Subgroups — #34443314

Order 20224.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773977	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765579	C₂	1	2	yes	maximal, complemented
4	#23765553	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765552	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765437	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765315	C₂	1	2	yes	maximal, complemented
8	#23765265	C₂	1	2	yes	maximal, complemented
9	#23765261	C₂	1	2	yes	maximal, complemented
10	#23765202	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2207	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	C₂²	1	1	yes	characteristic
13	#19099111	—	2	1	no	—
14	C₄×Q₈×D₇₉	—	2	1	no	—
15	C₂×#14579105	C₂²	1	2	yes	—
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098909	—	2	1	no	—
18	#19098796	—	2	1	no	—
19	#19098796	C₂²	1	2	yes	—
20	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
22	#19098657	C₂²	1	1	yes	characteristic
23	C₄×#11929614	—	2	1	no	—
24	#19098655	—	2	1	no	—
25	D₈×Dic₇₉	—	2	1	no	—
26	D₈×Dic₇₉	C₂²	1	2	yes	—
27	#107×D₇₉	C₂²	1	1	yes	characteristic
28	#103×D₇₉	—	2	1	no	—
29	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
30	#19098389	C₂²	1	2	yes	—
31	#19098388	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	#19098364	C₂²	1	2	yes	—
36	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
37	#19098338	C₂²	1	2	yes	—
38	C₂×SD₃₂×C₇₉	C₂²	1	2	yes	—
39	#493×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	#357×C₇₉	C₂²	1	2	yes	—
43	C₂×#11929623	—	4	1	no	—
44	C₂×Q₈×D₇₉	—	2	1	no	—
45	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
46	C₂×#11929614	—	2	1	no	—
47	Q₁₆×D₇₉	—	2	2	no	—
48	Q₁₆×D₇₉	—	2	2	no	—
49	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
50	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
51	#14579161	—	2	1	no	—
52	#14579148	—	4	1	no	—
53	#14579132	—	2	2	no	—
54	#14579132	—	4	1	no	—
55	Q₈×Dic₇₉	—	2	1	no	—
56	C₄×Dic₁₅₈	—	2	1	no	—
57	C₄×Dic₁₅₈	—	4	1	no	—
58	#14579126	—	2	1	no	—
59	#14579126	—	4	1	no	—
60	D₄×Dic₇₉	—	2	2	no	—
61	D₄×Dic₇₉	—	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	#14579105	—	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	#14579084	—	2	1	no	—
75	#14579082	—	2	2	no	—
76	#14579082	—	2	2	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	C₂³	1	1	yes	characteristic
81	C₂×D₈×C₇₉	—	2	1	no	—
82	C₂×D₈×C₇₉	C₂³	1	2	yes	—
83	#127×C₇₉	—	2	1	no	—
84	C₄×Q₈×C₇₉	—	2	1	no	—
85	SD₃₂×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	#11929623	—	8	4	no	—
91	C₂×Dic₁₅₈	—	2	1	no	—
92	C₂×Dic₁₅₈	—	4	1	no	—
93	C₂×Dic₁₅₈	—	4	1	no	—
94	Q₈×D₇₉	—	2	4	no	—
95	Q₈×D₇₉	—	4	2	no	—
96	C₂×C₂²⋊D₇₉	—	4	2	no	—
97	C₂²×Dic₇₉	—	4	2	no	—
98	C₂×C₄×D₇₉	—	4	1	no	—
99	C₂×C₄×D₇₉	—	4	1	no	—
100	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
101	C₂×C₄×D₇₉	D₈	1	1	yes	characteristic
102	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
103	Dic₃₁₆	—	2	1	no	—
