Subgroups — #34441735

Order 20224.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099006	C₂	1	1	yes	maximal, characteristic
2	#23772316	C₂	1	1	yes	maximal, characteristic
3	#23772267	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765290	C₂	1	8	yes	maximal, complemented
5	C₂×#19098378	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765135	C₂	1	1	yes	maximal, characteristic
7	#23765004	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₉×#2155	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
11	#19099329	—	2	1	no	—
12	#19099040	—	2	1	no	—
13	#19099006	C₄	1	4	yes	complemented
14	#19099006	C₂²	1	4	yes	—
15	C₂×Dic₆₃₂	C₄	1	2	yes	complemented
16	C₂×#14579106	C₄	1	2	yes	complemented
17	C₂×#14579101	C₂²	1	1	yes	characteristic
18	#19098684	C₂²	1	4	yes	—
19	#19098679	—	2	2	no	—
20	#19098669	C₂²	1	4	yes	complemented
21	#19098657	—	2	2	no	—
22	#19098579	—	2	1	no	—
23	#19098571	—	2	1	no	—
24	#19098518	C₂²	1	1	yes	characteristic
25	#19098378	C₂²	1	4	yes	complemented
26	#19098372	C₂²	1	4	yes	—
27	#19098340	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
30	#427×C₇₉	C₂²	1	1	yes	characteristic
31	#365×C₇₉	C₂²	1	4	yes	complemented
32	C₂×#11929624	—	2	1	no	—
33	C₂²×Dic₁₅₈	—	2	1	no	—
34	#14579183	—	2	1	no	—
35	#14579183	—	2	1	no	—
36	#14579183	C₂×C₄	1	2	yes	—
37	#14579183	C₂³	1	2	yes	—
38	C₂×Dic₃₁₆	—	2	1	no	—
39	C₂×Dic₃₁₆	—	2	1	no	—
40	C₂×Dic₃₁₆	—	2	1	no	—
41	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
42	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
44	C₂×Dic₃₁₆	C₂³	1	2	yes	—
45	C₂×#11929613	—	2	1	no	—
46	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
47	#14579169	—	2	4	no	—
48	#14579167	—	4	2	no	—
49	C₂×#11929608	D₄	1	1	yes	characteristic
50	M₁₆×D₇₉	—	4	2	no	—
51	C₄×Dic₁₅₈	—	4	2	no	—
52	#14579126	—	2	4	no	—
53	#14579120	D₄	1	1	yes	characteristic
54	#14579117	—	4	2	no	—
55	Dic₆₃₂	C₂×C₄	1	8	yes	—
56	#14579106	C₂×C₄	1	8	yes	—
57	#14579101	—	2	1	no	—
58	#14579101	C₂³	1	2	yes	—
59	C₂×C₇₉⋊C₁₆	—	2	1	no	—
60	#14579087	—	2	4	no	—
61	#14579084	—	2	4	no	—
62	#14579071	C₂³	1	2	yes	—
63	C₈×Dic₇₉	D₄	1	2	yes	—
64	C₂×M₁₆×C₇₉	—	2	1	no	—
65	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
66	#117×C₇₉	—	2	1	no	—
67	M₃₂×C₇₉	C₂×C₄	1	2	yes	—
68	M₃₂×C₇₉	C₂³	1	2	yes	—
69	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
70	#108×C₇₉	—	2	1	no	—
71	#108×C₇₉	C₂³	1	2	yes	—
72	#107×C₇₉	—	2	1	no	—
73	#106×C₇₉	C₂³	1	2	yes	—
74	#11929624	—	2	2	no	—
75	#11929624	—	2	2	no	—
76	#11929624	—	4	1	no	—
77	#11929624	—	4	1	no	—
78	C₂×Dic₁₅₈	—	2	2	no	—
79	C₂×Dic₁₅₈	—	2	2	no	—
80	C₂×Dic₁₅₈	—	2	1	no	—
81	C₂×Dic₁₅₈	—	2	2	no	—
82	C₂×Dic₁₅₈	—	4	1	no	—
83	C₂×Dic₁₅₈	—	4	1	no	—
84	C₂×Dic₁₅₈	—	4	1	no	—
85	C₂×D₃₁₆	—	2	1	no	—
