Subgroups — #34205332

Order 20224.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098368	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098365	C₂	1	1	yes	maximal, characteristic
5	#23764266	C₂	1	1	yes	maximal, characteristic
6	#23761492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
9	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₅₈	—	2	1	no	—
11	C₂×#14579106	—	2	2	no	—
12	C₂×#14579106	—	2	2	no	—
13	C₂×#14579106	C₄	1	4	yes	complemented
14	C₄×Dic₃₁₆	—	2	2	no	—
15	C₄×Dic₃₁₆	—	2	2	no	—
16	C₂×#14579095	C₂²	1	1	yes	characteristic
17	C₂×#14579095	C₂²	1	1	yes	characteristic
18	C₂×#14579086	—	2	1	no	—
19	C₂×#14579083	—	2	1	no	—
20	#19098465	—	2	1	no	—
21	#19098368	—	2	2	no	—
22	#19098365	—	2	2	no	—
23	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₅₈	—	2	1	no	—
27	C₂²×D₃₁₆	—	2	1	no	—
28	C₂×Dic₃₁₆	—	2	1	no	—
29	C₂×Dic₃₁₆	—	2	1	no	—
30	C₂×Dic₃₁₆	—	2	1	no	—
31	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
32	C₂×Dic₃₁₆	D₄	1	2	yes	—
33	C₂×Dic₃₁₆	D₄	1	2	yes	—
34	C₂×D₆₃₂	—	2	1	no	—
35	C₂×D₆₃₂	—	2	1	no	—
36	C₂×D₆₃₂	—	2	1	no	—
37	C₂×D₆₃₂	—	2	1	no	—
38	C₂×D₆₃₂	—	2	1	no	—
39	C₂×D₆₃₂	C₂×C₄	1	2	yes	—
40	C₄×Dic₁₅₈	—	2	4	no	—
41	C₄×Dic₁₅₈	—	4	2	no	—
42	C₂×C₄⋊Dic₇₉	—	2	1	no	—
43	C₂×C₄⋊Dic₇₉	—	2	1	no	—
44	C₂×Dic₇₉⋊C₄	—	4	1	no	—
45	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
46	C₂×(C₂×D₇₉)⋊C₄	—	4	1	no	—
47	C₂×C₄×Dic₇₉	—	4	1	no	—
48	#14579106	—	2	8	no	—
49	#14579095	—	2	1	no	—
50	#14579095	—	2	1	no	—
51	#14579095	—	2	1	no	—
52	#14579095	D₄	1	2	yes	—
53	#14579086	—	2	4	no	—
54	#14579083	—	4	2	no	—
55	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
58	C₂×C₁₆×C₇₉	—	2	1	no	—
59	C₂×C₁₆×C₇₉	—	2	1	no	—
60	C₂×C₁₆×C₇₉	—	2	1	no	—
61	C₂×C₁₆×C₇₉	—	2	1	no	—
62	C₂×C₁₆×C₇₉	—	2	1	no	—
63	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₇₉	—	2	1	no	—
65	C₄×C₈×C₇₉	—	2	1	no	—
66	C₂³×D₇₉	—	4	1	no	—
67	C₂×Dic₁₅₈	—	2	2	no	—
68	C₂×Dic₁₅₈	—	2	2	no	—
69	C₂×Dic₁₅₈	—	2	2	no	—
70	C₂×Dic₁₅₈	—	4	1	no	—
71	C₂×Dic₁₅₈	—	4	1	no	—
72	C₂×Dic₁₅₈	—	4	1	no	—
73	C₂×D₃₁₆	—	2	2	no	—
74	C₂×D₃₁₆	—	4	1	no	—
75	C₂×D₃₁₆	—	4	1	no	—
76	C₂×D₃₁₆	—	4	1	no	—
77	C₂×D₃₁₆	—	4	1	no	—
78	C₂×D₃₁₆	—	4	1	no	—
79	C₂²×Dic₇₉	—	4	1	no	—
80	C₂²×Dic₇₉	—	4	1	no	—
81	C₂²×Dic₇₉	—	4	1	no	—
82	Dic₃₁₆	—	2	2	no	—
83	Dic₃₁₆	—	2	2	no	—
84	Dic₃₁₆	—	2	2	no	—
85	Dic₃₁₆	C₂²⋊C₄	1	4	yes	complemented
86	D₆₃₂	—	2	2	no	—
87	D₆₃₂	—	2	2	no	—
88	D₆₃₂	—	4	1	no	—
89	D₆₃₂	—	4	1	no	—
90	C₄⋊Dic₇₉	—	2	2	no	—
91	C₄⋊Dic₇₉	—	4	1	no	—
92	C₄⋊Dic₇₉	—	4	1	no	—
93	C₄⋊Dic₇₉	—	4	1	no	—
94	Dic₇₉⋊C₄	—	4	4	no	—
95	(C₂×D₇₉)⋊C₄	—	8	2	no	—
96	(C₂×D₇₉)⋊C₄	—	8	2	no	—
97	C₄×Dic₇₉	—	4	4	no	—
98	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
99	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
100	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₉	—	2	1	no	—
