Subgroups — #34271336

Order 20224.

842 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23774318	C₂	1	2	yes	maximal, complemented
2	#23772113	C₂	1	1	yes	maximal, characteristic, complemented
3	#23762876	C₂	1	4	yes	maximal, complemented
4	#23762752	C₂	1	4	yes	maximal, complemented
5	#23762148	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762068	C₂	1	2	yes	maximal, complemented
7	C₂×#493×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579167	—	2	1	no	—
9	#19099113	C₂²	1	1	yes	characteristic
10	#19098939	—	2	2	no	—
11	#19098909	—	2	2	no	—
12	#19098755	C₂²	1	8	yes	complemented
13	#19098697	—	2	4	no	—
14	#19098642	—	2	1	no	—
15	#19098642	C₂²	1	2	yes	complemented
16	C₈×D₃₁₆	—	2	1	no	—
17	C₈×D₃₁₆	C₂²	1	2	yes	complemented
18	#19098616	C₂²	1	2	yes	—
19	#19098587	C₂²	1	1	yes	characteristic
20	#19098529	—	2	1	no	—
21	#19098316	C₂²	1	2	yes	complemented
22	#19098314	C₂²	1	2	yes	complemented
23	#19098307	C₂²	1	8	yes	complemented
24	C₂²×D₈×C₇₉	—	2	1	no	—
25	C₂²×D₈×C₇₉	—	2	1	no	—
26	C₂×#127×C₇₉	C₂²	1	2	yes	—
27	#493×C₇₉	—	2	1	no	—
28	#493×C₇₉	—	2	1	no	—
29	#493×C₇₉	C₂²	1	2	yes	—
30	#493×C₇₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	C₂×#11929624	—	2	1	no	—
33	C₂×#11929609	—	2	8	no	—
34	#14579167	—	4	4	no	—
35	C₂×#11929608	—	2	1	no	—
36	C₂×#11929607	—	2	2	no	—
37	C₂×C₈×D₇₉	—	2	2	no	—
38	#14579161	D₄	1	1	yes	characteristic
39	#14579158	D₄	1	1	yes	characteristic
40	#14579149	—	2	1	no	—
41	#14579135	—	2	1	no	—
42	#14579135	D₄	1	2	yes	—
43	C₄×D₃₁₆	—	2	1	no	—
44	C₄×D₃₁₆	C₂³	1	2	yes	—
45	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
47	C₄⋊(C₇₉⋊C₈)	C₂³	1	4	yes	—
48	#14579082	—	2	8	no	—
49	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
50	(C₂×D₇₉)⋊C₈	—	2	2	no	—
51	(C₂×D₇₉)⋊C₈	—	2	2	no	—
52	C₄×C₇₉⋊C₈	—	2	2	no	—
53	C₂²×D₄×C₇₉	—	2	2	no	—
54	C₂²×D₄×C₇₉	—	2	2	no	—
55	C₂²×D₄×C₇₉	—	2	2	no	—
56	C₂×D₈×C₇₉	—	2	2	no	—
57	C₂×D₈×C₇₉	—	2	2	no	—
58	C₂×D₈×C₇₉	—	2	2	no	—
59	C₂×D₈×C₇₉	—	2	2	no	—
60	C₂×D₈×C₇₉	—	4	4	no	—
61	C₂×D₈×C₇₉	—	4	4	no	—
62	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
64	#127×C₇₉	—	2	2	no	—
65	#127×C₇₉	—	2	2	no	—
66	#127×C₇₉	—	2	2	no	—
67	#127×C₇₉	C₂³	1	4	yes	—
68	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
70	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
73	#11929624	—	4	2	no	—
74	#11929624	—	4	2	no	—
75	C₂×Dic₁₅₈	—	2	1	no	—
76	C₂×D₃₁₆	—	2	1	no	—
77	C₂×D₃₁₆	—	2	1	no	—
78	C₂×D₃₁₆	C₂×D₄	1	2	yes	—
79	C₂×C₂²⋊D₇₉	—	2	2	no	—
80	C₂×C₄×D₇₉	—	2	2	no	—
81	#11929609	—	8	8	no	—
82	#11929608	—	4	2	no	—
83	#11929607	—	4	4	no	—
84	C₂×C₇₉⋊C₈	—	2	4	no	—
85	C₂×C₇₉⋊C₈	—	2	2	no	—
86	C₈×D₇₉	—	4	4	no	—
87	C₄⋊Dic₇₉	—	2	1	no	—
