Subgroups — #2248×D₇₉ #34443659

Order 20224.

918 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₃₂×D₇₉	C₂	1	4	yes	maximal, complemented
2	#494×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×D₇₉	C₂	1	1	yes	maximal, characteristic
4	#23765657	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765656	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765655	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765654	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#2248	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₇₉	—	2	1	no	—
11	C₂×Q₁₆×D₇₉	C₂²	1	2	yes	—
12	C₂×D₈×D₇₉	—	2	1	no	—
13	C₂×D₈×D₇₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₇₉	—	2	1	no	—
15	#127×D₇₉	—	2	1	no	—
16	C₂×#14579106	C₂²	1	4	yes	complemented
17	C₂×#14579105	C₂²	1	4	yes	complemented
18	C₂×#14579104	C₂²	1	4	yes	complemented
19	SD₃₂×D₇₉	—	2	16	no	—
20	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
21	#19098943	C₂²	1	1	yes	characteristic, complemented
22	#19098942	—	2	1	no	—
23	#19098941	—	2	1	no	—
24	#19098941	C₂²	1	2	yes	complemented
25	#19098940	C₂²	1	1	yes	characteristic
26	#19098939	—	2	1	no	—
27	#19098938	—	2	1	no	—
28	#19098938	C₂²	1	2	yes	—
29	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
30	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
31	C₁₆×Dic₇₉	C₂²	1	2	yes	complemented
32	C₂×SD₃₂×C₇₉	C₂²	1	4	yes	complemented
33	#494×C₇₉	C₂²	1	1	yes	characteristic, complemented
34	#493×C₇₉	C₂²	1	1	yes	characteristic
35	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
36	C₂×Q₈×D₇₉	—	2	2	no	—
37	C₂×D₄×D₇₉	—	2	2	no	—
38	C₂×D₄×D₇₉	—	4	2	no	—
39	C₂×Dic₃₁₆	—	2	1	no	—
40	C₂×Dic₃₁₆	C₂³	1	2	yes	—
41	C₂×#11929614	—	2	2	no	—
42	C₂×#11929614	—	2	2	no	—
43	Q₁₆×D₇₉	—	2	8	no	—
44	Q₁₆×D₇₉	—	4	4	no	—
45	C₂×D₆₃₂	—	2	1	no	—
46	C₂×D₆₃₂	C₂³	1	2	yes	—
47	C₂×#11929609	—	2	2	no	—
48	C₂×#11929609	—	2	2	no	—
49	D₈×D₇₉	—	2	8	no	—
50	D₈×D₇₉	—	4	4	no	—
51	C₂×C₈×D₇₉	D₄	1	1	yes	characteristic
52	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
53	#14579161	—	2	1	no	—
54	#14579160	—	4	2	no	—
55	#14579159	—	2	2	no	—
56	#14579158	—	2	1	no	—
57	#14579157	—	2	2	no	—
58	#14579157	—	4	2	no	—
59	C₄⋊C₄×D₇₉	—	4	2	no	—
60	C₄²×D₇₉	D₄	1	1	yes	characteristic
61	#14579106	—	2	8	no	—
62	#14579105	—	2	8	no	—
63	#14579104	—	2	8	no	—
64	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
65	C₁₆×D₇₉	D₄	1	8	yes	complemented
66	C₄×C₇₉⋊C₈	D₄	1	1	yes	characteristic
67	C₈×Dic₇₉	D₄	1	1	yes	characteristic
68	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₇₉	—	2	1	no	—
70	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
71	C₂×D₈×C₇₉	—	2	1	no	—
72	C₂×D₈×C₇₉	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₇₉	—	2	1	no	—
74	#127×C₇₉	—	2	1	no	—
75	SD₃₂×C₇₉	—	2	8	no	—
76	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
77	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	C₂³×D₇₉	—	4	2	no	—
79	C₂×Dic₁₅₈	—	2	2	no	—
