Subgroups — #34205236

Order 20224.

512 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772000	C₂	1	1	yes	maximal, characteristic, complemented
2	#23765116	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098364	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19098347	C₂	1	1	yes	maximal, characteristic, complemented
5	#23761510	C₂	1	1	yes	maximal, characteristic, complemented
6	#23761497	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#472×C₇₉	C₂	1	1	yes	maximal, characteristic
8	#19099046	—	2	1	no	—
9	C₂×#14579101	C₂²	1	1	yes	characteristic
10	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#19098605	—	2	2	no	—
12	#19098605	—	2	1	no	—
13	#19098601	—	2	1	no	—
14	#19098601	C₂²	1	1	yes	characteristic
15	C₂×#14579095	—	2	1	no	—
16	C₂×#14579095	C₂²	1	1	yes	characteristic
17	#19098364	—	2	2	no	—
18	#19098364	—	2	1	no	—
19	#19098364	—	2	1	no	—
20	#19098347	—	2	2	no	—
21	C₂×AGL(1,ℤ/8)×C₇₉	—	2	1	no	—
22	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
23	C₂×#131×C₇₉	C₂²	1	1	yes	characteristic
24	#472×C₇₉	—	2	2	no	—
25	#472×C₇₉	C₄	1	4	yes	complemented
26	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
27	C₂×#100×C₇₉	—	2	1	no	—
28	Q₈×Dic₇₉	—	4	1	no	—
29	D₄×Dic₇₉	—	4	2	no	—
30	D₄×Dic₇₉	—	4	1	no	—
31	#14579120	—	2	2	no	—
32	#14579120	—	2	1	no	—
33	C₂×C₄⋊Dic₇₉	—	2	2	no	—
34	C₂×C₄⋊Dic₇₉	—	2	1	no	—
35	#14579101	—	2	1	no	—
36	#14579101	—	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	—	2	1	no	—
41	#14579095	—	2	2	no	—
42	#14579095	—	2	1	no	—
43	#14579095	D₄	1	1	yes	characteristic
44	#14579095	D₄	1	2	yes	—
45	#14579095	D₄	1	1	yes	characteristic
46	#14579093	—	2	2	no	—
47	#14579093	—	2	1	no	—
48	C₂×#41×C₇₉	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₇₉	—	2	1	no	—
50	AGL(1,ℤ/8)×C₇₉	—	2	4	no	—
51	AGL(1,ℤ/8)×C₇₉	—	4	2	no	—
52	C₂×SD₁₆×C₇₉	—	4	1	no	—
53	C₂×D₈×C₇₉	—	2	1	no	—
54	C₂×D₈×C₇₉	—	2	1	no	—
55	C₂×D₈×C₇₉	—	2	1	no	—
56	C₂×D₈×C₇₉	—	2	1	no	—
57	C₂×D₈×C₇₉	—	2	1	no	—
58	C₂×D₈×C₇₉	—	4	1	no	—
59	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	complemented
60	#131×C₇₉	—	2	1	no	—
61	#131×C₇₉	—	2	1	no	—
62	#131×C₇₉	C₂×C₄	1	2	yes	—
63	#131×C₇₉	D₄	1	2	yes	—
64	C₂×M₁₆×C₇₉	—	2	1	no	—
65	C₂×M₁₆×C₇₉	—	2	1	no	—
66	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₇₉	—	2	1	no	—
68	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
69	#108×C₇₉	—	2	1	no	—
70	#108×C₇₉	C₂×C₄	1	2	yes	complemented
71	#100×C₇₉	—	2	4	no	—
72	C₂²×Dic₇₉	—	4	2	no	—
73	C₂²×Dic₇₉	—	4	1	no	—
74	C₄⋊Dic₇₉	—	2	1	no	—
75	C₄⋊Dic₇₉	—	2	2	no	—
76	C₄⋊Dic₇₉	—	2	1	no	—
77	C₄⋊Dic₇₉	—	4	2	no	—
78	C₄⋊Dic₇₉	—	4	1	no	—
79	C₂²⋊Dic₇₉	—	4	2	no	—
80	C₂²⋊Dic₇₉	—	4	1	no	—
81	C₄×Dic₇₉	—	4	2	no	—
82	C₄×Dic₇₉	—	4	1	no	—
83	C₇₉⋊C₁₆	—	4	1	no	—
84	C₇₉⋊C₁₆	—	4	1	no	—
85	C₂⁴×C₇₉	—	4	1	no	—
86	#41×C₇₉	—	2	1	no	—
87	#41×C₇₉	—	2	1	no	—
88	#41×C₇₉	C₂²⋊C₄	1	2	yes	—