88	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
89	(C₂×D₇₉)⋊C₄	—	2	2	no	—
90	(C₂×D₇₉)⋊C₄	—	2	2	no	—
91	C₂⁴×C₇₉	—	4	2	no	—
92	C₂×D₄×C₇₉	—	2	4	no	—
93	C₂×D₄×C₇₉	—	2	4	no	—
94	C₂×D₄×C₇₉	—	2	4	no	—
95	C₂×D₄×C₇₉	—	4	2	no	—
96	C₂×D₄×C₇₉	—	4	2	no	—
97	C₂×D₄×C₇₉	—	4	2	no	—
98	C₂×D₄×C₇₉	—	4	4	no	—
99	C₂×D₄×C₇₉	—	4	4	no	—
100	C₂×D₄×C₇₉	—	8	4	no	—
101	C₂×D₄×C₇₉	—	8	4	no	—
102	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
105	D₈×C₇₉	—	4	4	no	—
106	D₈×C₇₉	—	4	4	no	—
107	D₈×C₇₉	—	4	4	no	—
108	D₈×C₇₉	—	4	4	no	—
109	C₂×C₈×C₇₉	—	2	2	no	—
110	C₂×C₈×C₇₉	D₈	1	4	yes	complemented
111	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
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	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇₉	#41	1	1	yes	characteristic
118	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
119	C₂³×C₇₉	—	4	4	no	—
120	C₂³×C₇₉	—	8	4	no	—
121	C₂³×C₇₉	—	8	4	no	—
122	C₂³×C₇₉	—	8	4	no	—
123	C₂³×C₇₉	#121	1	1	yes	characteristic
124	C₂²×D₇₉	—	2	2	no	—
125	D₄×C₇₉	—	4	4	no	—
126	D₄×C₇₉	—	4	4	no	—
127	D₄×C₇₉	—	4	8	no	—
128	D₄×C₇₉	—	8	4	no	—
129	D₄×C₇₉	—	8	4	no	—
130	D₄×C₇₉	—	8	4	no	—
131	D₄×C₇₉	—	8	4	no	—
132	C₂×C₄×C₇₉	—	2	2	no	—
133	C₂×C₄×C₇₉	—	2	2	no	—
134	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
138	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
139	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
142	C₂²⋊D₇₉	—	4	4	no	—
143	C₂×Dic₇₉	—	2	2	no	—
144	D₃₁₆	—	2	4	no	—
145	D₃₁₆	—	4	2	no	—
146	D₃₁₆	—	4	2	no	—
147	C₄×D₇₉	—	4	2	no	—
148	C₄×D₇₉	—	4	2	no	—
149	Dic₁₅₈	—	4	2	no	—
150	C₈×C₇₉	—	2	1	no	—
151	C₈×C₇₉	—	2	1	no	—
152	C₈×C₇₉	C₂×D₈	1	2	yes	—
153	C₈×C₇₉	C₂×D₈	1	2	yes	—
154	C₇₉⋊C₈	—	4	2	no	—
155	C₇₉⋊C₈	—	4	4	no	—
156	C₂²×C₇₉	—	2	1	no	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	—	8	4	no	—
159	C₂²×C₇₉	—	8	4	no	—
160	C₂²×C₇₉	—	8	4	no	—
161	C₂²×C₇₉	—	16	8	no	—
162	C₂²×C₇₉	#459	1	2	yes	—
163	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
164	C₂×D₇₉	—	4	2	no	—
165	C₂×D₇₉	—	4	2	no	—
166	C₂×D₇₉	—	4	2	no	—
167	C₄×C₇₉	—	2	1	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	#459	1	2	yes	—
170	C₄×C₇₉	#472	1	2	yes	—
171	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₄×C₇₉	#576	1	2	yes	—
174	Dic₇₉	—	4	2	no	—
175	#19187	—	79	1	no	maximal, Sylow
176	C₂×C₇₉	—	2	2	no	—
177	C₂×C₇₉	—	16	4	no	—
178	C₂×C₇₉	C₂×#459	1	1	yes	characteristic
179	C₂×C₇₉	#3058	1	1	yes	characteristic
180	C₂×C₇₉	C₂×#472	1	1	yes	characteristic
181	D₇₉	—	8	2	no	—
182	#3205	—	79	2	no	—
183	C₂×#493	D₇₉	1	1	yes	characteristic, complemented
184	#2931	—	79	1	no	—
185	#1687	—	79	4	no	—
186	#1667	—	79	4	no	—
187	#1564	—	79	1	no	—
188	#1545	—	79	2	no	—
189	C₇₉	#19187	1	1	yes	characteristic, complemented, Sylow
