Subgroups — #34443718

Order 20224.

858 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	C₂×#19099000	C₂	1	2	yes	maximal, complemented
2	C₂×#19098996	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₁₆×D₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#23773882	C₂	1	2	yes	maximal, complemented
5	#23765667	C₂	1	2	yes	maximal, complemented
6	#23765665	C₂	1	2	yes	maximal, complemented
7	#23765662	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765650	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765648	C₂	1	1	yes	maximal, characteristic, complemented
10	#23764995	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579188	—	2	2	no	—
13	C₂×#14579184	C₂²	1	2	yes	—
14	C₂×D₈×D₇₉	C₂²	1	2	yes	—
15	#19099131	—	2	2	no	—
16	#19099000	—	2	8	no	—
17	#19098996	—	2	4	no	—
18	C₂×#14579104	C₂²	1	2	yes	complemented
19	C₂×#14579104	C₂²	1	2	yes	complemented
20	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
21	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic, complemented
22	C₂×D₁₂₆₄	C₂²	1	2	yes	complemented
23	C₂×#14579102	C₂²	1	2	yes	—
24	C₂×#14579102	C₂²	1	2	yes	complemented
25	D₁₆×D₇₉	—	2	4	no	—
26	C₂×#14579100	C₂²	1	1	yes	characteristic
27	C₂×C₁₆×D₇₉	C₂²	1	1	yes	characteristic
28	#19098953	—	2	2	no	—
29	#19098949	—	2	2	no	—
30	#19098948	C₂²	1	2	yes	complemented
31	#19098940	C₂²	1	2	yes	complemented
32	#19098938	C₂²	1	2	yes	complemented
33	#19098517	C₂²	1	1	yes	characteristic
34	#19098339	C₂²	1	1	yes	characteristic, complemented
35	#19098338	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic, complemented
37	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic, complemented
38	C₂×SD₃₂×C₇₉	C₂²	1	2	yes	complemented
39	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#495×C₇₉	C₂²	1	2	yes	complemented
41	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic, complemented
42	C₂×#11929625	—	2	2	no	—
43	C₂×D₄×D₇₉	—	2	2	no	—
44	#14579188	—	4	8	no	—
45	#14579184	—	2	4	no	—
46	#14579184	—	2	4	no	—
47	C₂×#11929611	—	2	2	no	—
48	C₂×#11929611	—	2	2	no	—
49	C₂×D₆₃₂	—	2	2	no	—
50	C₂×D₆₃₂	C₂³	1	2	yes	—
51	C₂×D₆₃₂	C₂³	1	2	yes	—
52	C₂×#11929609	—	2	2	no	—
53	C₂×#11929609	—	2	2	no	—
54	D₈×D₇₉	—	2	4	no	—
55	D₈×D₇₉	—	2	4	no	—
56	C₂×#11929607	D₄	1	1	yes	characteristic
57	C₂×C₈×D₇₉	C₂³	1	1	yes	characteristic
58	#14579158	—	2	2	no	—
59	#14579157	—	2	2	no	—
60	#14579150	—	2	2	no	—
61	#14579134	—	4	4	no	—
62	#14579131	—	4	4	no	—
63	#14579118	D₄	1	1	yes	characteristic
64	#14579104	—	2	4	no	—
65	#14579104	—	2	4	no	—
66	D₁₂₆₄	—	2	2	no	—
67	D₁₂₆₄	—	2	2	no	—
68	D₁₂₆₄	—	2	4	no	—
69	#14579102	—	2	4	no	—
70	#14579102	—	2	4	no	—
71	#14579100	D₄	1	4	yes	complemented
72	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
74	C₁₆×D₇₉	D₄	1	4	yes	complemented
75	#14579072	D₄	1	1	yes	characteristic
76	#14579071	D₄	1	1	yes	characteristic
77	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
78	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₇₉	—	2	2	no	—
80	C₂×D₈×C₇₉	C₂³	1	2	yes	—
81	#124×C₇₉	—	2	2	no	—
82	Q₃₂×C₇₉	—	2	2	no	—
83	SD₃₂×C₇₉	—	2	4	no	—
84	D₁₆×C₇₉	—	2	2	no	—
