	C₄⋊C₄×C₇₉	—	2	2	no	—
135	C₄⋊C₄×C₇₉	—	4	1	no	—
136	C₄⋊C₄×C₇₉	—	4	1	no	—
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	Q₈×C₇₉	—	4	4	no	—
144	Q₈×C₇₉	—	8	2	no	—
145	C₂×C₄×C₇₉	—	2	2	no	—
146	C₂×C₄×C₇₉	—	2	2	no	—
147	C₂×C₄×C₇₉	—	4	2	no	—
148	C₂×C₄×C₇₉	—	8	2	no	—
149	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
153	C₂×C₄×C₇₉	#137	1	2	yes	—
154	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
157	C₂²⋊D₇₉	—	4	4	no	—
158	C₂×Dic₇₉	—	2	2	no	—
159	C₂×Dic₇₉	—	4	2	no	—
160	C₂×Dic₇₉	—	8	2	no	—
161	D₃₁₆	—	2	4	no	—
162	D₃₁₆	—	4	2	no	—
163	D₃₁₆	—	4	2	no	—
164	C₄×D₇₉	—	4	2	no	—
165	C₄×D₇₉	—	4	2	no	—
166	Dic₁₅₈	—	4	2	no	—
167	Dic₁₅₈	—	4	4	no	—
168	Dic₁₅₈	—	8	2	no	—
169	C₈×C₇₉	—	4	2	no	—
170	C₈×C₇₉	—	4	2	no	—
171	C₇₉⋊C₈	—	2	1	no	—
172	C₇₉⋊C₈	—	2	1	no	—
173	C₇₉⋊C₈	—	4	2	no	—
174	C₇₉⋊C₈	C₂×SD₁₆	1	2	yes	—
175	C₇₉⋊C₈	C₂×SD₁₆	1	2	yes	—
176	C₂²×C₇₉	—	2	1	no	—
177	C₂²×C₇₉	—	2	1	no	—
178	C₂²×C₇₉	#461	1	2	yes	—
179	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
180	C₂×D₇₉	—	4	2	no	—
181	C₂×D₇₉	—	4	2	no	—
182	C₂×D₇₉	—	4	2	no	—
183	C₄×C₇₉	—	2	1	no	—
184	C₄×C₇₉	—	2	1	no	—
185	C₄×C₇₉	—	8	2	no	—
186	C₄×C₇₉	#461	1	2	yes	—
187	C₄×C₇₉	#473	1	2	yes	—
188	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
189	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
190	C₄×C₇₉	#576	1	2	yes	—
191	Dic₇₉	—	4	2	no	—
192	Dic₇₉	—	8	2	no	—
193	#19192	—	79	1	no	maximal, Sylow
194	C₂×C₇₉	—	2	2	no	—
195	C₂×C₇₉	C₂×#461	1	1	yes	characteristic
196	C₂×C₇₉	#3058	1	1	yes	characteristic
197	C₂×C₇₉	C₂×#473	1	1	yes	characteristic
198	D₇₉	—	8	2	no	—
199	#3206	—	79	1	no	—
200	#3206	—	79	1	no	—
201	C₂×#500	—	79	1	no	—
202	#2931	—	79	1	no	—
203	#1693	—	79	4	no	—
204	#1673	—	79	4	no	—
205	#1564	—	79	1	no	—
206	#1549	—	79	1	no	—
207	#1549	D₇₉	1	1	yes	characteristic, complemented
208	C₇₉	#19192	1	1	yes	characteristic, complemented, Sylow
209	C₂×#131	—	158	1	no	—
210	#532	—	79	1	no	—
211	C₂×C₄⋊Q₈	—	79	1	no	—
212	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
213	#500	—	79	2	no	—
214	#500	—	79	2	no	—
215	#500	—	158	1	no	—
216	#500	—	158	1	no	—
217	#492	—	158	1	no	—
218	#492	—	158	1	no	—
219	#487	—	158	1	no	—
220	#487	—	158	1	no	—
221	#461	—	79	4	no	—
222	#461	—	79	4	no	—
223	#448	—	158	2	no	—
224	#448	—	158	2	no	—
225	#436	—	79	2	no	—
226	#436	—	158	1	no	—
227	C₈×D₄	—	79	2	no	—
228	C₈×D₄	—	158	1	no	—
229	#431	—	79	1	no	—
230	#431	C₂×D₇₉	1	1	yes	characteristic
231	C₂×#107	—	158	1	no	—
232	C₂×#107	—	158	1	no	—
233	#423	C₂×D₇₉	1	1	yes	characteristic
234	#408	—	158	1	no	—
235	C₂×C₄×C₈	—	79	1	no	—
236	#335	—	79	2	no	—
237	#334	—	79	2	no	—
238	#332	—	79	4	no	—
239	#332	C₂×D₇₉	1	4	yes	—
240	C₂×#41	—	158	1	no	—
241	C₂²×Q₈	—	2	1	no	—
242	C₂²×Q₈	—	158	1	no	—
243	C₂×SD₁₆	—	158	4	no	—
244	C₂×SD₁₆	—	158	4	no	—
245	#131	—	316	4	no	—
246	C₂×M₁₆	—	2	1	no	—
247	C₂×M₁₆	—	158	2	no	—
248	C₂²×C₈	—	79	1	no	—
249	C₂²×C₈	—	79	1	no	—
250	C₂²×C₈	—	158	2	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	79	1	no	—
255	C₄⋊Q₈	—	79	2	no	—
256	C₄⋊Q₈	—	158	1	no	—
257	C₄⋊Q₈	—	158	1	no	—
258	C₄⋊Q₈	—	158	1	no	—
259	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
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	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	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	2	no	—
272	#107	—	158	2	no	—
273	#107	—	158	2	no	—
274	#107	—	158	2	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	79	2	no	—
277	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
278	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
279	#103	—	158	4	no	—
280	#103	—	158	4	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	158	2	no	—
283	C₂²⋊C₈	—	158	2	no	—
