Subgroups — #34279792

Order 20224.

606 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776265	C₂	1	1	yes	maximal, characteristic, complemented
2	#23775716	C₂	1	2	yes	maximal, complemented
3	#23763009	C₂	1	1	yes	maximal, characteristic, complemented
4	#23762973	C₂	1	2	yes	maximal, complemented
5	#23762799	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762791	C₂	1	1	yes	maximal, characteristic, complemented
7	#23762613	C₂	1	2	yes	maximal, complemented
8	#23762551	C₂	1	2	yes	maximal, complemented
9	#23762285	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11929612	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3334	C₂	1	1	yes	maximal, characteristic
12	#19099270	C₂²	1	2	yes	—
13	#19098968	C₂²	1	1	yes	characteristic
14	#19098964	—	2	1	no	—
15	#19098915	C₂²	1	2	yes	—
16	#19098913	C₂²	1	2	yes	—
17	#19098891	—	2	2	no	—
18	#19098861	—	2	2	no	—
19	#19098853	C₂²	1	2	yes	—
20	#19098853	C₂²	1	2	yes	—
21	#19098819	—	2	2	no	—
22	#19098770	C₂²	1	2	yes	—
23	#19098725	—	2	2	no	—
24	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
25	C₄×#11929612	C₂²	1	1	yes	characteristic
26	D₈×Dic₇₉	—	2	1	no	—
27	#19098641	C₂²	1	2	yes	—
28	#19098564	—	2	2	no	—
29	#19098316	C₂²	1	1	yes	characteristic
30	#19098313	C₂²	1	1	yes	characteristic
31	#19098311	C₂²	1	2	yes	—
32	#19098309	C₂²	1	2	yes	—
33	#19098293	C₂²	1	1	yes	characteristic
34	D₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
35	D₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
36	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
37	#547×C₇₉	C₂²	1	2	yes	—
38	#500×C₇₉	C₂²	1	1	yes	characteristic
39	#480×C₇₉	—	2	2	no	—
40	#474×C₇₉	C₂²	1	2	yes	—
41	#466×C₇₉	—	2	1	no	—
42	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
43	C₈×D₄×C₇₉	C₂²	1	1	yes	characteristic
44	C₂×#107×C₇₉	—	2	1	no	—
45	C₂×#11929614	—	2	2	no	—
46	C₂×#11929612	D₄	1	1	yes	characteristic
47	C₂×#11929608	—	2	2	no	—
48	#14579161	C₂³	1	2	yes	—
49	#14579152	—	2	2	no	—
50	#14579140	—	2	2	no	—
51	#14579138	—	4	2	no	—
52	C₄×Dic₁₅₈	C₂³	1	1	yes	characteristic
53	D₄×Dic₇₉	—	2	2	no	—
54	C₂×C₄⋊Dic₇₉	—	2	2	no	—
55	#14579095	—	2	1	no	—
56	#14579095	—	2	1	no	—
57	#14579092	—	2	2	no	—
58	#14579092	—	2	2	no	—
59	#14579092	D₄	1	1	yes	characteristic
60	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
61	C₄⋊(C₇₉⋊C₈)	C₂³	1	1	yes	characteristic
62	Dic₇₉⋊C₈	—	2	1	no	—
63	#14579085	—	2	2	no	—
64	#14579085	—	2	2	no	—
65	#14579085	—	2	2	no	—
66	#14579084	—	4	2	no	—
67	#14579084	D₄	1	1	yes	characteristic
68	#14579082	—	2	2	no	—
69	#14579082	—	2	2	no	—
70	#14579082	D₄	1	1	yes	characteristic
71	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
72	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
73	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
74	C₈×Dic₇₉	—	2	1	no	—
75	C₂×#41×C₇₉	—	2	2	no	—
76	C₂×D₈×C₇₉	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₇₉	—	2	1	no	—