80	C₂×Dic₁₅₈	—	4	2	no	—
81	Q₈×D₇₉	—	4	4	no	—
82	Q₈×D₇₉	—	4	4	no	—
83	C₂×D₃₁₆	—	2	2	no	—
84	C₂×D₃₁₆	—	4	2	no	—
85	C₂×C₂²⋊D₇₉	—	4	4	no	—
86	C₂×C₂²⋊D₇₉	—	4	2	no	—
87	D₄×D₇₉	—	4	4	no	—
88	D₄×D₇₉	—	4	4	no	—
89	D₄×D₇₉	—	8	8	no	—
90	C₂×C₄×D₇₉	—	4	2	no	—
91	C₂×C₄×D₇₉	D₈	1	2	yes	—
92	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
93	Dic₃₁₆	—	2	4	no	—
94	Dic₃₁₆	—	4	2	no	—
95	#11929614	—	4	4	no	—
96	#11929614	—	4	4	no	—
97	D₆₃₂	—	2	4	no	—
98	D₆₃₂	—	4	2	no	—
99	#11929609	—	4	4	no	—
100	#11929609	—	4	4	no	—
101	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
103	C₈×D₇₉	D₈	1	4	yes	—
104	C₈×D₇₉	C₂×D₄	1	4	yes	—
105	C₄⋊Dic₇₉	—	4	2	no	—
106	Dic₇₉⋊C₄	—	4	4	no	—
107	C₄×Dic₇₉	D₈	1	2	yes	—
108	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
109	C₇₉⋊C₁₆	C₂×D₄	1	4	yes	complemented
110	C₂×Q₈×C₇₉	—	2	2	no	—
111	C₂×D₄×C₇₉	—	2	2	no	—
112	C₂×D₄×C₇₉	—	4	2	no	—
113	Q₁₆×C₇₉	—	2	4	no	—
114	Q₁₆×C₇₉	—	4	2	no	—
115	D₈×C₇₉	—	2	4	no	—
116	D₈×C₇₉	—	4	2	no	—
117	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₇₉	—	4	2	no	—
120	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₁₆×C₇₉	C₂×D₄	1	4	yes	complemented
122	C₂³×C₇₉	—	4	2	no	—
123	C₂²×D₇₉	—	4	2	no	—
124	C₂²×D₇₉	—	8	4	no	—
125	C₂²×D₇₉	—	8	8	no	—
126	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
127	Q₈×C₇₉	—	4	2	no	—
128	Q₈×C₇₉	—	4	2	no	—
129	D₄×C₇₉	—	4	2	no	—
130	D₄×C₇₉	—	4	2	no	—
131	D₄×C₇₉	—	8	4	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₇₉	—	8	4	no	—
136	C₂²⋊D₇₉	—	8	8	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
139	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
140	D₃₁₆	—	4	2	no	—
141	D₃₁₆	—	4	2	no	—
142	D₃₁₆	—	8	4	no	—
143	C₄×D₇₉	—	8	4	no	—
144	C₄×D₇₉	SD₃₂	1	8	yes	complemented
145	C₄×D₇₉	#127	1	2	yes	—
146	C₄×D₇₉	C₂×D₈	1	2	yes	—
147	Dic₁₅₈	—	4	2	no	—
148	Dic₁₅₈	—	4	2	no	—
149	Dic₁₅₈	—	8	4	no	—
150	C₈×C₇₉	C₂×D₈	1	2	yes	—
151	C₈×C₇₉	C₂²×D₄	1	2	yes	—
152	C₇₉⋊C₈	#127	1	2	yes	—
153	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
154	C₂²×C₇₉	—	8	2	no	—
155	C₂²×C₇₉	—	8	4	no	—
156	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₇₉	—	8	2	no	—
158	C₂×D₇₉	—	8	4	no	—
159	C₂×D₇₉	—	16	8	no	—
160	C₂×D₇₉	#493	1	2	yes	—
161	C₂×D₇₉	C₂×SD₃₂	1	4	yes	—
162	C₄×C₇₉	—	8	2	no	—
163	C₄×C₇₉	C₂×SD₃₂	1	4	yes	complemented
164	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
165	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
166	Dic₇₉	—	8	2	no	—
167	Dic₇₉	#493	1	2	yes	—
168	Dic₇₉	C₂×SD₃₂	1	4	yes	complemented
169	C₂×#2248	—	79	1	no	maximal, Sylow
170	C₂×C₇₉	—	16	2	no	—
171	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
172	C₂×C₇₉	C₂²×SD₃₂	1	2	yes	—
173	D₇₉	—	16	2	no	—
174	D₇₉	#2248	1	4	yes	direct product factor
175	C₂²×SD₃₂	—	79	4	no	—
176	C₂×#494	—	79	1	no	—
177	C₂×#493	—	79	1	no	—
178	#2248	—	79	4	no	—
179	#2248	—	79	1	no	—
180	#2248	—	79	1	no	—
181	#2248	—	79	1	no	—
182	#2248	D₇₉	1	1	yes	characteristic, direct product factor