85	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
86	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
88	C₂³×D₇₉	—	4	2	no	—
89	#11929625	—	4	4	no	—
90	#11929625	—	4	4	no	—
91	C₂×D₃₁₆	—	2	2	no	—
92	C₂×D₃₁₆	—	2	2	no	—
93	C₂×D₃₁₆	—	4	2	no	—
94	C₂×D₃₁₆	—	4	4	no	—
95	C₂×C₂²⋊D₇₉	—	4	2	no	—
96	C₂×C₂²⋊D₇₉	—	4	4	no	—
97	D₄×D₇₉	—	4	4	no	—
98	D₄×D₇₉	—	4	4	no	—
99	C₂×C₄×D₇₉	—	4	2	no	—
100	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
101	#11929611	—	4	4	no	—
102	#11929611	—	4	4	no	—
103	D₆₃₂	—	2	2	no	—
104	D₆₃₂	—	2	2	no	—
105	D₆₃₂	—	2	2	no	—
106	D₆₃₂	—	2	2	no	—
107	D₆₃₂	—	4	4	no	—
108	#11929609	—	4	4	no	—
109	#11929609	—	4	4	no	—
110	#11929607	D₈	1	4	yes	—
111	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₈×D₇₉	C₂×D₄	1	2	yes	—
114	C₈×D₇₉	C₂×D₄	1	2	yes	—
115	Dic₇₉⋊C₄	D₈	1	2	yes	—
116	(C₂×D₇₉)⋊C₄	—	4	4	no	—
117	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
118	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
119	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
120	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₇₉	—	2	2	no	—
122	C₂×D₄×C₇₉	—	2	2	no	—
123	Q₁₆×C₇₉	—	2	2	no	—
124	Q₁₆×C₇₉	—	2	2	no	—
125	SD₁₆×C₇₉	—	4	4	no	—
126	D₈×C₇₉	—	2	2	no	—
127	D₈×C₇₉	—	2	2	no	—
128	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊C₄×C₇₉	—	4	4	no	—
131	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
132	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
133	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
134	C₂³×C₇₉	—	4	2	no	—
135	C₂²×D₇₉	—	4	2	no	—
136	C₂²×D₇₉	—	4	2	no	—
137	C₂²×D₇₉	—	8	4	no	—
138	C₂²×D₇₉	—	16	4	no	—
139	C₂²×D₇₉	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₇₉	—	4	2	no	—
141	Q₈×C₇₉	—	4	2	no	—
142	D₄×C₇₉	—	4	2	no	—
143	D₄×C₇₉	—	4	2	no	—
144	C₂×C₄×C₇₉	—	4	2	no	—
145	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₇₉	—	8	4	no	—
148	C₂²⋊D₇₉	—	16	4	no	—
149	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
150	C₂×Dic₇₉	C₂×D₈	1	1	yes	characteristic
151	D₃₁₆	—	4	2	no	—
152	D₃₁₆	—	4	2	no	—
153	D₃₁₆	—	4	2	no	—
154	D₃₁₆	—	4	2	no	—
155	D₃₁₆	—	8	4	no	—
156	D₃₁₆	—	16	4	no	—
157	C₄×D₇₉	—	8	4	no	—
158	C₄×D₇₉	#127	1	2	yes	—
159	C₄×D₇₉	C₂×D₈	1	2	yes	—
160	C₈×C₇₉	C₂×D₈	1	2	yes	—
161	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
163	C₇₉⋊C₈	#127	1	2	yes	—
164	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
165	C₂²×C₇₉	—	8	2	no	—
166	C₂²×C₇₉	—	8	4	no	—
167	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₇₉	—	2	2	no	—
169	C₂×D₇₉	—	8	2	no	—
170	C₂×D₇₉	—	8	2	no	—
171	C₂×D₇₉	—	8	4	no	—
172	C₂×D₇₉	—	8	4	no	—
173	C₂×D₇₉	—	16	8	no	—
174	C₂×D₇₉	#493	1	2	yes	—
175	C₄×C₇₉	—	2	2	no	—
176	C₄×C₇₉	—	8	2	no	—
177	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
178	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
179	Dic₇₉	—	2	2	no	—
180	Dic₇₉	#493	1	2	yes	—
181	#33720	—	79	1	no	maximal, Sylow
182	C₂×C₇₉	—	16	2	no	—
183	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
184	C₂×C₇₉	#3415	1	2	yes	—
185	D₇₉	—	2	2	no	—
186	D₇₉	—	16	2	no	—
187	D₇₉	—	16	2	no	—
188	C₂×#509	—	79	2	no	—
189	C₂×#508	—	79	1	no	—
