Subgroups — #23768413

Order 10112.

544 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14579140	C₂	1	1	yes	maximal, characteristic
2	C₂×#14579138	C₂	1	1	yes	maximal, characteristic
3	#19098497	C₂	1	1	yes	maximal, characteristic, complemented
4	#19098496	C₂	1	1	yes	maximal, characteristic, complemented
5	#19098484	C₂	1	1	yes	maximal, characteristic, complemented
6	#19098483	C₂	1	1	yes	maximal, characteristic, complemented
7	#19098467	C₂	1	1	yes	maximal, characteristic, complemented
8	#19098467	C₂	1	1	yes	maximal, characteristic, complemented
9	#19098462	C₂	1	1	yes	maximal, characteristic, complemented
10	#19098459	C₂	1	1	yes	maximal, characteristic, complemented
11	#19098438	C₂	1	1	yes	maximal, characteristic
12	#19098421	C₂	1	1	yes	maximal, characteristic
13	C₂²⋊C₄×Dic₇₉	C₂	1	1	yes	maximal, characteristic
14	C₂×C₇₉⋊#95	C₂	1	1	yes	maximal, characteristic
15	#385×C₇₉	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₅₈	C₂²	1	1	yes	characteristic
18	C₂³×Dic₇₉	C₂²	1	1	yes	characteristic
19	C₂³×Dic₇₉	C₂²	1	1	yes	characteristic
20	#14579140	—	2	4	no	—
21	#14579138	—	2	4	no	—
22	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₇₉	C₂²	1	1	yes	characteristic
26	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×Dic₇₉⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
33	C₂×C₂²⋊Dic₇₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₇₉	C₂²	1	1	yes	characteristic
36	C₇₉⋊#95	C₂²	1	1	yes	characteristic
37	C₇₉⋊#95	C₂²	1	1	yes	characteristic
38	C₇₉⋊#95	C₂²	1	1	yes	characteristic
39	C₇₉⋊#95	C₂²	1	1	yes	characteristic
40	C₇₉⋊#95	C₂²	1	1	yes	characteristic
41	C₇₉⋊#95	C₂²	1	1	yes	characteristic
42	C₇₉⋊#95	C₂²	1	1	yes	characteristic
43	C₇₉⋊#95	C₂²	1	1	yes	characteristic
44	C₇₉⋊#95	C₂²	1	1	yes	characteristic
45	C₇₉⋊#95	C₂²	1	1	yes	characteristic
46	C₂³×C₄×C₇₉	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
51	#95×C₇₉	C₂²	1	1	yes	characteristic
52	#95×C₇₉	C₂²	1	1	yes	characteristic
53	C₂×Dic₁₅₈	—	2	2	no	—
54	C₂×Dic₁₅₈	—	2	2	no	—
55	C₂×Dic₁₅₈	—	2	2	no	—
56	C₂×Dic₁₅₈	—	2	2	no	—
57	C₂×Dic₁₅₈	—	4	1	no	—
58	C₂×Dic₁₅₈	—	4	1	no	—
59	C₂²×Dic₇₉	—	2	2	no	—
60	C₂²×Dic₇₉	—	2	2	no	—
61	C₂²×Dic₇₉	—	4	1	no	—
62	C₂²×Dic₇₉	—	4	1	no	—
63	C₂²×Dic₇₉	—	4	1	no	—
64	C₂²×Dic₇₉	Q₈	1	4	yes	complemented
65	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
68	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
69	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
70	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
71	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
72	C₂²×Dic₇₉	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₇₉	—	2	2	no	—
