Subgroups — #34443606

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773886	C₂	1	1	yes	maximal, characteristic
2	#23765643	C₂	1	2	yes	maximal, complemented
3	#23765639	C₂	1	2	yes	maximal, complemented
4	#23765634	C₂	1	2	yes	maximal, complemented
5	#23765625	C₂	1	2	yes	maximal, complemented
6	#23765115	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765110	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765105	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19098363	C₂	1	1	yes	maximal, characteristic, complemented
10	#23764997	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579189	—	2	1	no	—
13	C₂×#14579183	C₂²	1	1	yes	characteristic
14	C₂²×D₆₃₂	C₂²	1	1	yes	characteristic
15	#19099132	—	2	1	no	—
16	C₂×#14579101	C₂²	1	1	yes	characteristic
17	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
18	#19098952	—	2	2	no	—
19	#19098947	C₂²	1	2	yes	—
20	#19098940	C₂²	1	2	yes	—
21	#19098366	C₂²	1	1	yes	characteristic, complemented
22	#19098366	C₂²	1	1	yes	characteristic, complemented
23	#19098366	C₂²	1	2	yes	complemented
24	#19098363	C₂²	1	2	yes	—
25	#19098363	C₂²	1	2	yes	—
26	#19098363	C₂²	1	2	yes	—
27	#19098363	C₂²	1	2	yes	—
28	#19098363	C₂²	1	1	yes	characteristic, complemented
29	#19098363	C₂²	1	1	yes	characteristic, complemented
30	#19098363	C₂²	1	2	yes	complemented
31	#19098339	C₂²	1	2	yes	complemented
32	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	complemented
33	#544×C₇₉	—	2	1	no	—
34	#501×C₇₉	—	2	2	no	—
35	#500×C₇₉	C₂²	1	2	yes	complemented
36	#499×C₇₉	C₂²	1	2	yes	complemented
37	#428×C₇₉	—	2	1	no	—
38	#427×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
40	#405×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×#11929624	—	2	1	no	—
42	C₂²×D₃₁₆	—	2	1	no	—
43	#14579189	—	4	4	no	—
44	#14579183	—	2	2	no	—
45	#14579183	—	2	2	no	—
46	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
47	C₂×#11929613	—	2	2	no	—
48	C₂×#11929613	—	2	1	no	—
49	C₂×D₆₃₂	—	2	2	no	—
50	C₂×D₆₃₂	—	2	2	no	—
51	C₂×D₆₃₂	—	2	2	no	—
52	C₂×D₆₃₂	—	2	1	no	—
53	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
54	C₂×D₆₃₂	C₂³	1	2	yes	—
55	#14579158	—	2	2	no	—
56	#14579149	—	2	2	no	—
57	#14579136	—	4	1	no	—
58	#14579136	—	4	1	no	—
59	#14579131	—	4	2	no	—
60	#14579101	—	2	2	no	—
61	C₂×C₇₉⋊C₁₆	—	2	2	no	—
62	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
65	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₇₉	—	2	2	no	—
68	C₄⋊Q₈×C₇₉	—	4	1	no	—
69	#125×C₇₉	—	2	2	no	—
70	#125×C₇₉	—	4	1	no	—
71	C₂²⋊Q₈×C₇₉	—	4	2	no	—
72	#117×C₇₉	—	2	1	no	—
73	#117×C₇₉	D₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
75	#107×C₇₉	—	2	1	no	—
76	#107×C₇₉	—	2	2	no	—
77	#107×C₇₉	C₂³	1	2	yes	—
78	#107×C₇₉	C₂³	1	1	yes	characteristic
79	#107×C₇₉	C₂³	1	1	yes	characteristic
80	#106×C₇₉	—	2	1	no	—
81	#106×C₇₉	—	2	2	no	—
82	#97×C₇₉	D₄	1	2	yes	—
83	C₄×C₈×C₇₉	C₂³	1	2	yes	—
84	C₂³×D₇₉	—	4	1	no	—
85	#11929624	—	4	2	no	—
86	#11929624	—	4	2	no	—
87	C₂×Dic₁₅₈	—	2	1	no	—