78	C₄⋊Q₈×C₇₉	C₂³	1	2	yes	—
79	#123×C₇₉	—	4	2	no	—
80	C₂²⋊Q₈×C₇₉	—	2	2	no	—
81	#121×C₇₉	—	2	2	no	—
82	C₄×D₄×C₇₉	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₇₉	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₇₉	—	2	2	no	—
85	#107×C₇₉	—	2	1	no	—
86	#107×C₇₉	—	2	1	no	—
87	#107×C₇₉	—	2	2	no	—
88	#107×C₇₉	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
90	#102×C₇₉	—	2	2	no	—
91	#102×C₇₉	D₄	1	2	yes	—
92	C₂²⋊C₈×C₇₉	—	2	1	no	—
93	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₅₈	—	2	2	no	—
95	C₂×Dic₁₅₈	C₂×D₄	1	1	yes	characteristic
96	C₂²×Dic₇₉	—	4	2	no	—
97	#11929614	—	8	2	no	—
98	#11929612	—	2	2	no	—
99	#11929608	—	8	2	no	—
100	C₂×C₇₉⋊C₈	—	2	1	no	—
101	C₂×C₇₉⋊C₈	—	2	2	no	—
102	C₂×C₇₉⋊C₈	—	2	1	no	—
103	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
104	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
105	C₄⋊Dic₇₉	—	2	2	no	—
106	C₄⋊Dic₇₉	—	2	2	no	—
107	C₄⋊Dic₇₉	—	4	2	no	—
108	C₄⋊Dic₇₉	C₂×D₄	1	1	yes	characteristic
109	Dic₇₉⋊C₄	—	2	1	no	—
110	Dic₇₉⋊C₄	—	4	2	no	—
111	C₂²⋊Dic₇₉	—	4	2	no	—
112	C₄×Dic₇₉	—	2	1	no	—
113	#41×C₇₉	—	4	4	no	—
114	#41×C₇₉	—	8	2	no	—
115	C₂×Q₈×C₇₉	—	2	2	no	—
116	C₂×D₄×C₇₉	—	4	2	no	—
117	C₂×D₄×C₇₉	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₇₉	—	2	1	no	—
120	C₂²×C₄×C₇₉	—	2	2	no	—
121	C₂²×C₄×C₇₉	—	4	2	no	—
122	D₈×C₇₉	—	2	2	no	—
123	C₂×C₈×C₇₉	—	2	1	no	—
124	C₂×C₈×C₇₉	—	4	2	no	—
125	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₄⋊C₄×C₇₉	—	2	2	no	—
128	C₄⋊C₄×C₇₉	—	2	2	no	—
129	C₄⋊C₄×C₇₉	—	4	2	no	—
130	C₄⋊C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₇₉	C₂×D₄	1	2	yes	—
132	C₂²⋊C₄×C₇₉	—	2	1	no	—
133	C₂²⋊C₄×C₇₉	—	2	2	no	—
134	C₂²⋊C₄×C₇₉	—	4	2	no	—
135	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₇₉	—	2	1	no	—
137	C₂³×C₇₉	—	2	2	no	—
138	Q₈×C₇₉	—	4	2	no	—
139	Q₈×C₇₉	—	4	2	no	—
140	D₄×C₇₉	—	2	2	no	—
141	D₄×C₇₉	—	2	1	no	—
142	D₄×C₇₉	—	2	2	no	—
143	D₄×C₇₉	—	8	4	no	—
144	D₄×C₇₉	#135	1	2	yes	—
145	C₂×C₄×C₇₉	—	2	2	no	—
146	C₂×C₄×C₇₉	—	2	1	no	—
147	C₂×C₄×C₇₉	—	4	2	no	—
148	C₂×C₄×C₇₉	—	4	1	no	—
149	C₂×C₄×C₇₉	—	4	2	no	—
150	C₂×C₄×C₇₉	—	4	1	no	—
151	C₂×C₄×C₇₉	—	4	2	no	—
152	C₂×C₄×C₇₉	—	8	4	no	—
153	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×Dic₇₉	—	2	1	no	—
157	C₂×Dic₇₉	—	2	1	no	—
158	C₂×Dic₇₉	—	4	2	no	—
159	C₂×Dic₇₉	—	16	2	no	—
160	Dic₁₅₈	—	2	1	no	—
161	Dic₁₅₈	—	2	1	no	—
162	Dic₁₅₈	—	8	2	no	—
163	C₈×C₇₉	—	2	1	no	—
164	C₈×C₇₉	—	4	1	no	—
165	C₈×C₇₉	#137	1	2	yes	—
166	C₇₉⋊C₈	—	2	1	no	—
167	C₇₉⋊C₈	—	2	1	no	—
168	C₇₉⋊C₈	—	4	1	no	—
169	C₇₉⋊C₈	—	4	2	no	—
170	C₇₉⋊C₈	—	4	1	no	—
171	C₂²×C₇₉	—	2	2	no	—
172	C₂²×C₇₉	—	4	1	no	—
173	C₂²×C₇₉	—	4	1	no	—
174	C₂²×C₇₉	—	4	2	no	—
175	C₂²×C₇₉	—	4	2	no	—
176	C₂²×C₇₉	—	4	2	no	—
177	C₂²×C₇₉	D₄²	1	1	yes	characteristic
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	4	1	no	—
180	C₄×C₇₉	—	4	2	no	—
181	C₄×C₇₉	—	8	2	no	—
182	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
