Subgroups — #34444982

Order 20224.

674 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#23775135	C₂	1	2	yes	maximal, complemented
3	#23765658	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765491	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765483	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765460	C₂	1	2	yes	maximal, complemented
7	#23765340	C₂	1	2	yes	maximal, complemented
8	#23765315	C₂	1	2	yes	maximal, complemented
9	Q₃₂×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₆×C₂²⋊D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#508×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	C₂²	1	2	yes	—
13	#19099220	—	2	2	no	—
14	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
15	C₂×#14579107	C₂²	1	2	yes	—
16	Q₃₂×D₇₉	—	2	4	no	—
17	C₂×#14579105	C₂²	1	2	yes	—
18	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
19	#19098946	C₂²	1	2	yes	—
20	#19098804	C₂²	1	2	yes	—
21	#19098796	C₂²	1	2	yes	—
22	#19098783	—	2	2	no	—
23	#19098725	—	2	2	no	—
24	#19098700	—	2	2	no	—
25	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
26	C₈×C₂²⋊D₇₉	C₂²	1	1	yes	characteristic
27	#19098389	C₂²	1	1	yes	characteristic
28	Dic₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
29	#19098368	C₂²	1	2	yes	—
30	#19098367	C₂²	1	2	yes	—
31	#19098364	C₂²	1	2	yes	—
32	C₂²⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
33	(C₂×D₇₉)⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
35	C₂×#135×C₇₉	C₂²	1	2	yes	—
36	#508×C₇₉	—	2	4	no	—
37	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
38	C₂×SD₃₂×C₇₉	C₂²	1	2	yes	—
39	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₇₉	—	2	2	no	—
42	C₂×Dic₃₁₆	C₂³	1	2	yes	—
43	C₂×#11929614	—	2	2	no	—
44	Q₁₆×D₇₉	—	2	4	no	—
45	Q₁₆×D₇₉	—	4	2	no	—
46	C₂×#11929612	—	2	2	no	—
47	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
48	#14579148	—	2	2	no	—
49	#14579148	—	4	2	no	—
50	#14579140	—	2	2	no	—
51	#14579140	—	4	2	no	—
52	#14579132	—	2	2	no	—
53	#14579132	—	4	2	no	—
54	Q₈×Dic₇₉	—	2	2	no	—
55	C₄×C₂²⋊D₇₉	—	4	2	no	—
56	C₄×C₂²⋊D₇₉	D₄	1	1	yes	characteristic
57	Dic₆₃₂	—	2	2	no	—
58	#14579107	—	2	4	no	—
59	#14579105	—	4	2	no	—
60	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
62	C₁₆×D₇₉	—	2	2	no	—
63	#14579095	C₂³	1	2	yes	—
64	Dic₇₉⋊C₈	D₄	1	1	yes	characteristic
65	#14579086	—	2	2	no	—
66	#14579085	—	2	2	no	—
67	#14579082	—	2	2	no	—
68	C₂²⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
69	(C₂×D₇₉)⋊C₈	D₄	1	1	yes	characteristic
70	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
71	C₂×#41×C₇₉	—	2	2	no	—
72	#135×C₇₉	—	2	4	no	—
73	#135×C₇₉	—	4	2	no	—
74	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
75	C₂×SD₁₆×C₇₉	—	2	2	no	—
76	C₂×D₈×C₇₉	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	Q₃₂×C₇₉	D₄	1	4	yes	complemented
79	SD₃₂×C₇₉	—	2	4	no	—
80	D₁₆×C₇₉	—	2	2	no	—
81	C₂×C₁₆×C₇₉	—	2	2	no	—
