Subgroups — #34259698

Order 20224.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772412	C₂	1	2	yes	maximal, complemented
2	C₄×#14579183	C₂	1	1	yes	maximal, characteristic
3	#23762296	C₂	1	2	yes	maximal, complemented
4	#23762235	C₂	1	4	yes	maximal, complemented
5	#23762194	C₂	1	2	yes	maximal, complemented
6	#23761832	C₂	1	2	yes	maximal, complemented
7	C₈×#11929608	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₄	1	1	yes	characteristic
10	C₄×#11929624	C₂²	1	2	yes	—
11	#19098690	C₂²	1	4	yes	—
12	C₄×Dic₃₁₆	C₄	1	1	yes	characteristic
13	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
14	C₄×#11929613	C₄	1	2	yes	—
15	C₄×#11929613	C₂²	1	2	yes	—
16	C₄×D₆₃₂	C₄	1	1	yes	characteristic
17	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
18	#19098637	—	2	4	no	—
19	#19098628	C₂²	1	4	yes	—
20	#19098618	—	2	2	no	—
21	#19098616	C₂²	1	1	yes	characteristic
22	#19098601	C₄	1	1	yes	characteristic
23	#19098560	C₄	1	2	yes	—
24	C₄×#11929608	C₂²	1	1	yes	characteristic
25	#97×D₇₉	—	2	2	no	—
26	#19098293	C₂²	1	4	yes	—
27	#19098290	C₂²	1	4	yes	—
28	#19098285	C₂²	1	2	yes	—
29	C₈×C₇₉⋊C₈	C₂²	1	2	yes	—
30	#432×C₇₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	#335×C₇₉	C₂²	1	2	yes	—
33	#334×C₇₉	C₂²	1	2	yes	—
34	C₂×#11929624	C₂×C₄	1	2	yes	—
35	#14579183	—	2	4	no	—
36	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#11929613	C₂×C₄	1	2	yes	—
38	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#11929608	D₄	1	2	yes	—
40	C₂×C₈×D₇₉	—	2	4	no	—
41	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₅₈	C₂³	1	2	yes	—
43	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
44	C₄×D₃₁₆	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₇₉	—	2	2	no	—
46	C₄×C₂²⋊D₇₉	—	2	2	no	—
47	#14579120	C₂×C₄	1	2	yes	—
48	#14579118	—	2	2	no	—
49	C₄²×D₇₉	—	2	2	no	—
50	#14579095	C₂×C₄	1	1	yes	characteristic
51	#14579095	C₂×C₄	1	1	yes	characteristic
52	#14579093	C₂×C₄	1	2	yes	—
53	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
54	Dic₇₉⋊C₈	—	2	4	no	—
55	#14579086	C₂×C₄	1	2	yes	—
56	#14579086	C₂×C₄	1	2	yes	—
57	#14579083	C₂×C₄	1	2	yes	—
58	#14579083	C₂×C₄	1	2	yes	—
59	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
60	(C₂×D₇₉)⋊C₈	—	2	4	no	—
61	#14579072	D₄	1	1	yes	characteristic
62	#14579072	D₄	1	1	yes	characteristic
63	#14579071	—	2	4	no	—
64	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
65	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
66	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₇₉	C₂³	1	4	yes	—
70	C₂²⋊C₈×C₇₉	—	2	2	no	—
71	#97×C₇₉	—	2	2	no	—
72	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
76	#11929624	—	2	4	no	—
77	#11929624	—	4	2	no	—
78	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
79	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
80	C₂×C₂²⋊D₇₉	—	2	2	no	—
81	C₂×C₄×D₇₉	—	2	2	no	—
82	C₂×C₄×D₇₉	—	2	2	no	—
83	C₂×C₄×D₇₉	—	2	2	no	—
84	Dic₃₁₆	—	2	2	no	—
85	#11929613	—	2	4	no	—
86	D₆₃₂	M₁₆	1	4	yes	complemented
87	#11929608	—	2	4	no	—
88	C₂×C₇₉⋊C₈	—	2	2	no	—
89	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
91	C₈×D₇₉	—	4	8	no	—
92	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
93	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
94	Dic₇₉⋊C₄	—	2	2	no	—
95	Dic₇₉⋊C₄	—	2	2	no	—