183	C₂×C₄×C₁₆	—	79	1	no	—
184	C₇₉	C₂×#2248	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₁₆	—	79	2	no	—
186	C₂²×Q₁₆	—	158	1	no	—
187	C₂²×D₈	—	79	2	no	—
188	C₂²×D₈	—	158	1	no	—
189	C₂×C₄⋊Q₈	—	158	1	no	—
190	C₂×#127	—	158	1	no	—
191	C₂×SD₃₂	—	79	4	no	—
192	C₂×SD₃₂	—	79	4	no	—
193	C₂×SD₃₂	—	79	4	no	—
194	C₂×SD₃₂	—	158	16	no	—
195	C₂×SD₃₂	C₂×D₇₉	1	4	yes	complemented
196	C₂²×C₁₆	—	79	2	no	—
197	#494	—	79	2	no	—
198	#494	—	79	1	no	—
199	#494	—	158	1	no	—
200	#494	—	158	1	no	—
201	#494	C₂×D₇₉	1	1	yes	characteristic, complemented
202	#493	—	79	2	no	—
203	#493	—	79	1	no	—
204	#493	—	158	1	no	—
205	#493	—	158	1	no	—
206	#493	C₂×D₇₉	1	1	yes	characteristic
207	C₂×C₄×C₈	—	79	1	no	—
208	C₄×C₁₆	—	79	2	no	—
209	C₄×C₁₆	—	79	1	no	—
210	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
211	C₂²×Q₈	—	158	2	no	—
212	C₂²×D₄	—	316	2	no	—
213	C₂²×D₄	—	158	2	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	C₂×Q₁₆	—	316	4	no	—
216	C₂×Q₁₆	—	79	2	no	—
217	C₂×Q₁₆	—	158	8	no	—
218	C₂×Q₁₆	—	158	2	no	—
219	C₂×Q₁₆	—	158	2	no	—
220	C₂×Q₁₆	—	158	1	no	—
221	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	316	4	no	—
224	C₂×D₈	—	79	2	no	—
225	C₂×D₈	—	158	8	no	—
226	C₂×D₈	—	158	2	no	—
227	C₂×D₈	—	158	2	no	—
228	C₂×D₈	—	158	1	no	—
229	C₂×D₈	C₂²×D₇₉	1	2	yes	—
230	C₂²×C₈	—	79	1	no	—
231	C₂²×C₈	—	79	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	316	2	no	—
234	C₄⋊Q₈	—	158	2	no	—
235	C₄⋊Q₈	—	158	1	no	—
236	#127	—	2	1	no	—
237	#127	—	316	2	no	—
238	#127	—	158	2	no	—
239	#127	—	158	1	no	—
240	C₂×C₄⋊C₄	—	316	2	no	—
241	C₂×C₄²	—	79	1	no	—
242	SD₃₂	—	2	8	no	—
243	SD₃₂	—	158	8	no	—
244	SD₃₂	—	158	8	no	—
245	SD₃₂	—	158	8	no	—
246	C₂×C₁₆	—	79	2	no	—
247	C₂×C₁₆	—	79	8	no	—
248	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
249	C₄×C₈	—	79	1	no	—
250	C₄×C₈	—	79	1	no	—
251	C₄×C₈	—	79	1	no	—
252	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
253	C₂⁴	—	316	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	316	2	no	—
256	C₂×Q₈	—	316	4	no	—
257	C₂×Q₈	—	316	4	no	—
258	C₂×Q₈	—	158	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	2	no	—
261	C₂×D₄	—	316	4	no	—
262	C₂×D₄	—	316	4	no	—
263	C₂×D₄	—	316	2	no	—
264	C₂×D₄	—	316	4	no	—
265	C₂×D₄	—	316	2	no	—
266	C₂×D₄	—	632	8	no	—
267	C₂×D₄	—	158	2	no	—
268	C₂²×C₄	—	316	2	no	—
269	C₂²×C₄	—	79	2	no	—
270	C₂²×C₄	—	79	1	no	—
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	4	2	no	—
273	Q₁₆	—	316	4	no	—
274	Q₁₆	—	316	4	no	—
275	Q₁₆	—	316	2	no	—
276	Q₁₆	—	158	4	no	—
277	D₈	—	2	4	no	—
278	D₈	—	4	2	no	—
279	D₈	—	316	4	no	—
280	D₈	—	316	4	no	—
281	D₈	—	316	2	no	—
282	D₈	—	158	4	no	—
283	C₂×C₈	—	79	1	no	—
284	C₂×C₈	—	79	1	no	—
285	C₂×C₈	—	79	4	no	—
286	C₂×C₈	—	79	4	no	—
287	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
288	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
289	C₄⋊C₄	—	4	2	no	—
290	C₄⋊C₄	—	316	4	no	—
291	C₄⋊C₄	—	316	2	no	—
292	C₄²	—	79	2	no	—
