Subgroups — #34296459

Order 20224.

738 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773746	C₂	1	1	yes	maximal, characteristic
2	#23772337	C₂	1	1	yes	maximal, characteristic, complemented
3	#23772117	C₂	1	1	yes	maximal, characteristic
4	#23764376	C₂	1	1	yes	maximal, characteristic, complemented
5	#23764227	C₂	1	1	yes	maximal, characteristic
6	#23764109	C₂	1	2	yes	maximal, complemented
7	#23764102	C₂	1	2	yes	maximal, complemented
8	#23763243	C₂	1	2	yes	maximal, complemented
9	#23763236	C₂	1	2	yes	maximal, complemented
10	#97×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#1838	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579169	C₂²	1	2	yes	—
13	#19099120	C₂²	1	1	yes	characteristic
14	#19099113	C₂²	1	1	yes	characteristic
15	#19098932	C₄	1	2	yes	complemented
16	#19098928	C₄	1	2	yes	complemented
17	#19098923	C₄	1	4	yes	complemented
18	#19098692	—	2	4	no	—
19	#19098633	—	2	4	no	—
20	#19098591	C₂²	1	2	yes	—
21	C₂×#14579087	C₂²	1	2	yes	—
22	C₂×#14579087	C₂²	1	2	yes	—
23	#19098560	C₂²	1	2	yes	—
24	#19098560	C₂²	1	2	yes	—
25	#19098531	C₂²	1	2	yes	—
26	#19098521	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	#423×C₇₉	C₂²	1	1	yes	characteristic
36	#423×C₇₉	C₂²	1	1	yes	characteristic
37	C₂×#97×C₇₉	C₂²	1	1	yes	characteristic
38	#341×C₇₉	C₂²	1	2	yes	—
39	#341×C₇₉	C₂²	1	2	yes	—
40	C₂×#11929624	C₂³	1	2	yes	—
41	#14579169	—	2	4	no	—
42	#14579169	—	2	4	no	—
43	C₂×#11929607	—	2	2	no	—
44	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
45	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₇₉	—	2	2	no	—
47	#14579161	C₂×C₄	1	1	yes	characteristic
48	#14579161	C₂×C₄	1	2	yes	—
49	#14579158	C₂×C₄	1	1	yes	characteristic
50	#14579153	C₂×C₄	1	2	yes	—
51	#14579149	C₂×C₄	1	2	yes	—
52	#14579140	—	2	2	no	—
53	#14579135	—	2	2	no	—
54	C₄×Dic₁₅₈	—	2	2	no	—
55	C₄×D₃₁₆	—	2	2	no	—
56	#14579120	C₂³	1	2	yes	—
57	C₂×C₄×Dic₇₉	—	2	2	no	—
58	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
59	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
60	Dic₇₉⋊C₈	—	2	4	no	—
61	#14579087	—	2	4	no	—
62	#14579087	—	2	4	no	—
63	#14579086	C₂×C₄	1	4	yes	—
64	#14579086	C₂×C₄	1	4	yes	—
65	#14579083	C₂×C₄	1	4	yes	—
66	#14579083	C₂×C₄	1	4	yes	—
67	(C₂×D₇₉)⋊C₈	—	2	4	no	—
68	#14579072	D₄	1	4	yes	—
69	#14579071	—	2	2	no	—
70	#14579071	—	2	2	no	—
71	#14579071	—	2	2	no	—
72	C₈×Dic₇₉	—	2	2	no	—
73	C₈×Dic₇₉	—	2	2	no	—
74	C₈×Dic₇₉	—	2	2	no	—
75	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
76	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₇₉	—	2	2	no	—
81	C₄⋊C₈×C₇₉	—	2	2	no	—
82	#97×C₇₉	C₂×C₄	1	2	yes	—
83	#97×C₇₉	C₂×C₄	1	2	yes	—
84	#11929624	—	2	2	no	—
85	#11929624	—	2	2	no	—
86	#11929624	—	2	2	no	—
87	#11929624	—	2	2	no	—
88	C₂×Dic₁₅₈	—	2	2	no	—
89	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
90	C₂×D₃₁₆	—	2	2	no	—