190	C₂²×D₁₆	—	79	1	no	—
191	#3155	—	79	2	no	—
192	#2250	—	79	2	no	—
193	#2250	—	79	2	no	—
194	#2249	—	79	1	no	—
195	#2249	D₇₉	1	1	yes	characteristic, complemented
196	#2246	—	79	1	no	—
197	#2246	—	79	1	no	—
198	#2108	—	79	1	no	—
199	C₇₉	#33720	1	1	yes	characteristic, complemented, Sylow
200	C₂×AGL(1,ℤ/8)	—	158	2	no	—
201	C₂×#135	—	79	2	no	—
202	C₂²×D₈	—	79	2	no	—
203	#535	—	158	2	no	—
204	#509	—	158	8	no	—
205	#508	—	158	4	no	—
206	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic, complemented
207	C₂×SD₃₂	—	79	2	no	—
208	C₂×SD₃₂	—	79	2	no	—
209	C₂×SD₃₂	C₂×D₇₉	1	2	yes	complemented
210	C₂×D₁₆	—	79	2	no	—
211	C₂×D₁₆	—	79	2	no	—
212	C₂×D₁₆	—	79	1	no	—
213	C₂×D₁₆	—	79	1	no	—
214	C₂×D₁₆	—	79	2	no	—
215	C₂×D₁₆	—	158	4	no	—
216	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
217	C₂×M₃₂	—	79	1	no	—
218	C₂²×C₁₆	—	79	1	no	—
219	#496	—	158	2	no	—
220	#496	—	158	2	no	—
221	#495	—	79	2	no	—
222	#495	C₂×D₇₉	1	2	yes	complemented
223	#493	—	79	2	no	—
224	#493	—	79	2	no	—
225	#405	—	79	1	no	—
226	#346	—	79	1	no	—
227	#346	—	79	1	no	—
228	C₄×C₁₆	—	79	1	no	—
229	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic, complemented
230	C₂×#41	—	158	2	no	—
231	C₂²×D₄	—	158	2	no	—
232	AGL(1,ℤ/8)	—	316	8	no	—
233	#135	—	158	4	no	—
234	#135	—	158	4	no	—
235	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	158	2	no	—
238	C₂×SD₁₆	—	158	2	no	—
239	C₂×D₈	—	79	2	no	—
240	C₂×D₈	—	79	2	no	—
241	C₂×D₈	—	158	4	no	—
242	C₂×D₈	—	158	4	no	—
243	C₂×D₈	—	158	2	no	—
244	C₂×D₈	—	158	2	no	—
245	C₂×D₈	—	158	2	no	—
246	C₂×D₈	C₂²×D₇₉	1	2	yes	—
247	C₂×M₁₆	—	79	1	no	—
248	C₂²×C₈	—	79	1	no	—
249	#127	—	158	2	no	—
250	#127	—	158	2	no	—
251	#124	—	2	2	no	—
252	#124	—	158	2	no	—
253	#121	—	316	4	no	—
254	#120	—	316	4	no	—
255	#117	—	79	1	no	—
256	Q₃₂	—	2	2	no	—
257	SD₃₂	—	2	4	no	—
258	SD₃₂	—	158	4	no	—
259	SD₃₂	—	158	4	no	—
260	D₁₆	—	2	2	no	—
261	D₁₆	—	158	4	no	—
262	D₁₆	—	158	4	no	—
263	D₁₆	—	158	2	no	—
264	D₁₆	—	158	2	no	—
265	D₁₆	—	158	4	no	—
266	M₃₂	—	79	4	no	—
267	C₂×C₁₆	—	79	1	no	—
268	C₂×C₁₆	—	79	1	no	—
269	C₂×C₁₆	—	79	4	no	—
270	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
271	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
272	#97	—	79	1	no	—
273	#97	—	79	1	no	—
274	C₄×C₈	—	79	1	no	—
275	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
276	C₂⁴	—	316	2	no	—
277	#41	—	316	4	no	—
278	#41	—	316	4	no	—
279	C₂×Q₈	—	2	2	no	—
280	C₂×D₄	—	2	2	no	—
281	C₂×D₄	—	316	2	no	—
282	C₂×D₄	—	316	4	no	—
283	C₂×D₄	—	316	4	no	—
284	C₂×D₄	—	316	2	no	—
285	C₂×D₄	—	316	4	no	—
286	C₂×D₄	—	316	4	no	—
287	C₂×D₄	—	158	2	no	—
288	C₂×D₄	—	158	2	no	—
289	C₂²×C₄	—	316	2	no	—
290	C₂²×C₄	—	79	1	no	—
291	Q₁₆	—	2	2	no	—
292	Q₁₆	—	2	2	no	—
293	SD₁₆	—	4	4	no	—
294	SD₁₆	—	316	4	no	—
295	SD₁₆	—	316	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	316	4	no	—
299	D₈	—	316	4	no	—
300	D₈	—	316	4	no	—
301	D₈	—	158	2	no	—
302	D₈	—	158	2	no	—
303	D₈	—	158	2	no	—
304	D₈	—	158	2	no	—
305	M₁₆	—	79	4	no	—
306	C₂×C₈	—	79	1	no	—
307	C₂×C₈	—	79	1	no	—
308	C₂×C₈	—	79	2	no	—
309	C₂×C₈	—	79	2	no	—
