Subgroups — #34292699

Order 20224.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772202	C₂	1	1	yes	maximal, characteristic
2	#23772052	C₂	1	2	yes	maximal, complemented
3	#23764291	C₂	1	1	yes	maximal, characteristic
4	#23764286	C₂	1	1	yes	maximal, characteristic
5	#23764151	C₂	1	1	yes	maximal, characteristic, complemented
6	#23764146	C₂	1	1	yes	maximal, characteristic, complemented
7	#23764109	C₂	1	2	yes	maximal, complemented
8	#23764102	C₂	1	2	yes	maximal, complemented
9	C₄×#14579087	C₂	1	2	yes	maximal, complemented
10	#97×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1778	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	1	yes	characteristic
13	C₄×#11929624	C₂²	1	2	yes	—
14	#19098676	C₄	1	2	yes	complemented
15	#19098671	C₄	1	2	yes	complemented
16	#19098665	C₄	1	4	yes	complemented
17	#19098622	—	2	4	no	—
18	#19098618	—	2	4	no	—
19	#19098601	C₂²	1	1	yes	characteristic
20	#19098591	C₂²	1	2	yes	—
21	#19098591	C₂²	1	2	yes	—
22	C₂×#14579087	C₂²	1	2	yes	—
23	C₂×#14579087	C₂²	1	2	yes	—
24	#19098560	C₂²	1	2	yes	—
25	#19098531	C₂²	1	2	yes	—
26	#19098531	C₂²	1	2	yes	—
27	C₂×#14579072	C₂²	1	1	yes	characteristic
28	C₂×#14579071	C₂²	1	1	yes	characteristic
29	C₂×#14579071	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
32	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
33	#19098324	C₂²	1	2	yes	—
34	#19098324	C₂²	1	2	yes	—
35	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₇₉	C₂²	1	2	yes	—
38	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
39	#341×C₇₉	C₂²	1	2	yes	—
40	C₂×#11929624	C₂³	1	2	yes	—
41	#14579183	—	2	2	no	—
42	#14579183	—	2	2	no	—
43	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×#11929613	C₂×C₄	1	2	yes	—
45	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
47	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
48	C₄×Dic₁₅₈	C₂×C₄	1	4	yes	—
49	C₄×D₃₁₆	C₂×C₄	1	4	yes	—
50	C₄×C₂²⋊D₇₉	—	2	4	no	—
51	#14579120	C₂³	1	2	yes	—
52	#14579118	—	2	2	no	—
53	C₂×C₄×Dic₇₉	—	2	2	no	—
54	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
56	C₄²×D₇₉	—	2	2	no	—
57	#14579095	C₂×C₄	1	2	yes	—
58	#14579093	C₂×C₄	1	2	yes	—
59	Dic₇₉⋊C₈	—	2	4	no	—
60	Dic₇₉⋊C₈	—	2	4	no	—
61	#14579087	—	2	4	no	—
62	#14579087	—	2	4	no	—
63	#14579086	C₂×C₄	1	4	yes	—
64	#14579083	C₂×C₄	1	4	yes	—
65	(C₂×D₇₉)⋊C₈	—	2	4	no	—
66	(C₂×D₇₉)⋊C₈	—	2	4	no	—
67	#14579072	D₄	1	4	yes	—
68	#14579071	—	2	2	no	—
69	#14579071	—	2	2	no	—
70	C₈×Dic₇₉	—	2	2	no	—
71	C₈×Dic₇₉	—	2	2	no	—
72	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
73	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
77	#108×C₇₉	—	2	2	no	—
78	#108×C₇₉	—	2	2	no	—
79	#97×C₇₉	—	2	2	no	—
80	#97×C₇₉	C₂×C₄	1	2	yes	—
81	#97×C₇₉	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₇₉	—	2	2	no	—
83	#11929624	—	2	2	no	—
84	#11929624	—	2	2	no	—
85	#11929624	—	2	2	no	—