82	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
83	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
84	C₂×Dic₁₅₈	—	2	2	no	—
85	C₂×Dic₁₅₈	—	4	2	no	—
86	Q₈×D₇₉	—	4	4	no	—
87	Q₈×D₇₉	—	8	2	no	—
88	C₂×C₂²⋊D₇₉	—	4	2	no	—
89	C₂×C₂²⋊D₇₉	D₈	1	1	yes	characteristic
90	C₂×C₄×D₇₉	—	4	2	no	—
91	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
92	Dic₃₁₆	—	2	2	no	—
93	Dic₃₁₆	—	2	2	no	—
94	#11929614	—	4	4	no	—
95	#11929612	—	8	2	no	—
96	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
98	C₈×D₇₉	—	2	1	no	—
99	C₈×D₇₉	—	2	1	no	—
100	C₄⋊Dic₇₉	—	2	2	no	—
101	C₄⋊Dic₇₉	—	4	2	no	—
102	Dic₇₉⋊C₄	—	4	2	no	—
103	Dic₇₉⋊C₄	D₈	1	1	yes	characteristic
104	C₂²⋊Dic₇₉	—	4	2	no	—
105	C₂²⋊Dic₇₉	—	4	2	no	—
106	C₂²⋊Dic₇₉	—	4	2	no	—
107	C₂²⋊Dic₇₉	—	4	2	no	—
108	C₂²⋊Dic₇₉	D₈	1	1	yes	characteristic
109	(C₂×D₇₉)⋊C₄	D₈	1	1	yes	characteristic
110	C₄×Dic₇₉	—	4	2	no	—
111	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
112	C₇₉⋊C₁₆	—	2	1	no	—
113	C₇₉⋊C₁₆	—	2	1	no	—
114	#41×C₇₉	—	4	4	no	—
115	#41×C₇₉	—	8	2	no	—
116	C₂×Q₈×C₇₉	—	2	2	no	—
117	C₂×D₄×C₇₉	—	2	2	no	—
118	C₂×D₄×C₇₉	—	4	2	no	—
119	C₂²×C₄×C₇₉	—	4	2	no	—
120	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
121	Q₁₆×C₇₉	—	2	2	no	—
122	Q₁₆×C₇₉	C₂×D₄	1	4	yes	—
123	SD₁₆×C₇₉	—	4	4	no	—
124	D₈×C₇₉	—	2	2	no	—
125	D₈×C₇₉	—	2	2	no	—
126	C₂×C₈×C₇₉	—	2	1	no	—
127	C₂×C₈×C₇₉	—	2	1	no	—
128	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
130	C₁₆×C₇₉	—	2	1	no	—
131	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
132	C₂³×C₇₉	—	4	2	no	—
133	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
134	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
135	Q₈×C₇₉	—	2	4	no	—
136	Q₈×C₇₉	—	4	2	no	—
137	D₄×C₇₉	—	4	2	no	—
138	D₄×C₇₉	—	4	2	no	—
139	D₄×C₇₉	—	8	4	no	—
140	C₂×C₄×C₇₉	—	2	1	no	—
141	C₂×C₄×C₇₉	—	2	1	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	—	8	4	no	—
144	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₇₉	—	2	2	no	—
147	C₂²⋊D₇₉	—	8	4	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	4	2	no	—
150	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
151	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
152	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
153	C₄×D₇₉	—	2	1	no	—
154	C₄×D₇₉	—	2	1	no	—
155	C₄×D₇₉	—	8	4	no	—
156	Dic₁₅₈	—	4	2	no	—
157	Dic₁₅₈	—	4	2	no	—
158	Dic₁₅₈	—	8	4	no	—
159	C₈×C₇₉	—	2	1	no	—
160	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₇₉⋊C₈	—	2	1	no	—
163	C₇₉⋊C₈	—	2	1	no	—
164	C₂²×C₇₉	—	2	1	no	—
165	C₂²×C₇₉	—	2	1	no	—
166	C₂²×C₇₉	—	2	1	no	—
167	C₂²×C₇₉	—	8	2	no	—
168	C₂²×C₇₉	—	8	2	no	—
169	C₂²×C₇₉	—	8	2	no	—
170	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₇₉	—	2	1	no	—
172	C₂×D₇₉	—	2	1	no	—
173	C₂×D₇₉	#508	1	2	yes	complemented
174	C₄×C₇₉	—	2	1	no	—
175	C₄×C₇₉	—	4	4	no	—
176	C₄×C₇₉	D₄²	1	1	yes	characteristic
177	C₄×C₇₉	#576	1	1	yes	characteristic
178	Dic₇₉	—	2	1	no	—