190	C₂²×D₈	—	2	1	no	—
191	C₂²×D₈	—	2	1	no	—
192	C₂×#131	—	158	1	no	—
193	#532	—	79	1	no	—
194	C₂×#127	C₂×D₇₉	1	2	yes	—
195	#493	—	2	1	no	—
196	#493	—	2	1	no	—
197	#493	—	158	2	no	—
198	#493	C₂×D₇₉	1	2	yes	—
199	#493	C₂×D₇₉	1	2	yes	—
200	#486	—	158	2	no	—
201	#459	—	79	8	no	—
202	#447	—	158	4	no	—
203	#436	—	79	2	no	—
204	#436	—	158	1	no	—
205	C₈×D₄	—	79	2	no	—
206	C₈×D₄	—	158	1	no	—
207	#431	—	79	2	no	—
208	#423	—	79	1	no	—
209	#408	—	158	1	no	—
210	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
211	#335	—	79	2	no	—
212	#334	—	79	2	no	—
213	#331	—	79	8	no	—
214	C₂×#41	—	158	1	no	—
215	C₂²×D₄	—	2	2	no	—
216	C₂²×D₄	—	2	2	no	—
217	C₂²×D₄	—	2	2	no	—
218	C₂×D₈	—	2	2	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	—	4	4	no	—
223	C₂×D₈	—	4	4	no	—
224	C₂×D₈	—	158	8	no	—
225	#131	—	316	4	no	—
226	C₂×M₁₆	—	158	1	no	—
227	C₂×M₁₆	—	158	2	no	—
228	C₂²×C₈	—	158	2	no	—
229	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
230	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
231	C₄⋊Q₈	—	79	1	no	—
232	#127	—	2	2	no	—
233	#127	—	2	2	no	—
234	#127	—	2	2	no	—
235	#127	—	79	1	no	—
236	#127	C₂²×D₇₉	1	4	yes	—
237	#124	—	158	1	no	—
238	#121	—	79	2	no	—
239	#121	—	158	1	no	—
240	C₄×D₄	—	79	2	no	—
241	C₄×D₄	—	158	1	no	—
242	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
243	C₄⋊C₈	—	79	2	no	—
244	C₄⋊C₈	—	79	4	no	—
245	C₄⋊C₈	—	158	1	no	—
246	#102	—	158	8	no	—
247	C₂²⋊C₈	—	158	2	no	—
248	C₂²⋊C₈	—	158	2	no	—
249	C₂²⋊C₈	—	158	2	no	—
250	C₄×C₈	—	158	2	no	—
251	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
252	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
254	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
255	C₂⁴	—	4	2	no	—
256	#41	—	316	2	no	—
257	#41	—	316	2	no	—
258	C₂×Q₈	—	158	1	no	—
259	C₂×D₄	—	2	4	no	—
260	C₂×D₄	—	2	4	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	2	no	—
265	C₂×D₄	—	4	4	no	—
266	C₂×D₄	—	4	4	no	—
267	C₂×D₄	—	8	4	no	—
268	C₂×D₄	—	8	4	no	—
269	C₂×D₄	—	79	2	no	—
270	C₂×D₄	—	158	2	no	—
271	C₂×D₄	—	158	1	no	—
272	C₂×D₄	—	158	1	no	—
273	C₂²×C₄	—	158	2	no	—
274	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
276	C₂²×C₄	#11929623	1	1	yes	characteristic
277	D₈	—	4	4	no	—
278	D₈	—	4	4	no	—
279	D₈	—	4	4	no	—
280	D₈	—	4	4	no	—
281	D₈	—	632	8	no	—
282	M₁₆	—	316	2	no	—
283	M₁₆	—	316	4	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	316	4	no	—
286	C₂×C₈	—	158	4	no	—
287	C₂×C₈	—	158	2	no	—
288	C₂×C₈	#11929609	1	4	yes	complemented
289	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
293	C₄⋊C₄	—	79	2	no	—
294	C₄⋊C₄	—	158	1	no	—
295	C₂²⋊C₄	—	158	2	no	—
296	C₂²⋊C₄	—	158	2	no	—
297	C₄²	D₄×D₇₉	1	1	yes	characteristic
