Subgroups — #34443617

Order 20224.

570 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773894	C₂	1	1	yes	maximal, characteristic
2	#23765647	C₂	1	2	yes	maximal, complemented
3	#23765639	C₂	1	2	yes	maximal, complemented
4	#23765634	C₂	1	2	yes	maximal, complemented
5	#23765629	C₂	1	2	yes	maximal, complemented
6	#23765119	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765110	C₂	1	1	yes	maximal, characteristic
8	#23765105	C₂	1	1	yes	maximal, characteristic
9	C₂×#19098366	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×#14579194	—	2	1	no	—
13	C₂×#14579183	C₂²	1	1	yes	characteristic
14	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
15	#19099139	—	2	1	no	—
16	C₂×#14579101	C₂²	1	1	yes	characteristic
17	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
18	#19098957	—	2	2	no	—
19	#19098947	C₂²	1	2	yes	—
20	#19098943	C₂²	1	2	yes	—
21	#19098366	C₂²	1	2	yes	—
22	#19098366	C₂²	1	2	yes	—
23	#19098366	C₂²	1	1	yes	characteristic
24	#19098366	C₂²	1	1	yes	characteristic
25	#19098366	C₂²	1	2	yes	—
26	#19098366	C₂²	1	2	yes	—
27	#19098366	C₂²	1	2	yes	—
28	#19098363	C₂²	1	1	yes	characteristic
29	#19098363	C₂²	1	1	yes	characteristic
30	#19098363	C₂²	1	2	yes	—
31	#19098339	C₂²	1	2	yes	—
32	C₄×C₇₉⋊C₁₆	C₂²	1	2	yes	—
33	#544×C₇₉	—	2	1	no	—
34	#501×C₇₉	—	2	2	no	—
35	#500×C₇₉	C₂²	1	2	yes	—
36	#499×C₇₉	C₂²	1	2	yes	—
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×Dic₁₅₈	—	2	1	no	—
43	#14579194	—	4	4	no	—
44	#14579183	—	2	2	no	—
45	#14579183	—	2	2	no	—
46	C₂×Dic₃₁₆	—	2	2	no	—
47	C₂×Dic₃₁₆	—	2	2	no	—
48	C₂×Dic₃₁₆	—	2	2	no	—
49	C₂×Dic₃₁₆	—	2	1	no	—
50	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
51	C₂×Dic₃₁₆	C₂³	1	2	yes	—
52	C₂×#11929613	—	2	2	no	—
53	C₂×#11929613	—	2	1	no	—
54	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
55	#14579161	—	2	2	no	—
56	#14579149	—	2	2	no	—
57	#14579145	—	4	2	no	—
58	#14579141	—	4	1	no	—
59	#14579141	—	4	1	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	#11929624	—	4	2	no	—
85	#11929624	—	4	2	no	—
86	C₂×Dic₁₅₈	—	2	2	no	—
87	C₂×Dic₁₅₈	—	2	1	no	—
88	C₂×Dic₁₅₈	—	4	2	no	—
89	C₂×Dic₁₅₈	—	4	1	no	—
90	C₂×Dic₁₅₈	—	4	2	no	—
91	C₂×D₃₁₆	—	2	1	no	—
92	C₂×C₂²⋊D₇₉	—	4	1	no	—
93	C₂²×Dic₇₉	—	4	1	no	—
94	C₂×C₄×D₇₉	—	4	1	no	—
95	Dic₃₁₆	—	2	2	no	—
96	Dic₃₁₆	—	2	2	no	—
97	Dic₃₁₆	—	2	1	no	—
98	Dic₃₁₆	—	2	1	no	—
99	Dic₃₁₆	—	4	2	no	—
100	Dic₃₁₆	—	4	4	no	—
101	#11929613	—	4	2	no	—
102	#11929613	—	4	4	no	—
103	D₆₃₂	—	2	1	no	—
104	D₆₃₂	—	2	1	no	—
105	C₄⋊Dic₇₉	—	4	2	no	—
106	C₄⋊Dic₇₉	—	4	2	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	1	no	—
136	Q₈×C₇₉	—	16	2	no	—
137	C₂×C₄×C₇₉	—	4	1	no	—
138	C₂×C₄×C₇₉	—	4	1	no	—
139	C₂×C₄×C₇₉	—	4	2	no	—
140	C₂×C₄×C₇₉	—	16	1	no	—
141	C₂×C₄×C₇₉	#124	1	2	yes	—
142	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₇₉	—	8	2	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	4	1	no	—
150	C₂×Dic₇₉	—	8	2	no	—
151	D₃₁₆	—	4	1	no	—
152	D₃₁₆	—	4	1	no	—
153	C₄×D₇₉	—	8	2	no	—
154	Dic₁₅₈	—	4	1	no	—
155	Dic₁₅₈	—	4	2	no	—
156	Dic₁₅₈	—	4	2	no	—
157	Dic₁₅₈	—	4	1	no	—
158	Dic₁₅₈	—	8	2	no	—
159	C₈×C₇₉	#124	1	2	yes	—
160	C₈×C₇₉	C₂×SD₁₆	1	4	yes	—