88	C₂×D₃₁₆	—	2	1	no	—
89	C₂×D₃₁₆	—	2	2	no	—
90	C₂×D₃₁₆	—	4	2	no	—
91	C₂×D₃₁₆	—	4	1	no	—
92	C₂×D₃₁₆	—	4	2	no	—
93	C₂×D₃₁₆	—	4	2	no	—
94	C₂×D₃₁₆	—	4	2	no	—
95	C₂×C₂²⋊D₇₉	—	4	1	no	—
96	C₂×C₄×D₇₉	—	4	1	no	—
97	Dic₃₁₆	—	2	1	no	—
98	Dic₃₁₆	—	2	1	no	—
99	#11929613	—	4	2	no	—
100	#11929613	—	4	4	no	—
101	D₆₃₂	—	2	2	no	—
102	D₆₃₂	—	2	2	no	—
103	D₆₃₂	—	2	1	no	—
104	D₆₃₂	—	2	1	no	—
105	D₆₃₂	—	4	2	no	—
106	D₆₃₂	—	4	4	no	—
107	(C₂×D₇₉)⋊C₄	—	4	2	no	—
108	(C₂×D₇₉)⋊C₄	—	4	2	no	—
109	C₇₉⋊C₁₆	—	2	1	no	—
110	C₇₉⋊C₁₆	—	2	1	no	—
111	C₇₉⋊C₁₆	—	2	2	no	—
112	C₂×Q₈×C₇₉	—	4	2	no	—
113	C₂²×C₄×C₇₉	—	4	1	no	—
114	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
115	M₁₆×C₇₉	SD₁₆	1	4	yes	—
116	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₇₉	#41	1	2	yes	—
119	C₂×C₈×C₇₉	#41	1	2	yes	—
120	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₇₉	—	2	1	no	—
122	C₄⋊C₄×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	2	1	no	—
124	C₄⋊C₄×C₇₉	—	4	1	no	—
125	C₄⋊C₄×C₇₉	—	4	2	no	—
126	C₄⋊C₄×C₇₉	—	4	2	no	—
127	C₄⋊C₄×C₇₉	—	4	2	no	—
128	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
129	C₄⋊C₄×C₇₉	D₈	1	1	yes	characteristic
130	C₂²⋊C₄×C₇₉	—	4	1	no	—
131	C₂²⋊C₄×C₇₉	D₈	1	2	yes	—
132	C₄²×C₇₉	—	4	1	no	—
133	C₄²×C₇₉	C₂×D₄	1	2	yes	—
134	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₇₉	—	4	2	no	—
136	C₂²×D₇₉	—	4	1	no	—
137	C₂²×D₇₉	—	8	2	no	—
138	C₂²×D₇₉	—	16	1	no	—
139	C₂²×D₇₉	—	16	1	no	—
140	Q₈×C₇₉	—	16	2	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	—	16	1	no	—
145	C₂×C₄×C₇₉	#124	1	2	yes	—
146	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
150	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₇₉	—	8	2	no	—
152	C₂×Dic₇₉	—	4	1	no	—
153	D₃₁₆	—	4	2	no	—
154	D₃₁₆	—	4	1	no	—
155	D₃₁₆	—	4	1	no	—
156	D₃₁₆	—	4	2	no	—
157	D₃₁₆	—	8	2	no	—
158	D₃₁₆	—	16	2	no	—
159	D₃₁₆	—	16	2	no	—
160	C₄×D₇₉	—	8	2	no	—
161	Dic₁₅₈	—	4	1	no	—
162	Dic₁₅₈	—	4	1	no	—
163	C₈×C₇₉	#124	1	2	yes	—
164	C₈×C₇₉	C₂×SD₁₆	1	4	yes	—
165	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
166	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
167	C₂²×C₇₉	—	2	1	no	—
168	C₂²×C₇₉	—	2	1	no	—
169	C₂²×C₇₉	#486	1	2	yes	—
170	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₇₉	—	8	1	no	—
172	C₂×D₇₉	—	8	2	no	—
173	C₂×D₇₉	—	8	2	no	—
174	C₂×D₇₉	—	8	1	no	—
175	C₂×D₇₉	—	8	2	no	—
176	C₂×D₇₉	—	8	1	no	—
177	C₂×D₇₉	—	16	4	no	—
178	C₄×C₇₉	—	2	2	no	—
179	C₄×C₇₉	—	2	2	no	—
180	C₄×C₇₉	—	8	2	no	—
181	C₄×C₇₉	—	8	1	no	—
182	C₄×C₇₉	—	8	1	no	—
183	C₄×C₇₉	#486	1	2	yes	—
184	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
185	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
186	Dic₇₉	—	8	1	no	—
187	#33702	—	79	1	no	maximal, Sylow
188	C₂×C₇₉	—	2	2	no	—
189	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
190	C₂×C₇₉	#3415	1	1	yes	characteristic
191	C₂×C₇₉	#3416	1	1	yes	characteristic
192	D₇₉	—	16	1	no	—
193	D₇₉	—	16	2	no	—
194	#3164	D₇₉	1	1	yes	characteristic, complemented
195	#3155	—	79	1	no	—