89	#41×C₇₉	SD₁₆	1	2	yes	—
90	C₂×Q₈×C₇₉	D₈	1	1	yes	characteristic
91	C₂×D₄×C₇₉	—	2	1	no	—
92	C₂×D₄×C₇₉	—	2	2	no	—
93	C₂×D₄×C₇₉	—	4	1	no	—
94	C₂×D₄×C₇₉	—	4	1	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	D₈	1	1	yes	characteristic
99	C₂²×C₄×C₇₉	—	2	1	no	—
100	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
101	SD₁₆×C₇₉	—	4	4	no	—
102	D₈×C₇₉	—	2	2	no	—
103	D₈×C₇₉	—	2	2	no	—
104	D₈×C₇₉	—	4	1	no	—
105	D₈×C₇₉	—	4	1	no	—
106	D₈×C₇₉	—	4	4	no	—
107	M₁₆×C₇₉	—	2	2	no	—
108	M₁₆×C₇₉	—	2	2	no	—
109	M₁₆×C₇₉	—	2	2	no	—
110	M₁₆×C₇₉	—	2	1	no	—
111	M₁₆×C₇₉	—	4	1	no	—
112	M₁₆×C₇₉	C₂²⋊C₄	1	2	yes	—
113	C₂×C₈×C₇₉	—	2	2	no	—
114	C₂×C₈×C₇₉	—	2	2	no	—
115	C₂×C₈×C₇₉	—	2	1	no	—
116	C₂×C₈×C₇₉	—	4	1	no	—
117	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
122	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
123	C₂³×C₇₉	—	2	1	no	—
124	C₂³×C₇₉	—	4	2	no	—
125	C₂³×C₇₉	—	8	1	no	—
126	C₂³×C₇₉	—	8	1	no	—
127	C₂³×C₇₉	—	16	1	no	—
128	C₂³×C₇₉	—	16	1	no	—
129	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	Q₈×C₇₉	—	2	1	no	—
131	Q₈×C₇₉	#102	1	2	yes	—
132	D₄×C₇₉	—	2	2	no	—
133	D₄×C₇₉	—	2	2	no	—
134	D₄×C₇₉	—	2	1	no	—
135	D₄×C₇₉	—	4	2	no	—
136	D₄×C₇₉	—	4	2	no	—
137	D₄×C₇₉	—	8	1	no	—
138	D₄×C₇₉	—	8	1	no	—
139	D₄×C₇₉	#102	1	2	yes	—
140	C₂×C₄×C₇₉	—	2	1	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	2	2	no	—
143	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₇₉	#120	1	1	yes	characteristic
145	C₂×C₄×C₇₉	#123	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂×SD₁₆	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	C₂×Dic₇₉	—	4	1	no	—
152	C₂×Dic₇₉	—	16	1	no	—
153	C₂×Dic₇₉	—	16	2	no	—
154	C₈×C₇₉	—	2	2	no	—
155	C₈×C₇₉	—	4	2	no	—
156	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
157	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
158	C₈×C₇₉	#121	1	1	yes	characteristic
159	C₈×C₇₉	#121	1	1	yes	characteristic
160	C₂²×C₇₉	—	2	1	no	—
161	C₂²×C₇₉	—	2	1	no	—
162	C₂²×C₇₉	—	2	2	no	—
163	C₂²×C₇₉	—	4	1	no	—
164	C₂²×C₇₉	—	4	1	no	—
165	C₂²×C₇₉	—	8	2	no	—
166	C₂²×C₇₉	—	16	2	no	—
167	C₂²×C₇₉	—	16	2	no	—
168	C₂²×C₇₉	—	16	1	no	—
169	C₂²×C₇₉	—	16	1	no	—
170	C₂²×C₇₉	#417	1	1	yes	characteristic
171	C₂²×C₇₉	#447	1	1	yes	characteristic
172	C₂²×C₇₉	#481	1	1	yes	characteristic
173	C₄×C₇₉	—	2	2	no	—
174	C₄×C₇₉	#386	1	1	yes	characteristic
175	C₄×C₇₉	C₂×#102	1	1	yes	characteristic
176	C₄×C₇₉	#448	1	1	yes	characteristic
177	C₄×C₇₉	#480	1	1	yes	characteristic
178	Dic₇₉	—	8	2	no	—
179	Dic₇₉	—	8	1	no	—
180	Dic₇₉	—	8	1	no	—
181	#12996	—	79	1	no	maximal, Sylow
182	C₂×C₇₉	—	2	1	no	—
183	C₂×C₇₉	—	2	1	no	—
184	C₂×C₇₉	—	4	2	no	—
185	C₂×C₇₉	—	16	2	no	—
186	C₂×C₇₉	#1874	1	1	yes	characteristic
187	C₂×C₇₉	#2192	1	1	yes	characteristic
188	C₂×C₇₉	#2193	1	1	yes	characteristic
189	C₂×#472	D₇₉	1	1	yes	characteristic
190	#2913	—	79	1	no	—
191	#2140	—	79	1	no	—
