Subgroups — #34445073

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776550	C₂	1	1	yes	maximal, characteristic, complemented
2	#23775182	C₂	1	2	yes	maximal, complemented
3	#23765658	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765525	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765517	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765467	C₂	1	2	yes	maximal, complemented
7	#23765371	C₂	1	2	yes	maximal, complemented
8	#23765361	C₂	1	2	yes	maximal, complemented
9	#23765281	C₂	1	1	yes	maximal, characteristic, complemented
10	#23765235	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3415	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099377	C₂²	1	2	yes	complemented
13	#19099374	—	2	2	no	—
14	#19099003	C₂²	1	2	yes	complemented
15	#19099002	—	2	2	no	—
16	#19098992	—	2	1	no	—
17	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic, complemented
18	Q₃₂×D₇₉	—	2	1	no	—
19	C₂×#14579105	C₂²	1	2	yes	complemented
20	#19098979	C₂²	1	1	yes	characteristic, complemented
21	#19098946	C₂²	1	2	yes	complemented
22	#19098838	C₂²	1	2	yes	—
23	#19098830	C₂²	1	2	yes	complemented
24	#19098824	—	2	2	no	—
25	#19098741	—	2	2	no	—
26	#19098734	—	2	2	no	—
27	#19098680	C₂²	1	2	yes	complemented
28	#19098678	C₂²	1	1	yes	characteristic
29	#19098393	C₂²	1	1	yes	characteristic, complemented
30	#19098383	C₂²	1	1	yes	characteristic, complemented
31	#19098380	C₂²	1	2	yes	complemented
32	#19098376	C₂²	1	2	yes	complemented
33	#19098370	C₂²	1	2	yes	complemented
34	#19098348	C₂²	1	1	yes	characteristic, complemented
35	#19098347	C₂²	1	1	yes	characteristic
36	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic, complemented
37	#576×C₇₉	C₂²	1	2	yes	—
38	#509×C₇₉	—	2	2	no	—
39	#509×C₇₉	C₂²	1	2	yes	—
40	#508×C₇₉	—	2	1	no	—
41	#508×C₇₉	C₂²	1	1	yes	characteristic
42	C₂×D₁₆×C₇₉	—	2	1	no	—
43	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
44	#504×C₇₉	C₂²	1	1	yes	characteristic
45	#14579208	—	2	2	no	—
46	#14579191	—	2	2	no	—
47	#14579191	—	2	2	no	—
48	#14579191	—	4	2	no	—
49	#14579190	—	4	2	no	—
50	#14579187	—	2	2	no	—
51	#14579187	—	4	2	no	—
52	#14579181	—	4	2	no	—
53	#14579181	D₄	1	1	yes	characteristic
54	#14579180	—	2	2	no	—
55	C₂×Dic₃₁₆	C₂³	1	2	yes	—
56	Q₁₆×D₇₉	—	2	2	no	—
57	C₂×#11929612	—	2	2	no	—
58	C₂×#11929612	—	4	2	no	—
59	#14579168	—	2	2	no	—
60	#14579167	C₂³	1	1	yes	characteristic
61	#14579148	—	2	2	no	—
62	Dic₆₃₂	—	2	1	no	—
63	Dic₆₃₂	—	2	1	no	—
64	#14579107	—	2	2	no	—
65	#14579105	—	2	2	no	—
66	#14579105	—	4	2	no	—
67	#14579101	C₂³	1	1	yes	characteristic
68	#14579100	—	2	1	no	—
69	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₁₆×D₇₉	—	2	1	no	—
