Subgroups — #34184956

Order 20224.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772118	C₂	1	1	yes	maximal, characteristic, complemented
2	#23764225	C₂	1	1	yes	maximal, characteristic, complemented
3	#23762068	C₂	1	1	yes	maximal, characteristic, complemented
4	#23761960	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761259	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761230	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#3205	C₂	1	1	yes	maximal, characteristic
8	C₂×#14579168	—	2	1	no	—
9	#19098635	—	2	2	no	—
10	#19098635	—	2	2	no	—
11	#19098616	C₂²	1	1	yes	characteristic
12	#19098587	C₂²	1	1	yes	characteristic
13	C₂×#14579082	—	2	1	no	—
14	C₂×#14579082	—	2	1	no	—
15	#19098530	—	2	1	no	—
16	C₂×#14579072	C₂²	1	1	yes	characteristic
17	#19098429	C₂²	1	1	yes	characteristic
18	#19098307	—	2	2	no	—
19	#19098297	—	2	2	no	—
20	#532×C₇₉	C₂²	1	1	yes	characteristic
21	C₂×#127×C₇₉	C₂²	1	1	yes	characteristic
22	#493×C₇₉	C₄	1	2	yes	complemented
23	#486×C₇₉	C₄	1	2	yes	complemented
24	#459×C₇₉	—	2	2	no	—
25	#447×C₇₉	—	2	2	no	—
26	#431×C₇₉	C₂²	1	1	yes	characteristic
27	#14579168	—	4	2	no	—
28	#14579168	—	4	2	no	—
29	C₂×#11929608	—	2	2	no	—
30	C₂×#11929608	—	2	1	no	—
31	C₂²×C₇₉⋊C₈	—	2	2	no	—
32	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
34	C₂×C₄⋊Dic₇₉	—	2	1	no	—
35	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
36	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₇₉⋊C₈)	D₄	1	2	yes	—
38	#14579082	—	4	2	no	—
39	#14579082	—	4	2	no	—
40	C₂²⋊(C₇₉⋊C₈)	—	2	4	no	—
41	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
42	#14579072	—	2	1	no	—
43	#14579072	—	2	1	no	—
44	C₄×C₇₉⋊C₈	—	2	1	no	—
45	C₇₉⋊#95	—	2	1	no	—
46	C₇₉⋊#95	—	2	1	no	—
47	C₂×#41×C₇₉	—	2	1	no	—
48	C₂²×D₄×C₇₉	—	2	1	no	—
49	C₂²×D₄×C₇₉	—	2	1	no	—
50	C₂²×D₄×C₇₉	—	2	1	no	—
51	C₂×D₈×C₇₉	—	2	4	no	—
52	C₄⋊Q₈×C₇₉	C₂×C₄	1	1	yes	characteristic
53	#127×C₇₉	—	2	1	no	—
54	#127×C₇₉	—	2	1	no	—
55	#127×C₇₉	—	2	1	no	—
56	#127×C₇₉	C₂×C₄	1	2	yes	—
57	#127×C₇₉	C₂×C₄	1	1	yes	characteristic
58	#124×C₇₉	—	2	1	no	—
59	#121×C₇₉	—	2	1	no	—
60	#121×C₇₉	D₄	1	2	yes	—
61	C₄×D₄×C₇₉	—	2	1	no	—
62	C₄×D₄×C₇₉	D₄	1	2	yes	—
63	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
64	C₄⋊C₈×C₇₉	—	2	1	no	—
65	#102×C₇₉	—	2	4	no	—
66	C₂²⋊C₈×C₇₉	—	2	1	no	—
67	C₄×C₈×C₇₉	C₂×C₄	1	2	yes	—
68	C₂²×Dic₇₉	—	4	1	no	—
69	#11929608	—	4	4	no	—
70	#11929608	—	4	1	no	—
71	#11929608	—	4	1	no	—
72	C₂×C₇₉⋊C₈	—	2	1	no	—
73	C₂×C₇₉⋊C₈	—	2	1	no	—
74	C₂×C₇₉⋊C₈	—	2	2	no	—
75	C₂×C₇₉⋊C₈	—	2	1	no	—
76	C₂×C₇₉⋊C₈	—	2	1	no	—
77	C₂×C₇₉⋊C₈	—	2	1	no	—
78	C₂×C₇₉⋊C₈	—	4	4	no	—
79	C₂×C₇₉⋊C₈	—	4	1	no	—
80	C₂×C₇₉⋊C₈	D₈	1	2	yes	complemented
81	C₄⋊Dic₇₉	—	4	1	no	—
82	C₄⋊Dic₇₉	—	4	1	no	—
83	C₂⁴×C₇₉	—	4	1	no	—
84	#41×C₇₉	—	4	1	no	—
85	#41×C₇₉	—	4	1	no	—
86	#41×C₇₉	—	4	1	no	—