179	Dic₇₉	—	2	1	no	—
180	Dic₇₉	—	2	1	no	—
181	Dic₇₉	—	8	2	no	—
182	Dic₇₉	—	8	2	no	—
183	#33851	—	79	1	no	maximal, Sylow
184	C₂×C₇₉	—	4	1	no	—
185	C₂×C₇₉	—	16	2	no	—
186	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
187	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
188	C₂×C₇₉	#3417	1	1	yes	characteristic
189	D₇₉	—	2	2	no	—
190	C₂×#508	D₇₉	1	1	yes	characteristic, complemented
191	C₂²×Q₃₂	—	79	1	no	—
192	#3285	—	79	2	no	—
193	#2249	—	79	1	no	—
194	#2216	—	79	1	no	—
195	#2215	—	79	1	no	—
196	#2211	—	79	2	no	—
197	#2189	—	79	2	no	—
198	#2185	—	79	2	no	—
199	C₄×Q₃₂	—	79	1	no	—
200	D₄×C₁₆	—	79	1	no	—
201	C₇₉	#33851	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	C₂×D₇₉	1	2	yes	—
203	C₂²×Q₁₆	—	79	2	no	—
204	#548	—	158	2	no	—
205	#508	—	2	4	no	—
206	C₂×Q₃₂	—	79	2	no	—
207	C₂×Q₃₂	—	79	1	no	—
208	C₂×Q₃₂	—	158	4	no	—
209	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
210	C₂×SD₃₂	—	79	2	no	—
211	C₂×SD₃₂	C₂×D₇₉	1	2	yes	—
212	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
213	C₂²×C₁₆	—	79	1	no	—
214	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
215	#495	—	79	2	no	—
216	#467	—	79	2	no	—
217	#466	—	79	2	no	—
218	#464	—	158	2	no	—
219	#452	—	158	2	no	—
220	#448	—	158	2	no	—
221	C₄×Q₁₆	—	79	2	no	—
222	C₈×D₄	—	79	1	no	—
223	#366	—	79	1	no	—
224	C₄⋊C₁₆	—	79	1	no	—
225	#358	—	79	2	no	—
226	#358	—	79	2	no	—
227	#357	—	79	2	no	—
228	C₂²⋊C₁₆	—	79	1	no	—
229	C₂²⋊C₁₆	—	79	1	no	—
230	C₄×C₁₆	—	79	1	no	—
231	C₂×#41	—	2	2	no	—
232	C₂²×Q₈	—	158	2	no	—
233	#135	—	2	4	no	—
234	#135	—	4	2	no	—
235	C₂×Q₁₆	—	316	2	no	—
236	C₂×Q₁₆	—	79	2	no	—
237	C₂×Q₁₆	—	158	2	no	—
238	C₂×Q₁₆	—	158	4	no	—
239	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
240	C₂×SD₁₆	—	2	2	no	—
241	C₂×SD₁₆	—	158	2	no	—
242	C₂×D₈	C₂²×D₇₉	1	2	yes	—
243	C₂²×C₈	—	79	1	no	—
244	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
245	#124	—	316	2	no	—
246	#124	—	158	2	no	—
247	C₂²⋊Q₈	—	316	2	no	—
248	C₂²⋊Q₈	—	158	2	no	—
249	#121	—	316	2	no	—
250	#121	—	158	2	no	—
251	C₄×Q₈	—	158	2	no	—
252	C₄×D₄	—	316	2	no	—
253	C₄×D₄	—	79	1	no	—
254	Q₃₂	—	158	4	no	—
255	Q₃₂	—	158	2	no	—
256	Q₃₂	C₂²⋊D₇₉	1	4	yes	complemented
257	SD₃₂	—	2	4	no	—
258	SD₃₂	—	316	2	no	—
259	D₁₆	—	2	2	no	—
260	C₂×C₁₆	—	2	2	no	—
261	C₂×C₁₆	—	79	1	no	—
262	C₂×C₁₆	—	79	1	no	—
263	C₂×C₁₆	—	158	2	no	—
264	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
266	#107	—	79	2	no	—
267	C₄⋊C₈	—	79	1	no	—
268	#103	—	158	2	no	—
269	#103	—	158	2	no	—
270	#102	—	158	2	no	—
271	C₂²⋊C₈	—	79	1	no	—
272	C₂²⋊C₈	—	79	1	no	—
273	C₄×C₈	—	79	1	no	—
274	#41	—	4	4	no	—
275	#41	—	8	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	316	2	no	—
278	C₂×Q₈	—	316	4	no	—
279	C₂×Q₈	—	632	2	no	—
280	C₂×Q₈	—	158	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	—	316	2	no	—
284	C₂×D₄	—	79	1	no	—
285	C₂²×C₄	—	4	2	no	—
286	C₂²×C₄	—	316	2	no	—
287	C₂²×C₄	—	79	1	no	—
