Subgroups — #34205141

Order 20224.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776503	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765028	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098347	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765014	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761489	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₉×#2955	C₂	1	1	yes	maximal, characteristic
8	#19098980	—	2	2	no	—
9	#19098980	—	2	2	no	—
10	C₂×#14579101	—	2	2	no	—
11	C₂×#14579101	C₂²	1	1	yes	characteristic
12	C₂×#14579101	C₂²	1	1	yes	characteristic
13	C₂×#14579101	C₂²	1	1	yes	characteristic
14	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	#19098347	—	2	2	no	—
16	#19098347	—	2	2	no	—
17	C₂²⋊(C₇₉⋊C₁₆)	—	2	2	no	—
18	#575×C₇₉	C₄	1	1	yes	characteristic
19	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
20	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
21	#444×C₇₉	—	2	2	no	—
22	#443×C₇₉	—	2	2	no	—
23	#430×C₇₉	C₄	1	1	yes	characteristic
24	#421×C₇₉	C₄	1	1	yes	characteristic
25	#421×C₇₉	C₄	1	1	yes	characteristic
26	#405×C₇₉	C₂²	1	1	yes	characteristic
27	#14579101	—	2	4	no	—
28	#14579101	—	2	1	no	—
29	#14579101	—	2	1	no	—
30	#14579101	—	2	1	no	—
31	#14579101	—	2	1	no	—
32	#14579101	—	2	1	no	—
33	#14579101	D₄	1	2	yes	—
34	C₂×C₇₉⋊C₁₆	—	2	4	no	—
35	C₂×C₇₉⋊C₁₆	—	2	1	no	—
36	C₂×C₇₉⋊C₁₆	—	2	1	no	—
37	C₂×C₇₉⋊C₁₆	—	2	1	no	—
38	C₂×C₇₉⋊C₁₆	—	2	1	no	—
39	C₂×C₇₉⋊C₁₆	—	2	1	no	—
40	C₂×C₇₉⋊C₁₆	D₄	1	2	yes	—
41	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
42	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
43	#137×C₇₉	—	2	1	no	—
44	#137×C₇₉	C₈	1	2	yes	—
45	AGL(1,ℤ/8)×C₇₉	—	2	1	no	—
46	AGL(1,ℤ/8)×C₇₉	C₈	1	2	yes	—
47	#135×C₇₉	—	2	2	no	—
48	#131×C₇₉	—	2	1	no	—
49	#131×C₇₉	—	2	1	no	—
50	#131×C₇₉	—	2	1	no	—
51	#131×C₇₉	—	2	1	no	—
52	#131×C₇₉	—	2	1	no	—
53	#131×C₇₉	—	2	1	no	—
54	#131×C₇₉	D₄	1	2	yes	—
55	#131×C₇₉	D₄	1	2	yes	—
56	C₂×M₁₆×C₇₉	—	2	1	no	—
57	C₂×M₁₆×C₇₉	—	2	1	no	—
58	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
59	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
60	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₇₉	—	2	1	no	—
62	C₂²×C₈×C₇₉	—	2	1	no	—
63	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
64	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
65	#108×C₇₉	—	2	2	no	—
66	#104×C₇₉	—	2	2	no	—
67	#104×C₇₉	—	2	2	no	—
68	#97×C₇₉	—	2	1	no	—
69	C₄×C₈×C₇₉	—	2	1	no	—
70	C₇₉⋊C₁₆	—	2	2	no	—
71	C₇₉⋊C₁₆	—	2	2	no	—
72	C₇₉⋊C₁₆	—	4	1	no	—
73	C₇₉⋊C₁₆	—	4	1	no	—
74	#41×C₇₉	—	2	1	no	—
75	#41×C₇₉	—	2	1	no	—
76	#41×C₇₉	—	2	1	no	—
77	#41×C₇₉	—	2	1	no	—
78	#41×C₇₉	—	2	1	no	—
79	#41×C₇₉	—	2	1	no	—
80	#41×C₇₉	C₂²⋊C₄	1	2	yes	—
81	#41×C₇₉	C₂×C₈	1	2	yes	—
82	C₂×Q₈×C₇₉	C₂×C₈	1	1	yes	characteristic
83	C₂×Q₈×C₇₉	M₁₆	1	1	yes	characteristic
84	C₂×D₄×C₇₉	—	2	1	no	—
85	C₂×D₄×C₇₉	—	2	1	no	—
86	C₂×D₄×C₇₉	C₂×C₈	1	1	yes	characteristic
87	C₂×D₄×C₇₉	M₁₆	1	1	yes	characteristic
88	C₂²×C₄×C₇₉	—	2	1	no	—
89	C₂²×C₄×C₇₉	—	2	1	no	—
90	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
91	Q₁₆×C₇₉	—	2	2	no	—
92	SD₁₆×C₇₉	—	2	2	no	—
93	SD₁₆×C₇₉	—	2	2	no	—
94	D₈×C₇₉	—	2	2	no	—
95	M₁₆×C₇₉	—	2	2	no	—
96	M₁₆×C₇₉	—	2	2	no	—
97	M₁₆×C₇₉	—	2	1	no	—
98	M₁₆×C₇₉	—	2	1	no	—
99	M₁₆×C₇₉	—	2	1	no	—
100	M₁₆×C₇₉	—	2	1	no	—
101	M₁₆×C₇₉	—	4	1	no	—
