Subgroups — #34445144

Order 20224.

580 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775748	C₂	1	1	yes	maximal, characteristic
2	#23765627	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765595	C₂	1	2	yes	maximal, complemented
4	#23765542	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765498	C₂	1	2	yes	maximal, complemented
6	#23765442	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765317	C₂	1	2	yes	maximal, complemented
8	C₄×#14579104	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19098367	C₂	1	2	yes	maximal, complemented
11	D₄×Q₁₆×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099217	—	2	1	no	—
13	C₂×#14579104	C₂²	1	1	yes	characteristic
14	C₂×#14579104	C₂²	1	2	yes	—
15	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
16	#19098968	C₂²	1	1	yes	characteristic
17	#19098877	—	2	2	no	—
18	#19098844	—	2	1	no	—
19	#19098799	C₂²	1	2	yes	—
20	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
21	C₂×#14579095	C₂²	1	2	yes	—
22	#19098390	C₂²	1	1	yes	characteristic
23	#19098390	C₂²	1	2	yes	—
24	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
25	#19098367	C₂²	1	1	yes	characteristic
26	#19098367	C₂²	1	2	yes	—
27	#19098367	C₂²	1	2	yes	—
28	#19098367	C₂²	1	2	yes	—
29	#19098367	C₂²	1	2	yes	—
30	#19098363	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
32	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₂²×Q₁₆×C₇₉	C₂²	1	2	yes	—
34	D₄×Q₈×C₇₉	—	2	1	no	—
35	#494×C₇₉	C₂²	1	1	yes	characteristic
36	#467×C₇₉	C₂²	1	2	yes	—
37	#462×C₇₉	—	2	1	no	—
38	#451×C₇₉	—	2	2	no	—
39	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×#11929623	—	4	1	no	—
42	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
43	#14579161	—	2	1	no	—
44	#14579145	—	4	2	no	—
45	#14579141	—	4	1	no	—
46	#14579135	—	2	2	no	—
47	C₄×D₃₁₆	—	2	1	no	—
48	C₂×C₄⋊Dic₇₉	—	2	2	no	—
49	#14579104	—	2	2	no	—
50	#14579104	—	4	2	no	—
51	C₂×C₇₉⋊C₁₆	—	4	2	no	—
52	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
53	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
55	#14579095	—	2	1	no	—
56	#14579095	—	2	2	no	—
57	#14579095	C₂³	1	1	yes	characteristic
58	#14579095	C₂³	1	2	yes	—
59	#14579083	—	2	1	no	—
60	#14579083	—	2	2	no	—
61	C₂²×Q₈×C₇₉	—	2	2	no	—
62	C₂×Q₁₆×C₇₉	—	2	4	no	—
63	C₂×Q₁₆×C₇₉	—	2	2	no	—
64	C₂×Q₁₆×C₇₉	—	2	1	no	—
65	C₂×Q₁₆×C₇₉	—	4	2	no	—
66	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₉	—	2	1	no	—
70	C₄⋊Q₈×C₇₉	—	4	1	no	—
71	C₂²⋊Q₈×C₇₉	—	2	2	no	—
72	C₂²⋊Q₈×C₇₉	—	4	2	no	—
73	C₄×Q₈×C₇₉	—	2	1	no	—
74	C₄×D₄×C₇₉	—	4	1	no	—
75	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
76	#107×C₇₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
78	#103×C₇₉	—	2	1	no	—
79	#103×C₇₉	—	2	2	no	—
80	C₂²⋊C₈×C₇₉	D₄	1	2	yes	—
81	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
