Subgroups — #34443322

Order 20224.

542 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773984	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic
3	#23765596	C₂	1	2	yes	maximal
4	#23765543	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765542	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765447	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765332	C₂	1	2	yes	maximal
8	#23765271	C₂	1	2	yes	maximal, complemented
9	#23765268	C₂	1	2	yes	maximal, complemented
10	#23765202	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#2209	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	C₂²	1	1	yes	characteristic
13	#19099118	—	2	1	no	—
14	C₄×Q₈×D₇₉	—	2	1	no	—
15	C₂×#14579107	C₂²	1	2	yes	—
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098913	—	2	1	no	—
18	#19098801	—	2	1	no	—
19	#19098801	C₂²	1	2	yes	—
20	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₇₉	—	2	1	no	—
22	Q₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
24	#19098657	C₂²	1	1	yes	characteristic
25	C₄×#11929614	—	2	1	no	—
26	#19098655	—	2	1	no	—
27	#107×D₇₉	C₂²	1	1	yes	characteristic
28	#103×D₇₉	—	2	1	no	—
29	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
30	#19098391	C₂²	1	2	yes	—
31	#19098390	C₂²	1	2	yes	—
32	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
33	#19098368	C₂²	1	2	yes	—
34	#19098367	C₂²	1	2	yes	—
35	#19098367	C₂²	1	2	yes	—
36	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
37	#19098338	C₂²	1	2	yes	—
38	C₂×Q₃₂×C₇₉	C₂²	1	2	yes	—
39	#494×C₇₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	#358×C₇₉	C₂²	1	2	yes	—
43	C₂×Q₈×D₇₉	—	2	1	no	—
44	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
45	C₂×#11929614	—	2	1	no	—
46	Q₁₆×D₇₉	—	2	2	no	—
47	Q₁₆×D₇₉	—	2	2	no	—
48	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
49	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
50	#14579161	—	2	1	no	—
51	#14579152	—	4	1	no	—
52	#14579141	—	4	1	no	—
53	#14579137	—	2	2	no	—
54	Q₈×Dic₇₉	—	2	1	no	—
55	Q₈×Dic₇₉	—	2	2	no	—
56	C₄×Dic₁₅₈	—	2	1	no	—
57	C₄×Dic₁₅₈	—	4	1	no	—
58	#14579126	—	2	1	no	—
59	#14579126	—	4	1	no	—
60	#14579126	—	4	1	no	—
61	#14579119	—	4	1	no	—
62	C₄⋊C₄×D₇₉	—	2	1	no	—
63	C₄⋊C₄×D₇₉	—	4	1	no	—
64	C₄²×D₇₉	—	4	1	no	—
65	C₄²×D₇₉	D₄	1	1	yes	characteristic
66	#14579107	—	2	4	no	—
67	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
68	#14579095	—	2	1	no	—
69	#14579095	C₂³	1	2	yes	—
70	#14579095	C₂³	1	1	yes	characteristic
71	#14579094	—	2	1	no	—
72	#14579086	—	2	1	no	—
73	#14579085	—	2	1	no	—
74	#14579085	—	2	2	no	—
75	#14579085	—	2	2	no	—
76	#14579084	—	2	1	no	—
77	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
78	C₈×Dic₇₉	D₄	1	1	yes	characteristic
79	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₇₉	—	2	1	no	—
81	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
82	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₇₉	—	2	1	no	—
84	C₄×Q₈×C₇₉	—	2	1	no	—
85	Q₃₂×C₇₉	—	2	4	no	—
86	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
