Subgroups — #34204874

Order 20224.

490 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	C₂×C₄×D₆₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098390	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098363	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098363	C₂	1	1	yes	maximal, characteristic
5	#23761491	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761478	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₁₆	—	2	1	no	—
10	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₄×D₆₃₂	—	2	2	no	—
13	C₄×D₆₃₂	—	2	2	no	—
14	C₂×#14579095	C₂²	1	1	yes	characteristic
15	C₂×#14579083	—	2	1	no	—
16	#19098390	—	2	2	no	—
17	#19098363	—	2	2	no	—
18	#19098363	C₄	1	4	yes	complemented
19	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
22	#389×C₇₉	—	2	1	no	—
23	#340×C₇₉	—	2	1	no	—
24	C₂²×D₃₁₆	—	2	1	no	—
25	C₂²×C₄×D₇₉	—	4	1	no	—
26	C₂×D₆₃₂	—	2	1	no	—
27	C₂×D₆₃₂	—	2	1	no	—
28	C₂×D₆₃₂	—	2	1	no	—
29	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
30	C₂×D₆₃₂	D₄	1	2	yes	—
31	C₂×D₆₃₂	Q₈	1	2	yes	—
32	C₄×D₃₁₆	—	2	4	no	—
33	C₄×D₃₁₆	—	4	2	no	—
34	C₂×C₄⋊Dic₇₉	—	2	1	no	—
35	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
36	C₂×C₇₉⋊C₁₆	—	2	1	no	—
37	C₂×C₇₉⋊C₁₆	—	2	1	no	—
38	C₂×C₇₉⋊C₁₆	—	2	1	no	—
39	C₂×C₇₉⋊C₁₆	—	2	1	no	—
40	C₂×C₇₉⋊C₁₆	—	2	1	no	—
41	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	complemented
42	#14579095	—	2	1	no	—
43	#14579095	D₄	1	2	yes	—
44	#14579083	—	2	4	no	—
45	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
48	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
50	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
51	#107×C₇₉	—	2	1	no	—
52	#107×C₇₉	—	2	1	no	—
53	#107×C₇₉	—	2	1	no	—
54	#107×C₇₉	C₂×C₄	1	2	yes	complemented
55	C₄×C₈×C₇₉	—	2	1	no	—
56	C₄×C₈×C₇₉	—	2	1	no	—
57	#95×C₇₉	—	4	1	no	—
58	C₂³×D₇₉	—	4	1	no	—
59	C₂×D₃₁₆	—	2	2	no	—
60	C₂×D₃₁₆	—	2	2	no	—
61	C₂×D₃₁₆	—	2	2	no	—
62	C₂×D₃₁₆	—	4	1	no	—
63	C₂×D₃₁₆	—	4	1	no	—
64	C₂×D₃₁₆	—	4	1	no	—
65	C₂²×Dic₇₉	—	4	1	no	—
66	C₂×C₄×D₇₉	—	4	4	no	—
67	C₂×C₄×D₇₉	—	8	2	no	—
68	C₂×C₄×D₇₉	—	8	2	no	—
69	D₆₃₂	—	2	2	no	—
70	D₆₃₂	—	2	2	no	—
71	D₆₃₂	—	2	2	no	—
72	D₆₃₂	C₄⋊C₄	1	4	yes	complemented
73	C₄⋊Dic₇₉	—	2	2	no	—
74	C₄⋊Dic₇₉	—	4	1	no	—
75	(C₂×D₇₉)⋊C₄	—	4	4	no	—
76	C₇₉⋊C₁₆	—	2	2	no	—
77	C₇₉⋊C₁₆	—	4	1	no	—
78	C₂²×C₄×C₇₉	—	4	1	no	—
79	C₂²×C₄×C₇₉	—	4	1	no	—
80	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₇₉	—	2	1	no	—
84	C₂×C₈×C₇₉	—	2	1	no	—
85	C₂×C₈×C₇₉	—	2	1	no	—
86	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
91	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
92	C₄⋊C₄×C₇₉	—	2	2	no	—
93	C₄⋊C₄×C₇₉	—	4	1	no	—
94	C₄⋊C₄×C₇₉	—	4	1	no	—
95	C₄⋊C₄×C₇₉	—	4	1	no	—
96	C₄⋊C₄×C₇₉	—	8	2	no	—
97	C₄²×C₇₉	—	2	1	no	—
98	C₄²×C₇₉	—	2	1	no	—
99	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₇₉	—	4	2	no	—
101	C₂²×D₇₉	—	4	2	no	—
102	C₂²×D₇₉	—	4	2	no	—
103	C₂²×D₇₉	—	8	1	no	—
104	C₂²×D₇₉	—	8	1	no	—
105	C₂²×D₇₉	—	8	1	no	—
106	C₂²×D₇₉	—	8	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	4	2	no	—
113	C₂×C₄×C₇₉	—	8	1	no	—
114	C₂×C₄×C₇₉	—	8	1	no	—
115	C₂×C₄×C₇₉	—	8	1	no	—
116	C₂×C₄×C₇₉	—	8	1	no	—
117	C₂×C₄×C₇₉	—	8	1	no	—
118	C₂×C₄×C₇₉	SD₃₂	1	2	yes	—
119	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
123	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