82	#11929623	—	8	4	no	—
83	C₂×Dic₁₅₈	—	4	1	no	—
84	C₂×D₃₁₆	—	2	1	no	—
85	C₂×C₂²⋊D₇₉	—	4	2	no	—
86	C₂²×Dic₇₉	—	4	2	no	—
87	C₂×C₄×D₇₉	—	4	1	no	—
88	D₆₃₂	—	2	1	no	—
89	D₆₃₂	—	2	1	no	—
90	C₄⋊Dic₇₉	—	2	1	no	—
91	C₄⋊Dic₇₉	—	2	2	no	—
92	C₄⋊Dic₇₉	—	4	2	no	—
93	C₄⋊Dic₇₉	—	4	1	no	—
94	(C₂×D₇₉)⋊C₄	—	4	1	no	—
95	(C₂×D₇₉)⋊C₄	—	4	2	no	—
96	C₇₉⋊C₁₆	—	2	1	no	—
97	C₇₉⋊C₁₆	—	2	1	no	—
98	C₇₉⋊C₁₆	—	2	2	no	—
99	C₂×Q₈×C₇₉	—	2	1	no	—
100	C₂×Q₈×C₇₉	—	2	2	no	—
101	C₂×Q₈×C₇₉	—	4	4	no	—
102	C₂×Q₈×C₇₉	—	4	2	no	—
103	C₂×Q₈×C₇₉	—	8	2	no	—
104	C₂×D₄×C₇₉	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₇₉	—	4	2	no	—
106	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
107	Q₁₆×C₇₉	—	2	1	no	—
108	Q₁₆×C₇₉	—	2	2	no	—
109	Q₁₆×C₇₉	—	2	2	no	—
110	Q₁₆×C₇₉	—	4	4	no	—
111	Q₁₆×C₇₉	—	8	1	no	—
112	Q₁₆×C₇₉	#41	1	2	yes	complemented
113	C₂×C₈×C₇₉	—	2	2	no	—
114	C₂×C₈×C₇₉	—	2	2	no	—
115	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₇₉	—	2	1	no	—
119	C₄⋊C₄×C₇₉	—	4	1	no	—
120	C₄⋊C₄×C₇₉	—	4	1	no	—
121	C₄⋊C₄×C₇₉	—	4	2	no	—
122	C₄⋊C₄×C₇₉	—	4	1	no	—
123	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
124	C₂²⋊C₄×C₇₉	—	4	2	no	—
125	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
126	C₄²×C₇₉	—	4	1	no	—
127	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₇₉	C₂×D₈	1	2	yes	—
129	C₂²×D₇₉	—	4	1	no	—
130	Q₈×C₇₉	—	2	2	no	—
131	Q₈×C₇₉	—	4	1	no	—
132	Q₈×C₇₉	—	4	2	no	—
133	Q₈×C₇₉	—	4	2	no	—
134	Q₈×C₇₉	—	8	4	no	—
135	D₄×C₇₉	SD₃₂	1	4	yes	complemented
136	C₂×C₄×C₇₉	—	2	2	no	—
137	C₂×C₄×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	—	4	1	no	—
139	C₂×C₄×C₇₉	—	4	1	no	—
140	C₂×C₄×C₇₉	—	4	2	no	—
141	C₂×C₄×C₇₉	—	8	4	no	—
142	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₇₉	—	8	4	no	—
147	C₂×Dic₇₉	—	4	1	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	8	4	no	—
150	D₃₁₆	—	4	1	no	—
151	D₃₁₆	—	4	1	no	—
152	C₄×D₇₉	—	8	2	no	—
153	Dic₁₅₈	—	8	2	no	—
154	C₈×C₇₉	—	2	2	no	—
155	C₈×C₇₉	—	2	1	no	—
156	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
157	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂²×C₇₉	—	2	2	no	—
159	C₂²×C₇₉	—	2	2	no	—
160	C₂²×C₇₉	C₂×SD₃₂	1	4	yes	—
161	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₇₉	—	8	1	no	—
163	C₂×D₇₉	—	8	1	no	—
164	C₂×D₇₉	—	8	1	no	—
165	C₄×C₇₉	—	2	1	no	—
166	C₄×C₇₉	—	2	2	no	—
167	C₄×C₇₉	—	4	2	no	—
168	C₄×C₇₉	—	8	1	no	—
169	C₄×C₇₉	—	8	2	no	—
170	C₄×C₇₉	C₂×SD₃₂	1	2	yes	—
171	C₄×C₇₉	#546	1	1	yes	characteristic
172	C₄×C₇₉	#577	1	1	yes	characteristic
173	Dic₇₉	—	8	2	no	—
174	Dic₇₉	—	8	1	no	—
175	#33865	—	79	1	no	maximal, Sylow
176	C₂×C₇₉	—	2	4	no	—
177	C₂×C₇₉	#3294	1	1	yes	characteristic
178	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
179	C₂×C₇₉	#3417	1	1	yes	characteristic
180	D₇₉	—	16	1	no	—
181	#3334	—	79	1	no	—
