Subgroups — #34443673

Order 20224.

834 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098993	C₂	1	4	yes	maximal, complemented
2	#23773828	C₂	1	1	yes	maximal, characteristic, complemented
3	#23765659	C₂	1	4	yes	maximal, complemented
4	#23765657	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765655	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765652	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765649	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₉×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579183	—	2	1	no	—
11	C₂×#14579183	C₂²	1	2	yes	—
12	#19099113	—	2	1	no	—
13	#19098993	—	2	16	no	—
14	C₂×#14579107	C₂²	1	4	yes	—
15	C₂×#14579105	C₂²	1	4	yes	—
16	C₂×#14579104	C₂²	1	4	yes	—
17	C₂×#14579102	C₂²	1	4	yes	—
18	C₂²×C₇₉⋊C₁₆	C₂²	1	2	yes	—
19	#19098947	—	2	1	no	—
20	#19098947	C₂²	1	2	yes	—
21	#19098943	C₂²	1	1	yes	characteristic
22	#19098940	C₂²	1	1	yes	characteristic
23	#19098936	—	2	1	no	—
24	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
27	C₂×#135×C₇₉	—	2	1	no	—
28	C₂×#135×C₇₉	C₂²	1	2	yes	—
29	#532×C₇₉	—	2	1	no	—
30	#495×C₇₉	—	2	1	no	—
31	#495×C₇₉	C₂²	1	2	yes	—
32	#494×C₇₉	C₂²	1	1	yes	characteristic
33	#493×C₇₉	C₂²	1	1	yes	characteristic
34	#492×C₇₉	—	2	1	no	—
35	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#11929624	—	2	2	no	—
37	#14579183	—	2	8	no	—
38	#14579183	—	4	4	no	—
39	C₂×Dic₃₁₆	—	2	1	no	—
40	C₂×Dic₃₁₆	C₂³	1	2	yes	—
41	C₂×#11929613	—	2	2	no	—
42	C₂×#11929613	—	2	2	no	—
43	C₂×D₆₃₂	—	2	1	no	—
44	C₂×D₆₃₂	C₂³	1	2	yes	—
45	#14579161	—	2	1	no	—
46	#14579158	—	2	1	no	—
47	#14579149	—	2	2	no	—
48	#14579135	—	4	2	no	—
49	C₄×D₃₁₆	—	4	2	no	—
50	#14579107	—	2	8	no	—
51	#14579105	—	2	8	no	—
52	#14579104	—	2	8	no	—
53	#14579102	—	2	8	no	—
54	C₂×C₇₉⋊C₁₆	D₄	1	8	yes	complemented
55	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
57	C₂×#41×C₇₉	—	2	2	no	—
58	#135×C₇₉	—	2	8	no	—
59	#135×C₇₉	—	4	4	no	—
60	C₂×Q₁₆×C₇₉	—	2	1	no	—
61	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₇₉	—	2	2	no	—
63	C₂×SD₁₆×C₇₉	—	2	2	no	—
64	C₂×D₈×C₇₉	—	2	1	no	—
65	C₂×D₈×C₇₉	C₂³	1	2	yes	—
66	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₇₉	—	2	1	no	—
69	#127×C₇₉	—	2	1	no	—
70	#124×C₇₉	—	2	2	no	—
71	#121×C₇₉	—	4	2	no	—
72	C₄×D₄×C₇₉	—	4	2	no	—
73	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
74	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
75	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	#11929624	—	4	4	no	—
79	#11929624	—	4	4	no	—
80	C₂×Dic₁₅₈	—	2	2	no	—
81	C₂×D₃₁₆	—	2	2	no	—
82	C₂×D₃₁₆	—	4	2	no	—
83	C₂×C₂²⋊D₇₉	—	4	2	no	—
84	C₂×C₄×D₇₉	—	4	2	no	—
85	Dic₃₁₆	—	2	4	no	—
86	Dic₃₁₆	—	4	2	no	—
87	#11929613	—	4	4	no	—
88	#11929613	—	4	4	no	—
89	D₆₃₂	—	2	4	no	—
90	D₆₃₂	—	4	2	no	—
91	C₄⋊Dic₇₉	—	4	2	no	—
92	(C₂×D₇₉)⋊C₄	—	4	4	no	—
93	C₇₉⋊C₁₆	C₂×D₄	1	4	yes	—
94	C₇₉⋊C₁₆	C₂×D₄	1	4	yes	—
95	#41×C₇₉	—	4	4	no	—
96	#41×C₇₉	—	4	4	no	—
97	C₂×Q₈×C₇₉	—	2	2	no	—
98	C₂×D₄×C₇₉	—	2	2	no	—
99	C₂×D₄×C₇₉	—	4	2	no	—
100	C₂×D₄×C₇₉	—	4	2	no	—
101	C₂²×C₄×C₇₉	—	4	2	no	—
102	C₂²×C₄×C₇₉	D₈	1	2	yes	—
103	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₇₉	—	2	4	no	—
105	Q₁₆×C₇₉	—	4	2	no	—
106	SD₁₆×C₇₉	—	4	4	no	—
107	SD₁₆×C₇₉	—	4	4	no	—
108	D₈×C₇₉	—	2	4	no	—
109	D₈×C₇₉	—	4	2	no	—
110	C₂×C₈×C₇₉	D₈	1	4	yes	—