71	#14579097	C₂³	1	2	yes	—
72	#14579096	D₄	1	1	yes	characteristic
73	#14579088	—	2	2	no	—
74	#14579088	D₄	1	1	yes	characteristic
75	#14579087	D₄	1	1	yes	characteristic
76	#14579079	—	2	2	no	—
77	#14579078	—	2	2	no	—
78	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
79	H(5,𝔽₂)×C₇₉	—	2	2	no	—
80	AGL(1,ℤ/8)×C₇₉	—	2	2	no	—
81	AGL(1,ℤ/8)×C₇₉	—	4	2	no	—
82	#135×C₇₉	—	2	2	no	—
83	#135×C₇₉	C₂³	1	2	yes	—
84	C₂×D₈×C₇₉	—	2	2	no	—
85	C₂×D₈×C₇₉	C₂³	1	2	yes	—
86	#131×C₇₉	C₂³	1	1	yes	characteristic
87	Q₃₂×C₇₉	D₄	1	1	yes	characteristic
88	SD₃₂×C₇₉	—	2	2	no	—
89	SD₃₂×C₇₉	D₄	1	2	yes	—
90	D₁₆×C₇₉	—	2	2	no	—
91	D₁₆×C₇₉	—	2	1	no	—
92	D₁₆×C₇₉	—	2	1	no	—
93	D₁₆×C₇₉	D₄	1	1	yes	characteristic
94	M₃₂×C₇₉	—	2	1	no	—
95	M₃₂×C₇₉	C₂³	1	1	yes	characteristic
96	C₂×C₁₆×C₇₉	—	2	1	no	—
97	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
98	#11929624	—	4	2	no	—
99	#11929624	C₂×D₄	1	1	yes	characteristic
100	#11929623	—	4	2	no	—
101	#11929623	—	8	2	no	—
102	C₂×Dic₁₅₈	—	2	2	no	—
103	C₂×Dic₁₅₈	—	4	2	no	—
104	Q₈×D₇₉	—	4	2	no	—
105	Dic₃₁₆	—	2	2	no	—
106	Dic₃₁₆	—	2	2	no	—
107	#11929614	—	4	2	no	—
108	#11929614	—	4	2	no	—
109	#11929614	D₈	1	1	yes	characteristic
110	#11929612	—	4	2	no	—
111	#11929612	—	4	2	no	—
112	#11929612	—	4	2	no	—
113	#11929612	—	4	2	no	—
114	#11929612	—	4	2	no	—
115	#11929612	D₈	1	1	yes	characteristic
116	#11929611	D₈	1	1	yes	characteristic
117	#11929609	—	4	2	no	—
118	#11929609	D₈	1	1	yes	characteristic
119	#11929608	—	2	2	no	—
120	#11929608	—	4	2	no	—
121	#11929608	C₂×D₄	1	1	yes	characteristic
122	#11929607	—	2	1	no	—
123	C₂×C₇₉⋊C₈	—	4	2	no	—
124	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₈×D₇₉	—	2	1	no	—
126	C₂²⋊Dic₇₉	—	8	2	no	—
127	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
128	C₇₉⋊C₁₆	—	2	1	no	—
129	C₇₉⋊C₁₆	—	2	1	no	—
130	#41×C₇₉	—	2	2	no	—
131	#41×C₇₉	—	4	2	no	—
132	#41×C₇₉	—	4	2	no	—
133	#41×C₇₉	C₂×D₄	1	1	yes	characteristic
134	C₂×D₄×C₇₉	—	2	2	no	—
135	C₂×D₄×C₇₉	—	4	2	no	—
136	C₂×D₄×C₇₉	—	4	2	no	—
137	C₂×D₄×C₇₉	—	8	2	no	—
138	Q₁₆×C₇₉	C₂×D₄	1	2	yes	—
139	SD₁₆×C₇₉	—	2	2	no	—
140	SD₁₆×C₇₉	—	4	2	no	—
141	D₈×C₇₉	—	2	2	no	—
142	D₈×C₇₉	—	2	2	no	—
143	D₈×C₇₉	—	4	2	no	—
144	D₈×C₇₉	C₂×D₄	1	2	yes	—
145	M₁₆×C₇₉	—	2	1	no	—
146	M₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
147	C₂×C₈×C₇₉	—	2	1	no	—
148	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
149	C₁₆×C₇₉	—	2	1	no	—
150	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
151	C₁₆×C₇₉	C₂×D₄	1	1	yes	characteristic
152	C₂³×C₇₉	—	4	2	no	—
153	C₂³×C₇₉	—	8	2	no	—
154	Q₈×C₇₉	—	2	2	no	—
155	Q₈×C₇₉	C₂×D₈	1	1	yes	characteristic
156	D₄×C₇₉	—	2	1	no	—
157	D₄×C₇₉	—	2	2	no	—
158	D₄×C₇₉	—	4	2	no	—