183	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
184	C₄×C₇₉	#576	1	1	yes	characteristic
185	C₄×C₇₉	#578	1	1	yes	characteristic
186	Dic₇₉	—	4	1	no	—
187	Dic₇₉	—	4	1	no	—
188	Dic₇₉	—	8	2	no	—
189	#19903	—	79	1	no	maximal, Sylow
190	C₂×C₇₉	—	4	2	no	—
191	C₂×C₇₉	—	8	2	no	—
192	C₂×C₇₉	#3276	1	1	yes	characteristic
193	C₂×C₇₉	#3280	1	1	yes	characteristic
194	C₂×C₇₉	#3290	1	1	yes	characteristic
195	#3380	—	79	1	no	—
196	#3334	D₇₉	1	1	yes	characteristic
197	#3331	—	79	2	no	—
198	#1710	—	79	1	no	—
199	#1703	—	79	2	no	—
200	#1675	—	79	1	no	—
201	#1674	—	79	1	no	—
202	#1644	—	79	2	no	—
203	#1635	—	79	2	no	—
204	#1589	—	79	1	no	—
205	C₈×SD₁₆	—	79	1	no	—
206	C₇₉	#19903	1	1	yes	characteristic, complemented, Sylow
207	#554	—	79	2	no	—
208	#547	C₂×D₇₉	1	2	yes	—
209	#500	—	79	1	no	—
210	#500	C₂×D₇₉	1	1	yes	characteristic
211	#499	—	158	1	no	—
212	#487	—	79	2	no	—
213	#487	—	79	2	no	—
214	#483	—	158	2	no	—
215	#480	—	2	2	no	—
216	#478	—	158	2	no	—
217	#476	—	79	2	no	—
218	#476	—	79	2	no	—
219	#474	C₂×D₇₉	1	2	yes	—
220	#470	—	158	2	no	—
221	#466	—	2	1	no	—
222	#462	—	79	2	no	—
223	#452	—	158	2	no	—
224	C₈×Q₈	—	79	1	no	—
225	C₄×SD₁₆	—	79	1	no	—
226	C₄×D₈	—	158	1	no	—
227	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
228	#436	—	79	2	no	—
229	C₈×D₄	C₂×D₇₉	1	1	yes	characteristic
230	C₂×#107	—	2	1	no	—
231	#417	—	158	2	no	—
232	#335	—	79	1	no	—
233	#334	—	79	1	no	—
234	#333	—	79	2	no	—
235	#332	—	79	2	no	—
236	Q₈⋊C₈	—	79	1	no	—
237	D₄⋊C₈	—	79	1	no	—
238	D₄⋊C₈	—	79	2	no	—
239	C₈²	—	79	1	no	—
240	C₂×#41	—	2	2	no	—
241	C₂×Q₁₆	—	158	2	no	—
242	C₂×SD₁₆	—	79	1	no	—
243	C₂×D₈	C₂²⋊D₇₉	1	1	yes	characteristic
244	C₂×M₁₆	—	158	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	79	2	no	—
247	C₄⋊Q₈	C₂²×D₇₉	1	2	yes	—
248	#125	—	158	2	no	—
249	#123	—	4	2	no	—
250	C₂²⋊Q₈	—	2	2	no	—
251	C₂²⋊Q₈	—	316	2	no	—
252	C₂²⋊Q₈	—	158	2	no	—
253	#121	—	2	2	no	—
254	C₄×Q₈	—	79	1	no	—
255	C₄×D₄	—	158	2	no	—
256	C₄×D₄	C₂²×D₇₉	1	1	yes	characteristic
257	C₄×D₄	C₂²×D₇₉	1	2	yes	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	158	2	no	—
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	#107	—	2	2	no	—
263	#107	—	158	1	no	—
264	#107	—	158	1	no	—
265	#107	C₂²⋊D₇₉	1	1	yes	characteristic
266	#106	—	79	1	no	—
267	#106	—	158	2	no	—
268	#106	—	158	2	no	—
269	C₄⋊C₈	—	79	1	no	—
270	C₄⋊C₈	—	79	2	no	—
271	C₄⋊C₈	—	158	1	no	—
272	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
273	#103	—	316	2	no	—
274	#103	—	79	1	no	—
275	#103	—	158	2	no	—
276	#103	—	158	2	no	—
277	#103	—	158	2	no	—
278	#102	—	2	2	no	—
279	#102	—	79	1	no	—
280	#102	—	158	2	no	—
281	#102	—	158	2	no	—
282	#102	C₂²⋊D₇₉	1	2	yes	—
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	158	2	no	—
285	C₄×C₈	—	79	1	no	—
286	C₄×C₈	—	79	1	no	—
287	C₄×C₈	—	158	1	no	—
288	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
289	#41	—	4	4	no	—
290	#41	—	8	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	79	1	no	—