87	#41×C₇₉	—	4	1	no	—
88	C₂×Q₈×C₇₉	—	2	1	no	—
89	C₂×D₄×C₇₉	—	2	2	no	—
90	C₂×D₄×C₇₉	—	2	2	no	—
91	C₂×D₄×C₇₉	—	2	1	no	—
92	C₂×D₄×C₇₉	—	2	2	no	—
93	C₂×D₄×C₇₉	—	2	1	no	—
94	C₂×D₄×C₇₉	—	2	2	no	—
95	C₂×D₄×C₇₉	—	4	1	no	—
96	C₂×D₄×C₇₉	—	4	1	no	—
97	C₂×D₄×C₇₉	—	4	1	no	—
98	C₂×D₄×C₇₉	—	4	1	no	—
99	C₂×D₄×C₇₉	—	4	1	no	—
100	C₂×D₄×C₇₉	—	4	1	no	—
101	C₂×D₄×C₇₉	—	4	1	no	—
102	C₂×D₄×C₇₉	—	8	2	no	—
103	C₂×D₄×C₇₉	—	8	2	no	—
104	C₂×D₄×C₇₉	C₂²⋊C₄	1	2	yes	—
105	C₂²×C₄×C₇₉	—	2	2	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	D₈×C₇₉	—	8	4	no	—
110	C₂×C₈×C₇₉	—	2	2	no	—
111	C₄⋊C₄×C₇₉	—	2	1	no	—
112	C₄⋊C₄×C₇₉	C₂²⋊C₄	1	2	yes	—
113	C₂²⋊C₄×C₇₉	—	2	2	no	—
114	C₂²⋊C₄×C₇₉	—	2	2	no	—
115	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₇₉	#41	1	1	yes	characteristic
119	C₂³×C₇₉	—	2	2	no	—
120	C₂³×C₇₉	—	4	2	no	—
121	C₂³×C₇₉	—	8	2	no	—
122	C₂³×C₇₉	—	8	2	no	—
123	C₂³×C₇₉	—	8	1	no	—
124	C₂³×C₇₉	—	8	1	no	—
125	C₂³×C₇₉	#121	1	1	yes	characteristic
126	Q₈×C₇₉	—	4	1	no	—
127	Q₈×C₇₉	—	4	1	no	—
128	D₄×C₇₉	—	4	2	no	—
129	D₄×C₇₉	—	4	4	no	—
130	D₄×C₇₉	—	4	2	no	—
131	D₄×C₇₉	—	4	2	no	—
132	D₄×C₇₉	—	4	1	no	—
133	D₄×C₇₉	—	4	2	no	—
134	D₄×C₇₉	—	4	1	no	—
135	D₄×C₇₉	—	4	2	no	—
136	D₄×C₇₉	—	8	1	no	—
137	D₄×C₇₉	—	8	2	no	—
138	D₄×C₇₉	—	8	2	no	—
139	D₄×C₇₉	—	8	2	no	—
140	D₄×C₇₉	—	8	1	no	—
141	D₄×C₇₉	—	8	1	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	—	4	1	no	—
144	C₂×C₄×C₇₉	—	4	1	no	—
145	C₂×C₄×C₇₉	—	4	2	no	—
146	C₂×C₄×C₇₉	—	4	2	no	—
147	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
151	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
152	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
153	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₇₉	#135	1	1	yes	characteristic
155	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×Dic₇₉	—	8	1	no	—
158	C₂×Dic₇₉	—	8	1	no	—
159	C₂×Dic₇₉	—	8	1	no	—
160	C₈×C₇₉	—	4	2	no	—
161	C₇₉⋊C₈	—	2	2	no	—
162	C₇₉⋊C₈	—	4	2	no	—
163	C₇₉⋊C₈	—	4	1	no	—
164	C₇₉⋊C₈	—	8	1	no	—
165	C₂²×C₇₉	—	2	1	no	—
166	C₂²×C₇₉	—	2	1	no	—
167	C₂²×C₇₉	—	4	2	no	—
168	C₂²×C₇₉	—	4	2	no	—
169	C₂²×C₇₉	—	4	2	no	—
170	C₂²×C₇₉	—	8	2	no	—
171	C₂²×C₇₉	—	8	2	no	—
172	C₂²×C₇₉	—	8	2	no	—
173	C₂²×C₇₉	—	16	2	no	—
174	C₂²×C₇₉	—	16	2	no	—
175	C₂²×C₇₉	#386	1	1	yes	characteristic
176	C₂²×C₇₉	#459	1	1	yes	characteristic
177	C₂²×C₇₉	#463	1	1	yes	characteristic
178	C₄×C₇₉	—	2	1	no	—
179	C₄×C₇₉	—	2	1	no	—
180	C₄×C₇₉	—	2	1	no	—
181	C₄×C₇₉	—	2	1	no	—
182	C₄×C₇₉	—	4	2	no	—
183	C₄×C₇₉	C₄×D₈	1	1	yes	characteristic
184	C₄×C₇₉	#442	1	1	yes	characteristic
185	C₄×C₇₉	#447	1	1	yes	characteristic
186	C₄×C₇₉	#449	1	1	yes	characteristic
187	Dic₇₉	—	16	1	no	—
188	#9293	—	79	1	no	maximal, Sylow
189	C₂×C₇₉	—	2	1	no	—
190	C₂×C₇₉	—	2	1	no	—
191	C₂×C₇₉	—	8	2	no	—
192	C₂×C₇₉	—	16	2	no	—
193	C₂×C₇₉	#1879	1	1	yes	characteristic
194	C₂×C₇₉	#1903	1	1	yes	characteristic