102	C₂×C₈×C₇₉	—	2	1	no	—
103	C₂×C₈×C₇₉	—	2	1	no	—
104	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
110	C₄²×C₇₉	—	2	1	no	—
111	C₄²×C₇₉	—	2	1	no	—
112	C₁₆×C₇₉	—	2	2	no	—
113	C₁₆×C₇₉	—	4	1	no	—
114	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₇₉	—	4	2	no	—
116	C₂²×D₇₉	—	8	1	no	—
117	C₂²×D₇₉	—	8	1	no	—
118	C₂²×D₇₉	—	8	1	no	—
119	C₂²×D₇₉	—	8	1	no	—
120	C₂²×D₇₉	—	8	1	no	—
121	C₂²×D₇₉	—	8	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	2	1	no	—
126	C₂×C₄×C₇₉	—	2	1	no	—
127	C₂×C₄×C₇₉	SD₃₂	1	2	yes	complemented
128	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
130	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
132	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×Dic₇₉	—	4	2	no	—
135	C₂×Dic₇₉	—	4	2	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	C₂×Dic₇₉	—	8	1	no	—
139	C₂×Dic₇₉	—	8	1	no	—
140	C₂×Dic₇₉	—	8	1	no	—
141	C₂×Dic₇₉	—	8	1	no	—
142	C₂×Dic₇₉	—	8	1	no	—
143	D₃₁₆	—	4	2	no	—
144	D₃₁₆	—	4	2	no	—
145	D₃₁₆	—	8	1	no	—
146	D₃₁₆	—	8	1	no	—
147	Dic₁₅₈	—	2	4	no	—
148	Dic₁₅₈	—	4	2	no	—
149	Dic₁₅₈	—	4	2	no	—
150	Dic₁₅₈	—	4	2	no	—
151	C₈×C₇₉	—	4	1	no	—
152	C₈×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
154	C₈×C₇₉	#121	1	1	yes	characteristic
155	C₈×C₇₉	#123	1	1	yes	characteristic
156	C₂²×C₇₉	C₄×D₈	1	1	yes	characteristic
157	C₂²×C₇₉	#447	1	1	yes	characteristic
158	C₂²×C₇₉	#459	1	1	yes	characteristic
159	C₂²×C₇₉	C₂×SD₃₂	1	1	yes	characteristic
160	C₂²×C₇₉	#508	1	1	yes	characteristic
161	C₂²×C₇₉	#509	1	1	yes	characteristic
162	C₂²×C₇₉	#510	1	1	yes	characteristic
163	C₂×D₇₉	—	8	2	no	—
164	C₂×D₇₉	—	8	2	no	—
165	C₂×D₇₉	—	8	2	no	—
166	C₂×D₇₉	—	16	1	no	—
167	C₂×D₇₉	—	16	1	no	—
168	C₂×D₇₉	—	16	1	no	—
169	C₂×D₇₉	—	16	1	no	—
170	C₄×C₇₉	—	2	2	no	—
171	C₄×C₇₉	—	4	1	no	—
172	C₄×C₇₉	#386	1	1	yes	characteristic
173	C₄×C₇₉	#442	1	1	yes	characteristic
174	C₄×C₇₉	#449	1	1	yes	characteristic
175	C₄×C₇₉	#463	1	1	yes	characteristic
176	Dic₇₉	—	4	4	no	—
177	Dic₇₉	—	8	2	no	—
178	Dic₇₉	—	16	1	no	—
179	#13013	—	79	1	no	maximal, Sylow
180	C₂×C₇₉	#1879	1	1	yes	characteristic
181	C₂×C₇₉	C₄×SD₃₂	1	1	yes	characteristic
182	C₂×C₇₉	#2175	1	1	yes	characteristic
183	C₂×C₇₉	#2185	1	1	yes	characteristic
184	C₂×C₇₉	#2186	1	1	yes	characteristic
185	C₂×C₇₉	#2207	1	1	yes	characteristic
186	C₂×C₇₉	#2211	1	1	yes	characteristic
187	D₇₉	—	16	2	no	—
188	C₂²×SD₃₂	—	79	1	no	—
189	C₂×C₄×Q₁₆	—	79	1	no	—
190	C₂×#358	—	79	1	no	—
191	C₂×#357	—	79	1	no	—
192	#1970	—	79	1	no	—
193	#1380	—	79	1	no	—
194	#1374	D₇₉	1	1	yes	characteristic, complemented
195	C₇₉	#13013	1	1	yes	characteristic, complemented, Sylow
196	C₂²×Q₁₆	—	79	1	no	—
197	C₂²×D₈	—	79	1	no	—
198	C₂×C₄×Q₈	—	158	1	no	—
199	C₂×SD₃₂	—	79	4	no	—
200	C₂×SD₃₂	—	158	2	no	—
201	C₂×SD₃₂	—	158	2	no	—