284	C₂²⋊C₈	—	158	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	79	1	no	—
287	C₄×C₈	—	79	1	no	—
288	C₄×C₈	—	79	1	no	—
289	C₄×C₈	—	79	1	no	—
290	C₄×C₈	—	158	1	no	—
291	#41	—	316	2	no	—
292	#41	—	316	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	—	8	2	no	—
296	C₂×Q₈	—	316	1	no	—
297	C₂×Q₈	—	632	2	no	—
298	C₂×Q₈	—	158	2	no	—
299	C₂×Q₈	—	158	1	no	—
300	C₂×D₄	—	79	2	no	—
301	C₂×D₄	—	158	2	no	—
302	C₂×D₄	—	158	1	no	—
303	C₂×D₄	—	158	1	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	—	316	1	no	—
306	C₂²×C₄	—	158	2	no	—
307	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
308	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
309	C₂²×C₄	#11929624	1	1	yes	characteristic
310	SD₁₆	—	632	4	no	—
311	SD₁₆	—	632	4	no	—
312	M₁₆	—	4	2	no	—
313	M₁₆	—	316	4	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	316	4	no	—
317	C₂×C₈	—	79	4	no	—
318	C₂×C₈	—	79	1	no	—
319	C₂×C₈	—	79	1	no	—
320	C₂×C₈	—	79	1	no	—
321	C₂×C₈	—	79	1	no	—
322	C₂×C₈	—	158	2	no	—
323	C₂×C₈	—	158	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	316	1	no	—
329	C₄⋊C₄	—	316	1	no	—
330	C₄⋊C₄	—	79	2	no	—
331	C₄⋊C₄	—	158	2	no	—
332	C₄⋊C₄	—	158	2	no	—
333	C₄⋊C₄	—	158	1	no	—
334	C₂²⋊C₄	—	158	2	no	—
335	C₂²⋊C₄	—	158	2	no	—
336	C₄²	D₄×D₇₉	1	1	yes	characteristic
337	C₄²	D₄×D₇₉	1	1	yes	characteristic
338	C₄²	#11929624	1	1	yes	characteristic
339	C₄²	C₂³×D₇₉	1	1	yes	characteristic
340	C₂³	—	158	2	no	—
341	C₂³	#14579134	1	1	yes	characteristic
342	Q₈	—	4	4	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	316	2	no	—
345	Q₈	—	316	4	no	—
346	Q₈	—	632	2	no	—
347	D₄	—	316	4	no	—
348	D₄	—	316	2	no	—
349	D₄	—	316	2	no	—
350	D₄	—	158	4	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	—	316	2	no	—
356	C₂×C₄	—	316	2	no	—
357	C₂×C₄	—	316	2	no	—
358	C₂×C₄	—	632	2	no	—
359	C₂×C₄	—	158	2	no	—
360	C₂×C₄	#14579134	1	1	yes	characteristic
361	C₂×C₄	#14579134	1	1	yes	characteristic
362	C₂×C₄	#14579134	1	1	yes	characteristic
363	C₂×C₄	SD₁₆×D₇₉	1	2	yes	—
364	C₂×C₄	#14579193	1	2	yes	—
365	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
366	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
367	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	316	2	no	—
371	C₈	—	79	2	no	—
372	C₈	—	79	2	no	—
373	C₈	—	158	1	no	—
374	C₈	—	158	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	1	no	—
377	C₂²	—	316	2	no	—
378	C₂²	—	316	2	no	—
379	C₂²	—	316	2	no	—
380	C₂²	#19098767	1	2	yes	—
381	C₂²	C₂×#14579134	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	8	2	no	—
385	C₄	—	316	2	no	—
386	C₄	—	632	2	no	—
387	C₄	#19098767	1	2	yes	—
388	C₄	#19098837	1	2	yes	—
389	C₄	C₂×SD₁₆×D₇₉	1	1	yes	characteristic
390	C₄	C₂×#14579193	1	1	yes	characteristic
391	C₄	#19099367	1	2	yes	—
392	C₂	—	2	2	no	—
393	C₂	—	632	2	no	—
394	C₂	C₂×#19098767	1	1	yes	characteristic
395	C₂	#23773047	1	1	yes	characteristic
396	C₂	C₂×#19098837	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579134
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1549	C₂
socle	C₂²×C₇₉	C₂×#121
prime radical	#19099113	C₂²
hypercenter	#1549	D₇₉
hypocenter	C₇₉	#19192
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1549

Complements

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

N	H ≅ G/N	type
3. C₂×#19098966	C₂	semidirect
5. #23762912	C₂	semidirect
6. #23762788	C₂	semidirect
7. #23762146	C₂	semidirect
8. #23762086	C₂	semidirect
9. C₇₉×#1549	C₂	semidirect
109. C₂×C₇₉⋊C₈	SD₁₆	semidirect
207. #1549	D₇₉	semidirect
208. C₇₉	#19192	semidirect

Maximal subgroups

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

subgroup	index
#23774327	2
#23774325	2
C₂×#19098966	2
#23772111	2
#23762912	2
#23762788	2
#23762146	2
#23762086	2
C₇₉×#1549	2
#19192	79