104	Dic₃₁₆	#41	1	2	yes	complemented
105	#11929614	—	2	4	no	—
106	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₈×D₇₉	—	2	1	no	—
109	C₈×D₇₉	—	2	1	no	—
110	C₈×D₇₉	—	4	1	no	—
111	C₄⋊Dic₇₉	—	2	2	no	—
112	C₄⋊Dic₇₉	—	2	1	no	—
113	C₄⋊Dic₇₉	—	4	1	no	—
114	C₄⋊Dic₇₉	—	4	1	no	—
115	Dic₇₉⋊C₄	—	4	1	no	—
116	Dic₇₉⋊C₄	—	4	1	no	—
117	Dic₇₉⋊C₄	—	4	1	no	—
118	C₂²⋊Dic₇₉	—	4	2	no	—
119	C₂²⋊Dic₇₉	—	4	2	no	—
120	C₄×Dic₇₉	—	4	1	no	—
121	C₄×Dic₇₉	—	4	1	no	—
122	C₄×Dic₇₉	—	4	1	no	—
123	C₄×Dic₇₉	D₈	1	1	yes	characteristic
124	C₄×Dic₇₉	D₈	1	1	yes	characteristic
125	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
126	C₇₉⋊C₁₆	—	2	2	no	—
127	C₂×Q₈×C₇₉	—	2	1	no	—
128	C₂×D₄×C₇₉	—	2	2	no	—
129	C₂×D₄×C₇₉	—	4	1	no	—
130	C₂×D₄×C₇₉	—	4	1	no	—
131	Q₁₆×C₇₉	—	2	1	no	—
132	Q₁₆×C₇₉	#41	1	2	yes	complemented
133	D₈×C₇₉	—	2	2	no	—
134	D₈×C₇₉	—	2	2	no	—
135	D₈×C₇₉	—	4	2	no	—
136	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
138	C₄⋊C₄×C₇₉	—	2	1	no	—
139	C₄⋊C₄×C₇₉	—	4	1	no	—
140	C₄²×C₇₉	—	4	1	no	—
141	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
142	C₁₆×C₇₉	—	2	2	no	—
143	C₂³×C₇₉	—	4	2	no	—
144	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
145	Q₈×C₇₉	—	2	2	no	—
146	Q₈×C₇₉	—	4	1	no	—
147	D₄×C₇₉	—	4	2	no	—
148	D₄×C₇₉	—	4	2	no	—
149	D₄×C₇₉	—	8	2	no	—
150	D₄×C₇₉	—	8	2	no	—
151	C₂×C₄×C₇₉	—	4	1	no	—
152	C₂×C₄×C₇₉	—	4	1	no	—
153	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
156	C₂²⋊D₇₉	—	8	4	no	—
157	C₂×Dic₇₉	—	4	1	no	—
158	C₂×Dic₇₉	—	4	1	no	—
159	C₂×Dic₇₉	—	4	2	no	—
160	C₂×Dic₇₉	—	8	4	no	—
161	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
162	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
163	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
164	C₄×D₇₉	—	2	2	no	—
165	C₄×D₇₉	—	2	1	no	—
166	C₄×D₇₉	—	2	1	no	—
167	C₄×D₇₉	—	2	2	no	—
168	C₄×D₇₉	—	4	4	no	—
169	C₄×D₇₉	—	8	2	no	—
170	Dic₁₅₈	—	2	2	no	—
171	Dic₁₅₈	—	4	4	no	—
172	Dic₁₅₈	—	4	1	no	—
173	Dic₁₅₈	—	8	2	no	—
174	C₈×C₇₉	—	2	1	no	—
175	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
176	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
177	C₇₉⋊C₈	—	2	1	no	—
178	C₇₉⋊C₈	—	2	1	no	—
179	C₂²×C₇₉	—	8	2	no	—
180	C₂²×C₇₉	—	8	2	no	—
181	C₂²×C₇₉	—	8	2	no	—
182	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₇₉	—	2	1	no	—
184	C₂×D₇₉	—	2	1	no	—
185	C₂×D₇₉	#509	1	2	yes	—
186	C₄×C₇₉	—	2	1	no	—
187	C₄×C₇₉	—	4	2	no	—
188	C₄×C₇₉	—	8	1	no	—
189	C₄×C₇₉	#510	1	2	yes	—
190	C₄×C₇₉	#525	1	1	yes	characteristic
191	C₄×C₇₉	#575	1	1	yes	characteristic
192	Dic₇₉	—	2	1	no	—
193	Dic₇₉	—	2	1	no	—
194	Dic₇₉	—	2	1	no	—
195	Dic₇₉	—	4	2	no	—
196	Dic₇₉	—	8	2	no	—
197	Dic₇₉	—	8	1	no	—
198	#33672	—	79	1	no	maximal, Sylow
199	C₂×C₇₉	—	16	2	no	—
200	C₂×C₇₉	#3100	1	1	yes	characteristic
201	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
202	C₂×C₇₉	C₂×#510	1	1	yes	characteristic
203	D₇₉	—	2	2	no	—
204	#3169	—	79	1	no	—
205	C₂×C₄×Q₁₆	—	79	1	no	—
206	#2232	—	79	2	no	—
207	#2227	—	79	1	no	—
208	#2227	—	79	1	no	—