102	M₁₆×C₇₉	—	4	1	no	—
103	M₁₆×C₇₉	C₂²⋊C₄	1	2	yes	—
104	M₁₆×C₇₉	C₂×C₈	1	2	yes	—
105	C₂×C₈×C₇₉	—	2	2	no	—
106	C₂×C₈×C₇₉	—	2	2	no	—
107	C₂×C₈×C₇₉	—	2	1	no	—
108	C₂×C₈×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	—	4	1	no	—
111	C₂×C₈×C₇₉	—	4	1	no	—
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
117	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
118	C₄⋊C₄×C₇₉	—	2	1	no	—
119	C₂²⋊C₄×C₇₉	—	2	1	no	—
120	C₄²×C₇₉	—	2	1	no	—
121	C₂³×C₇₉	—	2	1	no	—
122	C₂³×C₇₉	—	2	1	no	—
123	C₂³×C₇₉	#131	1	1	yes	characteristic
124	Q₈×C₇₉	—	2	1	no	—
125	Q₈×C₇₉	—	2	1	no	—
126	Q₈×C₇₉	—	2	1	no	—
127	Q₈×C₇₉	C₂²⋊C₈	1	2	yes	—
128	D₄×C₇₉	—	2	2	no	—
129	D₄×C₇₉	—	2	2	no	—
130	D₄×C₇₉	—	2	1	no	—
131	D₄×C₇₉	—	2	1	no	—
132	D₄×C₇₉	—	2	1	no	—
133	D₄×C₇₉	—	4	1	no	—
134	D₄×C₇₉	—	4	1	no	—
135	D₄×C₇₉	C₂²⋊C₈	1	2	yes	—
136	C₂×C₄×C₇₉	—	2	1	no	—
137	C₂×C₄×C₇₉	—	2	2	no	—
138	C₂×C₄×C₇₉	—	2	2	no	—
139	C₂×C₄×C₇₉	—	2	1	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	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
148	C₂×C₄×C₇₉	#131	1	1	yes	characteristic
149	C₈×C₇₉	—	2	2	no	—
150	C₈×C₇₉	—	2	2	no	—
151	C₈×C₇₉	—	4	1	no	—
152	C₈×C₇₉	#120	1	1	yes	characteristic
153	C₈×C₇₉	#121	1	1	yes	characteristic
154	C₈×C₇₉	#121	1	1	yes	characteristic
155	C₈×C₇₉	#123	1	1	yes	characteristic
156	C₂²×C₇₉	—	2	2	no	—
157	C₂²×C₇₉	—	4	1	no	—
158	C₂²×C₇₉	—	4	2	no	—
159	C₂²×C₇₉	—	4	1	no	—
160	C₂²×C₇₉	—	8	1	no	—
161	C₂²×C₇₉	#408	1	1	yes	characteristic
162	C₂²×C₇₉	C₈×D₄	1	1	yes	characteristic
163	C₂²×C₇₉	#435	1	1	yes	characteristic
164	C₄×C₇₉	—	2	2	no	—
165	C₄×C₇₉	—	4	1	no	—
166	C₄×C₇₉	—	8	1	no	—
167	C₄×C₇₉	#386	1	1	yes	characteristic
168	C₄×C₇₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₄×C₇₉	C₈×D₄	1	1	yes	characteristic
170	C₄×C₇₉	#435	1	1	yes	characteristic
171	#12979	—	79	1	no	maximal, Sylow
172	C₂×C₇₉	—	2	1	no	—
173	C₂×C₇₉	—	2	1	no	—
174	C₂×C₇₉	—	2	1	no	—
175	C₂×C₇₉	—	4	2	no	—
176	C₂×C₇₉	—	8	1	no	—
177	C₂×C₇₉	#1869	1	1	yes	characteristic
178	#3407	—	79	1	no	—
179	#2955	D₇₉	1	1	yes	characteristic
180	#2117	—	79	1	no	—
181	C₂×#350	—	79	1	no	—
182	#2113	—	79	1	no	—
183	#1379	—	79	1	no	—
184	#1378	—	79	1	no	—
185	C₇₉	#12979	1	1	yes	characteristic, complemented, Sylow
186	#575	Dic₇₉	1	1	yes	characteristic
187	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
188	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
189	#504	—	158	2	no	—
190	#504	—	158	2	no	—
191	C₂×M₃₂	—	79	1	no	—
192	C₂×M₃₂	—	79	1	no	—
193	C₂×M₃₂	—	79	1	no	—
194	C₂×M₃₂	—	158	2	no	—
195	C₂²×C₁₆	—	79	1	no	—
196	#444	—	2	2	no	—
197	#443	—	2	2	no	—
198	#430	Dic₇₉	1	1	yes	characteristic
199	#421	Dic₇₉	1	1	yes	characteristic
200	#421	Dic₇₉	1	1	yes	characteristic
201	#405	C₂×D₇₉	1	1	yes	characteristic
202	#350	—	158	2	no	—
203	#350	—	158	2	no	—
204	C₂²⋊C₁₆	—	158	2	no	—
205	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
206	C₂×#41	C₂×Dic₇₉	1	1	yes	characteristic
207	#137	—	2	1	no	—
208	#137	C₇₉⋊C₈	1	2	yes	—
209	AGL(1,ℤ/8)	—	2	1	no	—
210	AGL(1,ℤ/8)	C₇₉⋊C₈	1	2	yes	—
211	#135	—	2	2	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	2	1	no	—
216	#131	—	2	1	no	—
217	#131	—	2	1	no	—
218	#131	C₂²⋊D₇₉	1	2	yes	—
219	#131	C₂²⋊D₇₉	1	2	yes	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
223	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