87	#107×C₇₉	C₂³	1	1	yes	characteristic
88	#103×C₇₉	—	2	1	no	—
89	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
90	C₂×Dic₁₅₈	—	2	1	no	—
91	C₂×Dic₁₅₈	—	4	1	no	—
92	C₂×Dic₁₅₈	—	4	1	no	—
93	Q₈×D₇₉	—	2	4	no	—
94	Q₈×D₇₉	—	4	2	no	—
95	C₂×C₄×D₇₉	—	4	1	no	—
96	C₂×C₄×D₇₉	—	4	1	no	—
97	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
98	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
99	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
100	Dic₃₁₆	—	2	1	no	—
101	Dic₃₁₆	#41	1	2	yes	—
102	#11929614	—	2	4	no	—
103	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₈×D₇₉	—	2	1	no	—
106	C₈×D₇₉	—	2	1	no	—
107	C₈×D₇₉	—	4	1	no	—
108	C₄⋊Dic₇₉	—	2	2	no	—
109	C₄⋊Dic₇₉	—	2	1	no	—
110	C₄⋊Dic₇₉	—	4	1	no	—
111	C₄⋊Dic₇₉	—	4	2	no	—
112	C₄⋊Dic₇₉	—	4	1	no	—
113	Dic₇₉⋊C₄	—	4	1	no	—
114	Dic₇₉⋊C₄	—	4	1	no	—
115	Dic₇₉⋊C₄	—	4	2	no	—
116	Dic₇₉⋊C₄	—	4	1	no	—
117	(C₂×D₇₉)⋊C₄	—	4	2	no	—
118	C₄×Dic₇₉	—	4	1	no	—
119	C₄×Dic₇₉	—	4	1	no	—
120	C₄×Dic₇₉	—	4	2	no	—
121	C₄×Dic₇₉	—	4	1	no	—
122	C₄×Dic₇₉	D₈	1	1	yes	characteristic
123	C₄×Dic₇₉	D₈	1	1	yes	characteristic
124	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
125	C₇₉⋊C₁₆	—	2	2	no	—
126	C₂×Q₈×C₇₉	—	2	1	no	—
127	C₂×Q₈×C₇₉	—	2	2	no	—
128	Q₁₆×C₇₉	—	2	1	no	—
129	Q₁₆×C₇₉	—	2	2	no	—
130	Q₁₆×C₇₉	—	2	2	no	—
131	Q₁₆×C₇₉	—	4	2	no	—
132	Q₁₆×C₇₉	#41	1	2	yes	—
133	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
135	C₄⋊C₄×C₇₉	—	2	1	no	—
136	C₄⋊C₄×C₇₉	—	4	1	no	—
137	C₄⋊C₄×C₇₉	—	4	1	no	—
138	C₄⋊C₄×C₇₉	—	4	1	no	—
139	C₄²×C₇₉	—	4	1	no	—
140	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
141	C₁₆×C₇₉	—	2	2	no	—
142	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
143	Q₈×C₇₉	—	2	2	no	—
144	Q₈×C₇₉	—	4	1	no	—
145	Q₈×C₇₉	—	4	2	no	—
146	Q₈×C₇₉	—	4	2	no	—
147	C₂×C₄×C₇₉	—	4	1	no	—
148	C₂×C₄×C₇₉	—	4	2	no	—
149	C₂×C₄×C₇₉	—	4	1	no	—
150	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×Dic₇₉	—	4	1	no	—
154	C₂×Dic₇₉	—	4	1	no	—
155	C₂×Dic₇₉	—	4	2	no	—
156	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
158	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
159	C₄×D₇₉	—	2	2	no	—
160	C₄×D₇₉	—	2	1	no	—
161	C₄×D₇₉	—	2	1	no	—
162	C₄×D₇₉	—	2	2	no	—
163	C₄×D₇₉	—	4	4	no	—
164	C₄×D₇₉	—	8	2	no	—
165	Dic₁₅₈	—	2	2	no	—
166	Dic₁₅₈	—	4	4	no	—
167	Dic₁₅₈	—	4	1	no	—
168	Dic₁₅₈	—	8	2	no	—
169	C₈×C₇₉	—	2	1	no	—
170	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
171	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
172	C₇₉⋊C₈	—	2	1	no	—
173	C₇₉⋊C₈	—	2	1	no	—
174	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₇₉	—	2	1	no	—
176	C₂×D₇₉	—	2	1	no	—
177	C₂×D₇₉	#510	1	2	yes	—
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	4	2	no	—
180	C₄×C₇₉	—	8	2	no	—
181	C₄×C₇₉	—	8	1	no	—
182	C₄×C₇₉	#510	1	2	yes	—
183	C₄×C₇₉	#525	1	1	yes	characteristic
184	C₄×C₇₉	#575	1	1	yes	characteristic
185	Dic₇₉	—	2	1	no	—
186	Dic₇₉	—	2	1	no	—
187	Dic₇₉	—	2	1	no	—
188	Dic₇₉	—	4	2	no	—
189	Dic₇₉	—	8	2	no	—
190	Dic₇₉	—	8	1	no	—