91	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
92	C₂×C₂²⋊D₇₉	—	2	2	no	—
93	C₂²×Dic₇₉	—	2	1	no	—
94	C₂²×Dic₇₉	—	2	1	no	—
95	C₂×C₄×D₇₉	—	2	2	no	—
96	#11929607	—	4	4	no	—
97	C₂×C₇₉⋊C₈	—	2	1	no	—
98	C₂×C₇₉⋊C₈	—	2	1	no	—
99	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
100	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
101	C₂×C₇₉⋊C₈	#41	1	2	yes	—
102	C₂×C₇₉⋊C₈	#41	1	2	yes	—
103	C₈×D₇₉	—	4	4	no	—
104	C₄⋊Dic₇₉	—	2	2	no	—
105	C₄⋊Dic₇₉	—	2	2	no	—
106	C₄⋊Dic₇₉	C₂²×C₄	1	4	yes	—
107	Dic₇₉⋊C₄	—	2	4	no	—
108	C₂²⋊Dic₇₉	—	2	2	no	—
109	(C₂×D₇₉)⋊C₄	—	2	4	no	—
110	C₄×Dic₇₉	—	2	2	no	—
111	C₄×Dic₇₉	—	2	1	no	—
112	C₄×Dic₇₉	—	2	1	no	—
113	C₄×Dic₇₉	—	4	4	no	—
114	C₄×Dic₇₉	#41	1	2	yes	—
115	C₄×Dic₇₉	#41	1	2	yes	—
116	C₂²×C₄×C₇₉	#41	1	2	yes	—
117	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
118	M₁₆×C₇₉	—	2	2	no	—
119	M₁₆×C₇₉	—	2	2	no	—
120	M₁₆×C₇₉	—	2	2	no	—
121	M₁₆×C₇₉	—	2	2	no	—
122	C₂×C₈×C₇₉	—	2	2	no	—
123	C₂×C₈×C₇₉	—	2	2	no	—
124	C₂×C₈×C₇₉	—	2	1	no	—
125	C₂×C₈×C₇₉	—	2	1	no	—
126	C₂×C₈×C₇₉	—	2	1	no	—
127	C₂×C₈×C₇₉	—	2	1	no	—
128	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
130	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
132	C₄²×C₇₉	C₂×D₄	1	2	yes	—
133	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
134	C₂²×D₇₉	—	2	2	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
137	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₇₉	—	4	2	no	—
145	C₂²⋊D₇₉	—	4	2	no	—
146	C₂×Dic₇₉	—	2	4	no	—
147	C₂×Dic₇₉	—	2	2	no	—
148	C₂×Dic₇₉	—	2	4	no	—
149	C₂×Dic₇₉	—	2	2	no	—
150	C₂×Dic₇₉	—	2	2	no	—
151	C₂×Dic₇₉	—	4	2	no	—
152	D₃₁₆	—	2	2	no	—
153	D₃₁₆	—	2	2	no	—
154	D₃₁₆	—	8	2	no	—
155	C₄×D₇₉	—	4	2	no	—
156	C₄×D₇₉	—	4	2	no	—
157	Dic₁₅₈	—	2	2	no	—
158	Dic₁₅₈	—	2	2	no	—
159	Dic₁₅₈	—	8	2	no	—
160	C₈×C₇₉	—	2	2	no	—
161	C₈×C₇₉	—	2	2	no	—
162	C₈×C₇₉	—	4	2	no	—
163	C₈×C₇₉	—	4	2	no	—
164	C₇₉⋊C₈	—	2	2	no	—
165	C₇₉⋊C₈	#124	1	4	yes	—
166	C₂²×C₇₉	—	2	2	no	—
167	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₇₉	#529	1	1	yes	characteristic
169	C₂²×C₇₉	#529	1	1	yes	characteristic
170	C₂×D₇₉	—	4	4	no	—
171	C₂×D₇₉	—	4	2	no	—
172	C₄×C₇₉	—	2	2	no	—
173	C₄×C₇₉	—	2	2	no	—
174	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
175	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
176	C₄×C₇₉	C₂×#124	1	1	yes	characteristic
177	C₄×C₇₉	#532	1	1	yes	characteristic
178	Dic₇₉	—	2	4	no	—
179	Dic₇₉	—	4	2	no	—
180	Dic₇₉	—	4	4	no	—
181	Dic₇₉	—	4	2	no	—
182	#21662	—	79	1	no	maximal, Sylow
183	C₂×C₇₉	—	2	1	no	—
184	C₂×C₇₉	—	2	1	no	—
185	C₂×C₇₉	C₄×#124	1	1	yes	characteristic
186	C₂×C₇₉	#2956	1	2	yes	—
187	D₇₉	—	8	2	no	—
188	#3135	—	79	1	no	—
189	#2961	—	79	1	no	—
190	#2932	—	79	1	no	—
191	#1994	—	79	1	no	—
192	#1962	—	79	1	no	—
193	#1938	—	79	2	no	—
194	#1937	—	79	2	no	—
195	#1838	D₇₉	1	1	yes	characteristic, complemented
