Subgroups — #34190827

Order 20224.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098911	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098763	C₂	1	1	yes	maximal, characteristic
3	#23764337	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098312	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19098306	C₂	1	1	yes	maximal, characteristic
6	#23761234	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#477×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11929611	—	2	1	no	—
9	C₂×#14579158	C₂²	1	1	yes	characteristic
10	#19098911	—	2	2	no	—
11	#19098911	—	2	2	no	—
12	#19098763	—	2	2	no	—
13	#19098763	—	2	2	no	—
14	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#14579081	—	2	2	no	—
17	C₂×#14579081	—	2	1	no	—
18	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
19	#19098318	—	2	2	no	—
20	#19098312	—	2	2	no	—
21	#19098306	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₇₉	C₂²	1	1	yes	characteristic
24	#477×C₇₉	—	2	2	no	—
25	#477×C₇₉	—	2	2	no	—
26	C₂×#107×C₇₉	—	2	1	no	—
27	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#103×C₇₉	—	2	1	no	—
29	C₂²×D₃₁₆	—	2	1	no	—
30	C₂²×D₃₁₆	—	2	1	no	—
31	C₂²×D₃₁₆	—	2	1	no	—
32	C₂×#11929611	—	2	4	no	—
33	C₂×#11929611	—	4	2	no	—
34	C₂×#11929611	—	4	2	no	—
35	C₂²×C₇₉⋊C₈	—	2	1	no	—
36	C₂²×C₇₉⋊C₈	—	2	1	no	—
37	C₂²×C₇₉⋊C₈	D₄	1	2	yes	—
38	#14579158	—	2	1	no	—
39	#14579158	—	2	1	no	—
40	#14579158	—	2	1	no	—
41	#14579158	D₄	1	2	yes	—
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)	D₄	1	2	yes	—
46	#14579081	—	2	8	no	—
47	#14579081	—	2	4	no	—
48	C₄×C₇₉⋊C₈	—	2	1	no	—
49	C₄×C₇₉⋊C₈	—	2	1	no	—
50	C₂²×Q₈×C₇₉	—	2	1	no	—
51	C₂²×Q₈×C₇₉	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₉	—	2	1	no	—
53	C₄⋊Q₈×C₇₉	—	2	1	no	—
54	C₄⋊Q₈×C₇₉	—	2	1	no	—
55	C₄⋊Q₈×C₇₉	—	2	1	no	—
56	C₄⋊Q₈×C₇₉	D₄	1	2	yes	—
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	—	2	1	no	—
61	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
63	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
65	C₂×C₄²×C₇₉	—	2	1	no	—
66	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
67	#107×C₇₉	—	2	4	no	—
68	C₄⋊C₈×C₇₉	—	2	1	no	—
69	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
70	#103×C₇₉	—	2	4	no	—
71	C₂³×D₇₉	—	4	1	no	—
72	C₂×D₃₁₆	—	2	2	no	—
73	C₂×D₃₁₆	—	2	2	no	—
74	C₂×D₃₁₆	—	2	2	no	—
75	C₂×D₃₁₆	—	4	1	no	—
76	C₂×D₃₁₆	—	4	1	no	—
77	C₂×D₃₁₆	—	4	1	no	—
78	C₂×D₃₁₆	—	4	1	no	—
79	C₂×D₃₁₆	—	4	1	no	—
80	C₂×D₃₁₆	—	4	2	no	—
81	C₂×D₃₁₆	—	4	1	no	—
82	C₂×D₃₁₆	—	4	1	no	—
83	C₂×D₃₁₆	—	4	2	no	—
84	C₂×D₃₁₆	—	8	2	no	—
85	#11929611	—	4	8	no	—
86	C₂×C₇₉⋊C₈	—	2	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	—	4	1	no	—
91	C₂×C₇₉⋊C₈	—	4	1	no	—
92	C₂×C₇₉⋊C₈	—	4	2	no	—
93	C₂×C₇₉⋊C₈	—	4	1	no	—
94	C₂×Q₈×C₇₉	—	2	1	no	—
95	C₂×Q₈×C₇₉	—	2	1	no	—
96	C₂×Q₈×C₇₉	—	2	1	no	—
97	C₂×Q₈×C₇₉	—	2	2	no	—
98	C₂×Q₈×C₇₉	—	2	1	no	—
99	C₂×Q₈×C₇₉	—	2	1	no	—