202	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
203	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
204	C₄×Q₁₆	—	158	2	no	—
205	C₄×Q₁₆	—	158	2	no	—
206	C₂×#107	—	79	1	no	—
207	C₂×#107	—	79	1	no	—
208	C₂×#103	—	158	1	no	—
209	C₂×#102	—	158	1	no	—
210	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
211	#390	—	158	1	no	—
212	#358	—	158	2	no	—
213	#357	—	158	2	no	—
214	C₂²×Q₈	—	158	1	no	—
215	C₂²×D₄	—	158	1	no	—
216	C₂×Q₁₆	—	79	2	no	—
217	C₂×Q₁₆	—	79	2	no	—
218	C₂×Q₁₆	—	79	2	no	—
219	C₂×Q₁₆	—	158	1	no	—
220	C₂×Q₁₆	—	158	1	no	—
221	C₂×Q₁₆	—	158	1	no	—
222	C₂×D₈	—	79	2	no	—
223	C₂×D₈	—	158	1	no	—
224	C₂×D₈	—	158	1	no	—
225	C₂×D₈	—	158	1	no	—
226	C₂×D₈	—	158	1	no	—
227	C₂×D₈	—	158	1	no	—
228	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
229	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
230	C₄×Q₈	—	316	2	no	—
231	C₄×Q₈	—	158	4	no	—
232	C₂×C₄⋊C₄	—	316	1	no	—
233	C₂×C₄⋊C₄	—	158	1	no	—
234	C₂×C₄⋊C₄	—	158	1	no	—
235	C₂×C₂²⋊C₄	—	316	1	no	—
236	C₂×C₂²⋊C₄	—	316	1	no	—
237	C₂×C₄²	—	316	1	no	—
238	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
239	SD₃₂	—	158	8	no	—
240	C₂×C₁₆	—	2	1	no	—
241	C₂×C₁₆	—	2	1	no	—
242	C₂×C₁₆	—	2	1	no	—
243	C₂×C₁₆	—	2	1	no	—
244	C₂×C₁₆	—	2	1	no	—
245	C₂×C₁₆	C₄×D₇₉	1	2	yes	—
246	#107	—	79	2	no	—
247	#107	—	158	1	no	—
248	#107	—	158	1	no	—
249	#107	—	158	1	no	—
250	#103	—	158	4	no	—
251	#102	—	316	2	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	2	1	no	—
254	C₂⁴	—	316	1	no	—
255	C₂×Q₈	—	316	1	no	—
256	C₂×Q₈	—	316	1	no	—
257	C₂×Q₈	—	316	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	—	316	1	no	—
263	C₂×D₄	—	316	1	no	—
264	C₂×D₄	—	316	1	no	—
265	C₂×D₄	—	316	1	no	—
266	C₂×D₄	—	158	2	no	—
267	C₂²×C₄	—	316	1	no	—
268	C₂²×C₄	—	316	1	no	—
269	C₂²×C₄	—	316	1	no	—
270	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
271	C₂²×C₄	D₆₃₂	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
273	Q₁₆	—	79	4	no	—
274	Q₁₆	—	158	2	no	—
275	Q₁₆	—	158	2	no	—
276	Q₁₆	—	158	2	no	—
277	D₈	—	316	1	no	—
278	D₈	—	316	1	no	—
279	D₈	—	158	2	no	—
280	D₈	—	158	2	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
285	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
286	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
287	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
288	C₂×C₈	#11929624	1	1	yes	characteristic
289	C₂×C₈	#11929625	1	1	yes	characteristic
290	C₄⋊C₄	—	316	4	no	—
291	C₄⋊C₄	—	316	1	no	—
292	C₄⋊C₄	—	316	1	no	—
293	C₄⋊C₄	—	316	1	no	—
294	C₄⋊C₄	—	158	2	no	—
295	C₂²⋊C₄	—	632	2	no	—
296	C₂²⋊C₄	—	632	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	2	1	no	—
299	C₄²	—	316	4	no	—
300	C₁₆	—	2	2	no	—
301	C₁₆	—	4	1	no	—
302	C₂³	—	316	2	no	—
303	C₂³	—	632	1	no	—
304	C₂³	—	632	1	no	—
305	C₂³	—	632	1	no	—