196	#1765	—	79	2	no	—
197	#1764	—	79	2	no	—
198	C₄×#97	—	79	1	no	—
199	C₇₉	#21662	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	79	2	no	—
201	#533	—	79	1	no	—
202	#532	—	79	1	no	—
203	#491	—	79	2	no	—
204	#490	—	79	2	no	—
205	#489	—	79	4	no	—
206	#446	—	158	4	no	—
207	#435	—	158	4	no	—
208	#424	—	79	2	no	—
209	#423	C₂×D₇₉	1	1	yes	characteristic
210	#423	C₂×D₇₉	1	1	yes	characteristic
211	C₂×#104	—	79	2	no	—
212	C₂×#104	—	79	2	no	—
213	#416	—	79	2	no	—
214	#416	—	79	2	no	—
215	#408	—	79	2	no	—
216	#405	—	79	2	no	—
217	C₂×#97	—	79	1	no	—
218	C₂×#97	—	79	1	no	—
219	C₂×#97	—	79	1	no	—
220	C₂×#97	C₂×D₇₉	1	1	yes	characteristic
221	C₂×C₄×C₈	—	79	1	no	—
222	C₂×C₄×C₈	—	79	1	no	—
223	C₄³	—	79	1	no	—
224	#341	C₂×D₇₉	1	2	yes	—
225	#341	C₂×D₇₉	1	2	yes	—
226	#339	—	79	2	no	—
227	#339	—	79	2	no	—
228	C₂×#41	—	79	2	no	—
229	#131	—	158	4	no	—
230	#131	—	158	4	no	—
231	C₂×M₁₆	—	158	2	no	—
232	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
233	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
234	C₂²×C₈	—	79	1	no	—
235	C₂²×C₈	—	79	1	no	—
236	C₂²×C₈	—	158	2	no	—
237	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
239	C₄⋊Q₈	—	79	2	no	—
240	C₄⋊Q₈	—	79	1	no	—
241	#127	—	79	1	no	—
242	#125	—	79	2	no	—
243	#124	—	79	2	no	—
244	C₂²⋊Q₈	—	158	2	no	—
245	#121	—	158	2	no	—
246	C₄×Q₈	—	158	2	no	—
247	C₄×D₄	—	158	2	no	—
248	#117	—	79	2	no	—
249	C₂×C₄²	—	79	1	no	—
250	C₂×C₄²	—	79	1	no	—
251	C₂×C₄²	—	158	2	no	—
252	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
253	C₄⋊C₈	—	2	2	no	—
254	C₄⋊C₈	—	2	2	no	—
255	C₄⋊C₈	—	158	4	no	—
256	#104	—	158	4	no	—
257	#104	—	158	4	no	—
258	#103	—	79	4	no	—
259	#103	—	79	4	no	—
260	#102	—	79	4	no	—
261	#102	—	79	4	no	—
262	C₂²⋊C₈	—	158	4	no	—
263	#97	—	79	4	no	—
264	#97	—	158	2	no	—
265	#97	—	158	2	no	—
266	#97	—	158	2	no	—
267	#97	C₄×D₇₉	1	2	yes	—
268	#97	C₄×D₇₉	1	2	yes	—
269	C₄×C₈	—	158	2	no	—
270	C₄×C₈	—	158	2	no	—
271	C₄×C₈	—	158	2	no	—
272	#41	—	158	2	no	—
273	#41	—	158	2	no	—
274	#41	—	158	2	no	—
275	#41	—	158	2	no	—
276	C₂×Q₈	—	79	2	no	—
277	C₂×Q₈	—	158	2	no	—
278	C₂×D₄	—	79	2	no	—
279	C₂×D₄	—	158	2	no	—
280	C₂×D₄	—	158	2	no	—
281	C₂²×C₄	—	158	1	no	—
282	C₂²×C₄	—	158	2	no	—
283	C₂²×C₄	—	158	1	no	—
284	C₂²×C₄	#11929625	1	2	yes	—
285	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
286	M₁₆	—	2	2	no	—
287	M₁₆	—	2	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	316	4	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	1	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	316	4	no	—
298	C₂×C₈	—	79	2	no	—
299	C₂×C₈	—	79	2	no	—
300	C₂×C₈	—	79	2	no	—
301	C₂×C₈	—	79	2	no	—
302	C₂×C₈	—	158	1	no	—
303	C₂×C₈	—	158	1	no	—
304	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
305	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
306	C₄⋊C₄	—	79	4	no	—
307	C₄⋊C₄	—	158	4	no	—