182	D₄×Q₁₆	D₇₉	1	1	yes	characteristic, complemented
183	#2241	—	79	1	no	—
184	#2235	—	79	2	no	—
185	#2225	—	79	1	no	—
186	#2217	—	79	2	no	—
187	#2208	—	79	1	no	—
188	#2185	—	79	2	no	—
189	C₄×SD₃₂	—	79	1	no	—
190	D₄×C₁₆	—	79	1	no	—
191	C₂×#358	—	79	2	no	—
192	C₇₉	#33865	1	1	yes	characteristic, complemented, Sylow
193	C₂²×Q₁₆	C₂×D₇₉	1	2	yes	—
194	D₄×Q₈	—	2	1	no	—
195	#547	—	158	1	no	—
196	C₂×SD₃₂	—	79	2	no	—
197	C₂×SD₃₂	—	79	1	no	—
198	C₂²×C₁₆	—	79	2	no	—
199	#500	—	79	1	no	—
200	#494	C₂×D₇₉	1	1	yes	characteristic
201	#480	—	158	2	no	—
202	#474	—	158	1	no	—
203	#467	C₂×D₇₉	1	2	yes	—
204	#466	—	79	2	no	—
205	#462	—	2	1	no	—
206	#451	—	2	2	no	—
207	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
208	C₄×D₈	—	79	1	no	—
209	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
210	C₂×#107	—	79	2	no	—
211	#367	—	79	1	no	—
212	#367	—	79	2	no	—
213	C₄⋊C₁₆	—	79	1	no	—
214	#358	—	79	1	no	—
215	#358	—	79	2	no	—
216	#358	—	79	2	no	—
217	#358	—	79	2	no	—
218	#358	—	79	2	no	—
219	#357	—	79	1	no	—
220	C₂²⋊C₁₆	—	79	2	no	—
221	C₄×C₁₆	—	79	1	no	—
222	C₂×#41	—	316	1	no	—
223	C₂²×Q₈	—	2	2	no	—
224	C₂×Q₁₆	—	2	4	no	—
225	C₂×Q₁₆	—	2	2	no	—
226	C₂×Q₁₆	—	2	1	no	—
227	C₂×Q₁₆	—	4	2	no	—
228	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
229	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
230	C₂×D₈	—	79	1	no	—
231	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	4	1	no	—
234	C₄⋊Q₈	—	158	1	no	—
235	#123	—	316	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₂²⋊Q₈	—	4	2	no	—
238	C₂²⋊Q₈	—	316	1	no	—
239	#121	—	158	2	no	—
240	C₄×Q₈	—	2	1	no	—
241	C₄×D₄	—	4	1	no	—
242	C₄×D₄	—	158	1	no	—
243	C₄×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
244	C₂×C₄⋊C₄	—	158	2	no	—
245	SD₃₂	—	316	2	no	—
246	SD₃₂	—	158	2	no	—
247	C₂×C₁₆	—	316	2	no	—
248	C₂×C₁₆	—	79	1	no	—
249	C₂×C₁₆	—	79	1	no	—
250	C₂×C₁₆	—	79	2	no	—
251	#107	—	79	1	no	—
252	#107	—	79	2	no	—
253	#107	—	158	2	no	—
254	#107	—	158	1	no	—
255	#107	C₂²×D₇₉	1	1	yes	characteristic
256	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
257	#103	—	2	1	no	—
258	#103	—	2	2	no	—
259	#102	—	158	1	no	—
260	#102	—	158	2	no	—
261	C₂²⋊C₈	C₂²⋊D₇₉	1	2	yes	—
262	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
263	#41	—	632	4	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	4	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	8	2	no	—
269	C₂×Q₈	—	316	1	no	—
270	C₂×D₄	—	316	2	no	—
271	C₂×D₄	—	158	1	no	—
272	C₂×D₄	#11929609	1	1	yes	characteristic
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	316	2	no	—
275	C₂²×C₄	—	316	1	no	—
276	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
277	Q₁₆	—	2	2	no	—
278	Q₁₆	—	2	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	4	4	no	—
281	Q₁₆	—	8	1	no	—