125	C₂×Dic₇₉	—	4	2	no	—
126	C₂×Dic₇₉	—	8	1	no	—
127	C₂×Dic₇₉	—	8	1	no	—
128	D₃₁₆	—	2	4	no	—
129	D₃₁₆	—	4	2	no	—
130	D₃₁₆	—	4	2	no	—
131	D₃₁₆	—	4	2	no	—
132	C₄×D₇₉	—	8	8	no	—
133	C₈×C₇₉	—	4	1	no	—
134	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₈×C₇₉	#123	1	1	yes	characteristic
138	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
139	C₂²×C₇₉	#447	1	1	yes	characteristic
140	C₂²×C₇₉	#474	1	1	yes	characteristic
141	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
142	C₂²×C₇₉	#508	1	1	yes	characteristic
143	C₂²×C₇₉	#509	1	1	yes	characteristic
144	C₂²×C₇₉	#509	1	1	yes	characteristic
145	C₂×D₇₉	—	4	4	no	—
146	C₂×D₇₉	—	8	2	no	—
147	C₂×D₇₉	—	8	2	no	—
148	C₂×D₇₉	—	8	2	no	—
149	C₂×D₇₉	—	8	2	no	—
150	C₂×D₇₉	—	8	2	no	—
151	C₂×D₇₉	—	8	2	no	—
152	C₄×C₇₉	—	2	2	no	—
153	C₄×C₇₉	—	4	1	no	—
154	C₄×C₇₉	—	8	2	no	—
155	C₄×C₇₉	—	16	1	no	—
156	C₄×C₇₉	#385	1	1	yes	characteristic
157	C₄×C₇₉	#440	1	1	yes	characteristic
158	C₄×C₇₉	#452	1	1	yes	characteristic
159	C₄×C₇₉	#478	1	1	yes	characteristic
160	Dic₇₉	—	8	2	no	—
161	#12937	—	79	1	no	maximal, Sylow
162	C₂×C₇₉	#1862	1	1	yes	characteristic
163	C₂×C₇₉	C₄×SD₃₂	1	1	yes	characteristic
164	C₂×C₇₉	#2177	1	1	yes	characteristic
165	C₂×C₇₉	#2186	1	1	yes	characteristic
166	C₂×C₇₉	#2187	1	1	yes	characteristic
167	C₂×C₇₉	#2226	1	1	yes	characteristic
168	C₂×C₇₉	#2228	1	1	yes	characteristic
169	D₇₉	—	8	4	no	—
170	C₂×C₄×D₈	—	79	1	no	—
171	C₂×#367	—	79	1	no	—
172	C₂×#357	—	79	1	no	—
173	C₂×#357	—	79	1	no	—
174	#1942	D₇₉	1	1	yes	characteristic, complemented
175	#1380	—	79	1	no	—
176	#1374	—	79	1	no	—
177	C₇₉	#12937	1	1	yes	characteristic, complemented, Sylow
178	C₂²×D₈	—	79	1	no	—
179	C₂×C₄×D₄	—	158	1	no	—
180	C₂²×C₁₆	—	79	1	no	—
181	C₂²×C₁₆	—	79	1	no	—
182	C₄×D₈	—	158	2	no	—
183	C₄×D₈	—	158	2	no	—
184	C₂×#107	—	79	1	no	—
185	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
186	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
187	C₂×#102	—	158	1	no	—
188	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
189	#389	—	2	1	no	—
190	#367	—	158	2	no	—
191	#357	—	79	4	no	—
192	#357	—	158	2	no	—
193	#340	—	2	1	no	—
194	C₂²×D₄	—	158	1	no	—
195	C₂³×C₄	—	316	1	no	—
196	C₂×D₈	—	79	2	no	—
197	C₂×D₈	—	79	2	no	—
198	C₂×D₈	—	79	2	no	—
199	C₂×D₈	—	158	1	no	—
200	C₂×D₈	—	158	1	no	—
201	C₂×D₈	—	158	1	no	—
202	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
204	C₄×D₄	—	316	2	no	—
205	C₄×D₄	—	158	4	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	4	1	no	—
209	C₂×C₄⋊C₄	—	158	1	no	—
210	C₂×C₂²⋊C₄	—	316	1	no	—
211	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
212	C₂×C₁₆	—	79	2	no	—
213	C₂×C₁₆	—	158	1	no	—
214	C₂×C₁₆	—	158	1	no	—
215	C₂×C₁₆	—	158	1	no	—
216	C₂×C₁₆	—	158	1	no	—
217	C₂×C₁₆	—	158	1	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	79	2	no	—
222	#107	—	158	1	no	—
223	#107	C₄×D₇₉	1	2	yes	complemented
224	#102	—	158	4	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	#95	—	4	1	no	—
228	C₂⁴	—	316	1	no	—
229	C₂×D₄	—	316	1	no	—
230	C₂×D₄	—	316	1	no	—
231	C₂×D₄	—	316	1	no	—
232	C₂×D₄	—	158	2	no	—
233	C₂×D₄	—	158	2	no	—
234	C₂×D₄	—	158	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	316	4	no	—
238	C₂²×C₄	—	316	1	no	—
239	C₂²×C₄	—	632	2	no	—
240	C₂²×C₄	—	632	2	no	—
241	C₂²×C₄	#11929609	1	1	yes	characteristic