191	#33673	—	79	1	no	maximal, Sylow
192	C₂×C₇₉	#3100	1	1	yes	characteristic
193	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
194	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
195	D₇₉	—	2	2	no	—
196	#3170	—	79	1	no	—
197	C₂×C₄×Q₁₆	—	79	1	no	—
198	#2235	—	79	2	no	—
199	#2225	—	79	1	no	—
200	#2225	—	79	1	no	—
201	#2209	—	79	1	no	—
202	#2209	D₇₉	1	1	yes	characteristic, complemented
203	#2188	—	79	2	no	—
204	#2176	—	79	2	no	—
205	#2176	—	79	2	no	—
206	#2163	—	79	1	no	—
207	C₇₉	#33673	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₁₆	—	79	1	no	—
209	#533	—	158	1	no	—
210	C₂×C₄×Q₈	—	158	1	no	—
211	C₂×Q₃₂	—	79	2	no	—
212	C₂×Q₃₂	C₂×D₇₉	1	2	yes	—
213	#500	—	79	1	no	—
214	#494	C₂×D₇₉	1	1	yes	characteristic
215	#487	—	158	1	no	—
216	#467	—	79	2	no	—
217	#467	—	158	1	no	—
218	C₄×Q₁₆	—	79	2	no	—
219	C₄×Q₁₆	—	79	1	no	—
220	C₄×Q₁₆	—	79	1	no	—
221	C₄×Q₁₆	—	79	1	no	—
222	C₄×Q₁₆	—	158	1	no	—
223	C₄×Q₁₆	—	158	1	no	—
224	C₄×Q₁₆	—	158	1	no	—
225	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
226	C₂×#107	—	79	1	no	—
227	C₂×#103	—	158	1	no	—
228	C₂×C₄×C₈	—	79	1	no	—
229	#367	—	79	2	no	—
230	#367	—	79	2	no	—
231	C₄⋊C₁₆	—	79	1	no	—
232	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
233	#358	—	79	2	no	—
234	#358	—	79	2	no	—
235	#358	—	79	2	no	—
236	#358	C₂×D₇₉	1	2	yes	—
237	C₂²⋊C₁₆	—	79	2	no	—
238	#346	—	79	2	no	—
239	C₂²×Q₈	—	158	1	no	—
240	C₂×Q₁₆	—	2	1	no	—
241	C₂×Q₁₆	—	79	1	no	—
242	C₂×Q₁₆	—	158	1	no	—
243	C₂×Q₁₆	—	158	2	no	—
244	C₂×Q₁₆	—	158	2	no	—
245	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
246	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
247	C₂²×C₈	—	79	1	no	—
248	C₂²×C₈	—	79	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	—	158	1	no	—
251	#125	—	316	1	no	—
252	C₂²⋊Q₈	—	316	1	no	—
253	C₂²⋊Q₈	—	158	2	no	—
254	C₄×Q₈	—	2	1	no	—
255	C₄×Q₈	—	316	1	no	—
256	C₄×Q₈	—	316	1	no	—
257	C₄×Q₈	—	316	1	no	—
258	C₄×Q₈	—	158	1	no	—
259	C₄×Q₈	—	158	1	no	—
260	C₄×Q₈	—	158	2	no	—
261	C₄×Q₈	—	158	1	no	—
262	#117	—	316	1	no	—
263	C₂×C₄⋊C₄	—	316	1	no	—
264	C₂×C₄⋊C₄	—	158	1	no	—
265	C₂×C₄²	—	316	1	no	—
266	C₂×C₄²	—	79	1	no	—
267	Q₃₂	—	2	4	no	—
268	Q₃₂	—	158	4	no	—
269	C₂×C₁₆	—	79	2	no	—
270	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
271	#107	—	79	1	no	—
272	#107	—	79	2	no	—
273	#107	—	158	1	no	—
274	#107	—	158	1	no	—
275	#107	C₂²×D₇₉	1	1	yes	characteristic
276	#103	—	2	1	no	—
277	#103	—	158	1	no	—
278	#103	—	158	1	no	—
279	#103	—	158	2	no	—
280	#103	—	158	2	no	—
281	#103	—	158	1	no	—
282	C₄×C₈	—	79	1	no	—
283	C₄×C₈	—	79	1	no	—
284	C₄×C₈	—	79	1	no	—
285	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
286	C₂×Q₈	—	2	1	no	—
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	316	1	no	—
289	C₂×Q₈	—	316	2	no	—
290	C₂×Q₈	—	316	1	no	—
291	C₂×Q₈	—	158	4	no	—
292	C₂×Q₈	—	158	1	no	—
293	C₂²×C₄	—	316	1	no	—
294	C₂²×C₄	—	316	1	no	—
295	C₂²×C₄	—	79	1	no	—
296	C₂²×C₄	—	79	1	no	—
297	C₂²×C₄	—	79	1	no	—
298	Q₁₆	—	2	1	no	—