74	C₄⋊Dic₇₉	—	2	2	no	—
75	C₄⋊Dic₇₉	—	2	2	no	—
76	C₄⋊Dic₇₉	—	4	1	no	—
77	Dic₇₉⋊C₄	—	2	2	no	—
78	Dic₇₉⋊C₄	—	2	2	no	—
79	Dic₇₉⋊C₄	—	2	2	no	—
80	Dic₇₉⋊C₄	—	2	2	no	—
81	Dic₇₉⋊C₄	—	4	1	no	—
82	Dic₇₉⋊C₄	—	4	1	no	—
83	C₂²⋊Dic₇₉	—	2	2	no	—
84	C₂²⋊Dic₇₉	D₄	1	4	yes	—
85	C₄×Dic₇₉	—	4	1	no	—
86	C₄×Dic₇₉	—	4	1	no	—
87	C₂⁴×C₇₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	—	2	2	no	—
89	C₂²×C₄×C₇₉	—	2	2	no	—
90	C₂²×C₄×C₇₉	—	4	1	no	—
91	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₇₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₇₉	—	2	2	no	—
98	C₄⋊C₄×C₇₉	—	4	1	no	—
99	C₂²⋊C₄×C₇₉	—	2	2	no	—
100	C₂²⋊C₄×C₇₉	Q₈	1	4	yes	complemented
101	C₂³×C₇₉	—	2	1	no	—
102	C₂³×C₇₉	—	2	1	no	—
103	C₂³×C₇₉	—	2	1	no	—
104	C₂³×C₇₉	—	2	1	no	—
105	C₂³×C₇₉	C₂×D₄	1	2	yes	—
106	C₂³×C₇₉	C₂×Q₈	1	2	yes	—
107	C₂³×C₇₉	C₂×Q₈	1	2	yes	—
108	C₂³×C₇₉	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	4	4	no	—
125	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
126	C₂×C₄×C₇₉	C₂×Q₈	1	2	yes	—
127	C₂×C₄×C₇₉	#41	1	2	yes	—
128	C₂×Dic₇₉	—	2	1	no	—
129	C₂×Dic₇₉	—	2	1	no	—
130	C₂×Dic₇₉	—	2	1	no	—
131	C₂×Dic₇₉	—	2	1	no	—
132	C₂×Dic₇₉	—	2	1	no	—
133	C₂×Dic₇₉	—	2	1	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	—	2	1	no	—
139	C₂×Dic₇₉	—	2	1	no	—
140	C₂×Dic₇₉	—	2	1	no	—
141	C₂×Dic₇₉	—	2	1	no	—
142	C₂×Dic₇₉	—	2	1	no	—
143	C₂×Dic₇₉	—	2	1	no	—
144	C₂×Dic₇₉	—	2	1	no	—
145	C₂×Dic₇₉	—	2	1	no	—
146	C₂×Dic₇₉	—	2	1	no	—
147	C₂×Dic₇₉	—	4	4	no	—
148	C₂×Dic₇₉	—	4	4	no	—
149	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
150	C₂×Dic₇₉	C₂×D₄	1	2	yes	—
151	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
152	C₂×Dic₇₉	C₂×Q₈	1	2	yes	—
153	C₂×Dic₇₉	#41	1	2	yes	—
154	Dic₁₅₈	—	4	4	no	—
155	Dic₁₅₈	—	4	4	no	—
156	C₂²×C₇₉	—	2	2	no	—
157	C₂²×C₇₉	—	2	2	no	—
158	C₂²×C₇₉	—	2	2	no	—
159	C₂²×C₇₉	—	2	2	no	—
160	C₂²×C₇₉	—	2	2	no	—
161	C₂²×C₇₉	—	2	2	no	—
162	C₂²×C₇₉	C₄⋊Q₈	1	4	yes	—
163	C₂²×C₇₉	C₂²×D₄	1	1	yes	characteristic
164	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
165	C₂²×C₇₉	C₂²×Q₈	1	1	yes	characteristic
166	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
167	C₂²×C₇₉	C₂×#41	1	1	yes	characteristic
168	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂²×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
170	C₄×C₇₉	—	2	2	no	—
171	C₄×C₇₉	—	2	2	no	—
172	C₄×C₇₉	—	2	2	no	—
173	C₄×C₇₉	—	4	1	no	—
174	C₄×C₇₉	—	4	1	no	—
175	C₄×C₇₉	—	4	1	no	—
176	Dic₇₉	—	2	2	no	—
177	Dic₇₉	—	2	2	no	—
178	Dic₇₉	—	2	2	no	—
179	Dic₇₉	—	2	2	no	—
180	Dic₇₉	—	2	2	no	—
181	Dic₇₉	—	4	1	no	—
182	Dic₇₉	—	4	1	no	—
183	Dic₇₉	—	4	1	no	—
184	C₂×C₇₉	—	2	4	no	—
185	C₂×C₇₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
