Subgroups — #34190868

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×#19098850	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098840	C₂	1	1	yes	maximal, characteristic
3	#23764194	C₂	1	1	yes	maximal, characteristic
4	C₂×#19098312	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098309	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761235	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#487×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁₆	C₂²	1	1	yes	characteristic
9	#19098850	—	2	2	no	—
10	#19098850	—	2	2	no	—
11	#19098840	—	2	2	no	—
12	#19098840	—	2	2	no	—
13	C₂×#14579094	—	2	1	no	—
14	C₂×#14579092	—	2	1	no	—
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×#14579081	—	2	1	no	—
19	C₂×#14579081	—	2	1	no	—
20	C₂×(C₂×D₇₉)⋊C₈	—	2	2	no	—
21	#19098312	—	2	2	no	—
22	#19098309	—	2	2	no	—
23	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
25	#487×C₇₉	—	2	2	no	—
26	#487×C₇₉	—	2	2	no	—
27	C₂×#103×C₇₉	—	2	2	no	—
28	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
29	C₂²×D₃₁₆	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₇₉	—	2	1	no	—
31	C₂²×C₇₉⋊C₈	—	2	1	no	—
32	C₂²×C₇₉⋊C₈	—	2	1	no	—
33	C₂²×C₇₉⋊C₈	—	2	1	no	—
34	C₄×D₃₁₆	—	2	1	no	—
35	C₄×D₃₁₆	—	2	1	no	—
36	C₄×D₃₁₆	—	2	1	no	—
37	C₄×D₃₁₆	—	2	1	no	—
38	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
39	C₂×(C₂×D₇₉)⋊C₄	—	2	1	no	—
40	#14579094	—	2	4	no	—
41	#14579092	—	2	4	no	—
42	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
43	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
45	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
46	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
47	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
48	#14579081	—	2	4	no	—
49	#14579081	—	2	4	no	—
50	(C₂×D₇₉)⋊C₈	—	4	4	no	—
51	C₂²×Q₈×C₇₉	—	2	1	no	—
52	C₂²×Q₈×C₇₉	—	2	1	no	—
53	C₂²×C₈×C₇₉	D₄	1	2	yes	—
54	C₄⋊Q₈×C₇₉	—	2	1	no	—
55	C₄⋊Q₈×C₇₉	—	2	1	no	—
56	C₄⋊Q₈×C₇₉	—	2	1	no	—
57	C₄⋊Q₈×C₇₉	—	2	1	no	—
58	C₄⋊Q₈×C₇₉	—	2	1	no	—
59	C₄⋊Q₈×C₇₉	—	2	1	no	—
60	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
61	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
63	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
65	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
66	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
67	#103×C₇₉	—	2	8	no	—
68	C₄×C₈×C₇₉	—	2	1	no	—
69	C₄×C₈×C₇₉	—	2	1	no	—
70	C₂³×D₇₉	—	2	1	no	—
71	C₂×D₃₁₆	—	2	1	no	—
72	C₂×D₃₁₆	—	2	1	no	—
73	C₂×D₃₁₆	—	2	1	no	—
74	C₂×D₃₁₆	—	2	1	no	—
75	C₂×D₃₁₆	—	2	1	no	—
76	C₂×D₃₁₆	Q₁₆	1	2	yes	—
77	C₂²×Dic₇₉	—	2	1	no	—
78	C₂×C₄×D₇₉	—	4	1	no	—
79	C₂×C₄×D₇₉	—	4	1	no	—
80	C₂×C₄×D₇₉	—	4	2	no	—
81	C₂×C₄×D₇₉	—	4	2	no	—
82	C₂×C₇₉⋊C₈	—	2	2	no	—
83	C₂×C₇₉⋊C₈	—	2	2	no	—
84	C₂×C₇₉⋊C₈	—	4	1	no	—
85	C₂×C₇₉⋊C₈	—	4	1	no	—
86	C₂×C₇₉⋊C₈	—	4	1	no	—
87	C₂×C₇₉⋊C₈	—	4	2	no	—
88	C₂×C₇₉⋊C₈	—	4	1	no	—
89	C₂×C₇₉⋊C₈	—	4	1	no	—
90	C₄⋊Dic₇₉	—	2	1	no	—
91	C₄⋊Dic₇₉	—	2	1	no	—
92	(C₂×D₇₉)⋊C₄	—	4	1	no	—
93	(C₂×D₇₉)⋊C₄	—	4	1	no	—
94	C₂×Q₈×C₇₉	—	2	2	no	—
95	C₂×Q₈×C₇₉	—	2	2	no	—