86	C₂×C₂²⋊D₇₉	—	4	1	no	—
87	C₂²×Dic₇₉	—	4	1	no	—
88	C₂×C₄×D₇₉	—	4	1	no	—
89	Dic₃₁₆	—	2	2	no	—
90	Dic₃₁₆	—	2	2	no	—
91	Dic₃₁₆	—	2	1	no	—
92	Dic₃₁₆	—	2	2	no	—
93	Dic₃₁₆	C₂²×C₄	1	4	yes	—
94	Dic₃₁₆	C₂²×C₄	1	2	yes	—
95	#11929613	—	2	2	no	—
96	#11929613	—	2	2	no	—
97	D₆₃₂	—	2	1	no	—
98	D₆₃₂	C₂²×C₄	1	2	yes	—
99	#11929608	—	2	1	no	—
100	#11929608	—	2	1	no	—
101	#11929607	—	4	4	no	—
102	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
103	C₈×D₇₉	—	4	4	no	—
104	C₄⋊Dic₇₉	—	2	1	no	—
105	Dic₇₉⋊C₄	—	4	4	no	—
106	C₂²⋊Dic₇₉	—	2	1	no	—
107	C₄×Dic₇₉	—	4	4	no	—
108	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
109	C₇₉⋊C₁₆	—	2	2	no	—
110	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
111	M₁₆×C₇₉	—	2	2	no	—
112	M₁₆×C₇₉	—	4	1	no	—
113	C₂×C₈×C₇₉	—	4	1	no	—
114	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
118	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₇₉	—	2	2	no	—
121	C₂²⋊C₄×C₇₉	—	4	1	no	—
122	C₄²×C₇₉	—	4	1	no	—
123	C₁₆×C₇₉	C₂²×C₄	1	4	yes	—
124	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
125	C₂²×D₇₉	—	4	1	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₇₉	—	4	2	no	—
134	C₂²⋊D₇₉	—	4	2	no	—
135	C₂×Dic₇₉	—	2	2	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	4	1	no	—
138	C₂×Dic₇₉	—	4	2	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	D₃₁₆	—	2	2	no	—
141	D₃₁₆	—	4	1	no	—
142	C₄×D₇₉	—	4	2	no	—
143	C₄×D₇₉	—	4	2	no	—
144	Dic₁₅₈	—	2	2	no	—
145	Dic₁₅₈	—	2	4	no	—
146	Dic₁₅₈	—	4	2	no	—
147	Dic₁₅₈	—	4	2	no	—
148	Dic₁₅₈	—	4	2	no	—
149	Dic₁₅₈	—	4	1	no	—
150	C₈×C₇₉	—	4	2	no	—
151	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
152	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
153	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
154	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
155	C₇₉⋊C₈	—	2	2	no	—
156	C₂²×C₇₉	—	4	1	no	—
157	C₂²×C₇₉	#442	1	1	yes	characteristic
158	C₂²×C₇₉	#442	1	1	yes	characteristic
159	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂×D₇₉	—	4	2	no	—
161	C₂×D₇₉	—	16	1	no	—
162	C₄×C₇₉	—	8	2	no	—
163	C₄×C₇₉	#442	1	1	yes	characteristic
164	C₄×C₇₉	#442	1	1	yes	characteristic
165	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
166	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	Dic₇₉	—	4	2	no	—
168	Dic₇₉	—	4	2	no	—
169	Dic₇₉	—	4	4	no	—
170	#33470	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	—	2	1	no	—
172	C₂×C₇₉	—	2	1	no	—
173	C₂×C₇₉	—	2	1	no	—
174	C₂×C₇₉	C₂×#442	1	1	yes	characteristic
175	D₇₉	—	8	2	no	—
176	C₂×#510	—	79	1	no	—
177	#2957	—	79	1	no	—
178	#2950	—	79	1	no	—
179	#2180	—	79	8	no	—
180	#2155	D₇₉	1	1	yes	characteristic, complemented
181	C₂×#362	—	79	1	no	—
182	#2145	—	79	1	no	—
183	#2110	—	79	1	no	—