100	C₂×Q₈×C₇₉	—	4	1	no	—
101	C₂×Q₈×C₇₉	—	8	2	no	—
102	C₂×Q₈×C₇₉	SD₁₆	1	2	yes	—
103	C₂²×C₄×C₇₉	—	2	1	no	—
104	C₂²×C₄×C₇₉	—	2	1	no	—
105	C₂²×C₄×C₇₉	—	4	1	no	—
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
109	C₂×C₈×C₇₉	—	2	2	no	—
110	C₂×C₈×C₇₉	—	4	1	no	—
111	C₂×C₈×C₇₉	—	4	1	no	—
112	C₄⋊C₄×C₇₉	—	2	2	no	—
113	C₄⋊C₄×C₇₉	—	2	1	no	—
114	C₄⋊C₄×C₇₉	—	2	2	no	—
115	C₄⋊C₄×C₇₉	—	2	1	no	—
116	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₄⋊C₄×C₇₉	—	4	1	no	—
118	C₄⋊C₄×C₇₉	—	4	1	no	—
119	C₄⋊C₄×C₇₉	—	4	1	no	—
120	C₄²×C₇₉	—	4	1	no	—
121	C₄²×C₇₉	—	4	1	no	—
122	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
126	C₂³×C₇₉	#121	1	1	yes	characteristic
127	C₂²×D₇₉	—	4	2	no	—
128	C₂²×D₇₉	—	8	2	no	—
129	C₂²×D₇₉	—	8	1	no	—
130	C₂²×D₇₉	—	8	1	no	—
131	C₂²×D₇₉	—	8	2	no	—
132	Q₈×C₇₉	—	2	2	no	—
133	Q₈×C₇₉	—	2	2	no	—
134	Q₈×C₇₉	—	4	1	no	—
135	Q₈×C₇₉	—	4	1	no	—
136	Q₈×C₇₉	—	8	2	no	—
137	Q₈×C₇₉	—	8	2	no	—
138	C₂×C₄×C₇₉	—	2	2	no	—
139	C₂×C₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	4	1	no	—
141	C₂×C₄×C₇₉	—	4	1	no	—
142	C₂×C₄×C₇₉	—	4	1	no	—
143	C₂×C₄×C₇₉	—	4	2	no	—
144	C₂×C₄×C₇₉	—	4	2	no	—
145	C₂×C₄×C₇₉	—	8	2	no	—
146	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
147	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
148	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
149	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
150	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
151	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
153	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
154	C₂×C₄×C₇₉	#135	1	2	yes	—
155	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
157	C₂×C₄×C₇₉	#137	1	2	yes	—
158	D₃₁₆	—	4	2	no	—
159	D₃₁₆	—	4	2	no	—
160	D₃₁₆	—	4	4	no	—
161	D₃₁₆	—	8	2	no	—
162	D₃₁₆	—	8	2	no	—
163	D₃₁₆	—	8	1	no	—
164	D₃₁₆	—	8	1	no	—
165	D₃₁₆	—	8	1	no	—
166	D₃₁₆	—	8	1	no	—
167	C₈×C₇₉	—	4	2	no	—
168	C₇₉⋊C₈	—	4	2	no	—
169	C₇₉⋊C₈	—	4	2	no	—
170	C₇₉⋊C₈	—	8	1	no	—
171	C₂²×C₇₉	#392	1	1	yes	characteristic
172	C₂²×C₇₉	#464	1	2	yes	—
173	C₂²×C₇₉	#465	1	1	yes	characteristic
174	C₂²×C₇₉	#468	1	1	yes	characteristic
175	C₂²×C₇₉	#472	1	2	yes	—
176	C₂×D₇₉	—	8	2	no	—
177	C₂×D₇₉	—	8	2	no	—
178	C₂×D₇₉	—	8	2	no	—
179	C₂×D₇₉	—	16	2	no	—
180	C₂×D₇₉	—	16	2	no	—
181	C₄×C₇₉	—	2	1	no	—
182	C₄×C₇₉	—	2	1	no	—
183	C₄×C₇₉	—	4	2	no	—
184	C₄×C₇₉	—	8	1	no	—
185	C₄×C₇₉	—	8	2	no	—
186	C₄×C₇₉	#448	1	1	yes	characteristic
187	C₄×C₇₉	#449	1	1	yes	characteristic
188	C₄×C₇₉	#450	1	1	yes	characteristic
189	C₄×C₇₉	#452	1	1	yes	characteristic
190	C₄×C₇₉	#452	1	2	yes	—
191	C₄×C₇₉	#495	1	1	yes	characteristic
192	C₄×C₇₉	#497	1	1	yes	characteristic
193	#10269	—	79	1	no	maximal, Sylow
194	C₂×C₇₉	#1678	1	2	yes	—
195	C₂×C₇₉	#1698	1	2	yes	—
196	C₂×C₇₉	#2000	1	1	yes	characteristic
197	C₂×C₇₉	#2015	1	1	yes	characteristic
198	C₂×C₇₉	#2018	1	1	yes	characteristic
199	D₇₉	—	16	2	no	—
200	C₂×#486	—	79	1	no	—