96	C₂×Q₈×C₇₉	—	4	1	no	—
97	C₂×Q₈×C₇₉	—	4	1	no	—
98	C₂×Q₈×C₇₉	—	4	2	no	—
99	C₂×Q₈×C₇₉	—	4	2	no	—
100	C₂×Q₈×C₇₉	—	8	2	no	—
101	C₂²×C₄×C₇₉	—	4	1	no	—
102	C₂²×C₄×C₇₉	—	4	1	no	—
103	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
106	C₂×C₈×C₇₉	—	2	2	no	—
107	C₂×C₈×C₇₉	—	2	2	no	—
108	C₂×C₈×C₇₉	—	2	2	no	—
109	C₄⋊C₄×C₇₉	—	2	2	no	—
110	C₄⋊C₄×C₇₉	—	2	2	no	—
111	C₄⋊C₄×C₇₉	—	2	2	no	—
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₄⋊C₄×C₇₉	—	4	1	no	—
114	C₄⋊C₄×C₇₉	—	4	1	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	—	4	4	no	—
128	Q₈×C₇₉	—	8	2	no	—
129	Q₈×C₇₉	—	8	2	no	—
130	Q₈×C₇₉	—	8	1	no	—
131	Q₈×C₇₉	—	8	1	no	—
132	C₂×C₄×C₇₉	—	2	2	no	—
133	C₂×C₄×C₇₉	—	4	2	no	—
134	C₂×C₄×C₇₉	—	4	2	no	—
135	C₂×C₄×C₇₉	—	8	2	no	—
136	C₂×C₄×C₇₉	—	8	2	no	—
137	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
138	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂×Q₁₆	1	1	yes	characteristic
144	C₂×C₄×C₇₉	#135	1	2	yes	—
145	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
146	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
148	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
149	C₂×Dic₇₉	—	4	1	no	—
150	C₂×Dic₇₉	—	4	2	no	—
151	D₃₁₆	—	2	2	no	—
152	D₃₁₆	—	2	2	no	—
153	D₃₁₆	—	4	1	no	—
154	D₃₁₆	—	4	1	no	—
155	C₄×D₇₉	—	8	2	no	—
156	C₄×D₇₉	—	8	1	no	—
157	C₄×D₇₉	—	8	1	no	—
158	C₈×C₇₉	—	4	2	no	—
159	C₇₉⋊C₈	—	4	2	no	—
160	C₇₉⋊C₈	—	4	2	no	—
161	C₇₉⋊C₈	—	8	1	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	—	8	2	no	—
176	C₄×C₇₉	#449	1	2	yes	—
177	C₄×C₇₉	#451	1	1	yes	characteristic
178	C₄×C₇₉	#451	1	1	yes	characteristic
179	C₄×C₇₉	#452	1	1	yes	characteristic
180	C₄×C₇₉	#452	1	1	yes	characteristic
181	C₄×C₇₉	#495	1	1	yes	characteristic
182	C₄×C₇₉	#496	1	1	yes	characteristic
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	D₇₉	1	1	yes	characteristic, complemented
192	C₂×#476	—	79	1	no	—
193	C₂×#474	—	79	1	no	—
194	#1954	—	79	1	no	—
195	C₂×#333	—	79	1	no	—
196	C₂×#332	—	79	1	no	—
197	#1277	—	79	1	no	—
198	C₇₉	#10275	1	1	yes	characteristic, complemented, Sylow
199	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
200	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
201	C₂×C₄×D₄	—	79	1	no	—
202	#487	—	2	2	no	—
203	#487	—	2	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	—	79	1	no	—
212	C₂×C₄⋊C₈	—	79	1	no	—
213	C₂×#103	—	2	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	C₂×D₇₉	1	1	yes	characteristic
218	#333	—	158	2	no	—
219	#332	—	158	2	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×D₄	—	79	1	no	—
223	C₂³×C₄	—	158	1	no	—
224	C₂²×C₈	—	158	1	no	—
225	C₂²×C₈	—	158	1	no	—
226	C₂²×C₈	—	158	1	no	—
227	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
235	C₄⋊Q₈	C₂²⋊D₇₉	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	—	2	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	79	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	—	79	2	no	—
250	C₄⋊C₈	—	79	2	no	—
251	C₄⋊C₈	—	158	1	no	—
252	C₄⋊C₈	—	158	1	no	—
253	C₄⋊C₈	—	158	1	no	—
254	C₄⋊C₈	—	158	1	no	—
255	#103	—	2	8	no	—