224	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
228	#117	C₂×Dic₇₉	1	1	yes	characteristic
229	M₃₂	—	79	2	no	—
230	M₃₂	—	158	1	no	—
231	M₃₂	—	158	1	no	—
232	M₃₂	—	158	4	no	—
233	M₃₂	—	158	1	no	—
234	M₃₂	—	158	1	no	—
235	M₃₂	—	158	1	no	—
236	C₂×C₁₆	—	79	2	no	—
237	C₂×C₁₆	—	158	1	no	—
238	C₂×C₁₆	—	158	1	no	—
239	C₂×C₁₆	—	158	1	no	—
240	C₂×C₁₆	—	158	1	no	—
241	C₂×C₁₆	—	158	4	no	—
242	C₂×C₁₆	—	158	1	no	—
243	#108	—	2	2	no	—
244	#104	—	2	2	no	—
245	#104	—	2	2	no	—
246	#97	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	C₂²⋊Dic₇₉	1	2	yes	—
255	#41	C₂×C₇₉⋊C₈	1	2	yes	—
256	C₂×Q₈	C₂×C₇₉⋊C₈	1	1	yes	characteristic
257	C₂×Q₈	#11929608	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	C₂×C₇₉⋊C₈	1	1	yes	characteristic
261	C₂×D₄	#11929608	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
265	Q₁₆	—	2	2	no	—
266	SD₁₆	—	2	2	no	—
267	SD₁₆	—	2	2	no	—
268	D₈	—	2	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	C₂²⋊Dic₇₉	1	2	yes	—
278	M₁₆	C₂×C₇₉⋊C₈	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
287	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
288	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
289	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
290	C₂×C₈	#11929623	1	1	yes	characteristic
291	C₂×C₈	#11929623	1	1	yes	characteristic
292	C₄⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₁₆	—	316	1	no	—
296	C₁₆	—	316	1	no	—
297	C₁₆	—	158	2	no	—
298	C₁₆	—	158	2	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	#14579168	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	C₂²⋊(C₇₉⋊C₈)	1	2	yes	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
322	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
323	C₂×C₄	D₄×Dic₇₉	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₇₉⋊C₈	1	1	yes	characteristic
325	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
326	C₂×C₄	#14579168	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	4	1	no	—
330	C₈	#14579129	1	1	yes	characteristic
331	C₈	#14579133	1	1	yes	characteristic
332	C₈	#14579133	1	1	yes	characteristic
333	C₈	#14579142	1	1	yes	characteristic
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	#19098530	1	1	yes	characteristic
340	C₂²	D₄×C₇₉⋊C₈	1	1	yes	characteristic
341	C₂²	#19098635	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	8	1	no	—
345	C₄	#19098443	1	1	yes	characteristic
346	C₄	C₂×C₂²⋊(C₇₉⋊C₈)	1	1	yes	characteristic
347	C₄	D₄×C₇₉⋊C₈	1	1	yes	characteristic
348	C₄	#19098635	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	#23763693	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14579129
commutator	C₂×C₄×C₇₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2955	C₂
socle	C₂×C₇₉	#1869
prime radical	#575×C₇₉	C₄
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2955	D₇₉
hypocenter	C₇₉	#12979
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. #23776503	C₂	semidirect
2. #23765028	C₂	semidirect
3. C₂×#19098347	C₂	semidirect
4. #23765014	C₂	semidirect
5. #23761489	C₂	semidirect
6. #23761486	C₂	semidirect
185. C₇₉	#12979	semidirect

Maximal subgroups

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

subgroup	index
#23776503	2
#23765028	2
C₂×#19098347	2
#23765014	2
#23761489	2
#23761486	2
C₇₉×#2955	2
#12979	79