159	D₄×C₇₉	—	4	2	no	—
160	D₄×C₇₉	—	4	2	no	—
161	D₄×C₇₉	—	8	2	no	—
162	D₄×C₇₉	—	8	2	no	—
163	D₄×C₇₉	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₇₉	—	2	1	no	—
165	C₂×C₄×C₇₉	—	4	2	no	—
166	C₂×C₄×C₇₉	—	8	2	no	—
167	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
168	C₂²⋊D₇₉	—	2	1	no	—
169	C₂²⋊D₇₉	—	8	2	no	—
170	C₂×Dic₇₉	—	2	1	no	—
171	C₂×Dic₇₉	—	4	2	no	—
172	C₂×Dic₇₉	—	8	2	no	—
173	D₃₁₆	C₂×D₈	1	1	yes	characteristic
174	C₄×D₇₉	—	2	1	no	—
175	C₄×D₇₉	—	8	2	no	—
176	Dic₁₅₈	—	4	2	no	—
177	Dic₁₅₈	—	4	2	no	—
178	Dic₁₅₈	—	8	2	no	—
179	Dic₁₅₈	C₂×D₈	1	1	yes	characteristic
180	C₈×C₇₉	—	2	1	no	—
181	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
182	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
183	C₇₉⋊C₈	—	2	1	no	—
184	C₇₉⋊C₈	—	4	2	no	—
185	C₇₉⋊C₈	C₂×D₈	1	1	yes	characteristic
186	C₇₉⋊C₈	C₂×D₈	1	1	yes	characteristic
187	C₂²×C₇₉	—	2	1	no	—
188	C₂²×C₇₉	—	4	2	no	—
189	C₂²×C₇₉	—	8	2	no	—
190	C₂²×C₇₉	—	16	2	no	—
191	C₂²×C₇₉	—	32	2	no	—
192	C₂²×C₇₉	#576	1	1	yes	characteristic
193	C₂×D₇₉	—	2	1	no	—
194	C₄×C₇₉	—	2	1	no	—
195	C₄×C₇₉	—	4	2	no	—
196	C₄×C₇₉	D₄²	1	1	yes	characteristic
197	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
198	Dic₇₉	—	2	1	no	—
199	Dic₇₉	—	4	1	no	—
200	Dic₇₉	—	8	2	no	—
201	#33859	—	79	1	no	maximal, Sylow
202	C₂×C₇₉	—	2	1	no	—
203	C₂×C₇₉	—	4	1	no	—
204	C₂×C₇₉	—	8	2	no	—
205	C₂×C₇₉	—	16	2	no	—
206	C₂×C₇₉	D₄×D₈	1	1	yes	characteristic
207	D₇₉	—	4	1	no	—
208	#3417	—	79	1	no	—
209	#3415	D₇₉	1	1	yes	characteristic, complemented
210	#3290	—	79	2	no	—
211	#2249	—	79	1	no	—
212	#2222	—	79	1	no	—
213	#2221	—	79	1	no	—
214	#2212	—	79	2	no	—
215	#2194	—	79	2	no	—
216	#2193	—	79	2	no	—
217	#2178	—	79	1	no	—
218	#2170	—	79	1	no	—
219	C₇₉	#33859	1	1	yes	characteristic, complemented, Sylow
220	#578	—	79	2	no	—
221	#577	—	158	2	no	—
222	#576	C₂×D₇₉	1	2	yes	—
223	#510	—	79	2	no	—
224	#510	—	158	2	no	—
225	#509	—	2	2	no	—
226	#509	C₂×D₇₉	1	2	yes	—
227	#508	—	2	1	no	—
228	#508	—	158	1	no	—
229	#508	C₂×D₇₉	1	1	yes	characteristic
230	C₂×Q₃₂	—	79	1	no	—
231	C₂×Q₃₂	—	158	1	no	—
232	C₂×SD₃₂	—	79	2	no	—
233	C₂×D₁₆	—	2	1	no	—
234	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
235	#504	—	79	1	no	—
236	#504	C₂×D₇₉	1	1	yes	characteristic
237	#495	—	79	2	no	—
238	#473	—	79	2	no	—
239	#472	—	79	2	no	—
240	#471	—	158	2	no	—
241	#455	—	158	2	no	—
242	#454	—	158	2	no	—
243	#443	—	79	1	no	—
244	#443	—	79	2	no	—
245	#368	—	79	1	no	—
246	#364	—	79	1	no	—
247	#362	—	79	2	no	—
248	#361	—	79	2	no	—
249	#359	—	79	2	no	—
250	#350	—	79	1	no	—
251	#350	—	79	1	no	—
252	C₄×C₁₆	—	79	1	no	—
253	2₋¹⁺⁴	—	158	2	no	—
254	H(5,𝔽₂)	—	2	2	no	—
255	#137	—	316	2	no	—