192	C₂×#357	—	79	1	no	—
193	C₂×#350	—	79	1	no	—
194	#1387	—	79	1	no	—
195	#1382	—	79	1	no	—
196	C₇₉	#12996	1	1	yes	characteristic, complemented, Sylow
197	C₂×AGL(1,ℤ/8)	—	2	1	no	—
198	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
199	C₂×#131	C₂×D₇₉	1	1	yes	characteristic
200	#520	—	158	1	no	—
201	C₂×M₃₂	—	79	1	no	—
202	C₂²×C₁₆	—	79	1	no	—
203	#472	—	2	2	no	—
204	#472	Dic₇₉	1	4	yes	complemented
205	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
206	#428	—	158	2	no	—
207	#428	—	158	1	no	—
208	#427	—	79	1	no	—
209	#427	—	158	1	no	—
210	C₂×#107	—	79	1	no	—
211	C₂×#107	—	158	1	no	—
212	C₂×#100	—	2	1	no	—
213	#357	—	158	2	no	—
214	#357	—	158	1	no	—
215	#357	—	158	1	no	—
216	#350	—	158	2	no	—
217	C₂×#41	C₂²⋊D₇₉	1	1	yes	characteristic
218	C₂²×D₄	—	2	1	no	—
219	AGL(1,ℤ/8)	—	2	4	no	—
220	AGL(1,ℤ/8)	—	4	2	no	—
221	C₂×SD₁₆	—	4	1	no	—
222	C₂×D₈	—	2	1	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	—	2	1	no	—
225	C₂×D₈	—	2	1	no	—
226	C₂×D₈	—	2	1	no	—
227	C₂×D₈	—	4	1	no	—
228	C₂×D₈	C₂×Dic₇₉	1	2	yes	complemented
229	#131	—	2	1	no	—
230	#131	—	2	1	no	—
231	#131	C₂×Dic₇₉	1	2	yes	—
232	#131	C₂²⋊D₇₉	1	2	yes	—
233	C₂×M₁₆	—	2	1	no	—
234	C₂×M₁₆	—	2	1	no	—
235	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
236	C₂²×C₈	—	2	1	no	—
237	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
238	C₄×Q₈	—	316	1	no	—
239	C₄×D₄	—	316	2	no	—
240	C₄×D₄	—	316	1	no	—
241	#117	—	158	2	no	—
242	#117	—	158	1	no	—
243	C₂×C₄⋊C₄	—	158	2	no	—
244	C₂×C₄⋊C₄	—	158	1	no	—
245	M₃₂	—	158	1	no	—
246	M₃₂	—	158	1	no	—
247	C₂×C₁₆	—	158	1	no	—
248	C₂×C₁₆	—	158	1	no	—
249	C₂×C₁₆	—	158	1	no	—
250	C₂×C₁₆	—	158	1	no	—
251	#108	—	2	1	no	—
252	#108	C₂×Dic₇₉	1	2	yes	complemented
253	#107	—	79	2	no	—
254	#107	—	79	1	no	—
255	#107	—	79	1	no	—
256	#107	—	158	2	no	—
257	#107	—	158	1	no	—
258	#106	—	158	2	no	—
259	#106	—	158	1	no	—
260	#100	—	2	4	no	—
261	C₂⁴	—	4	1	no	—
262	#41	—	2	1	no	—
263	#41	—	2	1	no	—
264	#41	C₂²⋊Dic₇₉	1	2	yes	—
265	#41	#11929612	1	2	yes	—
266	C₂×Q₈	#11929609	1	1	yes	characteristic
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	#11929609	1	1	yes	characteristic
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	316	2	no	—
277	C₂²×C₄	—	316	1	no	—
278	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
279	SD₁₆	—	4	4	no	—
280	D₈	—	2	2	no	—
281	D₈	—	2	2	no	—
282	D₈	—	4	4	no	—
283	D₈	—	4	1	no	—
284	D₈	—	4	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	4	1	no	—
290	M₁₆	C₂²⋊Dic₇₉	1	2	yes	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
296	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
297	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
299	C₂×C₈	#11929623	1	1	yes	characteristic
300	C₂×C₈	#11929623	1	1	yes	characteristic
301	C₄⋊C₄	—	316	2	no	—
302	C₄⋊C₄	—	316	1	no	—
303	C₄⋊C₄	—	158	1	no	—
304	C₄⋊C₄	—	158	2	no	—
