Subgroups — #34445986

Order 20224.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776375	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765643	C₂	1	2	yes	maximal, complemented
3	#23765625	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765554	C₂	1	2	yes	maximal, complemented
5	#23765544	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765435	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765435	C₂	1	2	yes	maximal, complemented
8	#23765309	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765274	C₂	1	2	yes	maximal, complemented
10	C₄×#14579102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099235	—	2	1	no	—
13	C₂×#14579102	C₂²	1	2	yes	—
14	C₂×#14579102	C₂²	1	1	yes	characteristic
15	#19098940	C₂²	1	1	yes	characteristic
16	#19098940	C₂²	1	2	yes	—
17	#19098759	—	2	1	no	—
18	#19098759	—	2	2	no	—
19	C₄×D₆₃₂	C₂²	1	2	yes	—
20	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
21	#19098390	C₂²	1	2	yes	—
22	#19098388	C₂²	1	1	yes	characteristic
23	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	2	yes	—
25	#19098364	C₂²	1	1	yes	characteristic
26	#19098363	C₂²	1	2	yes	—
27	#19098363	C₂²	1	2	yes	—
28	#19098363	C₂²	1	2	yes	—
29	#19098363	C₂²	1	2	yes	—
30	#19098363	C₂²	1	1	yes	characteristic
31	#19098339	C₂²	1	2	yes	—
32	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
33	#554×C₇₉	—	2	1	no	—
34	#500×C₇₉	C₂²	1	1	yes	characteristic
35	#499×C₇₉	C₂²	1	2	yes	—
36	#478×C₇₉	—	2	2	no	—
37	#476×C₇₉	—	2	1	no	—
38	C₈×Q₈×C₇₉	C₂²	1	1	yes	characteristic
39	C₄×D₈×C₇₉	C₂²	1	2	yes	—
40	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×#11929625	—	4	1	no	—
42	C₂×D₆₃₂	—	2	2	no	—
43	C₂×D₆₃₂	—	2	1	no	—
44	C₂×D₆₃₂	C₂³	1	2	yes	—
45	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
46	#14579158	—	2	2	no	—
47	#14579158	—	2	1	no	—
48	#14579136	—	4	1	no	—
49	#14579136	—	4	2	no	—
50	C₄×D₃₁₆	—	2	2	no	—
51	C₄×D₃₁₆	—	2	1	no	—
52	#14579102	—	2	4	no	—
53	#14579102	—	2	2	no	—
54	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
57	#14579095	C₂³	1	1	yes	characteristic
58	#14579083	—	2	2	no	—
59	#14579083	—	2	1	no	—
60	C₂×D₈×C₇₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₇₉	—	2	1	no	—
62	#125×C₇₉	—	2	2	no	—
63	C₂²⋊Q₈×C₇₉	—	4	1	no	—
64	C₂²⋊Q₈×C₇₉	—	4	2	no	—
65	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
66	C₄×D₄×C₇₉	—	2	1	no	—
67	C₄×D₄×C₇₉	—	2	2	no	—
68	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
69	#107×C₇₉	—	2	1	no	—
70	#107×C₇₉	C₂³	1	1	yes	characteristic
71	#107×C₇₉	C₂³	1	2	yes	—
72	#106×C₇₉	—	2	2	no	—
73	C₄⋊C₈×C₇₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
75	#102×C₇₉	—	2	2	no	—