86	#11929624	—	2	2	no	—
87	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
88	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₇₉	—	2	2	no	—
90	C₂²×Dic₇₉	—	2	1	no	—
91	C₂²×Dic₇₉	—	2	1	no	—
92	C₂×C₄×D₇₉	—	2	4	no	—
93	C₂×C₄×D₇₉	—	2	2	no	—
94	Dic₃₁₆	—	2	2	no	—
95	#11929613	—	2	4	no	—
96	D₆₃₂	—	2	2	no	—
97	C₂×C₇₉⋊C₈	—	2	1	no	—
98	C₂×C₇₉⋊C₈	—	2	1	no	—
99	C₂×C₇₉⋊C₈	—	2	1	no	—
100	C₂×C₇₉⋊C₈	—	2	1	no	—
101	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
103	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
104	Dic₇₉⋊C₄	—	2	4	no	—
105	C₂²⋊Dic₇₉	—	2	2	no	—
106	(C₂×D₇₉)⋊C₄	—	2	4	no	—
107	C₄×Dic₇₉	—	2	4	no	—
108	C₄×Dic₇₉	—	2	2	no	—
109	C₄×Dic₇₉	—	2	1	no	—
110	C₄×Dic₇₉	—	2	1	no	—
111	C₄×Dic₇₉	—	4	4	no	—
112	C₄×Dic₇₉	#41	1	2	yes	—
113	C₄×Dic₇₉	#41	1	2	yes	—
114	C₂²×C₄×C₇₉	#41	1	2	yes	—
115	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
116	M₁₆×C₇₉	—	2	2	no	—
117	M₁₆×C₇₉	—	2	2	no	—
118	M₁₆×C₇₉	—	2	2	no	—
119	M₁₆×C₇₉	—	2	2	no	—
120	C₂×C₈×C₇₉	—	2	2	no	—
121	C₂×C₈×C₇₉	—	2	2	no	—
122	C₂×C₈×C₇₉	—	2	1	no	—
123	C₂×C₈×C₇₉	—	2	1	no	—
124	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
125	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₇₉	#41	1	2	yes	—
128	C₂×C₈×C₇₉	#41	1	2	yes	—
129	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
130	C₄²×C₇₉	C₂×D₄	1	2	yes	—
131	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
132	C₂²×D₇₉	—	2	2	no	—
133	C₂×C₄×C₇₉	—	4	2	no	—
134	C₂×C₄×C₇₉	C₄×D₄	1	4	yes	—
135	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
141	C₂²⋊D₇₉	—	4	2	no	—
142	C₂²⋊D₇₉	—	4	2	no	—
143	C₂×Dic₇₉	—	2	4	no	—
144	C₂×Dic₇₉	—	2	2	no	—
145	C₂×Dic₇₉	—	2	4	no	—
146	C₂×Dic₇₉	—	2	2	no	—
147	C₂×Dic₇₉	—	2	2	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	D₃₁₆	—	2	2	no	—
150	D₃₁₆	—	2	2	no	—
151	C₄×D₇₉	—	4	2	no	—
152	C₄×D₇₉	—	4	2	no	—
153	C₄×D₇₉	—	8	4	no	—
154	Dic₁₅₈	—	2	2	no	—
155	Dic₁₅₈	—	2	2	no	—
156	C₈×C₇₉	—	2	2	no	—
157	C₈×C₇₉	—	4	2	no	—
158	C₈×C₇₉	—	4	2	no	—
159	C₈×C₇₉	#117	1	2	yes	—
160	C₈×C₇₉	#117	1	2	yes	—
161	C₇₉⋊C₈	—	2	2	no	—
162	C₇₉⋊C₈	—	2	2	no	—
163	C₂²×C₇₉	—	2	2	no	—
164	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂²×C₇₉	#529	1	1	yes	characteristic
166	C₂²×C₇₉	#529	1	1	yes	characteristic
167	C₂×D₇₉	—	4	4	no	—
168	C₂×D₇₉	—	4	2	no	—
169	C₄×C₇₉	—	2	4	no	—
170	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
171	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
172	C₄×C₇₉	#532	1	1	yes	characteristic
173	C₄×C₇₉	#532	1	1	yes	characteristic
174	Dic₇₉	—	2	4	no	—
175	Dic₇₉	—	4	2	no	—
176	Dic₇₉	—	4	4	no	—
177	Dic₇₉	—	4	2	no	—
178	#21297	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	1	no	—
180	C₂×C₇₉	—	2	1	no	—
181	C₂×C₇₉	#2293	1	1	yes	characteristic
182	C₂×C₇₉	#2956	1	2	yes	—
183	D₇₉	—	8	2	no	—
184	#2943	—	79	1	no	—
185	#2923	—	79	2	no	—
186	#1975	—	79	1	no	—