111	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₇₉	—	4	2	no	—
117	C₂²⋊C₄×C₇₉	—	4	4	no	—
118	C₄²×C₇₉	D₈	1	2	yes	—
119	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₇₉	—	4	2	no	—
122	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₇₉	—	4	2	no	—
124	Q₈×C₇₉	—	4	2	no	—
125	Q₈×C₇₉	—	4	2	no	—
126	D₄×C₇₉	—	4	2	no	—
127	D₄×C₇₉	—	4	2	no	—
128	D₄×C₇₉	—	8	4	no	—
129	D₄×C₇₉	—	8	4	no	—
130	C₂×C₄×C₇₉	—	2	4	no	—
131	C₂×C₄×C₇₉	—	4	2	no	—
132	C₂×C₄×C₇₉	—	8	4	no	—
133	C₂×C₄×C₇₉	#127	1	2	yes	—
134	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₇₉	—	8	4	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	D₃₁₆	—	4	2	no	—
142	D₃₁₆	—	4	2	no	—
143	D₃₁₆	—	8	4	no	—
144	C₄×D₇₉	—	8	4	no	—
145	Dic₁₅₈	—	4	2	no	—
146	Dic₁₅₈	—	4	2	no	—
147	C₈×C₇₉	#127	1	2	yes	—
148	C₈×C₇₉	C₂×D₈	1	2	yes	—
149	C₈×C₇₉	C₂×D₈	1	2	yes	—
150	C₈×C₇₉	C₂²×D₄	1	2	yes	—
151	C₂²×C₇₉	—	2	2	no	—
152	C₂²×C₇₉	—	8	2	no	—
153	C₂²×C₇₉	—	8	4	no	—
154	C₂²×C₇₉	#493	1	2	yes	—
155	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₇₉	—	8	2	no	—
157	C₂×D₇₉	—	8	4	no	—
158	C₄×C₇₉	—	2	2	no	—
159	C₄×C₇₉	—	8	2	no	—
160	C₄×C₇₉	#493	1	2	yes	—
161	C₄×C₇₉	#508	1	4	yes	complemented
162	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
163	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
164	Dic₇₉	—	8	2	no	—
165	#33713	—	79	1	no	maximal, Sylow
166	C₂×C₇₉	—	2	2	no	—
167	C₂×C₇₉	—	16	2	no	—
168	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
169	C₂×C₇₉	C₂×#508	1	2	yes	—
170	D₇₉	—	16	2	no	—
171	C₂×#508	—	79	4	no	—
172	#3147	—	79	1	no	—
173	#3147	D₇₉	1	1	yes	characteristic, complemented
174	#2249	—	79	4	no	—
175	#2248	—	79	1	no	—
176	#2248	—	79	1	no	—
177	#2247	—	79	1	no	—
178	#2246	—	79	1	no	—
179	C₂×C₄×C₁₆	—	79	1	no	—
180	C₇₉	#33713	1	1	yes	characteristic, complemented, Sylow
181	C₂×#135	—	2	1	no	—
182	C₂×#135	—	79	2	no	—
183	C₂×#135	—	158	1	no	—
184	C₂×#135	C₂×D₇₉	1	2	yes	—
185	#532	—	2	1	no	—
186	#532	—	158	1	no	—
187	#508	—	158	16	no	—
188	C₂×Q₃₂	—	79	4	no	—
189	C₂×SD₃₂	—	79	4	no	—
190	C₂×SD₃₂	—	79	4	no	—
191	C₂×D₁₆	—	79	4	no	—
192	C₂²×C₁₆	—	79	2	no	—
193	#495	—	2	1	no	—
194	#495	—	79	2	no	—
195	#495	—	158	1	no	—
196	#495	C₂×D₇₉	1	2	yes	—
197	#494	—	79	1	no	—
198	#494	C₂×D₇₉	1	1	yes	characteristic
199	#493	—	79	1	no	—
200	#493	C₂×D₇₉	1	1	yes	characteristic
201	#492	—	2	1	no	—
202	#492	—	158	1	no	—
203	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
204	C₄×C₁₆	—	79	1	no	—
205	C₄×C₁₆	—	79	1	no	—
206	C₄×C₁₆	—	79	2	no	—
207	C₂×#41	—	2	2	no	—
208	C₂×#41	—	158	2	no	—
209	#135	—	2	8	no	—
210	#135	—	4	4	no	—
211	#135	—	316	4	no	—
212	#135	—	158	8	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	79	2	no	—
215	C₂×Q₁₆	—	158	1	no	—
216	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	158	2	no	—
220	C₂×SD₁₆	—	158	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	79	2	no	—
223	C₂×D₈	—	158	1	no	—
224	C₂×D₈	C₂²×D₇₉	1	2	yes	—
225	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	158	1	no	—
229	#127	—	2	1	no	—
230	#127	—	158	1	no	—
231	#124	—	2	2	no	—
232	#124	—	158	2	no	—
233	#121	—	4	2	no	—
234	#121	—	316	2	no	—
235	C₄×D₄	—	4	2	no	—
236	C₄×D₄	—	316	2	no	—
237	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
238	Q₃₂	—	158	8	no	—