209	#2207	—	79	1	no	—
210	#2207	D₇₉	1	1	yes	characteristic, complemented
211	#2185	—	79	2	no	—
212	#2175	—	79	2	no	—
213	#2175	—	79	2	no	—
214	#2163	—	79	1	no	—
215	C₇₉	#33672	1	1	yes	characteristic, complemented, Sylow
216	C₂²×Q₁₆	—	79	1	no	—
217	#532	—	158	1	no	—
218	C₂×C₄×Q₈	—	158	1	no	—
219	C₂×SD₃₂	—	79	2	no	—
220	C₂×SD₃₂	C₂×D₇₉	1	2	yes	—
221	#500	—	79	1	no	—
222	#493	C₂×D₇₉	1	1	yes	characteristic
223	#486	—	158	1	no	—
224	#466	—	79	2	no	—
225	#466	—	158	1	no	—
226	C₄×Q₁₆	—	79	1	no	—
227	C₄×Q₁₆	—	79	1	no	—
228	C₄×Q₁₆	—	79	1	no	—
229	C₄×Q₁₆	—	158	1	no	—
230	C₄×Q₁₆	—	158	1	no	—
231	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
232	C₄×D₈	—	79	2	no	—
233	C₄×D₈	—	158	1	no	—
234	C₂×#107	—	79	1	no	—
235	C₂×#103	—	158	1	no	—
236	C₂×C₄×C₈	—	79	1	no	—
237	#366	—	79	2	no	—
238	#366	—	79	2	no	—
239	C₄⋊C₁₆	—	79	1	no	—
240	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
241	#358	—	79	2	no	—
242	#358	—	79	2	no	—
243	#357	—	79	2	no	—
244	#357	C₂×D₇₉	1	2	yes	—
245	C₂²⋊C₁₆	—	79	2	no	—
246	#346	—	79	2	no	—
247	C₂×#41	—	316	1	no	—
248	C₂²×Q₈	—	158	1	no	—
249	C₂×Q₁₆	—	79	1	no	—
250	C₂×Q₁₆	—	158	1	no	—
251	C₂×Q₁₆	—	158	2	no	—
252	C₂×Q₁₆	—	158	2	no	—
253	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	C₂²×D₇₉	1	2	yes	—
256	C₂²×C₈	—	79	1	no	—
257	C₂²×C₈	—	79	1	no	—
258	C₄⋊Q₈	—	158	1	no	—
259	#127	—	2	1	no	—
260	#124	—	316	1	no	—
261	#121	—	316	1	no	—
262	#121	—	158	2	no	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	—	316	1	no	—
265	C₄×Q₈	—	316	1	no	—
266	C₄×Q₈	—	158	1	no	—
267	C₄×Q₈	—	158	1	no	—
268	C₄×Q₈	—	158	1	no	—
269	C₄×D₄	—	316	1	no	—
270	C₄×D₄	—	158	2	no	—
271	C₂×C₄⋊C₄	—	316	1	no	—
272	C₂×C₄⋊C₄	—	158	1	no	—
273	C₂×C₄²	—	316	1	no	—
274	C₂×C₄²	—	79	1	no	—
275	SD₃₂	—	2	4	no	—
276	SD₃₂	—	158	4	no	—
277	C₂×C₁₆	—	79	2	no	—
278	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
279	#107	—	79	1	no	—
280	#107	—	79	2	no	—
281	#107	—	158	1	no	—
282	#107	—	158	1	no	—
283	#107	C₂²×D₇₉	1	1	yes	characteristic
284	#103	—	2	1	no	—
285	#103	—	158	1	no	—
286	#103	—	158	1	no	—
287	#103	—	158	1	no	—
288	#102	—	158	2	no	—
289	#102	—	158	2	no	—
290	C₄×C₈	—	79	1	no	—
291	C₄×C₈	—	79	1	no	—
292	C₄×C₈	—	79	1	no	—
293	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
294	#41	—	632	4	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	316	1	no	—
297	C₂×Q₈	—	316	2	no	—
298	C₂×Q₈	—	316	1	no	—
299	C₂×Q₈	—	158	4	no	—
300	C₂×Q₈	—	158	1	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	—	316	2	no	—
305	C₂²×C₄	—	316	1	no	—
306	C₂²×C₄	—	316	1	no	—
307	C₂²×C₄	—	316	2	no	—
308	C₂²×C₄	—	79	1	no	—
309	C₂²×C₄	—	79	1	no	—
310	C₂²×C₄	—	79	1	no	—
311	Q₁₆	—	2	1	no	—
312	Q₁₆	—	79	2	no	—
313	Q₁₆	—	158	4	no	—
314	Q₁₆	—	158	1	no	—
315	Q₁₆	#11929623	1	2	yes	complemented
316	D₈	—	2	2	no	—
317	D₈	—	2	2	no	—
318	D₈	—	4	2	no	—
319	C₂×C₈	—	316	1	no	—
320	C₂×C₈	—	79	1	no	—
321	C₂×C₈	—	79	1	no	—
322	C₂×C₈	—	158	1	no	—
323	C₂×C₈	—	158	1	no	—