187	#1974	—	79	1	no	—
188	#1946	—	79	1	no	—
189	#1945	—	79	1	no	—
190	#1938	—	79	2	no	—
191	#1937	—	79	2	no	—
192	#1778	D₇₉	1	1	yes	characteristic, complemented
193	C₄×#104	—	79	2	no	—
194	C₄×#97	—	79	1	no	—
195	C₇₉	#21297	1	1	yes	characteristic, complemented, Sylow
196	C₂×#135	—	79	1	no	—
197	C₄×#41	—	79	2	no	—
198	#442	—	79	2	no	—
199	#441	—	79	2	no	—
200	#440	—	79	4	no	—
201	#433	—	158	4	no	—
202	#432	—	158	4	no	—
203	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
204	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
205	#427	—	79	1	no	—
206	#424	—	79	2	no	—
207	#424	—	79	2	no	—
208	C₂×#104	—	79	2	no	—
209	C₂×#104	—	79	2	no	—
210	#416	—	79	2	no	—
211	#408	—	79	2	no	—
212	#408	—	79	2	no	—
213	C₄×M₁₆	C₂×D₇₉	1	2	yes	—
214	C₂×#97	—	79	1	no	—
215	C₂×#97	—	79	1	no	—
216	C₂×#97	—	79	1	no	—
217	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
218	C₂×C₄×C₈	—	79	1	no	—
219	C₂×C₄×C₈	—	79	1	no	—
220	C₄³	—	79	1	no	—
221	#341	C₂×D₇₉	1	2	yes	—
222	#339	—	79	2	no	—
223	#339	—	79	2	no	—
224	C₂×#41	—	79	2	no	—
225	#135	—	158	2	no	—
226	#135	—	158	2	no	—
227	C₂×Q₁₆	—	79	1	no	—
228	C₂×SD₁₆	—	79	2	no	—
229	C₂×D₈	—	79	1	no	—
230	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
231	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
232	C₂²×C₈	—	79	1	no	—
233	C₂²×C₈	—	79	1	no	—
234	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
236	C₄×Q₈	—	79	4	no	—
237	C₄×D₄	—	79	4	no	—
238	C₄×D₄	—	158	4	no	—
239	#117	—	79	2	no	—
240	#117	—	158	2	no	—
241	C₂×C₄²	—	79	1	no	—
242	C₂×C₄²	—	79	1	no	—
243	C₂×C₄²	—	158	2	no	—
244	C₂×C₄²	—	158	2	no	—
245	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
246	#108	—	2	2	no	—
247	#108	—	2	2	no	—
248	#107	—	79	2	no	—
249	#106	—	79	2	no	—
250	C₄⋊C₈	—	158	4	no	—
251	C₄⋊C₈	—	158	4	no	—
252	#104	—	158	4	no	—
253	#104	—	158	4	no	—
254	#103	—	79	4	no	—
255	#102	—	79	4	no	—
256	C₂²⋊C₈	—	158	4	no	—
257	C₂²⋊C₈	—	158	4	no	—
258	#97	—	2	2	no	—
259	#97	—	79	4	no	—
260	#97	—	158	2	no	—
261	#97	—	158	2	no	—
262	#97	C₄×D₇₉	1	2	yes	—
263	#97	C₄×D₇₉	1	2	yes	—
264	C₄×C₈	—	2	2	no	—
265	C₄×C₈	—	158	2	no	—
266	C₄×C₈	—	158	2	no	—
267	#41	—	158	2	no	—
268	#41	—	158	2	no	—
269	#41	—	158	2	no	—
270	#41	—	158	2	no	—
271	C₂×Q₈	—	79	2	no	—
272	C₂×D₄	—	79	2	no	—
273	C₂×D₄	—	158	2	no	—
274	C₂²×C₄	—	158	1	no	—
275	C₂²×C₄	—	158	2	no	—
276	C₂²×C₄	—	158	4	no	—
277	C₂²×C₄	—	158	1	no	—
278	C₂²×C₄	#11929625	1	2	yes	—
279	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
280	Q₁₆	—	158	2	no	—
281	SD₁₆	—	158	4	no	—
282	D₈	—	158	2	no	—
283	M₁₆	—	2	2	no	—
284	M₁₆	—	2	2	no	—
285	M₁₆	—	2	2	no	—
286	M₁₆	—	2	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	79	2	no	—
292	C₂×C₈	—	79	2	no	—
293	C₂×C₈	—	158	1	no	—
294	C₂×C₈	—	158	1	no	—
295	C₂×C₈	—	158	1	no	—
296	C₂×C₈	—	158	1	no	—
297	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