186	C₂×C₇₉	#533	1	1	yes	characteristic
187	C₂×C₇₉	#543	1	1	yes	characteristic
188	C₂×C₇₉	#546	1	1	yes	characteristic
189	C₂×C₇₉	#546	1	1	yes	characteristic
190	C₂×C₇₉	#554	1	1	yes	characteristic
191	C₂×C₇₉	#554	1	1	yes	characteristic
192	#2549	—	79	1	no	maximal, Sylow
193	C₇₉	#2549	1	1	yes	characteristic, complemented, Sylow
194	C₂×C₂²⋊Q₈	—	79	1	no	—
195	C₂×C₂²⋊Q₈	—	79	1	no	—
196	#398	—	79	1	no	—
197	#398	—	79	1	no	—
198	#395	—	79	1	no	—
199	#395	—	79	1	no	—
200	#391	—	79	1	no	—
201	#391	—	79	1	no	—
202	#389	—	79	1	no	—
203	#389	—	79	1	no	—
204	#385	—	79	1	no	—
205	#385	D₇₉	1	1	yes	characteristic
206	#380	—	79	1	no	—
207	C₄×C₂²⋊C₄	—	79	1	no	—
208	C₂×#95	—	79	1	no	—
209	C₂²×Q₈	—	79	1	no	—
210	C₂²×Q₈	—	79	1	no	—
211	C₂³×C₄	—	79	1	no	—
212	C₂³×C₄	—	79	1	no	—
213	C₂³×C₄	C₂×D₇₉	1	1	yes	characteristic
214	C₂²⋊Q₈	—	158	4	no	—
215	C₂²⋊Q₈	—	158	4	no	—
216	C₂×C₄⋊C₄	—	79	1	no	—
217	C₂×C₄⋊C₄	—	79	1	no	—
218	C₂×C₄⋊C₄	—	79	1	no	—
219	C₂×C₄⋊C₄	—	79	1	no	—
220	C₂×C₄⋊C₄	—	79	1	no	—
221	C₂×C₄⋊C₄	—	79	1	no	—
222	C₂×C₄⋊C₄	—	79	1	no	—
223	C₂×C₄⋊C₄	—	79	1	no	—
224	C₂×C₄⋊C₄	—	79	1	no	—
225	C₂×C₄⋊C₄	—	79	1	no	—
226	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	79	1	no	—
229	C₂×C₂²⋊C₄	—	79	1	no	—
230	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	C₂×D₇₉	1	1	yes	characteristic
232	C₂×C₄²	—	79	1	no	—
233	C₂×C₄²	—	79	1	no	—
234	#95	—	79	1	no	—
235	#95	—	79	1	no	—
236	#95	—	79	1	no	—
237	#95	—	79	1	no	—
238	#95	—	79	1	no	—
239	#95	—	79	1	no	—
240	#95	—	79	1	no	—
241	#95	—	79	1	no	—
242	#95	—	79	1	no	—
243	#95	—	79	1	no	—
244	#95	C₂×D₇₉	1	1	yes	characteristic
245	#95	C₂×D₇₉	1	1	yes	characteristic
246	C₂⁴	C₂²×D₇₉	1	1	yes	characteristic
247	C₂×Q₈	—	316	1	no	—
248	C₂×Q₈	—	316	1	no	—
249	C₂×Q₈	—	158	2	no	—
250	C₂×Q₈	—	158	2	no	—
251	C₂×Q₈	—	158	2	no	—
252	C₂×Q₈	—	158	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	316	1	no	—
257	C₂²×C₄	—	316	1	no	—
258	C₂²×C₄	—	316	1	no	—
259	C₂²×C₄	—	79	4	no	—
260	C₂²×C₄	—	79	1	no	—
261	C₂²×C₄	—	79	1	no	—
262	C₂²×C₄	—	79	1	no	—
263	C₂²×C₄	—	79	1	no	—
264	C₂²×C₄	—	79	1	no	—
265	C₂²×C₄	—	79	1	no	—
266	C₂²×C₄	—	79	1	no	—
267	C₂²×C₄	—	79	1	no	—
268	C₂²×C₄	—	158	2	no	—
269	C₂²×C₄	—	158	2	no	—
270	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
271	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
272	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
274	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
275	C₂²×C₄	C₂²×D₇₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	316	1	no	—
279	C₄⋊C₄	—	316	1	no	—
280	C₄⋊C₄	—	316	1	no	—
281	C₄⋊C₄	—	158	2	no	—
282	C₄⋊C₄	—	158	2	no	—
283	C₄⋊C₄	—	158	2	no	—
284	C₄⋊C₄	—	158	2	no	—