76	#102×C₇₉	—	2	1	no	—
77	C₄×C₈×C₇₉	C₂³	1	2	yes	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	#11929625	—	8	4	no	—
80	C₂×D₃₁₆	—	2	2	no	—
81	C₂×D₃₁₆	—	2	1	no	—
82	C₂×D₃₁₆	—	4	2	no	—
83	C₂×D₃₁₆	—	4	1	no	—
84	C₂×D₃₁₆	—	4	1	no	—
85	C₂×D₃₁₆	—	4	2	no	—
86	C₂×C₄×D₇₉	—	4	2	no	—
87	C₂×C₄×D₇₉	—	4	1	no	—
88	D₆₃₂	—	2	2	no	—
89	D₆₃₂	—	2	1	no	—
90	D₆₃₂	—	2	2	no	—
91	D₆₃₂	—	2	1	no	—
92	D₆₃₂	—	8	2	no	—
93	D₆₃₂	—	8	1	no	—
94	C₄⋊Dic₇₉	—	2	1	no	—
95	(C₂×D₇₉)⋊C₄	—	4	1	no	—
96	(C₂×D₇₉)⋊C₄	—	4	2	no	—
97	C₇₉⋊C₁₆	—	2	1	no	—
98	C₇₉⋊C₁₆	—	2	2	no	—
99	C₇₉⋊C₁₆	—	2	1	no	—
100	C₂×Q₈×C₇₉	—	4	1	no	—
101	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
102	C₂×D₄×C₇₉	—	2	1	no	—
103	C₂²×C₄×C₇₉	—	4	1	no	—
104	C₂²×C₄×C₇₉	—	4	2	no	—
105	D₈×C₇₉	—	2	1	no	—
106	D₈×C₇₉	#41	1	2	yes	complemented
107	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₇₉	—	2	1	no	—
111	C₄⋊C₄×C₇₉	—	2	2	no	—
112	C₄⋊C₄×C₇₉	—	4	2	no	—
113	C₄⋊C₄×C₇₉	—	4	1	no	—
114	C₄⋊C₄×C₇₉	—	4	2	no	—
115	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
116	C₄⋊C₄×C₇₉	D₈	1	2	yes	—
117	C₂²⋊C₄×C₇₉	—	4	1	no	—
118	C₂²⋊C₄×C₇₉	—	4	2	no	—
119	C₄²×C₇₉	C₂×D₄	1	2	yes	—
120	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₇₉	—	4	1	no	—
122	C₂²×D₇₉	—	4	2	no	—
123	C₂²×D₇₉	—	4	1	no	—
124	Q₈×C₇₉	—	2	2	no	—
125	Q₈×C₇₉	—	16	1	no	—
126	D₄×C₇₉	—	2	2	no	—
127	D₄×C₇₉	—	4	1	no	—
128	C₂×C₄×C₇₉	—	4	1	no	—
129	C₂×C₄×C₇₉	—	4	2	no	—
130	C₂×C₄×C₇₉	—	8	2	no	—
131	C₂×C₄×C₇₉	—	16	2	no	—
132	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₇₉	—	4	1	no	—
138	D₃₁₆	—	4	2	no	—
139	D₃₁₆	—	4	1	no	—
140	D₃₁₆	—	4	1	no	—
141	D₃₁₆	—	4	2	no	—
142	D₃₁₆	—	8	4	no	—
143	D₃₁₆	—	8	2	no	—
144	D₃₁₆	—	16	2	no	—
145	D₃₁₆	—	16	1	no	—
146	C₄×D₇₉	—	8	4	no	—
147	C₄×D₇₉	—	8	2	no	—
148	C₈×C₇₉	—	2	2	no	—
149	C₈×C₇₉	—	2	1	no	—
150	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
151	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂²×C₇₉	—	4	2	no	—
153	C₂²×C₇₉	—	8	1	no	—
154	C₂²×C₇₉	—	8	1	no	—
155	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₇₉	—	8	2	no	—
157	C₂×D₇₉	—	8	1	no	—
158	C₂×D₇₉	—	8	2	no	—
159	C₂×D₇₉	—	8	1	no	—
160	C₂×D₇₉	—	8	1	no	—
161	C₂×D₇₉	—	8	2	no	—
162	C₄×C₇₉	—	2	2	no	—
163	C₄×C₇₉	—	2	2	no	—
164	C₄×C₇₉	—	2	1	no	—
165	C₄×C₇₉	—	8	1	no	—
166	C₄×C₇₉	—	8	2	no	—
167	C₄×C₇₉	#509	1	2	yes	—
168	C₄×C₇₉	#548	1	1	yes	characteristic
169	C₄×C₇₉	#577	1	1	yes	characteristic
170	Dic₇₉	—	8	1	no	—
171	#33933	—	79	1	no	maximal, Sylow
172	C₂×C₇₉	—	8	2	no	—
173	C₂×C₇₉	#3358	1	1	yes	characteristic
174	C₂×C₇₉	C₂×#509	1	1	yes	characteristic
175	C₂×C₇₉	#3415	1	1	yes	characteristic
176	D₇₉	—	16	2	no	—
177	D₇₉	—	16	1	no	—
178	#3391	—	79	1	no	—