298	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
300	C₂×C₈	#11929624	1	2	yes	—
301	C₂×C₈	#11929624	1	2	yes	—
302	C₄⋊C₄	—	79	4	no	—
303	C₄⋊C₄	—	158	4	no	—
304	C₂²⋊C₄	—	158	2	no	—
305	C₂²⋊C₄	—	158	4	no	—
306	C₄²	—	316	4	no	—
307	C₄²	—	79	2	no	—
308	C₄²	—	79	2	no	—
309	C₄²	—	158	1	no	—
310	C₄²	—	158	1	no	—
311	C₄²	—	158	4	no	—
312	C₄²	—	158	2	no	—
313	C₄²	C₂×C₄×D₇₉	1	2	yes	—
314	C₄²	D₄×D₇₉	1	2	yes	—
315	C₂³	—	158	2	no	—
316	C₂³	C₂×#11929625	1	1	yes	characteristic
317	Q₈	—	158	2	no	—
318	Q₈	—	158	2	no	—
319	D₄	—	316	2	no	—
320	D₄	—	316	2	no	—
321	D₄	—	158	2	no	—
322	D₄	—	158	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	316	2	no	—
325	C₂×C₄	—	316	2	no	—
326	C₂×C₄	—	316	2	no	—
327	C₂×C₄	—	632	4	no	—
328	C₂×C₄	—	158	4	no	—
329	C₂×C₄	—	158	2	no	—
330	C₂×C₄	—	158	2	no	—
331	C₂×C₄	—	158	4	no	—
332	C₂×C₄	—	158	2	no	—
333	C₂×C₄	#14579123	1	4	yes	—
334	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
336	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
337	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
338	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
339	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	—	158	2	no	—
344	C₈	—	158	2	no	—
345	C₈	#14579118	1	2	yes	—
346	C₈	#14579118	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	316	2	no	—
349	C₂²	—	316	4	no	—
350	C₂²	C₂×#14579123	1	1	yes	characteristic
351	C₂²	#19099100	1	1	yes	characteristic
352	C₂²	#19099100	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	316	2	no	—
355	C₄	—	316	2	no	—
356	C₄	—	316	4	no	—
357	C₄	—	158	4	no	—
358	C₄	C₂×#14579118	1	1	yes	characteristic
359	C₄	#19099031	1	1	yes	characteristic
360	C₄	#19099110	1	1	yes	characteristic
361	C₄	#19099110	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	632	2	no	—
365	C₂	#23765861	1	1	yes	characteristic
366	C₂	#23772309	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1778	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂×#14579183	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1778	D₇₉
hypocenter	C₇₉	#21297
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1778

Complements

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

N	H ≅ G/N	type
2. #23772052	C₂	semidirect
5. #23764151	C₂	semidirect
6. #23764146	C₂	semidirect
7. #23764109	C₂	semidirect
8. #23764102	C₂	semidirect
9. C₄×#14579087	C₂	semidirect
10. #97×Dic₇₉	C₂	semidirect
11. C₇₉×#1778	C₂	semidirect
14. #19098676	C₄	semidirect
15. #19098671	C₄	semidirect
16. #19098665	C₄	semidirect
192. #1778	D₇₉	semidirect
195. C₇₉	#21297	semidirect

Maximal subgroups

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

subgroup	index
#23772202	2
#23772052	2
#23764291	2
#23764286	2
#23764151	2
#23764146	2
#23764109	2
#23764102	2
C₄×#14579087	2
#97×Dic₇₉	2
C₇₉×#1778	2
#21297	79