293	C₂×Q₈	—	158	2	no	—
294	C₂×D₄	—	4	2	no	—
295	C₂×D₄	D₄×D₇₉	1	1	yes	characteristic
296	C₂×D₄	C₂×C₂²⋊D₇₉	1	2	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	316	2	no	—
301	Q₁₆	—	632	2	no	—
302	SD₁₆	—	158	2	no	—
303	D₈	—	2	2	no	—
304	M₁₆	—	632	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	—	79	1	no	—
308	C₂×C₈	—	79	1	no	—
309	C₂×C₈	—	158	1	no	—
310	C₂×C₈	—	158	2	no	—
311	C₂×C₈	—	158	1	no	—
312	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	—	316	2	no	—
318	C₄⋊C₄	—	316	2	no	—
319	C₄⋊C₄	—	79	1	no	—
320	C₄⋊C₄	—	158	2	no	—
321	C₄⋊C₄	—	158	2	no	—
322	C₄⋊C₄	—	158	1	no	—
323	C₄⋊C₄	D₄×D₇₉	1	1	yes	characteristic
324	C₄⋊C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	1	no	—
327	C₂²⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	316	2	no	—
329	C₄²	—	158	1	no	—
330	C₄²	C₂³×D₇₉	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	2	2	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	632	2	no	—
336	Q₈	—	158	1	no	—
337	Q₈	—	158	1	no	—
338	D₄	—	2	2	no	—
339	D₄	—	2	1	no	—
340	D₄	—	2	2	no	—
341	D₄	—	8	4	no	—
342	D₄	#14579180	1	2	yes	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	4	no	—
351	C₂×C₄	—	316	2	no	—
352	C₂×C₄	—	158	1	no	—
353	C₂×C₄	—	158	1	no	—
354	C₂×C₄	—	1264	2	no	—
355	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
357	C₂×C₄	#14579206	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	316	1	no	—
361	C₈	—	316	2	no	—
362	C₈	—	316	1	no	—
363	C₈	—	158	1	no	—
364	C₈	—	158	1	no	—
365	C₈	#14579191	1	2	yes	—
366	C₂²	—	2	2	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	D₄×C₂²⋊D₇₉	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	316	1	no	—
378	C₄	—	316	1	no	—
379	C₄	—	632	2	no	—
380	C₄	C₂×#14579180	1	1	yes	characteristic
381	C₄	C₂×#14579191	1	1	yes	characteristic
382	C₄	#19099364	1	1	yes	characteristic
383	C₄	#19099378	1	1	yes	characteristic
384	C₂	—	4	2	no	—
385	C₂	—	8	2	no	—
386	C₂	#23775042	1	1	yes	characteristic
387	C₂	#23775085	1	1	yes	characteristic
388	C₂	#23775186	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₇₉
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting, prime radical	C₇₉×#3334	C₂
socle	C₂²×C₇₉	D₄²
hypercenter	#3334	D₇₉
hypocenter	C₇₉	#19903
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3334

Complements

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

N	H ≅ G/N	type
1. #23776265	C₂	semidirect
2. #23775716	C₂	semidirect
3. #23763009	C₂	semidirect
4. #23762973	C₂	semidirect
5. #23762799	C₂	semidirect
6. #23762791	C₂	semidirect
7. #23762613	C₂	semidirect
8. #23762551	C₂	semidirect
9. #23762285	C₂	semidirect
10. C₈×#11929612	C₂	semidirect
206. C₇₉	#19903	semidirect

Maximal subgroups

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

subgroup	index
#23776265	2
#23775716	2
#23763009	2
#23762973	2
#23762799	2
#23762791	2
#23762613	2
#23762551	2
#23762285	2
C₈×#11929612	2
C₇₉×#3334	2
#19903	79