256	#137	—	316	2	no	—
257	#137	—	158	2	no	—
258	#137	—	158	2	no	—
259	AGL(1,ℤ/8)	—	2	2	no	—
260	AGL(1,ℤ/8)	—	4	2	no	—
261	AGL(1,ℤ/8)	—	316	2	no	—
262	AGL(1,ℤ/8)	—	158	2	no	—
263	#135	—	2	2	no	—
264	#135	—	316	2	no	—
265	#135	—	79	1	no	—
266	#135	—	158	2	no	—
267	#135	C₂²×D₇₉	1	2	yes	—
268	C₂×Q₁₆	—	79	2	no	—
269	C₂×Q₁₆	—	158	2	no	—
270	C₂×SD₁₆	—	316	2	no	—
271	C₂×SD₁₆	—	158	2	no	—
272	C₂×D₈	—	2	2	no	—
273	C₂×D₈	C₂²×D₇₉	1	2	yes	—
274	#131	—	79	1	no	—
275	#131	—	158	2	no	—
276	#131	C₂²×D₇₉	1	1	yes	characteristic
277	#124	—	158	2	no	—
278	Q₃₂	—	158	2	no	—
279	Q₃₂	—	158	1	no	—
280	Q₃₂	—	158	1	no	—
281	Q₃₂	C₂²⋊D₇₉	1	1	yes	characteristic
282	SD₃₂	—	2	2	no	—
283	SD₃₂	—	316	2	no	—
284	SD₃₂	—	158	2	no	—
285	SD₃₂	C₂²⋊D₇₉	1	2	yes	—
286	D₁₆	—	2	2	no	—
287	D₁₆	—	2	1	no	—
288	D₁₆	—	2	1	no	—
289	D₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
290	M₃₂	—	2	1	no	—
291	M₃₂	—	79	1	no	—
292	M₃₂	—	158	1	no	—
293	M₃₂	C₂²×D₇₉	1	1	yes	characteristic
294	C₂×C₁₆	—	2	1	no	—
295	C₂×C₁₆	—	79	1	no	—
296	C₂×C₁₆	—	158	1	no	—
297	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
298	#108	—	79	1	no	—
299	#108	—	79	2	no	—
300	#104	—	79	1	no	—
301	#104	—	79	1	no	—
302	#104	—	158	2	no	—
303	#101	—	158	2	no	—
304	#100	—	158	2	no	—
305	C₄×C₈	—	79	1	no	—
306	#41	—	2	2	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	316	2	no	—
310	#41	—	316	2	no	—
311	#41	—	79	1	no	—
312	#41	—	632	2	no	—
313	#41	D₄×D₇₉	1	1	yes	characteristic
314	C₂×Q₈	—	316	2	no	—
315	C₂×Q₈	—	316	2	no	—
316	C₂×Q₈	—	158	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	4	2	no	—
319	C₂×D₄	—	4	2	no	—
320	C₂×D₄	—	8	2	no	—
321	Q₁₆	—	316	2	no	—
322	Q₁₆	—	316	2	no	—
323	Q₁₆	—	79	1	no	—
324	Q₁₆	—	158	2	no	—
325	Q₁₆	—	158	2	no	—
326	Q₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
327	SD₁₆	—	2	2	no	—
328	SD₁₆	—	4	2	no	—
329	SD₁₆	—	316	2	no	—
330	SD₁₆	—	316	2	no	—
331	SD₁₆	—	316	2	no	—
332	SD₁₆	—	316	2	no	—
333	SD₁₆	—	316	2	no	—
334	SD₁₆	—	79	1	no	—
335	SD₁₆	—	79	1	no	—
336	D₈	—	2	2	no	—
337	D₈	—	2	2	no	—
338	D₈	—	4	2	no	—
339	D₈	—	316	2	no	—
340	D₈	—	79	1	no	—
341	D₈	C₂×C₂²⋊D₇₉	1	2	yes	—
342	M₁₆	—	2	1	no	—
343	M₁₆	—	316	2	no	—
344	M₁₆	—	79	1	no	—
345	M₁₆	—	158	2	no	—
346	M₁₆	—	158	1	no	—
347	M₁₆	D₄×D₇₉	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	316	2	no	—
350	C₂×C₈	—	79	1	no	—
351	C₂×C₈	—	158	1	no	—
352	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
353	C₂²⋊C₄	—	632	2	no	—
354	C₄²	—	79	1	no	—
355	C₁₆	—	2	1	no	—
356	C₁₆	—	158	1	no	—
357	C₁₆	—	158	1	no	—
358	C₁₆	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
359	C₁₆	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
360	C₂³	—	4	2	no	—