293	C₄²	—	79	1	no	—
294	C₄²	D₄×D₇₉	1	1	yes	characteristic
295	C₁₆	—	79	4	no	—
296	C₁₆	D₄×D₇₉	1	4	yes	complemented
297	C₂³	—	4	2	no	—
298	C₂³	—	316	2	no	—
299	C₂³	—	79	1	no	—
300	C₂³	—	632	4	no	—
301	C₂³	—	632	8	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	316	2	no	—
305	Q₈	—	316	2	no	—
306	Q₈	—	632	4	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	4	no	—
310	D₄	—	316	2	no	—
311	D₄	—	316	2	no	—
312	D₄	—	632	4	no	—
313	D₄	—	632	8	no	—
314	D₄	—	632	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	316	2	no	—
317	C₂×C₄	—	79	2	no	—
318	C₂×C₄	—	79	1	no	—
319	C₂×C₄	—	79	8	no	—
320	C₂×C₄	—	79	2	no	—
321	C₂×C₄	—	79	2	no	—
322	C₂×C₄	—	632	4	no	—
323	C₂×C₄	D₈×D₇₉	1	2	yes	—
324	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
325	C₈	—	79	2	no	—
326	C₈	—	79	2	no	—
327	C₈	D₈×D₇₉	1	2	yes	—
328	C₈	C₂×D₄×D₇₉	1	2	yes	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	4	no	—
331	C₂²	—	79	2	no	—
332	C₂²	—	79	4	no	—
333	C₂²	—	632	2	no	—
334	C₂²	—	632	4	no	—
335	C₂²	—	1264	8	no	—
336	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	79	2	no	—
339	C₄	—	79	4	no	—
340	C₄	—	632	2	no	—
341	C₄	SD₃₂×D₇₉	1	4	yes	complemented
342	C₄	#127×D₇₉	1	1	yes	characteristic
343	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
344	C₂	—	16	2	no	—
345	C₂	—	79	4	no	—
346	C₂	—	1264	2	no	—
347	C₂	#493×D₇₉	1	1	yes	characteristic
348	C₂	C₂×SD₃₂×D₇₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2248	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#493×D₇₉	C₂
hypercenter	#2248	D₇₉
hypocenter	C₇₉	C₂×#2248
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2248

Complements

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

N	H ≅ G/N	type
1. C₂×SD₃₂×D₇₉	C₂	semidirect
2. #494×D₇₉	C₂	semidirect
4. #23765657	C₂	semidirect
5. #23765656	C₂	semidirect
6. #23765655	C₂	semidirect
7. #23765654	C₂	semidirect
8. C₄×C₁₆×D₇₉	C₂	semidirect
9. C₇₉×#2248	C₂	semidirect
16. C₂×#14579106	C₂²	semidirect
17. C₂×#14579105	C₂²	semidirect
18. C₂×#14579104	C₂²	semidirect
21. #19098943	C₂²	semidirect
24. #19098941	C₂²	semidirect
30. C₄×C₇₉⋊C₁₆	C₂²	semidirect
31. C₁₆×Dic₇₉	C₂²	semidirect
32. C₂×SD₃₂×C₇₉	C₂²	semidirect
33. #494×C₇₉	C₂²	semidirect
35. C₄×C₁₆×C₇₉	C₂²	semidirect
65. C₁₆×D₇₉	D₄	semidirect
109. C₇₉⋊C₁₆	C₂×D₄	semidirect
121. C₁₆×C₇₉	C₂×D₄	semidirect
144. C₄×D₇₉	SD₃₂	semidirect
163. C₄×C₇₉	C₂×SD₃₂	semidirect
168. Dic₇₉	C₂×SD₃₂	semidirect
174. D₇₉	#2248	direct (N × H)
182. #2248	D₇₉	direct (N × H)
184. C₇₉	C₂×#2248	semidirect
195. C₂×SD₃₂	C₂×D₇₉	semidirect
201. #494	C₂×D₇₉	semidirect
210. C₄×C₁₆	C₂×D₇₉	semidirect
296. C₁₆	D₄×D₇₉	semidirect
341. C₄	SD₃₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₃₂×D₇₉	2
#494×D₇₉	2
#493×D₇₉	2
#23765657	2
#23765656	2
#23765655	2
#23765654	2
C₄×C₁₆×D₇₉	2
C₇₉×#2248	2
C₂×#2248	79

Subgroups — #2248×D79 #34443659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2248×D₇₉ #34443659