196	#2244	—	79	2	no	—
197	#2243	—	79	2	no	—
198	#2242	—	79	2	no	—
199	#2240	—	79	2	no	—
200	#2140	—	79	1	no	—
201	#2139	—	79	1	no	—
202	#2138	—	79	1	no	—
203	C₂×#357	—	79	1	no	—
204	#2108	—	79	1	no	—
205	C₇₉	#33702	1	1	yes	characteristic, complemented, Sylow
206	C₂×AGL(1,ℤ/8)	—	158	1	no	—
207	C₂×#135	—	79	1	no	—
208	C₂²×D₈	—	79	1	no	—
209	#544	—	2	1	no	—
210	#535	—	158	1	no	—
211	C₂×M₃₂	—	79	1	no	—
212	C₂²×C₁₆	—	79	1	no	—
213	#501	—	2	2	no	—
214	#500	C₂×D₇₉	1	2	yes	complemented
215	#499	C₂×D₇₉	1	2	yes	complemented
216	#496	—	158	2	no	—
217	#495	—	79	2	no	—
218	#493	—	79	2	no	—
219	#428	—	2	1	no	—
220	#427	C₂×D₇₉	1	1	yes	characteristic
221	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
222	#405	C₂×D₇₉	1	1	yes	characteristic
223	#358	—	79	1	no	—
224	#358	—	79	1	no	—
225	#358	—	79	2	no	—
226	#357	—	79	2	no	—
227	#357	—	79	2	no	—
228	#357	—	79	1	no	—
229	#357	—	79	1	no	—
230	#357	—	79	2	no	—
231	#357	—	79	2	no	—
232	#357	—	79	2	no	—
233	#346	—	79	2	no	—
234	C₄×C₁₆	—	79	2	no	—
235	C₂×#41	—	158	1	no	—
236	C₂²×D₄	—	158	1	no	—
237	AGL(1,ℤ/8)	—	316	4	no	—
238	#135	—	158	2	no	—
239	#135	—	158	2	no	—
240	C₂×Q₁₆	—	79	1	no	—
241	C₂×SD₁₆	—	158	1	no	—
242	C₂×SD₁₆	—	158	2	no	—
243	C₂×D₈	—	79	2	no	—
244	C₂×D₈	—	79	1	no	—
245	C₂×D₈	—	158	1	no	—
246	C₂×D₈	—	158	2	no	—
247	C₂×D₈	—	158	2	no	—
248	C₂×D₈	—	158	2	no	—
249	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
251	C₄⋊Q₈	—	2	2	no	—
252	C₄⋊Q₈	—	4	1	no	—
253	#127	—	158	2	no	—
254	#125	—	2	2	no	—
255	#125	—	4	1	no	—
256	#124	—	158	2	no	—
257	C₂²⋊Q₈	—	4	2	no	—
258	#121	—	316	1	no	—
259	#121	—	316	1	no	—
260	#120	—	316	2	no	—
261	#117	—	2	1	no	—
262	#117	C₂²⋊D₇₉	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	2	1	no	—
264	M₃₂	—	158	2	no	—
265	C₂×C₁₆	—	79	1	no	—
266	C₂×C₁₆	—	79	2	no	—
267	C₂×C₁₆	—	79	1	no	—
268	C₂×C₁₆	—	158	2	no	—
269	#107	—	2	1	no	—
270	#107	—	2	2	no	—
271	#107	C₂²×D₇₉	1	2	yes	—
272	#107	C₂²×D₇₉	1	1	yes	characteristic
273	#107	C₂²×D₇₉	1	1	yes	characteristic
274	#106	—	2	1	no	—
275	#106	—	2	2	no	—
276	#97	C₂²⋊D₇₉	1	2	yes	—
277	C₄×C₈	C₂²×D₇₉	1	2	yes	—
278	C₂⁴	—	316	1	no	—
279	#41	—	316	2	no	—
280	#41	—	316	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	—	158	1	no	—
283	C₂×D₄	—	316	1	no	—
284	C₂×D₄	—	316	2	no	—
285	C₂×D₄	—	316	1	no	—
286	C₂×D₄	—	316	2	no	—
287	C₂×D₄	—	316	2	no	—
288	C₂×D₄	—	316	2	no	—
289	C₂×D₄	—	158	2	no	—
290	C₂×D₄	—	158	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	316	1	no	—
293	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
294	Q₁₆	—	158	1	no	—
295	Q₁₆	—	158	1	no	—
296	SD₁₆	—	316	2	no	—
297	SD₁₆	—	316	4	no	—
298	D₈	—	316	2	no	—
299	D₈	—	316	4	no	—
300	D₈	—	158	2	no	—
301	D₈	—	158	2	no	—
302	D₈	—	158	1	no	—
303	D₈	—	158	1	no	—
304	M₁₆	#11929611	1	4	yes	—
305	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
307	C₂×C₈	#11929625	1	2	yes	—
308	C₂×C₈	#11929625	1	2	yes	—
309	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	—	4	2	no	—