184	C₇₉	#33470	1	1	yes	characteristic, complemented, Sylow
185	C₂×#135	—	79	1	no	—
186	C₂²×Q₁₆	—	79	1	no	—
187	#568	—	158	1	no	—
188	#519	—	158	1	no	—
189	#510	—	79	4	no	—
190	#510	—	79	4	no	—
191	C₂×Q₃₂	—	79	2	no	—
192	C₂×SD₃₂	—	79	2	no	—
193	C₂×M₃₂	—	79	1	no	—
194	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
195	#444	—	79	4	no	—
196	#443	—	158	2	no	—
197	#441	—	79	4	no	—
198	C₄×Q₁₆	—	158	2	no	—
199	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
200	#427	C₂×D₇₉	1	1	yes	characteristic
201	#421	—	158	1	no	—
202	#419	—	158	1	no	—
203	#405	—	79	1	no	—
204	#365	C₂×D₇₉	1	4	yes	complemented
205	#362	—	79	4	no	—
206	#360	—	79	4	no	—
207	#347	—	79	4	no	—
208	C₂×#41	—	158	1	no	—
209	C₂²×Q₈	—	158	1	no	—
210	#135	—	79	2	no	—
211	#135	—	79	2	no	—
212	#135	—	158	1	no	—
213	#135	—	158	1	no	—
214	C₂×Q₁₆	—	79	2	no	—
215	C₂×Q₁₆	—	79	2	no	—
216	C₂×Q₁₆	—	79	1	no	—
217	C₂×Q₁₆	—	79	2	no	—
218	C₂×Q₁₆	—	158	1	no	—
219	C₂×Q₁₆	—	158	1	no	—
220	C₂×Q₁₆	—	158	1	no	—
221	C₂×SD₁₆	—	158	1	no	—
222	C₂×D₈	—	79	1	no	—
223	#131	—	316	2	no	—
224	#131	—	158	4	no	—
225	C₂×M₁₆	—	2	1	no	—
226	C₂×M₁₆	—	316	2	no	—
227	C₂×M₁₆	—	79	1	no	—
228	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
229	C₄×Q₈	—	316	2	no	—
230	C₄×Q₈	—	158	4	no	—
231	#117	—	2	1	no	—
232	#117	—	316	2	no	—
233	#117	—	79	1	no	—
234	Q₃₂	—	79	8	no	—
235	SD₃₂	—	79	8	no	—
236	M₃₂	—	79	2	no	—
237	M₃₂	—	158	1	no	—
238	M₃₂	C₄×D₇₉	1	2	yes	—
239	M₃₂	C₂²×D₇₉	1	2	yes	—
240	C₂×C₁₆	—	158	1	no	—
241	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
242	#108	—	2	1	no	—
243	#108	C₂²×D₇₉	1	2	yes	—
244	#107	—	2	1	no	—
245	#106	C₂²×D₇₉	1	2	yes	—
246	#104	—	158	4	no	—
247	#103	—	158	4	no	—
248	#97	—	79	2	no	—
249	C₄×C₈	—	79	2	no	—
250	#41	—	316	1	no	—
251	#41	—	316	1	no	—
252	#41	—	158	2	no	—
253	#41	—	158	2	no	—
254	C₂×Q₈	—	316	1	no	—
255	C₂×Q₈	—	316	1	no	—
256	C₂×Q₈	—	316	1	no	—
257	C₂×Q₈	—	158	1	no	—
258	C₂×Q₈	—	158	2	no	—
259	C₂×Q₈	—	158	2	no	—
260	C₂×Q₈	—	158	2	no	—
261	C₂×D₄	—	316	1	no	—
262	C₂×D₄	—	158	1	no	—
263	C₂²×C₄	—	316	1	no	—
264	C₂²×C₄	—	316	1	no	—
265	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
266	Q₁₆	—	79	4	no	—
267	Q₁₆	—	79	2	no	—
268	Q₁₆	—	158	2	no	—
269	Q₁₆	—	158	2	no	—
270	Q₁₆	—	158	2	no	—
271	Q₁₆	—	158	1	no	—
272	SD₁₆	—	158	2	no	—
273	SD₁₆	—	158	2	no	—
274	D₈	—	79	2	no	—
275	D₈	—	158	1	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	316	4	no	—
279	M₁₆	—	158	1	no	—
280	M₁₆	—	158	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	316	4	no	—
283	C₂×C₈	—	79	2	no	—
284	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
285	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
286	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