299	Q₁₆	—	2	2	no	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	4	2	no	—
302	Q₁₆	—	79	2	no	—
303	Q₁₆	—	158	4	no	—
304	Q₁₆	—	158	1	no	—
305	Q₁₆	#11929623	1	2	yes	—
306	C₂×C₈	—	316	1	no	—
307	C₂×C₈	—	79	1	no	—
308	C₂×C₈	—	79	1	no	—
309	C₂×C₈	—	158	1	no	—
310	C₂×C₈	—	158	1	no	—
311	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
312	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	4	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	316	1	no	—
318	C₄⋊C₄	—	316	1	no	—
319	C₄⋊C₄	—	316	2	no	—
320	C₄⋊C₄	—	316	1	no	—
321	C₄⋊C₄	—	316	2	no	—
322	C₄⋊C₄	—	316	1	no	—
323	C₄⋊C₄	—	316	1	no	—
324	C₄⋊C₄	—	158	1	no	—
325	C₄⋊C₄	—	158	2	no	—
326	C₂²⋊C₄	—	316	2	no	—
327	C₄²	—	4	1	no	—
328	C₄²	—	316	1	no	—
329	C₄²	—	316	1	no	—
330	C₄²	—	316	2	no	—
331	C₄²	—	316	1	no	—
332	C₄²	—	79	1	no	—
333	C₄²	—	79	1	no	—
334	C₄²	—	79	1	no	—
335	C₄²	D₄×D₇₉	1	1	yes	characteristic
336	C₁₆	—	2	2	no	—
337	C₁₆	—	158	2	no	—
338	C₂³	—	79	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	1	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	316	4	no	—
344	Q₈	—	316	1	no	—
345	Q₈	—	632	2	no	—
346	Q₈	—	158	2	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	316	4	no	—
351	C₂×C₄	—	316	1	no	—
352	C₂×C₄	—	316	1	no	—
353	C₂×C₄	—	316	2	no	—
354	C₂×C₄	—	79	1	no	—
355	C₂×C₄	—	79	1	no	—
356	C₂×C₄	—	79	1	no	—
357	C₂×C₄	—	632	2	no	—
358	C₂×C₄	—	158	2	no	—
359	C₂×C₄	—	158	2	no	—
360	C₂×C₄	—	158	1	no	—
361	C₂×C₄	—	158	1	no	—
362	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
363	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
364	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
365	C₈	—	2	1	no	—
366	C₈	—	158	1	no	—
367	C₈	—	158	1	no	—
368	C₈	#41×D₇₉	1	1	yes	characteristic
369	C₈	C₂×#11929623	1	1	yes	characteristic
370	C₂²	—	79	2	no	—
371	C₂²	—	158	1	no	—
372	C₂²	—	158	1	no	—
373	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	316	2	no	—
379	C₄	—	632	1	no	—
380	C₄	—	632	2	no	—
381	C₄	—	158	1	no	—
382	C₄	—	158	1	no	—
383	C₄	—	158	1	no	—
384	C₄	#19099005	1	2	yes	—
385	C₄	#19099073	1	1	yes	characteristic
386	C₄	#19099359	1	1	yes	characteristic
387	C₂	—	158	2	no	—
388	C₂	#23773399	1	1	yes	characteristic
389	C₂	#510×D₇₉	1	1	yes	characteristic
390	C₂	C₂×#19099005	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2209	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂²×D₇₉	C₂×D₈
hypercenter	#2209	D₇₉
hypocenter	C₇₉	#33673
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2209

Complements

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

N	H ≅ G/N	type
4. #23765543	C₂	semidirect
5. #23765542	C₂	semidirect
6. #23765447	C₂	semidirect
8. #23765271	C₂	semidirect
9. #23765268	C₂	semidirect
11. C₇₉×#2209	C₂	semidirect
202. #2209	D₇₉	semidirect
207. C₇₉	#33673	semidirect

Maximal subgroups

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

subgroup	index
#23773984	2
C₄×Q₁₆×D₇₉	2
#23765596	2
#23765543	2
#23765542	2
#23765447	2
#23765332	2
#23765271	2
#23765268	2
#23765202	2
C₇₉×#2209	2
#33673	79