239	SD₃₂	—	158	8	no	—
240	SD₃₂	—	158	8	no	—
241	D₁₆	—	158	8	no	—
242	C₂×C₁₆	—	79	8	no	—
243	C₂×C₁₆	—	79	2	no	—
244	C₂×C₁₆	—	79	2	no	—
245	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
246	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
249	#41	—	4	4	no	—
250	#41	—	4	4	no	—
251	#41	—	316	4	no	—
252	#41	—	316	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	158	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	316	2	no	—
259	C₂×D₄	—	316	2	no	—
260	C₂×D₄	—	158	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	316	2	no	—
263	C₂²×C₄	#11929609	1	2	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	4	2	no	—
267	Q₁₆	—	316	2	no	—
268	Q₁₆	—	158	4	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	4	4	no	—
271	SD₁₆	—	316	4	no	—
272	SD₁₆	—	316	4	no	—
273	D₈	—	2	4	no	—
274	D₈	—	4	2	no	—
275	D₈	—	316	2	no	—
276	D₈	—	158	4	no	—
277	C₂×C₈	#11929609	1	4	yes	—
278	C₂×C₈	D₄×D₇₉	1	4	yes	—
279	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
282	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	316	2	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	316	4	no	—
287	C₄²	#11929609	1	2	yes	—
288	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
289	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
290	C₁₆	—	79	4	no	—
291	C₁₆	—	79	4	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	316	2	no	—
294	C₂³	C₂×#11929609	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	316	2	no	—
298	Q₈	—	316	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	316	2	no	—
304	D₄	—	316	2	no	—
305	D₄	—	632	4	no	—
306	D₄	—	632	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	316	2	no	—
311	C₂×C₄	—	632	4	no	—
312	C₂×C₄	#14579157	1	2	yes	—
313	C₂×C₄	C₂×#11929609	1	2	yes	—
314	C₂×C₄	C₂×#11929609	1	2	yes	—
315	C₂×C₄	C₂×#11929609	1	2	yes	—
316	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
318	C₈	#14579157	1	2	yes	—
319	C₈	C₂×#11929609	1	2	yes	—
320	C₈	C₂×#11929609	1	2	yes	—
321	C₈	C₂×D₄×D₇₉	1	2	yes	—
322	C₂²	—	2	2	no	—
323	C₂²	—	8	4	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	632	4	no	—
326	C₂²	—	632	2	no	—
327	C₂²	#19098939	1	2	yes	—
328	C₂²	C₂²×#11929609	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	632	2	no	—
332	C₄	#19098939	1	2	yes	—
333	C₄	#19098993	1	4	yes	complemented
334	C₄	C₂×#14579157	1	1	yes	characteristic
335	C₄	C₂²×#11929609	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	—	1264	2	no	—
339	C₂	C₂×#19098939	1	1	yes	characteristic
340	C₂	C₂×#19098993	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3147	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	#3147	D₇₉
hypocenter	C₇₉	#33713
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#19098993	C₂	semidirect
2. #23773828	C₂	semidirect
3. #23765659	C₂	semidirect
4. #23765657	C₂	semidirect
5. #23765655	C₂	semidirect
6. #23765652	C₂	semidirect
7. #23765649	C₂	semidirect
8. C₂×C₄×C₇₉⋊C₁₆	C₂	semidirect
9. C₇₉×#3147	C₂	semidirect
54. C₂×C₇₉⋊C₁₆	D₄	semidirect
161. C₄×C₇₉	#508	semidirect
173. #3147	D₇₉	semidirect
180. C₇₉	#33713	semidirect
333. C₄	#19098993	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098993	2
#23773828	2
#23765659	2
#23765657	2
#23765655	2
#23765652	2
#23765649	2
C₂×C₄×C₇₉⋊C₁₆	2
C₇₉×#3147	2
#33713	79