195	C₂×C₇₉	#1910	1	1	yes	characteristic
196	#3205	D₇₉	1	1	yes	characteristic
197	#2932	—	79	1	no	—
198	#1961	—	79	1	no	—
199	#1545	—	79	1	no	—
200	#1517	—	79	1	no	—
201	#1286	—	79	1	no	—
202	#1275	—	79	1	no	—
203	C₇₉	#9293	1	1	yes	characteristic, complemented, Sylow
204	C₂×#131	—	158	1	no	—
205	#532	C₂×D₇₉	1	1	yes	characteristic
206	C₂×#127	C₂×D₇₉	1	1	yes	characteristic
207	#493	Dic₇₉	1	2	yes	complemented
208	#486	Dic₇₉	1	2	yes	complemented
209	#459	—	2	2	no	—
210	#447	—	2	2	no	—
211	#435	—	158	2	no	—
212	#435	—	158	2	no	—
213	#431	—	79	1	no	—
214	#431	C₂×D₇₉	1	1	yes	characteristic
215	#423	—	79	1	no	—
216	C₂×#102	—	158	1	no	—
217	C₂×#102	—	158	1	no	—
218	#408	—	158	1	no	—
219	C₂×#97	—	79	1	no	—
220	#382	—	79	1	no	—
221	#331	—	158	2	no	—
222	#327	—	158	2	no	—
223	C₂×#41	—	2	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×D₄	—	2	1	no	—
226	C₂²×D₄	—	2	1	no	—
227	C₂×D₈	—	2	4	no	—
228	#131	—	316	2	no	—
229	#131	—	316	2	no	—
230	C₂×M₁₆	—	158	2	no	—
231	C₂×M₁₆	—	158	1	no	—
232	C₂²×C₈	—	79	1	no	—
233	C₂²×C₈	—	79	1	no	—
234	C₂²×C₈	—	158	2	no	—
235	C₄⋊Q₈	C₂×Dic₇₉	1	1	yes	characteristic
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#127	—	2	1	no	—
239	#127	C₂×Dic₇₉	1	2	yes	—
240	#127	C₂×Dic₇₉	1	1	yes	characteristic
241	#124	—	2	1	no	—
242	#121	—	2	1	no	—
243	#121	C₂²⋊D₇₉	1	2	yes	—
244	C₄×D₄	—	2	1	no	—
245	C₄×D₄	C₂²⋊D₇₉	1	2	yes	—
246	C₂×C₄⋊C₄	—	158	1	no	—
247	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	79	2	no	—
250	C₄⋊C₈	—	158	1	no	—
251	C₄⋊C₈	—	158	1	no	—
252	#102	—	2	4	no	—
253	#102	—	316	2	no	—
254	#102	—	316	2	no	—
255	C₂²⋊C₈	—	2	1	no	—
256	C₂²⋊C₈	—	158	1	no	—
257	C₂²⋊C₈	—	158	4	no	—
258	#97	—	158	1	no	—
259	#97	—	158	1	no	—
260	C₄×C₈	—	158	1	no	—
261	C₄×C₈	C₂×Dic₇₉	1	2	yes	—
262	#95	—	158	1	no	—
263	#95	—	158	1	no	—
264	C₂⁴	—	4	1	no	—
265	#41	—	4	1	no	—
266	#41	—	4	1	no	—
267	#41	—	4	1	no	—
268	#41	—	4	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	4	1	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	8	2	no	—
284	C₂×D₄	—	8	2	no	—
285	C₂×D₄	C₂²⋊Dic₇₉	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	316	1	no	—
288	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
290	C₂²×C₄	#11929623	1	1	yes	characteristic
291	D₈	—	8	4	no	—
292	M₁₆	—	316	1	no	—
293	M₁₆	—	316	1	no	—
294	M₁₆	—	316	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	316	4	no	—
297	C₂×C₈	—	316	1	no	—
298	C₂×C₈	—	79	2	no	—
299	C₂×C₈	—	158	1	no	—
300	C₂×C₈	—	158	2	no	—
301	C₂×C₈	—	158	1	no	—
302	C₂×C₈	—	158	1	no	—
303	C₂×C₈	—	158	1	no	—
304	C₂×C₈	—	158	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	316	1	no	—
307	C₄⋊C₄	—	316	1	no	—
308	C₄⋊C₄	C₂²⋊Dic₇₉	1	2	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₄²	C₂²×Dic₇₉	1	1	yes	characteristic