201	C₂×#477	D₇₉	1	1	yes	characteristic, complemented
202	C₂×#460	—	79	1	no	—
203	#1984	—	79	1	no	—
204	C₂×#333	—	79	1	no	—
205	C₂×#331	—	79	1	no	—
206	#1277	—	79	1	no	—
207	C₇₉	#10269	1	1	yes	characteristic, complemented, Sylow
208	C₂²×SD₁₆	—	158	1	no	—
209	C₂×C₄⋊Q₈	C₂×D₇₉	1	1	yes	characteristic
210	C₂×#127	—	79	1	no	—
211	C₂×C₄×Q₈	C₂×D₇₉	1	1	yes	characteristic
212	#486	—	158	2	no	—
213	#486	—	158	2	no	—
214	#477	—	2	2	no	—
215	#477	—	2	2	no	—
216	#460	—	158	2	no	—
217	#460	—	158	2	no	—
218	C₂×#107	—	2	1	no	—
219	C₂×C₄⋊C₈	—	79	1	no	—
220	C₂×C₄⋊C₈	—	79	1	no	—
221	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
222	C₂×#103	—	2	1	no	—
223	C₂×#102	—	158	2	no	—
224	C₂×#102	—	158	1	no	—
225	C₂×C₄×C₈	—	79	1	no	—
226	#336	—	158	2	no	—
227	#333	—	158	2	no	—
228	#331	—	158	2	no	—
229	C₂²×Q₈	—	2	1	no	—
230	C₂²×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
231	C₂²×D₄	—	158	1	no	—
232	C₂²×D₄	—	158	1	no	—
233	C₂²×D₄	—	158	1	no	—
234	C₂×SD₁₆	—	316	2	no	—
235	C₂×SD₁₆	—	316	2	no	—
236	C₂×SD₁₆	—	158	4	no	—
237	C₂²×C₈	—	2	1	no	—
238	C₂²×C₈	—	79	2	no	—
239	C₂²×C₈	—	158	1	no	—
240	C₂²×C₈	—	158	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	C₂²⋊D₇₉	1	2	yes	—
245	#127	—	79	2	no	—
246	#127	—	158	1	no	—
247	#127	—	158	1	no	—
248	#127	—	158	1	no	—
249	C₄×Q₈	—	2	1	no	—
250	C₄×Q₈	—	2	1	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	C₂²⋊D₇₉	1	1	yes	characteristic
257	C₂×C₄²	—	2	1	no	—
258	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
259	#107	—	2	4	no	—
260	C₄⋊C₈	—	2	1	no	—
261	C₄⋊C₈	—	79	2	no	—
262	C₄⋊C₈	—	158	1	no	—
263	C₄⋊C₈	—	158	1	no	—
264	C₄⋊C₈	—	158	1	no	—
265	C₄⋊C₈	C₂²⋊D₇₉	1	2	yes	—
266	#103	—	2	4	no	—
267	#102	—	158	4	no	—
268	#102	—	158	8	no	—
269	C₄×C₈	—	158	1	no	—
270	C₄×C₈	—	158	1	no	—
271	C₂⁴	—	316	1	no	—
272	C₂×Q₈	—	2	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	—	8	2	no	—
280	C₂×Q₈	#11929611	1	2	yes	—
281	C₂×D₄	—	316	1	no	—
282	C₂×D₄	—	316	1	no	—
283	C₂×D₄	—	316	1	no	—
284	C₂×D₄	—	316	1	no	—
285	C₂×D₄	—	316	1	no	—
286	C₂×D₄	—	316	2	no	—
287	C₂×D₄	—	316	1	no	—
288	C₂×D₄	—	316	1	no	—
289	C₂×D₄	—	316	2	no	—
290	C₂×D₄	—	632	2	no	—
291	C₂×D₄	—	158	2	no	—
292	C₂×D₄	—	158	2	no	—
293	C₂×D₄	—	158	2	no	—
294	C₂²×C₄	—	2	1	no	—
295	C₂²×C₄	—	2	1	no	—
296	C₂²×C₄	—	4	1	no	—
297	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
298	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
299	C₂²×C₄	#11929623	1	1	yes	characteristic
300	SD₁₆	—	316	8	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	4	1	no	—
303	C₂×C₈	—	4	1	no	—
304	C₂×C₈	—	316	1	no	—
305	C₂×C₈	—	316	1	no	—
306	C₂×C₈	—	316	1	no	—
307	C₂×C₈	—	316	2	no	—
308	C₂×C₈	—	158	2	no	—
309	C₂×C₈	—	158	2	no	—
310	C₂×C₈	—	158	2	no	—
311	C₂×C₈	—	158	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	1	no	—
316	C₄⋊C₄	—	4	1	no	—
317	C₄⋊C₄	—	4	1	no	—
318	C₄⋊C₄	—	4	1	no	—
319	C₄⋊C₄	—	4	1	no	—