287	C₂×C₈	#11929625	1	1	yes	characteristic
288	C₂×C₈	#11929625	1	1	yes	characteristic
289	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	316	4	no	—
292	C₄⋊C₄	—	158	1	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	158	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	316	4	no	—
297	C₄²	—	79	2	no	—
298	C₁₆	—	158	2	no	—
299	C₁₆	C₂×C₄×D₇₉	1	4	yes	—
300	C₂³	—	316	1	no	—
301	C₂³	#14579188	1	1	yes	characteristic
302	Q₈	—	316	2	no	—
303	Q₈	—	316	2	no	—
304	Q₈	—	316	1	no	—
305	Q₈	—	316	2	no	—
306	Q₈	—	158	2	no	—
307	Q₈	—	158	4	no	—
308	D₄	—	316	2	no	—
309	D₄	—	316	2	no	—
310	D₄	—	316	1	no	—
311	D₄	—	158	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	316	1	no	—
314	C₂×C₄	—	316	2	no	—
315	C₂×C₄	—	316	2	no	—
316	C₂×C₄	—	316	2	no	—
317	C₂×C₄	—	316	2	no	—
318	C₂×C₄	—	316	2	no	—
319	C₂×C₄	—	158	2	no	—
320	C₂×C₄	#14579123	1	1	yes	characteristic
321	C₂×C₄	#14579123	1	1	yes	characteristic
322	C₂×C₄	#14579188	1	1	yes	characteristic
323	C₂×C₄	#14579188	1	1	yes	characteristic
324	C₂×C₄	#14579188	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	158	2	no	—
328	C₈	#14579123	1	1	yes	characteristic
329	C₈	#14579123	1	1	yes	characteristic
330	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
331	C₈	C₂×#11929625	1	1	yes	characteristic
332	C₂²	—	4	1	no	—
333	C₂²	—	316	2	no	—
334	C₂²	—	1264	1	no	—
335	C₂²	#19098674	1	1	yes	characteristic
336	C₂²	#19098674	1	1	yes	characteristic
337	C₂²	C₂×#14579188	1	1	yes	characteristic
338	C₄	—	8	2	no	—
339	C₄	—	316	2	no	—
340	C₄	—	316	2	no	—
341	C₄	—	316	4	no	—
342	C₄	#19098674	1	1	yes	characteristic
343	C₄	#19098674	1	1	yes	characteristic
344	C₄	C₂×#14579123	1	1	yes	characteristic
345	C₄	C₂×#14579188	1	1	yes	characteristic
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	—	2	1	no	—
349	C₂	—	632	2	no	—
350	C₂	C₂×#19098674	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19098674
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2155	C₂
socle	C₂×C₇₉	C₂×#442
prime radical	C₂×#14579183	C₂²
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	#2155	D₇₉
hypocenter	C₇₉	#33470
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
3. #23772267	C₂	semidirect
4. #23765290	C₂	semidirect
5. C₂×#19098378	C₂	semidirect
7. #23765004	C₂	semidirect
8. C₇₉×#2155	C₂	semidirect
13. #19099006	C₄	semidirect
15. C₂×Dic₆₃₂	C₄	semidirect
16. C₂×#14579106	C₄	semidirect
20. #19098669	C₂²	semidirect
25. #19098378	C₂²	semidirect
27. #19098340	C₂²	semidirect
31. #365×C₇₉	C₂²	semidirect
180. #2155	D₇₉	semidirect
184. C₇₉	#33470	semidirect
204. #365	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099006	2
#23772316	2
#23772267	2
#23765290	2
C₂×#19098378	2
#23765135	2
#23765004	2
C₇₉×#2155	2
#33470	79