Subgroups — #34190869

Order 20224.

562 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098915	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098850	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098844	C₂	1	1	yes	maximal, characteristic
4	#23764193	C₂	1	1	yes	maximal, characteristic
5	C₂×#19098310	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761234	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#333×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14579161	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
10	#19098915	—	2	2	no	—
11	#19098915	—	2	2	no	—
12	#19098850	—	2	2	no	—
13	#19098850	—	2	2	no	—
14	#19098844	—	2	2	no	—
15	#19098844	—	2	2	no	—
16	C₂×#14579095	—	2	1	no	—
17	C₂×#14579092	—	2	1	no	—
18	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×#14579084	—	2	2	no	—
20	C₂×#14579083	—	2	1	no	—
21	C₂×#14579081	—	2	1	no	—
22	C₂×(C₂×D₇₉)⋊C₈	—	2	2	no	—
23	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
24	#19098310	—	2	2	no	—
25	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
28	#333×C₇₉	—	2	2	no	—
29	C₂²×Dic₁₅₈	—	2	1	no	—
30	C₂²×D₃₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₇₉	—	2	1	no	—
32	C₂²×C₇₉⋊C₈	—	2	1	no	—
33	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
34	#14579161	—	2	1	no	—
35	#14579161	—	2	1	no	—
36	#14579161	—	2	1	no	—
37	#14579161	D₄	1	2	yes	—
38	C₄×D₃₁₆	—	2	1	no	—
39	C₄×D₃₁₆	—	2	1	no	—
40	C₄×D₃₁₆	—	2	1	no	—
41	C₄×D₃₁₆	—	2	1	no	—
42	C₂×C₄⋊Dic₇₉	—	2	1	no	—
43	C₂×C₄⋊Dic₇₉	—	2	1	no	—
44	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
45	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
46	#14579095	—	2	4	no	—
47	#14579092	—	2	4	no	—
48	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
49	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
50	#14579084	—	2	8	no	—
51	#14579083	—	2	4	no	—
52	#14579081	—	2	4	no	—
53	(C₂×D₇₉)⋊C₈	—	4	4	no	—
54	C₄×C₇₉⋊C₈	—	2	1	no	—
55	C₄×C₇₉⋊C₈	—	2	1	no	—
56	C₂²×Q₈×C₇₉	—	2	1	no	—
57	C₂²×C₈×C₇₉	—	2	1	no	—
58	C₂²×C₈×C₇₉	—	2	1	no	—
59	C₄⋊Q₈×C₇₉	—	2	1	no	—
60	C₄⋊Q₈×C₇₉	—	2	1	no	—
61	C₄⋊Q₈×C₇₉	—	2	1	no	—
62	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
65	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₇₉	—	2	1	no	—
67	C₄⋊C₈×C₇₉	—	2	1	no	—
68	C₄⋊C₈×C₇₉	—	2	1	no	—
69	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
70	C₂³×D₇₉	—	2	1	no	—
71	C₂×Dic₁₅₈	—	2	2	no	—
72	C₂×Dic₁₅₈	—	4	1	no	—
73	C₂×Dic₁₅₈	—	4	2	no	—
74	C₂×Dic₁₅₈	—	4	2	no	—
75	C₂×D₃₁₆	—	2	1	no	—
76	C₂×D₃₁₆	—	2	1	no	—
77	C₂×D₃₁₆	—	2	1	no	—
78	C₂×D₃₁₆	—	2	1	no	—
79	C₂×D₃₁₆	—	2	1	no	—
80	C₂×D₃₁₆	Q₁₆	1	2	yes	—
81	C₂²×Dic₇₉	—	2	1	no	—
82	C₂²×Dic₇₉	—	4	1	no	—
83	C₂×C₄×D₇₉	—	4	1	no	—
84	C₂×C₄×D₇₉	—	4	1	no	—
85	C₂×C₄×D₇₉	—	4	2	no	—
86	C₂×C₄×D₇₉	—	4	2	no	—
87	C₂×C₇₉⋊C₈	—	2	2	no	—
88	C₂×C₇₉⋊C₈	—	2	2	no	—