282	Q₁₆	#11929623	1	2	yes	complemented
283	D₈	—	158	1	no	—
284	D₈	—	158	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
289	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	316	2	no	—
296	C₄⋊C₄	—	316	1	no	—
297	C₄⋊C₄	—	158	1	no	—
298	C₄⋊C₄	—	158	2	no	—
299	C₄⋊C₄	#11929609	1	1	yes	characteristic
300	C₂²⋊C₄	—	4	2	no	—
301	C₂²⋊C₄	—	316	1	no	—
302	C₂²⋊C₄	—	316	2	no	—
303	C₂²⋊C₄	#11929609	1	2	yes	—
304	C₄²	—	4	1	no	—
305	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₁₆	—	158	1	no	—
307	C₁₆	—	158	1	no	—
308	C₁₆	—	158	2	no	—
309	C₂³	—	316	1	no	—
310	C₂³	C₂×#11929609	1	2	yes	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	Q₈	—	8	4	no	—
316	Q₈	—	632	2	no	—
317	D₄	—	316	1	no	—
318	D₄	—	316	1	no	—
319	D₄	—	632	4	no	—
320	D₄	#14579104	1	4	yes	complemented
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	8	4	no	—
327	C₂×C₄	—	316	1	no	—
328	C₂×C₄	—	316	2	no	—
329	C₂×C₄	—	632	4	no	—
330	C₂×C₄	—	632	2	no	—
331	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
332	C₂×C₄	C₂×#11929609	1	2	yes	—
333	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
335	C₈	—	2	2	no	—
336	C₈	—	2	1	no	—
337	C₈	C₂×#11929623	1	1	yes	characteristic
338	C₈	#14579206	1	1	yes	characteristic
339	C₂²	—	2	2	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	632	1	no	—
342	C₂²	—	632	1	no	—
343	C₂²	—	632	1	no	—
344	C₂²	C₂×#14579104	1	4	yes	—
345	C₂²	C₂²×#11929609	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	2	no	—
348	C₄	—	4	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	2	no	—
351	C₄	—	632	1	no	—
352	C₄	—	632	2	no	—
353	C₄	C₂×#14579104	1	2	yes	—
354	C₄	#19099207	1	1	yes	characteristic
355	C₄	#19099374	1	1	yes	characteristic
356	C₂	—	2	4	no	—
357	C₂	—	1264	1	no	—
358	C₂	#23775213	1	1	yes	characteristic
359	C₂	C₂²×#14579104	1	1	yes	characteristic
360	C₂	#23776551	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	D₄×Q₁₆×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23775748	C₂
hypercenter	D₄×Q₁₆	D₇₉
hypocenter	C₇₉	#33865
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #23765627	C₂	semidirect
3. #23765595	C₂	semidirect
4. #23765542	C₂	semidirect
5. #23765498	C₂	semidirect
6. #23765442	C₂	semidirect
7. #23765317	C₂	semidirect
8. C₄×#14579104	C₂	semidirect
9. D₄×C₇₉⋊C₁₆	C₂	semidirect
10. C₂×#19098367	C₂	semidirect
11. D₄×Q₁₆×C₇₉	C₂	semidirect
112. Q₁₆×C₇₉	#41	semidirect
135. D₄×C₇₉	SD₃₂	semidirect
182. D₄×Q₁₆	D₇₉	semidirect
192. C₇₉	#33865	semidirect
282. Q₁₆	#11929623	semidirect
320. D₄	#14579104	semidirect

Maximal subgroups

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

subgroup	index
#23775748	2
#23765627	2
#23765595	2
#23765542	2
#23765498	2
#23765442	2
#23765317	2
C₄×#14579104	2
D₄×C₇₉⋊C₁₆	2
C₂×#19098367	2
D₄×Q₁₆×C₇₉	2
#33865	79