96	C₂²⋊Dic₇₉	—	2	2	no	—
97	(C₂×D₇₉)⋊C₄	—	2	2	no	—
98	(C₂×D₇₉)⋊C₄	—	2	2	no	—
99	C₄×Dic₇₉	—	2	2	no	—
100	C₄×Dic₇₉	—	2	2	no	—
101	C₄×Dic₇₉	—	2	2	no	—
102	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
105	C₂×C₈×C₇₉	—	2	2	no	—
106	C₂×C₈×C₇₉	—	2	4	no	—
107	C₂×C₈×C₇₉	—	4	1	no	—
108	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₉	#41	1	1	yes	characteristic
115	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
116	C₂³×C₇₉	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	2	no	—
121	C₂×C₄×C₇₉	—	4	1	no	—
122	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₇₉	—	4	4	no	—
129	C₂×Dic₇₉	—	2	2	no	—
130	C₂×Dic₇₉	—	2	2	no	—
131	C₂×Dic₇₉	—	2	2	no	—
132	D₃₁₆	—	2	2	no	—
133	D₃₁₆	C₂×M₁₆	1	4	yes	—
134	C₄×D₇₉	—	4	4	no	—
135	C₄×D₇₉	—	4	4	no	—
136	C₄×D₇₉	—	8	2	no	—
137	Dic₁₅₈	—	2	2	no	—
138	Dic₁₅₈	—	2	2	no	—
139	C₈×C₇₉	—	2	1	no	—
140	C₈×C₇₉	—	2	1	no	—
141	C₈×C₇₉	—	2	1	no	—
142	C₈×C₇₉	—	4	4	no	—
143	C₈×C₇₉	C₂×M₁₆	1	2	yes	—
144	C₇₉⋊C₈	—	2	2	no	—
145	C₇₉⋊C₈	—	4	2	no	—
146	C₇₉⋊C₈	#135	1	4	yes	complemented
147	C₂²×C₇₉	—	2	1	no	—
148	C₂²×C₇₉	—	2	1	no	—
149	C₂²×C₇₉	—	2	1	no	—
150	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×D₇₉	—	2	4	no	—
152	C₂×D₇₉	—	4	2	no	—
153	C₂×D₇₉	—	4	2	no	—
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	—	2	1	no	—
156	C₄×C₇₉	—	2	1	no	—
157	C₄×C₇₉	—	2	1	no	—
158	C₄×C₇₉	C₂²×M₁₆	1	1	yes	characteristic
159	C₄×C₇₉	#568	1	1	yes	characteristic
160	C₄×C₇₉	C₂×#135	1	2	yes	—
161	Dic₇₉	—	4	2	no	—
162	Dic₇₉	—	4	2	no	—
163	Dic₇₉	—	4	2	no	—
164	#18035	—	79	1	no	maximal, Sylow
165	C₂×C₇₉	—	4	1	no	—
166	C₂×C₇₉	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₇₉	C₄×#135	1	1	yes	characteristic
168	C₂×C₇₉	#2957	1	1	yes	characteristic
169	D₇₉	—	4	4	no	—
170	#2970	—	79	2	no	—
171	C₄×#135	—	79	1	no	—
172	#1591	—	79	2	no	—
173	#1582	—	79	4	no	—
174	C₈×D₈	—	79	2	no	—
175	#1568	D₇₉	1	1	yes	characteristic, complemented
176	#1488	—	79	2	no	—
177	C₈×M₁₆	—	79	1	no	—
178	C₇₉	#18035	1	1	yes	characteristic, complemented, Sylow
179	C₂×#135	—	79	1	no	—
180	C₄×#41	—	79	2	no	—
181	#446	—	79	4	no	—
182	C₄×Q₁₆	—	79	1	no	—
183	C₄×Q₁₆	—	79	1	no	—
184	C₄×SD₁₆	—	79	2	no	—
185	C₄×SD₁₆	—	79	2	no	—
186	C₄×D₈	—	79	1	no	—
187	C₄×D₈	—	79	1	no	—
188	#435	—	158	4	no	—
189	C₈×D₄	—	79	4	no	—
190	#432	—	158	2	no	—
191	#432	C₂×D₇₉	1	2	yes	—
192	#431	—	79	1	no	—
193	#427	—	79	1	no	—
194	#416	—	79	2	no	—
195	C₄×M₁₆	—	79	1	no	—
196	C₂×#97	—	158	2	no	—
197	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
198	#335	C₂×D₇₉	1	2	yes	—
199	#334	C₂×D₇₉	1	2	yes	—
200	Q₈⋊C₈	—	79	4	no	—
201	D₄⋊C₈	—	79	4	no	—
202	#322	—	79	2	no	—
203	C₈²	—	79	2	no	—
204	C₂×#41	—	79	2	no	—
205	#135	—	158	4	no	—
206	C₂×Q₁₆	—	79	1	no	—
207	C₂×SD₁₆	—	79	2	no	—
208	C₂×D₈	—	79	1	no	—
209	C₂×M₁₆	—	79	2	no	—
210	C₂²×C₈	—	158	4	no	—
211	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₇₉	1	1	yes	characteristic
213	C₄×Q₈	—	79	2	no	—
214	C₄×Q₈	—	79	2	no	—
215	C₄×D₄	—	79	2	no	—
216	C₄×D₄	—	79	2	no	—
217	C₄×D₄	—	158	2	no	—
218	C₄×D₄	—	158	2	no	—
219	#117	—	79	2	no	—
220	#117	—	158	2	no	—
221	C₂×C₄²	—	158	2	no	—
222	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
223	#107	—	79	1	no	—
224	#107	—	79	1	no	—
225	#106	—	79	2	no	—
226	C₄⋊C₈	—	158	4	no	—
227	C₄⋊C₈	—	158	1	no	—