298	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
299	C₄²	#11929623	1	1	yes	characteristic
300	C₄²	C₂³×D₇₉	1	1	yes	characteristic
301	C₂³	—	4	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	—	8	4	no	—
304	C₂³	—	8	4	no	—
305	C₂³	—	158	2	no	—
306	C₂³	#14579132	1	1	yes	characteristic
307	Q₈	—	316	2	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	D₄	—	4	8	no	—
311	D₄	—	8	4	no	—
312	D₄	—	8	4	no	—
313	D₄	—	8	4	no	—
314	D₄	—	8	4	no	—
315	D₄	—	316	4	no	—
316	D₄	—	316	2	no	—
317	D₄	—	316	2	no	—
318	D₄	—	158	4	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	316	2	no	—
322	C₂×C₄	—	316	2	no	—
323	C₂×C₄	—	158	2	no	—
324	C₂×C₄	#14579132	1	1	yes	characteristic
325	C₂×C₄	#14579132	1	1	yes	characteristic
326	C₂×C₄	#14579132	1	1	yes	characteristic
327	C₂×C₄	C₂×#11929609	1	2	yes	—
328	C₂×C₄	#14579187	1	2	yes	—
329	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
330	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
331	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	316	2	no	—
335	C₈	—	316	4	no	—
336	C₈	C₂×#11929609	1	2	yes	—
337	C₈	C₂×#11929609	1	2	yes	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	8	4	no	—
341	C₂²	—	8	4	no	—
342	C₂²	—	8	4	no	—
343	C₂²	—	16	8	no	—
344	C₂²	—	316	2	no	—
345	C₂²	—	316	2	no	—
346	C₂²	—	316	2	no	—
347	C₂²	#19098755	1	2	yes	—
348	C₂²	C₂×#14579132	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	316	2	no	—
352	C₄	#19098755	1	2	yes	—
353	C₄	#19098830	1	2	yes	—
354	C₄	C₂²×#11929609	1	1	yes	characteristic
355	C₄	C₂×#14579187	1	1	yes	characteristic
356	C₄	#19099364	1	2	yes	—
357	C₂	—	2	2	no	—
358	C₂	—	16	4	no	—
359	C₂	—	632	2	no	—
360	C₂	C₂×#19098755	1	1	yes	characteristic
361	C₂	#23773044	1	1	yes	characteristic
362	C₂	C₂×#19098830	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579132
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₂×#493×C₇₉	C₂
socle	C₂²×C₇₉	C₂×#121
hypercenter	C₂×#493	D₇₉
hypocenter	C₇₉	#19187
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. #23774318	C₂	semidirect
2. #23772113	C₂	semidirect
3. #23762876	C₂	semidirect
4. #23762752	C₂	semidirect
5. #23762148	C₂	semidirect
6. #23762068	C₂	semidirect
7. C₂×#493×C₇₉	C₂	semidirect
12. #19098755	C₂²	semidirect
15. #19098642	C₂²	semidirect
17. C₈×D₃₁₆	C₂²	semidirect
21. #19098316	C₂²	semidirect
22. #19098314	C₂²	semidirect
23. #19098307	C₂²	semidirect
110. C₂×C₈×C₇₉	D₈	semidirect
183. C₂×#493	D₇₉	semidirect
189. C₇₉	#19187	semidirect
288. C₂×C₈	#11929609	semidirect

Maximal subgroups

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

subgroup	index
#23774318	2
#23772113	2
#23762876	2
#23762752	2
#23762148	2
#23762068	2
C₂×#493×C₇₉	2
#19187	79