288	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	316	4	no	—
291	Q₁₆	—	158	2	no	—
292	Q₁₆	—	158	2	no	—
293	Q₁₆	C₂×C₂²⋊D₇₉	1	4	yes	—
294	SD₁₆	—	4	4	no	—
295	SD₁₆	—	632	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	79	1	no	—
301	C₂×C₈	—	79	1	no	—
302	C₂×C₈	—	158	1	no	—
303	C₂×C₈	—	158	1	no	—
304	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
305	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
306	C₄⋊C₄	—	316	2	no	—
307	C₄⋊C₄	—	316	2	no	—
308	C₄⋊C₄	—	79	1	no	—
309	C₄⋊C₄	—	158	2	no	—
310	C₂²⋊C₄	—	316	2	no	—
311	C₂²⋊C₄	—	316	2	no	—
312	C₂²⋊C₄	—	316	2	no	—
313	C₂²⋊C₄	—	316	2	no	—
314	C₂²⋊C₄	—	79	1	no	—
315	C₂²⋊C₄	—	79	1	no	—
316	C₄²	—	316	2	no	—
317	C₄²	—	79	1	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	158	1	no	—
320	C₁₆	—	158	1	no	—
321	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
322	C₂³	—	4	2	no	—
323	C₂³	—	79	1	no	—
324	C₂³	D₈×D₇₉	1	1	yes	characteristic
325	Q₈	—	2	4	no	—
326	Q₈	—	4	2	no	—
327	Q₈	—	316	2	no	—
328	Q₈	—	316	2	no	—
329	Q₈	—	632	4	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	8	4	no	—
333	D₄	—	632	4	no	—
334	D₄	—	158	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	8	4	no	—
339	C₂×C₄	—	316	2	no	—
340	C₂×C₄	—	316	2	no	—
341	C₂×C₄	—	79	1	no	—
342	C₂×C₄	—	79	1	no	—
343	C₂×C₄	—	79	1	no	—
344	C₂×C₄	—	632	4	no	—
345	C₂×C₄	—	158	1	no	—
346	C₂×C₄	—	158	1	no	—
347	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	158	1	no	—
351	C₈	—	158	1	no	—
352	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
353	C₈	#14579206	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	79	2	no	—
361	C₂²	—	158	1	no	—
362	C₂²	—	158	1	no	—
363	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	4	4	no	—
366	C₄	—	632	2	no	—
367	C₄	—	632	2	no	—
368	C₄	—	158	1	no	—
369	C₄	—	158	1	no	—
370	C₄	—	158	1	no	—
371	C₄	D₄×C₂²⋊D₇₉	1	1	yes	characteristic
372	C₄	#19099364	1	1	yes	characteristic
373	C₂	—	4	1	no	—
374	C₂	—	16	2	no	—
375	C₂	—	158	2	no	—
376	C₂	D₈×C₂²⋊D₇₉	1	1	yes	characteristic
377	C₂	#508×D₇₉	1	1	yes	characteristic
378	C₂	#23776550	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#508×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	C₂×#508	D₇₉
hypocenter	C₇₉	#33851
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₇₉	C₂	semidirect
2. #23775135	C₂	semidirect
3. #23765658	C₂	semidirect
4. #23765491	C₂	semidirect
5. #23765483	C₂	semidirect
6. #23765460	C₂	semidirect
7. #23765340	C₂	semidirect
8. #23765315	C₂	semidirect
9. Q₃₂×Dic₇₉	C₂	semidirect
10. C₁₆×C₂²⋊D₇₉	C₂	semidirect
11. C₂×#508×C₇₉	C₂	semidirect
78. Q₃₂×C₇₉	D₄	semidirect
173. C₂×D₇₉	#508	semidirect
190. C₂×#508	D₇₉	semidirect
201. C₇₉	#33851	semidirect
256. Q₃₂	C₂²⋊D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₇₉	2
#23775135	2
#23765658	2
#23765491	2
#23765483	2
#23765460	2
#23765340	2
#23765315	2
Q₃₂×Dic₇₉	2
C₁₆×C₂²⋊D₇₉	2
C₂×#508×C₇₉	2
#33851	79