Subgroups — #23775698

Order 10112.

420 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	#19099213	C₂	1	1	yes	maximal, characteristic, complemented
2	#19099212	C₂	1	1	yes	maximal, characteristic, complemented
3	#19098912	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098899	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098897	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098841	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098807	C₂	1	1	yes	maximal, characteristic, complemented
8	#19098806	C₂	1	1	yes	maximal, characteristic, complemented
9	#19098766	C₂	1	1	yes	maximal, characteristic, complemented
10	#19098705	C₂	1	1	yes	maximal, characteristic, complemented
11	D₈×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19098638	C₂	1	1	yes	maximal, characteristic, complemented
13	#19098564	C₂	1	1	yes	maximal, characteristic, complemented
14	#19098561	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11929623	—	2	1	no	—
17	C₂×#11929623	C₂²	1	1	yes	characteristic
18	C₂×#11929612	C₂²	1	1	yes	characteristic
19	C₂×#11929608	C₂²	1	1	yes	characteristic
20	C₂×#11929607	C₂²	1	1	yes	characteristic
21	#14579160	C₂²	1	1	yes	characteristic
22	#14579159	C₂²	1	1	yes	characteristic
23	#14579144	—	2	1	no	—
24	#14579143	—	2	1	no	—
25	#14579142	—	2	1	no	—
26	#14579142	—	2	1	no	—
27	#14579140	C₂²	1	1	yes	characteristic
28	#14579138	—	2	1	no	—
29	#14579137	C₂²	1	1	yes	characteristic
30	#14579133	—	2	1	no	—
31	#14579132	C₂²	1	1	yes	characteristic
32	D₄×Dic₇₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₇₉	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₇₉	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
36	C₂×Dic₇₉⋊C₄	—	2	1	no	—
37	C₄⋊C₄×D₇₉	C₂²	1	1	yes	characteristic
38	#14579095	C₂²	1	1	yes	characteristic
39	#14579095	C₂²	1	1	yes	characteristic
40	#14579094	C₂²	1	1	yes	characteristic
41	Dic₇₉⋊C₈	C₂²	1	1	yes	characteristic
42	#14579086	C₂²	1	1	yes	characteristic
43	#14579086	C₂²	1	1	yes	characteristic
44	#14579085	C₂²	1	1	yes	characteristic
45	#14579085	C₂²	1	1	yes	characteristic
46	#14579084	C₂²	1	1	yes	characteristic
47	#14579082	C₂²	1	1	yes	characteristic
48	#14579082	C₂²	1	1	yes	characteristic
49	#14579082	C₂²	1	1	yes	characteristic
50	(C₂×D₇₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₇₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₇₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₇₉	C₂²	1	1	yes	characteristic
55	#121×C₇₉	C₂²	1	1	yes	characteristic
56	#103×C₇₉	C₂²	1	1	yes	characteristic
57	#102×C₇₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
59	#11929623	—	2	2	no	—
60	#11929623	—	4	4	no	—
61	#11929623	—	4	1	no	—
62	C₂×Dic₁₅₈	—	2	1	no	—
63	C₂×Dic₁₅₈	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₉	—	2	1	no	—
65	C₂×C₂²⋊D₇₉	—	2	1	no	—
66	C₂×C₂²⋊D₇₉	D₄	1	1	yes	characteristic
67	C₂²×Dic₇₉	—	2	1	no	—