361	C₂³	—	8	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	316	2	no	—
364	Q₈	—	316	2	no	—
365	Q₈	—	79	1	no	—
366	Q₈	—	632	2	no	—
367	Q₈	D₈×D₇₉	1	1	yes	characteristic
368	D₄	—	2	1	no	—
369	D₄	—	2	2	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	2	no	—
375	D₄	—	79	1	no	—
376	D₄	—	632	2	no	—
377	D₄	—	158	1	no	—
378	D₄	D₈×D₇₉	1	1	yes	characteristic
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	8	2	no	—
382	C₂×C₄	—	316	2	no	—
383	C₂×C₄	—	632	2	no	—
384	C₂×C₄	—	632	2	no	—
385	C₂×C₄	—	158	1	no	—
386	C₂×C₄	—	158	1	no	—
387	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	316	2	no	—
390	C₈	—	79	1	no	—
391	C₈	—	79	1	no	—
392	C₈	—	158	1	no	—
393	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
394	C₈	#14579206	1	1	yes	characteristic
395	C₂²	—	2	1	no	—
396	C₂²	—	4	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	16	2	no	—
399	C₂²	—	32	2	no	—
400	C₂²	—	158	1	no	—
401	C₂²	#19099364	1	1	yes	characteristic
402	C₄	—	2	1	no	—
403	C₄	—	4	2	no	—
404	C₄	—	316	1	no	—
405	C₄	—	632	2	no	—
406	C₄	—	158	1	no	—
407	C₄	D₄×C₂²⋊D₇₉	1	1	yes	characteristic
408	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
409	C₂	—	2	1	no	—
410	C₂	—	4	1	no	—
411	C₂	—	8	2	no	—
412	C₂	—	16	2	no	—
413	C₂	—	316	1	no	—
414	C₂	D₈×C₂²⋊D₇₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×C₂²⋊D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3415	C₂
socle	C₂×C₇₉	D₄×D₈
hypercenter	#3415	D₇₉
hypocenter	C₇₉	#33859
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #3415

Complements

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

N	H ≅ G/N	type
1. #23776550	C₂	semidirect
2. #23775182	C₂	semidirect
3. #23765658	C₂	semidirect
4. #23765525	C₂	semidirect
5. #23765517	C₂	semidirect
6. #23765467	C₂	semidirect
7. #23765371	C₂	semidirect
8. #23765361	C₂	semidirect
9. #23765281	C₂	semidirect
10. #23765235	C₂	semidirect
11. C₇₉×#3415	C₂	semidirect
12. #19099377	C₂²	semidirect
14. #19099003	C₂²	semidirect
17. C₂×Dic₆₃₂	C₂²	semidirect
19. C₂×#14579105	C₂²	semidirect
20. #19098979	C₂²	semidirect
21. #19098946	C₂²	semidirect
23. #19098830	C₂²	semidirect
27. #19098680	C₂²	semidirect
29. #19098393	C₂²	semidirect
30. #19098383	C₂²	semidirect
31. #19098380	C₂²	semidirect
32. #19098376	C₂²	semidirect
33. #19098370	C₂²	semidirect
34. #19098348	C₂²	semidirect
36. C₁₆×Dic₇₉	C₂²	semidirect
209. #3415	D₇₉	semidirect
219. C₇₉	#33859	semidirect

Maximal subgroups

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

subgroup	index
#23776550	2
#23775182	2
#23765658	2
#23765525	2
#23765517	2
#23765467	2
#23765371	2
#23765361	2
#23765281	2
#23765235	2
C₇₉×#3415	2
#33859	79