312	C₄²	D₄×D₇₉	1	1	yes	characteristic
313	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
314	C₄²	#11929623	1	1	yes	characteristic
315	C₂³	—	2	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	8	2	no	—
318	C₂³	—	8	2	no	—
319	C₂³	—	8	1	no	—
320	C₂³	—	8	1	no	—
321	C₂³	#14579133	1	1	yes	characteristic
322	Q₈	—	4	1	no	—
323	Q₈	—	4	1	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	4	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	1	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	2	no	—
332	D₄	—	8	1	no	—
333	D₄	—	8	2	no	—
334	D₄	—	8	2	no	—
335	D₄	—	8	2	no	—
336	D₄	—	8	1	no	—
337	D₄	—	8	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	632	1	no	—
344	C₂×C₄	—	632	1	no	—
345	C₂×C₄	—	632	1	no	—
346	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
347	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
348	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
349	C₂×C₄	#14579129	1	1	yes	characteristic
350	C₂×C₄	#14579133	1	1	yes	characteristic
351	C₂×C₄	#14579142	1	1	yes	characteristic
352	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
353	C₂×C₄	#14579180	1	1	yes	characteristic
354	C₂×C₄	#14579185	1	1	yes	characteristic
355	C₂×C₄	#14579185	1	1	yes	characteristic
356	C₈	—	4	2	no	—
357	C₈	—	316	2	no	—
358	C₈	—	316	1	no	—
359	C₈	—	632	1	no	—
360	C₈	—	158	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	16	2	no	—
370	C₂²	—	16	2	no	—
371	C₂²	#19098443	1	1	yes	characteristic
372	C₂²	#19098756	1	1	yes	characteristic
373	C₂²	#19098776	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	1264	1	no	—
380	C₄	D₈×Dic₇₉	1	1	yes	characteristic
381	C₄	#19098675	1	1	yes	characteristic
382	C₄	#19098695	1	1	yes	characteristic
383	C₄	#19098705	1	1	yes	characteristic
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	8	2	no	—
387	C₂	—	16	2	no	—
388	C₂	#23763750	1	1	yes	characteristic
389	C₂	#23763882	1	1	yes	characteristic
390	C₂	#23763928	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098443
commutator	C₄²×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting, prime radical	C₇₉×#3205	C₂
socle	C₂²×C₇₉	#386
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#3205	D₇₉
hypocenter	C₇₉	#9293
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3205

Complements

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

N	H ≅ G/N	type
1. #23772118	C₂	semidirect
2. #23764225	C₂	semidirect
3. #23762068	C₂	semidirect
4. #23761960	C₂	semidirect
5. #23761259	C₂	semidirect
6. #23761230	C₂	semidirect
22. #493×C₇₉	C₄	semidirect
23. #486×C₇₉	C₄	semidirect
80. C₂×C₇₉⋊C₈	D₈	semidirect
203. C₇₉	#9293	semidirect
207. #493	Dic₇₉	semidirect
208. #486	Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23772118	2
#23764225	2
#23762068	2
#23761960	2
#23761259	2
#23761230	2
C₇₉×#3205	2
#9293	79