320	C₄²	—	4	1	no	—
321	C₄²	—	4	1	no	—
322	C₄²	D₄×D₇₉	1	1	yes	characteristic
323	C₄²	D₄×D₇₉	1	1	yes	characteristic
324	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
326	C₂³	—	316	2	no	—
327	C₂³	—	632	2	no	—
328	C₂³	—	632	1	no	—
329	C₂³	—	632	1	no	—
330	C₂³	—	632	2	no	—
331	C₂³	#14579133	1	1	yes	characteristic
332	Q₈	—	2	2	no	—
333	Q₈	—	2	2	no	—
334	Q₈	—	4	1	no	—
335	Q₈	—	4	1	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	8	2	no	—
338	D₄	—	316	2	no	—
339	D₄	—	316	2	no	—
340	D₄	—	316	4	no	—
341	D₄	—	632	2	no	—
342	D₄	—	632	2	no	—
343	D₄	—	632	1	no	—
344	D₄	—	632	1	no	—
345	D₄	—	632	1	no	—
346	D₄	—	632	1	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	4	2	no	—
354	C₂×C₄	—	8	2	no	—
355	C₂×C₄	#14579129	1	1	yes	characteristic
356	C₂×C₄	#14579129	1	1	yes	characteristic
357	C₂×C₄	#14579130	1	1	yes	characteristic
358	C₂×C₄	#14579132	1	1	yes	characteristic
359	C₂×C₄	#14579133	1	1	yes	characteristic
360	C₂×C₄	#14579157	1	1	yes	characteristic
361	C₂×C₄	C₂×#11929611	1	1	yes	characteristic
362	C₂×C₄	#14579181	1	1	yes	characteristic
363	C₂×C₄	#14579184	1	2	yes	—
364	C₂×C₄	#14579186	1	1	yes	characteristic
365	C₂×C₄	#14579192	1	1	yes	characteristic
366	C₂×C₄	#14579193	1	2	yes	—
367	C₈	—	4	2	no	—
368	C₈	—	316	2	no	—
369	C₈	—	316	2	no	—
370	C₈	—	632	1	no	—
371	C₂²	—	632	2	no	—
372	C₂²	—	632	2	no	—
373	C₂²	—	632	2	no	—
374	C₂²	—	1264	2	no	—
375	C₂²	—	1264	2	no	—
376	C₂²	#19098471	1	1	yes	characteristic
377	C₂²	#19098787	1	2	yes	—
378	C₂²	#19098794	1	1	yes	characteristic
379	C₂²	#19098811	1	1	yes	characteristic
380	C₂²	#19098831	1	2	yes	—
381	C₄	—	2	1	no	—
382	C₄	—	2	1	no	—
383	C₄	—	4	2	no	—
384	C₄	—	8	1	no	—
385	C₄	—	8	2	no	—
386	C₄	#19098702	1	1	yes	characteristic
387	C₄	#19098708	1	1	yes	characteristic
388	C₄	#19098713	1	1	yes	characteristic
389	C₄	#19098723	1	1	yes	characteristic
390	C₄	#19098724	1	2	yes	—
391	C₄	#19098948	1	1	yes	characteristic
392	C₄	#19098956	1	1	yes	characteristic
393	C₂	—	1264	2	no	—
394	C₂	#23762818	1	2	yes	—
395	C₂	#23762942	1	2	yes	—
396	C₂	#23764401	1	1	yes	characteristic
397	C₂	#23764491	1	1	yes	characteristic
398	C₂	#23764506	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14579133
commutator, cosocle	C₂×C₄²×C₇₉	C₂³
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₂×#477×C₇₉	C₂
socle	C₂³×C₇₉	#121
prime radical	C₂×#14579158	C₂²
hypercenter	C₂×#477	D₇₉
hypocenter	C₇₉	#10269
LCS	G ▷ C₂×C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#477

Complements

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

N	H ≅ G/N	type
3. #23764337	C₂	semidirect
4. C₂×#19098312	C₂	semidirect
6. #23761234	C₂	semidirect
7. C₂×#477×C₇₉	C₂	semidirect
201. C₂×#477	D₇₉	semidirect
207. C₇₉	#10269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098911	2
C₂×#19098763	2
#23764337	2
C₂×#19098312	2
C₂×#19098306	2
#23761234	2
C₂×#477×C₇₉	2
#10269	79