305	C₄⋊C₄	—	158	1	no	—
306	C₂²⋊C₄	—	316	2	no	—
307	C₂²⋊C₄	—	316	1	no	—
308	C₄²	—	316	2	no	—
309	C₄²	—	316	1	no	—
310	C₁₆	—	316	1	no	—
311	C₁₆	—	316	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	4	2	no	—
314	C₂³	—	8	1	no	—
315	C₂³	—	8	1	no	—
316	C₂³	—	16	1	no	—
317	C₂³	—	16	1	no	—
318	C₂³	#14579186	1	1	yes	characteristic
319	Q₈	—	2	1	no	—
320	Q₈	#14579082	1	2	yes	—
321	D₄	—	2	2	no	—
322	D₄	—	2	2	no	—
323	D₄	—	2	1	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	8	1	no	—
327	D₄	—	8	1	no	—
328	D₄	#14579082	1	2	yes	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	316	1	no	—
333	C₂×C₄	—	316	2	no	—
334	C₂×C₄	—	316	1	no	—
335	C₂×C₄	—	1264	1	no	—
336	C₂×C₄	—	1264	2	no	—
337	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
338	C₂×C₄	#14579129	1	1	yes	characteristic
339	C₂×C₄	#14579142	1	1	yes	characteristic
340	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
341	C₂×C₄	C₂×#11929612	1	1	yes	characteristic
342	C₂×C₄	#14579191	1	1	yes	characteristic
343	C₈	—	2	2	no	—
344	C₈	—	4	2	no	—
345	C₈	D₄×Dic₇₉	1	1	yes	characteristic
346	C₈	D₄×Dic₇₉	1	1	yes	characteristic
347	C₈	#14579133	1	1	yes	characteristic
348	C₈	#14579133	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	4	1	no	—
353	C₂²	—	4	1	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	16	2	no	—
356	C₂²	—	16	2	no	—
357	C₂²	—	16	1	no	—
358	C₂²	—	16	1	no	—
359	C₂²	#19098564	1	1	yes	characteristic
360	C₂²	#19098696	1	1	yes	characteristic
361	C₂²	#19098879	1	1	yes	characteristic
362	C₄	—	2	2	no	—
363	C₄	—	632	2	no	—
364	C₄	—	632	1	no	—
365	C₄	—	632	1	no	—
366	C₄	#19098443	1	1	yes	characteristic
367	C₄	C₂×#14579082	1	1	yes	characteristic
368	C₄	#19098701	1	1	yes	characteristic
369	C₄	#19098874	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	4	2	no	—
373	C₂	—	16	2	no	—
374	C₂	#23763718	1	1	yes	characteristic
375	C₂	#23765355	1	1	yes	characteristic
376	C₂	#23765362	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098696
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting, prime radical	C₂×#472×C₇₉	C₂
socle	C₂²×C₇₉	#447
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	C₂×#472	D₇₉
hypocenter	C₇₉	#12996
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#472

Complements

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

N	H ≅ G/N	type
1. #23772000	C₂	semidirect
2. #23765116	C₂	semidirect
3. C₂×#19098364	C₂	semidirect
4. C₂×#19098347	C₂	semidirect
5. #23761510	C₂	semidirect
6. #23761497	C₂	semidirect
25. #472×C₇₉	C₄	semidirect
59. C₂×D₈×C₇₉	C₂×C₄	semidirect
70. #108×C₇₉	C₂×C₄	semidirect
196. C₇₉	#12996	semidirect
204. #472	Dic₇₉	semidirect
228. C₂×D₈	C₂×Dic₇₉	semidirect
252. #108	C₂×Dic₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23772000	2
#23765116	2
C₂×#19098364	2
C₂×#19098347	2
#23761510	2
#23761497	2
C₂×#472×C₇₉	2
#12996	79