317	C₄⋊C₄	#11929609	1	1	yes	characteristic
318	C₄⋊C₄	#11929609	1	1	yes	characteristic
319	C₂²⋊C₄	—	4	1	no	—
320	C₂²⋊C₄	—	316	2	no	—
321	C₂²⋊C₄	—	316	2	no	—
322	C₂²⋊C₄	#11929609	1	2	yes	—
323	C₄²	—	4	1	no	—
324	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
325	C₁₆	—	158	1	no	—
326	C₁₆	—	158	1	no	—
327	C₁₆	—	158	2	no	—
328	C₂³	—	316	2	no	—
329	C₂³	—	316	1	no	—
330	C₂³	—	632	2	no	—
331	C₂³	—	1264	1	no	—
332	C₂³	—	1264	1	no	—
333	C₂³	C₂×#11929609	1	1	yes	characteristic
334	Q₈	—	16	2	no	—
335	Q₈	—	316	1	no	—
336	Q₈	—	316	1	no	—
337	D₄	—	316	1	no	—
338	D₄	—	316	2	no	—
339	D₄	—	316	2	no	—
340	D₄	—	316	1	no	—
341	D₄	—	632	2	no	—
342	D₄	—	632	2	no	—
343	D₄	—	1264	2	no	—
344	D₄	—	1264	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	16	1	no	—
349	C₂×C₄	—	316	1	no	—
350	C₂×C₄	—	632	2	no	—
351	C₂×C₄	#14579150	1	2	yes	—
352	C₂×C₄	C₂×#11929609	1	2	yes	—
353	C₂×C₄	C₂×#11929609	1	2	yes	—
354	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
355	C₂×C₄	C₂×#11929611	1	2	yes	—
356	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
357	C₈	#14579150	1	2	yes	—
358	C₈	C₂×#11929611	1	4	yes	—
359	C₈	C₂×#11929625	1	1	yes	characteristic
360	C₈	C₂×#11929625	1	1	yes	characteristic
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	632	1	no	—
364	C₂²	—	632	2	no	—
365	C₂²	—	632	1	no	—
366	C₂²	—	632	2	no	—
367	C₂²	—	632	1	no	—
368	C₂²	—	632	2	no	—
369	C₂²	—	1264	4	no	—
370	C₂²	#19098911	1	2	yes	—
371	C₂²	C₂²×#11929609	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	8	2	no	—
375	C₄	—	8	1	no	—
376	C₄	—	8	1	no	—
377	C₄	—	632	1	no	—
378	C₄	#19098911	1	2	yes	—
379	C₄	C₂×#14579150	1	1	yes	characteristic
380	C₄	C₂²×#11929611	1	1	yes	characteristic
381	C₂	—	2	2	no	—
382	C₂	—	1264	2	no	—
383	C₂	—	1264	1	no	—
384	C₂	C₂×#19098911	1	1	yes	characteristic
385	C₂	#23776539	1	1	yes	characteristic
386	C₂	#23776544	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#3164	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	#23773886	C₂
hypercenter	#3164	D₇₉
hypocenter	C₇₉	#33702
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #23765643	C₂	semidirect
3. #23765639	C₂	semidirect
4. #23765634	C₂	semidirect
5. #23765625	C₂	semidirect
6. #23765115	C₂	semidirect
7. #23765110	C₂	semidirect
8. #23765105	C₂	semidirect
9. C₂×#19098363	C₂	semidirect
10. #23764997	C₂	semidirect
11. C₇₉×#3164	C₂	semidirect
21. #19098366	C₂²	semidirect
22. #19098366	C₂²	semidirect
23. #19098366	C₂²	semidirect
28. #19098363	C₂²	semidirect
29. #19098363	C₂²	semidirect
30. #19098363	C₂²	semidirect
31. #19098339	C₂²	semidirect
32. C₄×C₇₉⋊C₁₆	C₂²	semidirect
35. #500×C₇₉	C₂²	semidirect
36. #499×C₇₉	C₂²	semidirect
194. #3164	D₇₉	semidirect
205. C₇₉	#33702	semidirect
214. #500	C₂×D₇₉	semidirect
215. #499	C₂×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23773886	2
#23765643	2
#23765639	2
#23765634	2
#23765625	2
#23765115	2
#23765110	2
#23765105	2
C₂×#19098363	2
#23764997	2
C₇₉×#3164	2
#33702	79