161	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
162	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
163	C₂²×C₇₉	—	2	1	no	—
164	C₂²×C₇₉	—	2	1	no	—
165	C₂²×C₇₉	#486	1	2	yes	—
166	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₇₉	—	8	1	no	—
168	C₂×D₇₉	—	8	1	no	—
169	C₂×D₇₉	—	8	1	no	—
170	C₄×C₇₉	—	2	2	no	—
171	C₄×C₇₉	—	2	2	no	—
172	C₄×C₇₉	—	8	2	no	—
173	C₄×C₇₉	—	8	1	no	—
174	C₄×C₇₉	—	8	1	no	—
175	C₄×C₇₉	#486	1	2	yes	—
176	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
177	C₄×C₇₉	C₂²×SD₁₆	1	1	yes	characteristic
178	Dic₇₉	—	8	1	no	—
179	Dic₇₉	—	8	2	no	—
180	#33703	—	79	1	no	maximal, Sylow
181	C₂×C₇₉	—	2	2	no	—
182	C₂×C₇₉	C₂×#486	1	1	yes	characteristic
183	C₂×C₇₉	#3416	1	1	yes	characteristic
184	C₂×C₇₉	#3417	1	1	yes	characteristic
185	D₇₉	—	16	1	no	—
186	#3164	D₇₉	1	1	yes	characteristic, complemented
187	#3156	—	79	1	no	—
188	#2245	—	79	2	no	—
189	#2243	—	79	2	no	—
190	#2242	—	79	2	no	—
191	#2241	—	79	2	no	—
192	#2141	—	79	1	no	—
193	#2139	—	79	1	no	—
194	#2138	—	79	1	no	—
195	C₂×#358	—	79	1	no	—
196	#2108	—	79	1	no	—
197	C₇₉	#33703	1	1	yes	characteristic, complemented, Sylow
198	C₂×#137	—	158	1	no	—
199	C₂×#135	—	79	1	no	—
200	C₂²×Q₁₆	—	79	1	no	—
201	#544	—	2	1	no	—
202	#536	—	158	1	no	—
203	C₂×M₃₂	—	79	1	no	—
204	C₂²×C₁₆	—	79	1	no	—
205	#501	—	2	2	no	—
206	#500	C₂×D₇₉	1	2	yes	—
207	#499	C₂×D₇₉	1	2	yes	—
208	#497	—	158	2	no	—
209	#495	—	79	2	no	—
210	#494	—	79	2	no	—
211	#428	—	2	1	no	—
212	#427	C₂×D₇₉	1	1	yes	characteristic
213	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
214	#405	C₂×D₇₉	1	1	yes	characteristic
215	#358	—	79	2	no	—
216	#358	—	79	2	no	—
217	#358	—	79	1	no	—
218	#358	—	79	1	no	—
219	#358	—	79	2	no	—
220	#358	—	79	2	no	—
221	#358	—	79	2	no	—
222	#357	—	79	1	no	—
223	#357	—	79	1	no	—
224	#357	—	79	2	no	—
225	#346	—	79	2	no	—
226	C₄×C₁₆	—	79	2	no	—
227	C₂×#41	—	158	1	no	—
228	C₂²×Q₈	—	158	1	no	—
229	#137	—	316	4	no	—
230	#135	—	158	2	no	—
231	#135	—	158	2	no	—
232	C₂×Q₁₆	—	79	2	no	—
233	C₂×Q₁₆	—	79	1	no	—
234	C₂×Q₁₆	—	158	1	no	—
235	C₂×Q₁₆	—	158	2	no	—
236	C₂×Q₁₆	—	158	2	no	—
237	C₂×Q₁₆	—	158	2	no	—
238	C₂×SD₁₆	—	158	1	no	—
239	C₂×SD₁₆	—	158	2	no	—
240	C₂×D₈	—	79	1	no	—
241	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
242	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
243	C₄⋊Q₈	—	2	2	no	—
244	C₄⋊Q₈	—	4	1	no	—
245	C₄⋊Q₈	—	158	2	no	—
246	#125	—	2	2	no	—
247	#125	—	4	1	no	—
248	#124	—	158	2	no	—
249	#123	—	316	2	no	—
250	C₂²⋊Q₈	—	4	2	no	—
251	C₂²⋊Q₈	—	316	1	no	—
252	C₂²⋊Q₈	—	316	1	no	—
253	#117	—	2	1	no	—
254	#117	C₂²⋊D₇₉	1	1	yes	characteristic
255	C₂×C₄⋊C₄	—	2	1	no	—
256	M₃₂	—	158	2	no	—
257	C₂×C₁₆	—	79	1	no	—
258	C₂×C₁₆	—	79	2	no	—
259	C₂×C₁₆	—	79	1	no	—
260	C₂×C₁₆	—	158	2	no	—
261	#107	—	2	1	no	—
262	#107	—	2	2	no	—
263	#107	C₂²×D₇₉	1	2	yes	—
264	#107	C₂²×D₇₉	1	1	yes	characteristic
265	#107	C₂²×D₇₉	1	1	yes	characteristic
266	#106	—	2	1	no	—
267	#106	—	2	2	no	—
268	#97	C₂²⋊D₇₉	1	2	yes	—
269	C₄×C₈	C₂²×D₇₉	1	2	yes	—
270	#41	—	316	2	no	—
271	#41	—	316	2	no	—
272	C₂×Q₈	—	4	2	no	—
273	C₂×Q₈	—	316	1	no	—
274	C₂×Q₈	—	316	2	no	—