89	C₂×C₇₉⋊C₈	—	2	2	no	—
90	C₂×C₇₉⋊C₈	—	2	2	no	—
91	C₂×C₇₉⋊C₈	—	4	1	no	—
92	C₂×C₇₉⋊C₈	—	4	1	no	—
93	C₄⋊Dic₇₉	—	2	2	no	—
94	C₄⋊Dic₇₉	—	2	1	no	—
95	C₄⋊Dic₇₉	—	2	1	no	—
96	C₄⋊Dic₇₉	—	2	2	no	—
97	C₄⋊Dic₇₉	—	4	1	no	—
98	C₄⋊Dic₇₉	—	4	1	no	—
99	(C₂×D₇₉)⋊C₄	—	4	1	no	—
100	(C₂×D₇₉)⋊C₄	—	4	1	no	—
101	C₂×Q₈×C₇₉	—	2	2	no	—
102	C₂×Q₈×C₇₉	—	4	1	no	—
103	C₂×Q₈×C₇₉	—	8	2	no	—
104	C₂²×C₄×C₇₉	—	4	1	no	—
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
108	C₂×C₈×C₇₉	—	2	2	no	—
109	C₂×C₈×C₇₉	—	4	1	no	—
110	C₂×C₈×C₇₉	—	4	1	no	—
111	C₂×C₈×C₇₉	—	4	2	no	—
112	C₂×C₈×C₇₉	—	4	1	no	—
113	C₄⋊C₄×C₇₉	—	2	2	no	—
114	C₄⋊C₄×C₇₉	—	2	2	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₇₉	#121	1	1	yes	characteristic
122	C₂²×D₇₉	—	2	2	no	—
123	C₂²×D₇₉	—	4	1	no	—
124	C₂²×D₇₉	—	4	2	no	—
125	C₂²×D₇₉	—	4	1	no	—
126	C₂²×D₇₉	—	4	2	no	—
127	Q₈×C₇₉	—	8	2	no	—
128	Q₈×C₇₉	—	8	2	no	—
129	C₂×C₄×C₇₉	—	2	2	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	8	2	no	—
132	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₇₉	#120	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	#127	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
139	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
140	C₂×C₄×C₇₉	#135	1	2	yes	—
141	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
142	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
143	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
144	C₂×Dic₇₉	—	4	1	no	—
145	C₂×Dic₇₉	—	4	2	no	—
146	C₂×Dic₇₉	—	4	2	no	—
147	C₂×Dic₇₉	—	8	2	no	—
148	D₃₁₆	—	2	2	no	—
149	D₃₁₆	—	2	2	no	—
150	D₃₁₆	—	4	1	no	—
151	D₃₁₆	—	4	1	no	—
152	C₄×D₇₉	—	8	2	no	—
153	C₄×D₇₉	—	8	1	no	—
154	C₄×D₇₉	—	8	1	no	—
155	Dic₁₅₈	—	4	4	no	—
156	Dic₁₅₈	—	8	1	no	—
157	Dic₁₅₈	—	8	1	no	—
158	C₈×C₇₉	—	4	2	no	—
159	C₈×C₇₉	—	8	1	no	—
160	C₇₉⋊C₈	—	4	2	no	—
161	C₇₉⋊C₈	—	4	2	no	—
162	C₂²×C₇₉	#392	1	1	yes	characteristic
163	C₂²×C₇₉	#463	1	2	yes	—
164	C₂²×C₇₉	#467	1	1	yes	characteristic
165	C₂²×C₇₉	#470	1	1	yes	characteristic
166	C₂²×C₇₉	#471	1	2	yes	—
167	C₂×D₇₉	—	4	4	no	—
168	C₂×D₇₉	—	4	2	no	—
169	C₂×D₇₉	—	8	2	no	—
170	C₂×D₇₉	—	8	1	no	—
171	C₂×D₇₉	—	8	1	no	—
172	C₄×C₇₉	—	2	1	no	—
173	C₄×C₇₉	—	2	1	no	—
174	C₄×C₇₉	—	8	2	no	—
175	C₄×C₇₉	#449	1	2	yes	—
176	C₄×C₇₉	#451	1	1	yes	characteristic
177	C₄×C₇₉	#451	1	1	yes	characteristic
178	C₄×C₇₉	#452	1	1	yes	characteristic
179	C₄×C₇₉	#452	1	1	yes	characteristic
180	C₄×C₇₉	#495	1	1	yes	characteristic
181	C₄×C₇₉	#496	1	1	yes	characteristic
182	Dic₇₉	—	8	2	no	—
183	Dic₇₉	—	8	1	no	—
184	#10275	—	79	1	no	maximal, Sylow
185	C₂×C₇₉	#1683	1	2	yes	—
186	C₂×C₇₉	#1703	1	2	yes	—
187	C₂×C₇₉	#2001	1	1	yes	characteristic
188	C₂×C₇₉	#2012	1	1	yes	characteristic
189	C₂×C₇₉	#2019	1	1	yes	characteristic
190	D₇₉	—	8	2	no	—
191	C₂×#487	—	79	1	no	—
192	C₂×#476	—	79	1	no	—
193	C₂×#474	—	79	1	no	—
194	#1954	—	79	1	no	—