310	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
311	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
312	C₄⋊C₄	—	79	2	no	—
313	C₂²⋊C₄	—	4	4	no	—
314	C₂²⋊C₄	—	316	4	no	—
315	C₂²⋊C₄	—	79	2	no	—
316	C₄²	—	79	1	no	—
317	C₄²	D₄×D₇₉	1	1	yes	characteristic
318	C₁₆	—	79	2	no	—
319	C₁₆	—	79	2	no	—
320	C₁₆	D₄×D₇₉	1	2	yes	—
321	C₁₆	D₄×D₇₉	1	2	yes	—
322	C₂³	—	4	2	no	—
323	C₂³	—	316	2	no	—
324	C₂³	—	316	2	no	—
325	C₂³	—	79	1	no	—
326	C₂³	—	632	4	no	—
327	C₂³	—	1264	4	no	—
328	Q₈	—	4	2	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	4	2	no	—
331	D₄	—	4	2	no	—
332	D₄	—	316	2	no	—
333	D₄	—	316	2	no	—
334	D₄	—	316	2	no	—
335	D₄	—	316	2	no	—
336	D₄	—	632	4	no	—
337	D₄	—	632	4	no	—
338	D₄	—	1264	4	no	—
339	D₄	—	1264	4	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	79	1	no	—
342	C₂×C₄	—	79	2	no	—
343	C₂×C₄	—	79	2	no	—
344	C₂×C₄	—	79	2	no	—
345	C₂×C₄	—	632	4	no	—
346	C₂×C₄	D₈×D₇₉	1	2	yes	—
347	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
348	C₈	—	79	2	no	—
349	C₈	—	79	2	no	—
350	C₈	D₈×D₇₉	1	2	yes	—
351	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
352	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
353	C₂²	—	8	4	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	79	2	no	—
356	C₂²	—	632	4	no	—
357	C₂²	—	632	4	no	—
358	C₂²	—	632	2	no	—
359	C₂²	—	632	2	no	—
360	C₂²	—	158	2	no	—
361	C₂²	—	1264	8	no	—
362	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	8	2	no	—
365	C₄	—	79	2	no	—
366	C₄	—	158	2	no	—
367	C₄	#127×D₇₉	1	1	yes	characteristic
368	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
369	C₂	—	16	2	no	—
370	C₂	—	158	2	no	—
371	C₂	—	1264	2	no	—
372	C₂	—	1264	2	no	—
373	C₂	#493×D₇₉	1	1	yes	characteristic
374	C₂	#23776538	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2249	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	#2249	D₇₉
hypocenter	C₇₉	#33720
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. C₂×#19099000	C₂	semidirect
2. C₂×#19098996	C₂	semidirect
3. C₂×D₁₆×D₇₉	C₂	semidirect
4. #23773882	C₂	semidirect
5. #23765667	C₂	semidirect
6. #23765665	C₂	semidirect
7. #23765662	C₂	semidirect
8. #23765650	C₂	semidirect
9. #23765648	C₂	semidirect
10. #23764995	C₂	semidirect
11. C₇₉×#2249	C₂	semidirect
18. C₂×#14579104	C₂²	semidirect
19. C₂×#14579104	C₂²	semidirect
21. C₂×D₁₂₆₄	C₂²	semidirect
22. C₂×D₁₂₆₄	C₂²	semidirect
24. C₂×#14579102	C₂²	semidirect
30. #19098948	C₂²	semidirect
31. #19098940	C₂²	semidirect
32. #19098938	C₂²	semidirect
34. #19098339	C₂²	semidirect
35. #19098338	C₂²	semidirect
36. C₁₆×Dic₇₉	C₂²	semidirect
37. C₂×Q₃₂×C₇₉	C₂²	semidirect
38. C₂×SD₃₂×C₇₉	C₂²	semidirect
40. #495×C₇₉	C₂²	semidirect
41. C₄×C₁₆×C₇₉	C₂²	semidirect
71. #14579100	D₄	semidirect
74. C₁₆×D₇₉	D₄	semidirect
195. #2249	D₇₉	semidirect
199. C₇₉	#33720	semidirect
206. C₂×Q₃₂	C₂×D₇₉	semidirect
209. C₂×SD₃₂	C₂×D₇₉	semidirect
222. #495	C₂×D₇₉	semidirect
229. C₄×C₁₆	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099000	2
C₂×#19098996	2
C₂×D₁₆×D₇₉	2
#23773882	2
#23765667	2
#23765665	2
#23765662	2
#23765650	2
#23765648	2
#23764995	2
C₇₉×#2249	2
#33720	79