228	C₄⋊C₈	C₂²×D₇₉	1	4	yes	—
229	#103	—	79	2	no	—
230	#103	—	79	2	no	—
231	#102	—	79	2	no	—
232	#102	—	79	2	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	158	4	no	—
235	C₂²⋊C₈	—	158	1	no	—
236	#97	—	2	2	no	—
237	#97	—	79	1	no	—
238	#97	—	79	1	no	—
239	#97	—	158	4	no	—
240	C₄×C₈	—	79	2	no	—
241	C₄×C₈	—	79	2	no	—
242	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
243	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
246	#41	—	316	2	no	—
247	#41	—	158	4	no	—
248	C₂×Q₈	—	79	2	no	—
249	C₂×D₄	—	79	2	no	—
250	C₂×D₄	—	158	2	no	—
251	C₂²×C₄	—	158	2	no	—
252	C₂²×C₄	—	158	2	no	—
253	C₂²×C₄	—	158	2	no	—
254	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
256	C₂²×C₄	#11929625	1	1	yes	characteristic
257	Q₁₆	—	158	2	no	—
258	SD₁₆	—	158	4	no	—
259	D₈	—	79	4	no	—
260	M₁₆	—	158	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	316	8	no	—
265	C₂×C₈	—	79	2	no	—
266	C₂×C₈	—	79	2	no	—
267	C₂×C₈	—	158	2	no	—
268	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
272	C₄⋊C₄	—	79	2	no	—
273	C₄⋊C₄	—	79	2	no	—
274	C₄⋊C₄	—	158	2	no	—
275	C₄⋊C₄	—	158	2	no	—
276	C₂²⋊C₄	—	158	2	no	—
277	C₂²⋊C₄	—	158	2	no	—
278	C₂²⋊C₄	—	158	2	no	—
279	C₄²	—	158	2	no	—
280	C₄²	—	158	2	no	—
281	C₄²	—	158	2	no	—
282	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
283	C₄²	D₄×D₇₉	1	1	yes	characteristic
284	C₄²	#11929625	1	1	yes	characteristic
285	C₄²	C₂³×D₇₉	1	1	yes	characteristic
286	C₂³	—	158	2	no	—
287	C₂³	#14579123	1	1	yes	characteristic
288	Q₈	—	158	2	no	—
289	Q₈	—	158	2	no	—
290	D₄	—	316	4	no	—
291	D₄	—	79	4	no	—
292	D₄	—	158	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	316	4	no	—
298	C₂×C₄	—	316	4	no	—
299	C₂×C₄	—	632	2	no	—
300	C₂×C₄	—	158	2	no	—
301	C₂×C₄	—	158	2	no	—
302	C₂×C₄	—	158	2	no	—
303	C₂×C₄	#14579123	1	1	yes	characteristic
304	C₂×C₄	#14579123	1	1	yes	characteristic
305	C₂×C₄	#14579123	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	316	2	no	—
314	C₈	—	79	4	no	—
315	C₈	—	158	2	no	—
316	C₈	M₁₆×D₇₉	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	316	2	no	—
321	C₂²	—	316	2	no	—
322	C₂²	—	158	4	no	—
323	C₂²	C₂×#14579123	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	316	2	no	—
329	C₄	—	316	2	no	—
330	C₄	—	316	2	no	—
331	C₄	C₂×M₁₆×D₇₉	1	1	yes	characteristic
332	C₄	#19099329	1	1	yes	characteristic
333	C₄	#135×D₇₉	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	316	4	no	—
336	C₂	#23772135	1	1	yes	characteristic
337	C₂	#23772171	1	1	yes	characteristic
338	C₂	#23772316	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1568	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂×#14579183	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#1568	D₇₉
hypocenter	C₇₉	#18035
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
1. #23772412	C₂	semidirect
3. #23762296	C₂	semidirect
4. #23762235	C₂	semidirect
5. #23762194	C₂	semidirect
6. #23761832	C₂	semidirect
7. C₈×#11929608	C₂	semidirect
8. C₇₉×#1568	C₂	semidirect
86. D₆₃₂	M₁₆	semidirect
146. C₇₉⋊C₈	#135	semidirect
175. #1568	D₇₉	semidirect
178. C₇₉	#18035	semidirect

Maximal subgroups

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

subgroup	index
#23772412	2
C₄×#14579183	2
#23762296	2
#23762235	2
#23762194	2
#23761832	2
C₈×#11929608	2
C₇₉×#1568	2
#18035	79