285	C₄⋊C₄	—	158	2	no	—
286	C₄⋊C₄	—	158	2	no	—
287	C₄⋊C₄	—	158	2	no	—
288	C₂²⋊C₄	—	2	2	no	—
289	C₂²⋊C₄	—	79	4	no	—
290	C₂²⋊C₄	—	158	2	no	—
291	C₂²⋊C₄	Dic₁₅₈	1	4	yes	complemented
292	C₄²	—	316	1	no	—
293	C₄²	—	316	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	D₄×D₇₉	1	2	yes	—
299	C₂³	Q₈×D₇₉	1	2	yes	—
300	C₂³	C₂×Dic₁₅₈	1	2	yes	—
301	C₂³	C₂³×D₇₉	1	1	yes	characteristic
302	Q₈	—	316	4	no	—
303	Q₈	—	316	4	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	316	4	no	—
321	C₂×C₄	—	316	4	no	—
322	C₂×C₄	—	79	2	no	—
323	C₂×C₄	—	79	2	no	—
324	C₂×C₄	—	79	2	no	—
325	C₂×C₄	—	79	2	no	—
326	C₂×C₄	—	79	2	no	—
327	C₂×C₄	—	158	1	no	—
328	C₂×C₄	—	158	1	no	—
329	C₂×C₄	—	158	1	no	—
330	C₂×C₄	—	158	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	—	158	1	no	—
338	C₂×C₄	—	158	1	no	—
339	C₂×C₄	—	158	1	no	—
340	C₂×C₄	—	158	1	no	—
341	C₂×C₄	—	158	1	no	—
342	C₂×C₄	—	158	1	no	—
343	C₂×C₄	—	158	1	no	—
344	C₂×C₄	—	158	1	no	—
345	C₂×C₄	—	158	1	no	—
346	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
347	C₂×C₄	C₂×Dic₁₅₈	1	2	yes	—
348	C₂×C₄	#11929623	1	2	yes	—
349	C₂²	—	2	2	no	—
350	C₂²	—	2	2	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	#14579160	1	4	yes	—
356	C₂²	C₂×D₄×D₇₉	1	1	yes	characteristic
357	C₂²	C₂×Q₈×D₇₉	1	1	yes	characteristic
358	C₂²	C₂²×Dic₁₅₈	1	1	yes	characteristic
359	C₂²	#41×D₇₉	1	1	yes	characteristic
360	C₂²	C₂×#11929623	1	1	yes	characteristic
361	C₂²	#14579206	1	1	yes	characteristic
362	C₂²	#14579207	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	316	1	no	—
370	C₄	—	316	1	no	—
371	C₄	—	316	1	no	—
372	C₄	—	158	2	no	—
373	C₄	—	158	2	no	—
374	C₄	—	158	2	no	—
375	C₄	—	158	2	no	—
376	C₄	—	158	2	no	—
377	C₂	—	2	4	no	—
378	C₂	C₂×#14579160	1	1	yes	characteristic
379	C₂	#19099115	1	1	yes	characteristic
380	C₂	#19099188	1	1	yes	characteristic
381	C₂	#19099200	1	1	yes	characteristic
382	C₂	#19099205	1	1	yes	characteristic
383	C₂	#19099270	1	1	yes	characteristic
384	C₂	#19099272	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₉
commutator, socle, cosocle	C₂³×C₇₉	C₂⁴
Fitting	#385×C₇₉	C₂
prime radical	C₂⁴×C₇₉	C₂³
hypercenter	#385	D₇₉
hypocenter	C₇₉	#2549
LCS	G ▷ C₂³×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ #385

Complements

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

N	H ≅ G/N	type
3. #19098497	C₂	semidirect
4. #19098496	C₂	semidirect
5. #19098484	C₂	semidirect
6. #19098483	C₂	semidirect
7. #19098467	C₂	semidirect
8. #19098467	C₂	semidirect
9. #19098462	C₂	semidirect
10. #19098459	C₂	semidirect
64. C₂²×Dic₇₉	Q₈	semidirect
100. C₂²⋊C₄×C₇₉	Q₈	semidirect
193. C₇₉	#2549	semidirect
291. C₂²⋊C₄	Dic₁₅₈	semidirect