256	#102	—	158	4	no	—
257	#102	—	158	4	no	—
258	C₂²⋊C₈	—	316	4	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₂⁴	—	158	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	8	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	—	4	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	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
284	C₂²×C₄	#11929623	1	1	yes	characteristic
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	316	1	no	—
289	C₂×C₈	—	316	1	no	—
290	C₂×C₈	—	316	1	no	—
291	C₂×C₈	—	316	1	no	—
292	C₂×C₈	—	316	1	no	—
293	C₂×C₈	—	316	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	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	158	1	no	—
305	C₄⋊C₄	—	158	1	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×C₂²⋊D₇₉	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	#14579132	1	1	yes	characteristic
318	Q₈	—	4	4	no	—
319	Q₈	—	8	2	no	—
320	Q₈	—	8	2	no	—
321	Q₈	—	8	1	no	—
322	Q₈	—	8	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	—	4	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	316	1	no	—
333	C₂×C₄	—	316	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	#14579129	1	1	yes	characteristic
339	C₂×C₄	#14579130	1	1	yes	characteristic
340	C₂×C₄	#14579133	1	1	yes	characteristic
341	C₂×C₄	#14579133	1	1	yes	characteristic
342	C₂×C₄	#14579157	1	1	yes	characteristic
343	C₂×C₄	Q₁₆×D₇₉	1	1	yes	characteristic
344	C₂×C₄	#14579181	1	2	yes	—
345	C₂×C₄	#14579184	1	1	yes	characteristic
346	C₂×C₄	#14579186	1	2	yes	—
347	C₂×C₄	#14579193	1	1	yes	characteristic
348	C₂×C₄	#14579193	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	316	2	no	—
351	C₈	—	316	2	no	—
352	C₈	—	632	1	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	#19098471	1	1	yes	characteristic
359	C₂²	#19098775	1	2	yes	—
360	C₂²	#19098801	1	1	yes	characteristic
361	C₂²	#19098818	1	1	yes	characteristic
362	C₂²	#19098823	1	2	yes	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	2	no	—
367	C₄	—	632	1	no	—
368	C₄	#19098708	1	2	yes	—
369	C₄	#19098717	1	1	yes	characteristic
370	C₄	#19098717	1	1	yes	characteristic
371	C₄	#19098724	1	1	yes	characteristic
372	C₄	#19098724	1	1	yes	characteristic
373	C₄	#19098945	1	1	yes	characteristic
374	C₄	#19098953	1	1	yes	characteristic
375	C₂	—	632	2	no	—
376	C₂	#23762848	1	2	yes	—
377	C₂	#23762972	1	2	yes	—
378	C₂	#23764407	1	1	yes	characteristic
379	C₂	#23764477	1	1	yes	characteristic
380	C₂	#23764512	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579132
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#487×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂²×D₃₁₆	D₄
hypercenter	C₂×#487	D₇₉
hypocenter	C₇₉	#10275
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
4. C₂×#19098312	C₂	semidirect
5. C₂×#19098309	C₂	semidirect
6. #23761235	C₂	semidirect
7. C₂×#487×C₇₉	C₂	semidirect
191. C₂×#487	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×#19098850	2
C₂×#19098840	2
#23764194	2
C₂×#19098312	2
C₂×#19098309	2
#23761235	2
C₂×#487×C₇₉	2
#10275	79