306	C₂³	—	632	1	no	—
307	C₂³	—	632	1	no	—
308	C₂³	—	632	1	no	—
309	C₂³	C₂×D₆₃₂	1	1	yes	characteristic
310	Q₈	—	316	2	no	—
311	Q₈	—	316	2	no	—
312	Q₈	—	316	2	no	—
313	Q₈	—	158	4	no	—
314	D₄	—	316	2	no	—
315	D₄	—	316	2	no	—
316	D₄	—	632	1	no	—
317	D₄	—	632	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	316	2	no	—
324	C₂×C₄	—	316	2	no	—
325	C₂×C₄	—	316	2	no	—
326	C₂×C₄	—	316	2	no	—
327	C₂×C₄	—	632	1	no	—
328	C₂×C₄	—	632	1	no	—
329	C₂×C₄	—	632	1	no	—
330	C₂×C₄	—	632	1	no	—
331	C₂×C₄	—	632	1	no	—
332	C₂×C₄	#14579106	1	2	yes	complemented
333	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
334	C₂×C₄	#14579131	1	1	yes	characteristic
335	C₂×C₄	#14579136	1	1	yes	characteristic
336	C₂×C₄	#14579183	1	1	yes	characteristic
337	C₂×C₄	#14579189	1	1	yes	characteristic
338	C₂×C₄	#14579189	1	1	yes	characteristic
339	C₈	—	4	1	no	—
340	C₈	C₂²⋊C₄×D₇₉	1	1	yes	characteristic
341	C₈	#14579123	1	1	yes	characteristic
342	C₈	#14579134	1	1	yes	characteristic
343	C₈	#14579146	1	1	yes	characteristic
344	C₂²	—	632	2	no	—
345	C₂²	—	632	2	no	—
346	C₂²	—	632	2	no	—
347	C₂²	—	1264	1	no	—
348	C₂²	—	1264	1	no	—
349	C₂²	—	1264	1	no	—
350	C₂²	—	1264	1	no	—
351	C₂²	C₄×D₆₃₂	1	1	yes	characteristic
352	C₂²	#19098699	1	1	yes	characteristic
353	C₂²	#19098759	1	1	yes	characteristic
354	C₂²	C₂×#14579106	1	1	yes	characteristic
355	C₂²	#19098995	1	1	yes	characteristic
356	C₂²	#19099001	1	1	yes	characteristic
357	C₂²	#19099006	1	1	yes	characteristic
358	C₄	—	2	2	no	—
359	C₄	—	4	1	no	—
360	C₄	—	316	4	no	—
361	C₄	—	632	2	no	—
362	C₄	—	1264	1	no	—
363	C₄	#19098441	1	1	yes	characteristic
364	C₄	#19098676	1	1	yes	characteristic
365	C₄	#19098710	1	1	yes	characteristic
366	C₄	#19098779	1	1	yes	characteristic
367	C₂	—	1264	2	no	—
368	C₂	#23763747	1	1	yes	characteristic
369	C₂	C₄×#14579106	1	1	yes	characteristic
370	C₂	#23765266	1	1	yes	characteristic
371	C₂	#23765319	1	1	yes	characteristic
372	C₂	#23765325	1	1	yes	characteristic
373	C₂	#23765441	1	1	yes	characteristic
374	C₂	#23765463	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₃₂
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#1374	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂²×D₆₃₂	C₂²
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#1374	D₇₉
hypocenter	C₇₉	#13013
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Dic₃₁₆	C₂	semidirect
3. C₂×#19098368	C₂	semidirect
6. #23761492	C₂	semidirect
7. C₇₉×#1374	C₂	semidirect
13. C₂×#14579106	C₄	semidirect
85. Dic₃₁₆	C₂²⋊C₄	semidirect
127. C₂×C₄×C₇₉	SD₃₂	semidirect
194. #1374	D₇₉	semidirect
195. C₇₉	#13013	semidirect
332. C₂×C₄	#14579106	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14579106	2
C₂×C₄×Dic₃₁₆	2
C₂×#19098368	2
C₂×#19098365	2
#23764266	2
#23761492	2
C₇₉×#1374	2
#13013	79