242	C₂²×C₄	#11929609	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
244	D₈	—	79	4	no	—
245	D₈	—	158	2	no	—
246	D₈	—	158	2	no	—
247	D₈	—	158	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
252	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
253	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
254	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
255	C₂×C₈	#11929624	1	1	yes	characteristic
256	C₂×C₈	#11929625	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄⋊C₄	—	316	1	no	—
263	C₄⋊C₄	—	158	2	no	—
264	C₂²⋊C₄	—	316	4	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₁₆	—	316	1	no	—
268	C₁₆	—	158	2	no	—
269	C₂³	—	316	2	no	—
270	C₂³	—	316	2	no	—
271	C₂³	—	316	2	no	—
272	C₂³	—	632	1	no	—
273	C₂³	—	632	1	no	—
274	C₂³	—	632	1	no	—
275	C₂³	—	632	1	no	—
276	C₂³	C₂×#11929609	1	1	yes	characteristic
277	D₄	—	316	2	no	—
278	D₄	—	316	2	no	—
279	D₄	—	316	2	no	—
280	D₄	—	158	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	316	2	no	—
293	C₂×C₄	—	632	8	no	—
294	C₂×C₄	—	632	1	no	—
295	C₂×C₄	—	632	1	no	—
296	C₂×C₄	#14579104	1	2	yes	—
297	C₂×C₄	C₄×C₂²⋊D₇₉	1	1	yes	characteristic
298	C₂×C₄	#14579129	1	1	yes	characteristic
299	C₂×C₄	#14579137	1	1	yes	characteristic
300	C₂×C₄	#14579181	1	1	yes	characteristic
301	C₂×C₄	#14579186	1	1	yes	characteristic
302	C₂×C₄	#14579192	1	1	yes	characteristic
303	C₈	—	4	1	no	—
304	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
305	C₈	#14579123	1	1	yes	characteristic
306	C₈	#14579139	1	1	yes	characteristic
307	C₈	#14579146	1	1	yes	characteristic
308	C₂²	—	316	4	no	—
309	C₂²	—	632	2	no	—
310	C₂²	—	632	2	no	—
311	C₂²	—	632	2	no	—
312	C₂²	—	632	2	no	—
313	C₂²	—	632	2	no	—
314	C₂²	—	632	2	no	—
315	C₂²	C₄×#11929609	1	1	yes	characteristic
316	C₂²	#19098696	1	1	yes	characteristic
317	C₂²	#19098840	1	1	yes	characteristic
318	C₂²	C₂×#14579104	1	1	yes	characteristic
319	C₂²	#19098993	1	1	yes	characteristic
320	C₂²	#19098998	1	1	yes	characteristic
321	C₂²	#19098999	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	4	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	16	1	no	—
326	C₄	—	632	2	no	—
327	C₄	#19098434	1	1	yes	characteristic
328	C₄	#19098661	1	1	yes	characteristic
329	C₄	#19098723	1	1	yes	characteristic
330	C₄	#19098860	1	1	yes	characteristic
331	C₂	—	632	4	no	—
332	C₂	#23763646	1	1	yes	characteristic
333	C₂	C₄×#14579104	1	1	yes	characteristic
334	C₂	#23765274	1	1	yes	characteristic
335	C₂	#23765321	1	1	yes	characteristic
336	C₂	#23765328	1	1	yes	characteristic
337	C₂	#23765544	1	1	yes	characteristic
338	C₂	#23765554	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1942	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1942	D₇₉
hypocenter	C₇₉	#12937
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1942

Complements

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

N	H ≅ G/N	type
2. C₂×#19098390	C₂	semidirect
5. #23761491	C₂	semidirect
6. #23761478	C₂	semidirect
7. C₇₉×#1942	C₂	semidirect
18. #19098363	C₄	semidirect
41. C₂×C₇₉⋊C₁₆	C₂×C₄	semidirect
54. #107×C₇₉	C₂×C₄	semidirect
72. D₆₃₂	C₄⋊C₄	semidirect
174. #1942	D₇₉	semidirect
177. C₇₉	#12937	semidirect
223. #107	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₆₃₂	2
C₂×#19098390	2
C₂×#19098363	2
#23761491	2
#23761478	2
C₇₉×#1942	2
#12937	79