68	C₂²×Dic₇₉	—	2	1	no	—
69	C₂²×Dic₇₉	—	2	1	no	—
70	C₂²×Dic₇₉	—	2	1	no	—
71	C₂×C₄×D₇₉	—	2	1	no	—
72	C₂×C₄×D₇₉	C₂³	1	1	yes	characteristic
73	#11929612	—	4	1	no	—
74	#11929608	—	4	1	no	—
75	#11929607	—	4	1	no	—
76	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₇₉	—	2	1	no	—
79	C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₇₉	C₂³	1	1	yes	characteristic
82	Dic₇₉⋊C₄	—	2	1	no	—
83	Dic₇₉⋊C₄	—	2	1	no	—
84	Dic₇₉⋊C₄	—	2	1	no	—
85	Dic₇₉⋊C₄	—	2	1	no	—
86	Dic₇₉⋊C₄	—	2	1	no	—
87	Dic₇₉⋊C₄	—	2	1	no	—
88	Dic₇₉⋊C₄	D₄	1	1	yes	characteristic
89	C₂²⋊Dic₇₉	—	2	1	no	—
90	C₂²⋊Dic₇₉	—	2	1	no	—
91	C₂²⋊Dic₇₉	—	2	1	no	—
92	C₂²⋊Dic₇₉	—	2	1	no	—
93	C₂²⋊Dic₇₉	D₄	1	1	yes	characteristic
94	(C₂×D₇₉)⋊C₄	—	2	1	no	—
95	(C₂×D₇₉)⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₇₉	C₂³	1	1	yes	characteristic
97	#41×C₇₉	—	2	2	no	—
98	#41×C₇₉	—	4	1	no	—
99	C₂×Q₈×C₇₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₇₉	—	2	1	no	—
101	C₂×D₄×C₇₉	—	2	1	no	—
102	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₇₉	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	—	2	1	no	—
105	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
106	SD₁₆×C₇₉	—	4	1	no	—
107	D₈×C₇₉	—	2	2	no	—
108	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
109	C₂×C₈×C₇₉	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₇₉	C₂³	1	1	yes	characteristic
111	C₂²⋊C₄×C₇₉	—	2	1	no	—
112	C₂³×C₇₉	—	2	1	no	—
113	C₂³×C₇₉	—	2	1	no	—
114	C₂³×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₇₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₇₉	—	2	1	no	—
117	Q₈×C₇₉	—	2	1	no	—
118	D₄×C₇₉	—	2	1	no	—
119	D₄×C₇₉	—	2	1	no	—
120	D₄×C₇₉	—	2	1	no	—
121	D₄×C₇₉	—	4	2	no	—
122	D₄×C₇₉	—	4	2	no	—
123	D₄×C₇₉	#41	1	2	yes	complemented
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	—	2	1	no	—
128	C₂×C₄×C₇₉	—	4	2	no	—
129	C₂×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
131	C₂²⋊D₇₉	—	2	2	no	—
132	C₂²⋊D₇₉	—	4	2	no	—
133	C₂²⋊D₇₉	—	4	2	no	—
134	C₂×Dic₇₉	—	2	1	no	—
135	C₂×Dic₇₉	—	2	1	no	—
136	C₂×Dic₇₉	—	2	1	no	—
137	C₂×Dic₇₉	—	2	1	no	—
138	C₂×Dic₇₉	—	4	2	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	C₂×Dic₇₉	—	4	2	no	—
142	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
144	C₂×Dic₇₉	C₂×D₄	1	1	yes	characteristic
145	C₄×D₇₉	—	2	1	no	—
146	C₄×D₇₉	—	2	1	no	—
147	C₄×D₇₉	—	4	2	no	—
148	Dic₁₅₈	—	2	1	no	—
149	Dic₁₅₈	—	2	1	no	—
150	Dic₁₅₈	—	4	2	no	—
151	C₈×C₇₉	—	2	1	no	—
152	C₈×C₇₉	—	2	1	no	—
153	C₇₉⋊C₈	—	2	1	no	—
154	C₇₉⋊C₈	—	2	1	no	—
155	C₂²×C₇₉	—	2	2	no	—
156	C₂²×C₇₉	—	2	1	no	—
157	C₂²×C₇₉	—	2	1	no	—
158	C₂²×C₇₉	—	2	1	no	—
159	C₂²×C₇₉	—	4	1	no	—
160	C₂²×C₇₉	—	4	1	no	—
161	C₂²×C₇₉	—	4	1	no	—
162	C₂²×C₇₉	—	4	1	no	—
163	C₂²×C₇₉	—	4	1	no	—
164	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×D₇₉	—	2	1	no	—
166	C₂×D₇₉	—	2	1	no	—
167	C₂×D₇₉	—	2	1	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	4	1	no	—