195	C₂×#333	D₇₉	1	1	yes	characteristic, complemented
196	C₂×#332	—	79	1	no	—
197	#1277	—	79	1	no	—
198	C₇₉	#10275	1	1	yes	characteristic, complemented, Sylow
199	C₂×C₄⋊Q₈	—	79	1	no	—
200	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
201	C₂×C₄×D₄	—	79	1	no	—
202	#487	—	158	2	no	—
203	#487	—	158	2	no	—
204	#476	—	158	2	no	—
205	#476	—	158	2	no	—
206	#474	—	158	2	no	—
207	#474	—	158	2	no	—
208	C₂×#107	—	158	1	no	—
209	C₂×#106	—	158	1	no	—
210	C₂×C₄⋊C₈	—	79	1	no	—
211	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
212	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
213	C₂×#103	—	158	2	no	—
214	C₂×#102	—	158	1	no	—
215	C₂×#102	—	158	1	no	—
216	C₂×C₂²⋊C₈	—	158	2	no	—
217	C₂×C₄×C₈	—	79	1	no	—
218	#333	—	2	2	no	—
219	#332	—	158	2	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	158	1	no	—
222	C₂²×D₄	—	79	1	no	—
223	C₂³×C₄	—	158	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	79	2	no	—
227	C₂²×C₈	—	158	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	79	2	no	—
232	C₄⋊Q₈	—	158	1	no	—
233	C₄⋊Q₈	—	158	1	no	—
234	C₄⋊Q₈	—	158	1	no	—
235	C₄⋊Q₈	D₃₁₆	1	2	yes	—
236	C₄×D₄	—	158	1	no	—
237	C₄×D₄	—	158	1	no	—
238	C₄×D₄	—	158	1	no	—
239	C₄×D₄	—	158	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	79	1	no	—
243	C₂×C₄⋊C₄	—	158	1	no	—
244	C₂×C₄⋊C₄	—	158	1	no	—
245	C₂×C₂²⋊C₄	—	158	1	no	—
246	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
247	#107	—	158	4	no	—
248	#106	—	158	4	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	79	2	no	—
253	C₄⋊C₈	—	158	1	no	—
254	C₄⋊C₈	C₂²⋊D₇₉	1	2	yes	—
255	#103	—	158	8	no	—
256	#102	—	158	4	no	—
257	#102	—	158	4	no	—
258	C₂²⋊C₈	—	316	4	no	—
259	C₄×C₈	—	158	1	no	—
260	C₄×C₈	—	158	1	no	—
261	C₂⁴	—	158	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	8	2	no	—
265	C₂×Q₈	—	316	1	no	—
266	C₂×Q₈	—	316	2	no	—
267	C₂×Q₈	—	316	2	no	—
268	C₂×Q₈	—	158	2	no	—
269	C₂×D₄	—	79	2	no	—
270	C₂×D₄	—	158	1	no	—
271	C₂×D₄	—	158	1	no	—
272	C₂×D₄	—	158	1	no	—
273	C₂×D₄	—	158	1	no	—
274	C₂×D₄	—	158	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	316	1	no	—
277	C₂²×C₄	—	316	1	no	—
278	C₂²×C₄	—	316	1	no	—
279	C₂²×C₄	—	316	2	no	—
280	C₂²×C₄	—	316	2	no	—
281	C₂²×C₄	—	158	1	no	—
282	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
283	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
284	C₂²×C₄	#11929624	1	1	yes	characteristic
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	1	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	1	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	—	316	1	no	—
291	C₂×C₈	—	316	1	no	—
292	C₂×C₈	—	158	2	no	—
293	C₂×C₈	—	158	2	no	—
294	C₂×C₈	—	158	2	no	—
295	C₂×C₈	—	158	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	316	1	no	—
301	C₄⋊C₄	—	316	1	no	—
302	C₄⋊C₄	—	158	2	no	—
303	C₄⋊C₄	—	158	1	no	—