308	C₄⋊C₄	—	158	2	no	—
309	C₄⋊C₄	—	158	2	no	—
310	C₂²⋊C₄	—	158	2	no	—
311	C₂²⋊C₄	—	158	4	no	—
312	C₄²	—	316	4	no	—
313	C₄²	—	79	2	no	—
314	C₄²	—	79	2	no	—
315	C₄²	—	158	1	no	—
316	C₄²	—	158	1	no	—
317	C₄²	—	158	2	no	—
318	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
319	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
320	C₄²	D₄×D₇₉	1	2	yes	—
321	C₂³	—	158	2	no	—
322	C₂³	C₂×#11929625	1	1	yes	characteristic
323	Q₈	—	632	2	no	—
324	Q₈	—	158	2	no	—
325	Q₈	—	158	2	no	—
326	D₄	—	316	2	no	—
327	D₄	—	316	2	no	—
328	D₄	—	632	2	no	—
329	D₄	—	158	2	no	—
330	D₄	—	158	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	316	2	no	—
333	C₂×C₄	—	316	2	no	—
334	C₂×C₄	—	316	2	no	—
335	C₂×C₄	—	158	4	no	—
336	C₂×C₄	—	158	2	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	—	158	4	no	—
339	C₂×C₄	—	158	2	no	—
340	C₂×C₄	#14579123	1	2	yes	—
341	C₂×C₄	#14579123	1	2	yes	—
342	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
344	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
345	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
346	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
347	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	2	2	no	—
350	C₈	—	4	2	no	—
351	C₈	—	4	2	no	—
352	C₈	—	79	4	no	—
353	C₈	—	158	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	316	4	no	—
356	C₂²	—	316	2	no	—
357	C₂²	C₂×#14579123	1	1	yes	characteristic
358	C₂²	#19099098	1	1	yes	characteristic
359	C₂²	#19099098	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	2	2	no	—
362	C₄	—	316	2	no	—
363	C₄	—	316	4	no	—
364	C₄	—	316	2	no	—
365	C₄	—	158	4	no	—
366	C₄	#19099031	1	1	yes	characteristic
367	C₄	#19099031	1	1	yes	characteristic
368	C₄	#124×D₇₉	1	1	yes	characteristic
369	C₄	#19099112	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	632	2	no	—
373	C₂	#23765939	1	1	yes	characteristic
374	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×#1838	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#19099113	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1838	D₇₉
hypocenter	C₇₉	#21662
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1838

Complements

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

N	H ≅ G/N	type
2. #23772337	C₂	semidirect
4. #23764376	C₂	semidirect
6. #23764109	C₂	semidirect
7. #23764102	C₂	semidirect
8. #23763243	C₂	semidirect
9. #23763236	C₂	semidirect
10. #97×Dic₇₉	C₂	semidirect
11. C₇₉×#1838	C₂	semidirect
15. #19098932	C₄	semidirect
16. #19098928	C₄	semidirect
17. #19098923	C₄	semidirect
195. #1838	D₇₉	semidirect
199. C₇₉	#21662	semidirect

Maximal subgroups

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

subgroup	index
#23773746	2
#23772337	2
#23772117	2
#23764376	2
#23764227	2
#23764109	2
#23764102	2
#23763243	2
#23763236	2
#97×Dic₇₉	2
C₇₉×#1838	2
#21662	79