324	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
325	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	316	1	no	—
329	C₄⋊C₄	—	316	1	no	—
330	C₄⋊C₄	—	316	1	no	—
331	C₄⋊C₄	—	316	1	no	—
332	C₄⋊C₄	—	316	1	no	—
333	C₄⋊C₄	—	158	2	no	—
334	C₄⋊C₄	—	158	1	no	—
335	C₂²⋊C₄	—	316	2	no	—
336	C₂²⋊C₄	—	316	2	no	—
337	C₄²	—	4	1	no	—
338	C₄²	—	316	1	no	—
339	C₄²	—	316	1	no	—
340	C₄²	—	316	1	no	—
341	C₄²	—	79	1	no	—
342	C₄²	—	79	1	no	—
343	C₄²	—	79	1	no	—
344	C₄²	D₄×D₇₉	1	1	yes	characteristic
345	C₁₆	—	2	2	no	—
346	C₁₆	—	158	2	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	79	1	no	—
349	Q₈	—	2	2	no	—
350	Q₈	—	4	1	no	—
351	Q₈	—	316	4	no	—
352	Q₈	—	316	1	no	—
353	Q₈	—	632	2	no	—
354	Q₈	—	158	2	no	—
355	D₄	—	4	2	no	—
356	D₄	—	4	2	no	—
357	D₄	—	8	2	no	—
358	D₄	—	8	2	no	—
359	D₄	—	632	4	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	316	4	no	—
363	C₂×C₄	—	316	1	no	—
364	C₂×C₄	—	316	1	no	—
365	C₂×C₄	—	316	2	no	—
366	C₂×C₄	—	79	1	no	—
367	C₂×C₄	—	79	1	no	—
368	C₂×C₄	—	79	1	no	—
369	C₂×C₄	—	632	2	no	—
370	C₂×C₄	—	632	4	no	—
371	C₂×C₄	—	158	2	no	—
372	C₂×C₄	—	158	2	no	—
373	C₂×C₄	—	158	1	no	—
374	C₂×C₄	—	158	1	no	—
375	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
376	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
377	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	158	1	no	—
380	C₈	—	158	1	no	—
381	C₈	#41×D₇₉	1	1	yes	characteristic
382	C₈	C₂×#11929623	1	1	yes	characteristic
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	79	2	no	—
387	C₂²	—	158	1	no	—
388	C₂²	—	158	1	no	—
389	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	8	1	no	—
393	C₄	—	316	2	no	—
394	C₄	—	632	2	no	—
395	C₄	—	632	1	no	—
396	C₄	—	158	1	no	—
397	C₄	—	158	1	no	—
398	C₄	—	158	1	no	—
399	C₄	#19099002	1	2	yes	—
400	C₄	#19099073	1	1	yes	characteristic
401	C₄	#19099359	1	1	yes	characteristic
402	C₂	—	16	2	no	—
403	C₂	—	158	2	no	—
404	C₂	#23773399	1	1	yes	characteristic
405	C₂	#509×D₇₉	1	1	yes	characteristic
406	C₂	C₂×#19099002	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×#2207	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23773977	C₂
hypercenter	#2207	D₇₉
hypocenter	C₇₉	#33672
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2207

Complements

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

N	H ≅ G/N	type
2. C₄×Q₁₆×D₇₉	C₂	semidirect
3. #23765579	C₂	semidirect
4. #23765553	C₂	semidirect
5. #23765552	C₂	semidirect
6. #23765437	C₂	semidirect
7. #23765315	C₂	semidirect
8. #23765265	C₂	semidirect
9. #23765261	C₂	semidirect
10. #23765202	C₂	semidirect
11. C₇₉×#2207	C₂	semidirect
104. Dic₃₁₆	#41	semidirect
132. Q₁₆×C₇₉	#41	semidirect
210. #2207	D₇₉	semidirect
215. C₇₉	#33672	semidirect
315. Q₁₆	#11929623	semidirect

Maximal subgroups

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

subgroup	index
#23773977	2
C₄×Q₁₆×D₇₉	2
#23765579	2
#23765553	2
#23765552	2
#23765437	2
#23765315	2
#23765265	2
#23765261	2
#23765202	2
C₇₉×#2207	2
#33672	79