179	#3380	D₇₉	1	1	yes	characteristic, complemented
180	#2244	—	79	2	no	—
181	#2240	—	79	1	no	—
182	#2228	—	79	2	no	—
183	#2226	—	79	1	no	—
184	#2206	—	79	2	no	—
185	#2206	—	79	1	no	—
186	#2183	—	79	1	no	—
187	#2177	—	79	2	no	—
188	C₄×D₁₆	—	79	1	no	—
189	C₇₉	#33933	1	1	yes	characteristic, complemented, Sylow
190	#554	—	2	1	no	—
191	#550	—	158	1	no	—
192	C₂×D₁₆	—	79	1	no	—
193	C₂×D₁₆	—	79	2	no	—
194	#500	C₂×D₇₉	1	1	yes	characteristic
195	#499	C₂×D₇₉	1	2	yes	—
196	#493	—	79	1	no	—
197	#493	—	79	2	no	—
198	#478	—	2	2	no	—
199	#476	—	2	1	no	—
200	#459	—	158	1	no	—
201	#459	—	158	2	no	—
202	C₈×Q₈	C₂×D₇₉	1	1	yes	characteristic
203	C₄×D₈	—	79	1	no	—
204	C₄×D₈	—	79	2	no	—
205	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
206	C₄×D₈	C₂×D₇₉	1	2	yes	—
207	#367	—	79	2	no	—
208	#366	—	79	1	no	—
209	C₄⋊C₁₆	—	79	1	no	—
210	C₄⋊C₁₆	—	79	2	no	—
211	#357	—	79	1	no	—
212	#357	—	79	1	no	—
213	#357	—	79	2	no	—
214	#357	—	79	2	no	—
215	#357	—	79	2	no	—
216	#357	—	79	2	no	—
217	#346	—	79	2	no	—
218	C₄×C₁₆	—	79	1	no	—
219	C₂×#41	—	316	1	no	—
220	C₂×D₈	—	79	1	no	—
221	C₂×D₈	—	79	2	no	—
222	C₂×D₈	—	158	2	no	—
223	C₂×D₈	—	158	1	no	—
224	C₂×D₈	C₂²×D₇₉	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	1	no	—
226	#127	—	158	2	no	—
227	#127	—	158	1	no	—
228	#125	—	2	2	no	—
229	C₂²⋊Q₈	—	4	1	no	—
230	C₂²⋊Q₈	—	4	2	no	—
231	#121	—	316	1	no	—
232	#121	—	316	2	no	—
233	C₄×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	2	2	no	—
236	C₄×D₄	—	158	2	no	—
237	C₄×D₄	—	158	1	no	—
238	C₂×C₄⋊C₄	—	4	1	no	—
239	D₁₆	—	158	2	no	—
240	D₁₆	—	158	4	no	—
241	C₂×C₁₆	—	79	1	no	—
242	C₂×C₁₆	—	79	1	no	—
243	C₂×C₁₆	—	79	2	no	—
244	#107	—	2	1	no	—
245	#107	—	79	1	no	—
246	#107	C₂²×D₇₉	1	1	yes	characteristic
247	#107	C₂²×D₇₉	1	2	yes	—
248	#106	—	2	2	no	—
249	C₄⋊C₈	C₂²⋊D₇₉	1	1	yes	characteristic
250	C₄⋊C₈	C₂²⋊D₇₉	1	2	yes	—
251	#102	—	2	2	no	—
252	#102	—	2	1	no	—
253	#102	—	158	2	no	—
254	#102	—	158	1	no	—
255	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
256	C₄×C₈	C₂²×D₇₉	1	2	yes	—
257	#41	—	632	4	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	#11929609	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	316	2	no	—
262	C₂×D₄	—	316	1	no	—
263	C₂×D₄	—	316	1	no	—
264	C₂×D₄	—	316	2	no	—
265	C₂×D₄	—	158	2	no	—
266	C₂×D₄	—	158	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	316	2	no	—
270	C₂²×C₄	—	316	1	no	—
271	D₈	—	2	1	no	—
272	D₈	—	632	2	no	—
273	D₈	—	632	1	no	—
274	D₈	—	158	1	no	—
275	D₈	—	158	2	no	—
276	D₈	—	158	1	no	—
277	D₈	—	158	2	no	—
278	D₈	#11929625	1	2	yes	complemented
279	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