275	C₂×Q₈	—	316	2	no	—
276	C₂×Q₈	—	158	2	no	—
277	C₂×Q₈	—	158	1	no	—
278	C₂×D₄	—	316	1	no	—
279	C₂×D₄	—	158	1	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	316	1	no	—
282	C₂²×C₄	—	316	1	no	—
283	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
284	Q₁₆	—	316	2	no	—
285	Q₁₆	—	316	4	no	—
286	Q₁₆	—	158	2	no	—
287	Q₁₆	—	158	2	no	—
288	Q₁₆	—	158	1	no	—
289	Q₁₆	—	158	1	no	—
290	SD₁₆	—	316	2	no	—
291	SD₁₆	—	316	4	no	—
292	D₈	—	158	1	no	—
293	D₈	—	158	1	no	—
294	M₁₆	#11929611	1	4	yes	—
295	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
297	C₂×C₈	#11929625	1	2	yes	—
298	C₂×C₈	#11929625	1	2	yes	—
299	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	316	2	no	—
308	C₄⋊C₄	—	316	2	no	—
309	C₄⋊C₄	#11929609	1	1	yes	characteristic
310	C₄⋊C₄	#11929609	1	1	yes	characteristic
311	C₂²⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	316	2	no	—
313	C₂²⋊C₄	—	316	2	no	—
314	C₂²⋊C₄	#11929609	1	2	yes	—
315	C₄²	—	4	1	no	—
316	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
317	C₁₆	—	158	1	no	—
318	C₁₆	—	158	1	no	—
319	C₁₆	—	158	2	no	—
320	C₂³	—	316	1	no	—
321	C₂³	C₂×#11929609	1	1	yes	characteristic
322	Q₈	—	16	2	no	—
323	Q₈	—	316	2	no	—
324	Q₈	—	316	1	no	—
325	Q₈	—	316	1	no	—
326	Q₈	—	316	2	no	—
327	Q₈	—	632	2	no	—
328	D₄	—	316	1	no	—
329	D₄	—	316	1	no	—
330	D₄	—	632	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	16	1	no	—
335	C₂×C₄	—	316	2	no	—
336	C₂×C₄	—	316	1	no	—
337	C₂×C₄	—	632	2	no	—
338	C₂×C₄	—	632	2	no	—
339	C₂×C₄	#14579150	1	2	yes	—
340	C₂×C₄	C₂×#11929609	1	2	yes	—
341	C₂×C₄	C₂×#11929609	1	2	yes	—
342	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
343	C₂×C₄	C₂×#11929611	1	2	yes	—
344	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
345	C₈	#14579150	1	2	yes	—
346	C₈	C₂×#11929611	1	4	yes	—
347	C₈	C₂×#11929625	1	1	yes	characteristic
348	C₈	C₂×#11929625	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	632	1	no	—
352	C₂²	—	632	1	no	—
353	C₂²	—	632	1	no	—
354	C₂²	#19098911	1	2	yes	—
355	C₂²	C₂²×#11929609	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	8	2	no	—
359	C₄	—	8	1	no	—
360	C₄	—	8	1	no	—
361	C₄	—	632	1	no	—
362	C₄	—	632	2	no	—
363	C₄	#19098911	1	2	yes	—
364	C₄	C₂×#14579150	1	1	yes	characteristic
365	C₄	C₂²×#11929611	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	1264	1	no	—
368	C₂	C₂×#19098911	1	1	yes	characteristic
369	C₂	#23776547	1	1	yes	characteristic
370	C₂	#23776551	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	C₂×#14579183	C₂²
hypercenter	#3164	D₇₉
hypocenter	C₇₉	#33703
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. #23765647	C₂	semidirect
3. #23765639	C₂	semidirect
4. #23765634	C₂	semidirect
5. #23765629	C₂	semidirect
6. #23765119	C₂	semidirect
9. C₂×#19098366	C₂	semidirect
10. #23764997	C₂	semidirect
11. C₇₉×#3164	C₂	semidirect
186. #3164	D₇₉	semidirect
197. C₇₉	#33703	semidirect

Maximal subgroups

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

subgroup	index
#23773894	2
#23765647	2
#23765639	2
#23765634	2
#23765629	2
#23765119	2
#23765110	2
#23765105	2
C₂×#19098366	2
#23764997	2
C₇₉×#3164	2
#33703	79