170	C₄×C₇₉	—	4	1	no	—
171	C₄×C₇₉	C₂×#41	1	1	yes	characteristic
172	C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₇₉	—	2	1	no	—
174	Dic₇₉	—	2	1	no	—
175	Dic₇₉	—	4	1	no	—
176	Dic₇₉	—	4	1	no	—
177	Dic₇₉	—	4	1	no	—
178	Dic₇₉	—	4	1	no	—
179	Dic₇₉	AGL(1,ℤ/8)	1	2	yes	—
180	C₂×C₇₉	—	4	1	no	—
181	C₂×C₇₉	—	4	2	no	—
182	C₂×C₇₉	—	8	1	no	—
183	C₂×C₇₉	#546	1	1	yes	characteristic
184	C₂×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₂×C₇₉	#578	1	1	yes	characteristic
186	D₇₉	—	4	1	no	—
187	#3330	—	79	1	no	maximal, Sylow
188	C₇₉	#3330	1	1	yes	characteristic, complemented, Sylow
189	#547	—	79	1	no	—
190	#547	—	79	1	no	—
191	#487	—	79	1	no	—
192	#484	—	79	1	no	—
193	#484	—	79	1	no	—
194	#474	—	79	1	no	—
195	#468	—	79	1	no	—
196	#468	—	79	1	no	—
197	#461	—	79	1	no	—
198	#449	—	79	1	no	—
199	#449	D₇₉	1	1	yes	characteristic, complemented
200	C₄×D₈	—	79	1	no	—
201	#436	—	79	1	no	—
202	#417	—	79	1	no	—
203	#417	—	79	1	no	—
204	C₂×#41	—	79	1	no	—
205	C₂×#41	—	158	1	no	—
206	C₂×#41	C₂×D₇₉	1	1	yes	characteristic
207	C₂×SD₁₆	—	79	1	no	—
208	C₂×SD₁₆	C₂×D₇₉	1	1	yes	characteristic
209	C₂×D₈	C₂×D₇₉	1	1	yes	characteristic
210	C₂×M₁₆	—	79	1	no	—
211	C₂×M₁₆	—	79	1	no	—
212	C₄⋊Q₈	—	79	1	no	—
213	C₄⋊Q₈	—	79	1	no	—
214	#123	—	158	1	no	—
215	#123	—	158	1	no	—
216	#123	—	158	1	no	—
217	#123	—	158	1	no	—
218	C₂²⋊Q₈	—	79	1	no	—
219	C₂²⋊Q₈	—	79	1	no	—
220	C₂²⋊Q₈	—	158	1	no	—
221	#121	—	79	1	no	—
222	#121	—	158	1	no	—
223	#121	C₂×D₇₉	1	1	yes	characteristic
224	C₄×D₄	—	79	1	no	—
225	C₄×D₄	—	79	1	no	—
226	C₄×D₄	—	79	1	no	—
227	C₂×C₄⋊C₄	—	79	1	no	—
228	C₂×C₄⋊C₄	—	79	1	no	—
229	C₂×C₄⋊C₄	—	158	1	no	—
230	#107	—	79	1	no	—
231	#107	—	79	1	no	—
232	#107	—	79	1	no	—
233	C₄⋊C₈	—	79	1	no	—
234	#103	—	79	1	no	—
235	#103	—	79	1	no	—
236	#103	—	79	1	no	—
237	#103	—	79	1	no	—
238	#103	—	79	1	no	—
239	#103	C₂×D₇₉	1	1	yes	characteristic
240	#102	—	79	1	no	—
241	#102	—	79	1	no	—
242	#102	—	79	1	no	—
243	#102	C₂×D₇₉	1	1	yes	characteristic
244	C₂²⋊C₈	—	79	1	no	—
245	C₂²⋊C₈	C₂×D₇₉	1	1	yes	characteristic
246	C₄×C₈	—	79	1	no	—
247	#41	—	2	2	no	—
248	#41	—	4	1	no	—
249	#41	—	316	4	no	—
250	#41	—	316	1	no	—
251	#41	—	158	2	no	—
252	C₂×Q₈	—	79	1	no	—
253	C₂×Q₈	—	158	1	no	—
254	C₂×Q₈	C₂²×D₇₉	1	1	yes	characteristic
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	79	1	no	—
258	C₂×D₄	—	158	1	no	—
259	C₂×D₄	—	158	1	no	—
260	C₂×D₄	C₂²×D₇₉	1	1	yes	characteristic
261	C₂×D₄	C₂²×D₇₉	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	79	1	no	—
264	C₂²×C₄	—	158	1	no	—
265	C₂²×C₄	—	158	1	no	—
266	C₂²×C₄	—	158	1	no	—
267	C₂²×C₄	—	158	1	no	—
268	C₂²×C₄	—	158	1	no	—
269	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
270	SD₁₆	—	4	1	no	—
271	SD₁₆	—	316	1	no	—
272	D₈	—	2	2	no	—
273	M₁₆	—	316	1	no	—
274	M₁₆	—	316	1	no	—
275	C₂×C₈	—	79	1	no	—
276	C₂×C₈	—	79	1	no	—
277	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