280	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	158	1	no	—
288	C₄⋊C₄	#11929609	1	1	yes	characteristic
289	C₄⋊C₄	#11929609	1	2	yes	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	316	1	no	—
293	C₂²⋊C₄	—	316	2	no	—
294	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
295	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
296	C₁₆	—	158	1	no	—
297	C₁₆	—	158	2	no	—
298	C₁₆	—	158	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	316	2	no	—
301	C₂³	—	316	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	16	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	316	1	no	—
307	D₄	—	316	2	no	—
308	D₄	—	316	1	no	—
309	D₄	—	316	2	no	—
310	D₄	—	632	4	no	—
311	D₄	—	632	2	no	—
312	D₄	—	1264	2	no	—
313	D₄	—	1264	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	16	2	no	—
318	C₂×C₄	—	316	1	no	—
319	C₂×C₄	—	632	4	no	—
320	C₂×C₄	—	632	2	no	—
321	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
322	C₂×C₄	C₂×#11929609	1	2	yes	—
323	C₂×C₄	C₂×#11929609	1	2	yes	—
324	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
325	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	C₂×#11929625	1	1	yes	characteristic
329	C₈	#14579209	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	632	2	no	—
334	C₂²	—	632	1	no	—
335	C₂²	—	632	1	no	—
336	C₂²	—	632	2	no	—
337	C₂²	—	632	2	no	—
338	C₂²	—	632	1	no	—
339	C₂²	C₂²×#11929609	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	632	1	no	—
346	C₄	#19098998	1	2	yes	—
347	C₄	#19099224	1	1	yes	characteristic
348	C₄	#19099371	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	1264	2	no	—
351	C₂	—	1264	1	no	—
352	C₂	#23776035	1	1	yes	characteristic
353	C₂	C₂×#19098998	1	1	yes	characteristic
354	C₂	#23776539	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3380	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	#3380	D₇₉
hypocenter	C₇₉	#33933
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3380

Complements

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

N	H ≅ G/N	type
1. #23776375	C₂	semidirect
2. #23765643	C₂	semidirect
3. #23765625	C₂	semidirect
4. #23765554	C₂	semidirect
5. #23765544	C₂	semidirect
6. #23765435	C₂	semidirect
7. #23765435	C₂	semidirect
8. #23765309	C₂	semidirect
9. #23765274	C₂	semidirect
10. C₄×#14579102	C₂	semidirect
11. C₇₉×#3380	C₂	semidirect
106. D₈×C₇₉	#41	semidirect
179. #3380	D₇₉	semidirect
189. C₇₉	#33933	semidirect
278. D₈	#11929625	semidirect

Maximal subgroups

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

subgroup	index
#23776375	2
#23765643	2
#23765625	2
#23765554	2
#23765544	2
#23765435	2
#23765309	2
#23765274	2
C₄×#14579102	2
C₇₉×#3380	2
#33933	79