304	C₄⋊C₄	—	158	1	no	—
305	C₄⋊C₄	—	158	2	no	—
306	C₂²⋊C₄	—	316	1	no	—
307	C₂²⋊C₄	—	316	1	no	—
308	C₄²	D₄×D₇₉	1	1	yes	characteristic
309	C₄²	D₄×D₇₉	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
311	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
312	C₂³	—	316	1	no	—
313	C₂³	—	316	2	no	—
314	C₂³	—	316	1	no	—
315	C₂³	—	316	2	no	—
316	C₂³	—	158	2	no	—
317	C₂³	#14579134	1	1	yes	characteristic
318	Q₈	—	8	2	no	—
319	Q₈	—	8	2	no	—
320	Q₈	—	316	4	no	—
321	Q₈	—	632	1	no	—
322	Q₈	—	632	1	no	—
323	D₄	—	316	1	no	—
324	D₄	—	316	1	no	—
325	D₄	—	158	2	no	—
326	D₄	—	158	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	316	1	no	—
331	C₂×C₄	—	316	2	no	—
332	C₂×C₄	—	316	2	no	—
333	C₂×C₄	—	632	2	no	—
334	C₂×C₄	—	632	1	no	—
335	C₂×C₄	—	632	1	no	—
336	C₂×C₄	—	632	2	no	—
337	C₂×C₄	#14579129	1	1	yes	characteristic
338	C₂×C₄	#14579131	1	1	yes	characteristic
339	C₂×C₄	#14579131	1	1	yes	characteristic
340	C₂×C₄	#14579134	1	1	yes	characteristic
341	C₂×C₄	#14579135	1	1	yes	characteristic
342	C₂×C₄	#14579157	1	1	yes	characteristic
343	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
344	C₂×C₄	#14579182	1	1	yes	characteristic
345	C₂×C₄	#14579184	1	2	yes	—
346	C₂×C₄	#14579188	1	2	yes	—
347	C₂×C₄	#14579190	1	1	yes	characteristic
348	C₂×C₄	#14579193	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₈	—	316	2	no	—
352	C₈	—	316	2	no	—
353	C₂²	—	316	4	no	—
354	C₂²	—	316	2	no	—
355	C₂²	—	632	2	no	—
356	C₂²	—	632	1	no	—
357	C₂²	—	632	1	no	—
358	C₂²	#19098472	1	1	yes	characteristic
359	C₂²	#19098781	1	2	yes	—
360	C₂²	#19098803	1	1	yes	characteristic
361	C₂²	#19098820	1	1	yes	characteristic
362	C₂²	#19098825	1	2	yes	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	632	1	no	—
367	C₄	—	632	2	no	—
368	C₄	#19098711	1	2	yes	—
369	C₄	#19098717	1	1	yes	characteristic
370	C₄	#19098720	1	1	yes	characteristic
371	C₄	#19098722	1	1	yes	characteristic
372	C₄	#19098728	1	1	yes	characteristic
373	C₄	#19098948	1	1	yes	characteristic
374	C₄	#19098949	1	1	yes	characteristic
375	C₂	—	632	2	no	—
376	C₂	#23762854	1	2	yes	—
377	C₂	#23762978	1	2	yes	—
378	C₂	#23764409	1	1	yes	characteristic
379	C₂	#23764478	1	1	yes	characteristic
380	C₂	#23764516	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579134
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#333×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂²×D₃₁₆	D₄
hypercenter	C₂×#333	D₇₉
hypocenter	C₇₉	#10275
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
1. C₂×#19098915	C₂	semidirect
5. C₂×#19098310	C₂	semidirect
6. #23761234	C₂	semidirect
7. C₂×#333×C₇₉	C₂	semidirect
195. C₂×#333	D₇₉	semidirect
198. C₇₉	#10275	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098915	2
C₂×#19098850	2
C₂×#19098844	2
#23764193	2
C₂×#19098310	2
#23761234	2
C₂×#333×C₇₉	2
#10275	79