278	C₂×C₈	C₂²×D₇₉	1	1	yes	characteristic
279	C₄⋊C₄	—	79	1	no	—
280	C₄⋊C₄	—	79	1	no	—
281	C₄⋊C₄	—	79	1	no	—
282	C₄⋊C₄	—	79	1	no	—
283	C₄⋊C₄	—	158	1	no	—
284	C₄⋊C₄	—	158	1	no	—
285	C₄⋊C₄	—	158	1	no	—
286	C₄⋊C₄	—	158	1	no	—
287	C₄⋊C₄	—	158	1	no	—
288	C₄⋊C₄	—	158	1	no	—
289	C₄⋊C₄	—	158	1	no	—
290	C₄⋊C₄	C₂²×D₇₉	1	1	yes	characteristic
291	C₂²⋊C₄	—	2	1	no	—
292	C₂²⋊C₄	—	79	1	no	—
293	C₂²⋊C₄	—	79	1	no	—
294	C₂²⋊C₄	—	158	1	no	—
295	C₂²⋊C₄	—	158	1	no	—
296	C₂²⋊C₄	—	158	1	no	—
297	C₂²⋊C₄	—	158	1	no	—
298	C₂²⋊C₄	—	158	1	no	—
299	C₄²	—	79	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	—	2	1	no	—
302	C₂³	—	79	1	no	—
303	C₂³	D₄×D₇₉	1	1	yes	characteristic
304	Q₈	—	2	1	no	—
305	Q₈	—	2	1	no	—
306	Q₈	—	316	2	no	—
307	Q₈	—	158	1	no	—
308	Q₈	—	158	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	2	1	no	—
312	D₄	—	4	2	no	—
313	D₄	—	4	2	no	—
314	D₄	—	316	2	no	—
315	D₄	—	316	2	no	—
316	D₄	—	158	2	no	—
317	D₄	#11929623	1	2	yes	complemented
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	—	4	2	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	—	316	2	no	—
328	C₂×C₄	—	79	1	no	—
329	C₂×C₄	—	79	1	no	—
330	C₂×C₄	—	79	1	no	—
331	C₂×C₄	—	158	1	no	—
332	C₂×C₄	—	158	1	no	—
333	C₂×C₄	—	158	1	no	—
334	C₂×C₄	—	158	1	no	—
335	C₂×C₄	—	158	1	no	—
336	C₂×C₄	—	158	1	no	—
337	C₂×C₄	D₄×D₇₉	1	1	yes	characteristic
338	C₂×C₄	C₂³×D₇₉	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	158	1	no	—
342	C₈	—	158	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	—	158	1	no	—
353	C₂²	—	158	1	no	—
354	C₂²	—	158	1	no	—
355	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	316	1	no	—
360	C₄	—	316	1	no	—
361	C₄	—	316	1	no	—
362	C₄	—	316	1	no	—
363	C₄	—	79	2	no	—
364	C₄	—	158	1	no	—
365	C₄	—	158	1	no	—
366	C₄	C₂×#11929623	1	1	yes	characteristic
367	C₄	#14579206	1	1	yes	characteristic
368	C₂	—	4	1	no	—
369	C₂	—	4	2	no	—
370	C₂	—	8	1	no	—
371	C₂	—	316	1	no	—
372	C₂	#19099200	1	1	yes	characteristic
373	C₂	AGL(1,ℤ/8)×D₇₉	1	1	yes	characteristic
374	C₂	#19099377	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₉
commutator, cosocle	C₂×C₄×C₇₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₉
Fitting	#449×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₄
hypercenter	#449	D₇₉
hypocenter	C₇₉	#3330
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #19099213	C₂	semidirect
2. #19099212	C₂	semidirect
3. #19098912	C₂	semidirect
4. #19098899	C₂	semidirect
5. #19098897	C₂	semidirect
6. #19098841	C₂	semidirect
7. #19098807	C₂	semidirect
8. #19098806	C₂	semidirect
9. #19098766	C₂	semidirect
10. #19098705	C₂	semidirect
11. D₈×Dic₇₉	C₂	semidirect
12. #19098638	C₂	semidirect
13. #19098564	C₂	semidirect
14. #19098561	C₂	semidirect
15. #449×C₇₉	C₂	semidirect
123. D₄×C₇₉	#41	semidirect
188. C₇₉	#3330	semidirect
199. #449	D₇₉	semidirect
317. D₄	#11929623	semidirect