Subgroups — #34257392

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23772517	C₂	1	1	yes	maximal, characteristic
2	#23772044	C₂	1	1	yes	maximal, characteristic
3	#23762188	C₂	1	2	yes	maximal, complemented
4	#23762183	C₂	1	2	yes	maximal, complemented
5	#23762178	C₂	1	2	yes	maximal, complemented
6	#23762170	C₂	1	2	yes	maximal, complemented
7	#23762153	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762150	C₂	1	1	yes	maximal, characteristic, complemented
9	#23762146	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19098313	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₉×#2988	C₂	1	1	yes	maximal, characteristic
12	#19098694	—	2	1	no	—
13	#19098641	—	2	1	no	—
14	#19098635	—	2	2	no	—
15	#19098624	C₄	1	4	yes	—
16	#19098620	C₂²	1	2	yes	—
17	#19098616	C₂²	1	1	yes	characteristic
18	#19098616	C₂²	1	2	yes	—
19	#19098616	C₂²	1	2	yes	—
20	#19098590	C₄	1	2	yes	—
21	#19098587	C₂²	1	2	yes	—
22	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
23	#19098530	C₄	1	2	yes	—
24	C₄×#11929608	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
26	#19098315	C₂²	1	2	yes	—
27	#19098315	C₂²	1	2	yes	—
28	#19098315	C₂²	1	1	yes	characteristic
29	#19098315	C₂²	1	1	yes	characteristic
30	#19098313	C₂²	1	1	yes	characteristic
31	#19098313	C₂²	1	2	yes	—
32	#19098313	C₂²	1	2	yes	—
33	#19098313	C₂²	1	2	yes	—
34	#19098313	C₂²	1	2	yes	—
35	#19098313	C₂²	1	1	yes	characteristic
36	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
37	C₈×Q₈×C₇₉	—	2	1	no	—
38	C₈×D₄×C₇₉	—	2	2	no	—
39	C₈×D₄×C₇₉	—	2	1	no	—
40	#431×C₇₉	C₂²	1	2	yes	—
41	#431×C₇₉	C₂²	1	1	yes	characteristic
42	C₂×C₄⋊C₈×C₇₉	C₂²	1	2	yes	—
43	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
44	C₂×#11929608	—	2	2	no	—
45	C₂×#11929608	C₂×C₄	1	2	yes	—
46	C₂²×C₇₉⋊C₈	—	2	1	no	—
47	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
48	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
49	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
50	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
51	C₄⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
52	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
53	C₄⋊(C₇₉⋊C₈)	C₂³	1	1	yes	characteristic
54	C₄⋊(C₇₉⋊C₈)	C₂³	1	1	yes	characteristic
55	C₂²⋊(C₇₉⋊C₈)	—	2	1	no	—
56	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
57	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
58	C₂²⋊(C₇₉⋊C₈)	C₂×C₄	1	4	yes	—
59	#14579072	—	2	2	no	—
60	#14579072	C₂×C₄	1	2	yes	—
61	C₄×C₇₉⋊C₈	—	2	1	no	—
62	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
63	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
64	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
65	C₄×C₇₉⋊C₈	C₂³	1	2	yes	—
66	C₂×#41×C₇₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₇₉	—	2	1	no	—
68	C₂²×C₈×C₇₉	—	2	2	no	—
69	C₄×Q₈×C₇₉	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₇₉	Q₈	1	1	yes	characteristic
71	C₄×D₄×C₇₉	D₄	1	2	yes	—
72	C₄×D₄×C₇₉	D₄	1	1	yes	characteristic
73	C₄×D₄×C₇₉	Q₈	1	2	yes	—
74	C₄×D₄×C₇₉	Q₈	1	1	yes	characteristic
75	#117×C₇₉	C₂×C₄	1	2	yes	—
76	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
79	C₄⋊C₈×C₇₉	—	2	1	no	—
80	C₄⋊C₈×C₇₉	—	2	2	no	—
81	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
83	C₄⋊C₈×C₇₉	C₂³	1	1	yes	characteristic
84	C₂²⋊C₈×C₇₉	—	2	1	no	—
85	C₂²⋊C₈×C₇₉	—	2	2	no	—
86	C₄×C₈×C₇₉	—	2	1	no	—
87	C₄×C₈×C₇₉	—	2	2	no	—
88	#11929608	—	2	4	no	—
89	#11929608	—	8	2	no	—
90	C₂×C₇₉⋊C₈	—	2	4	no	—
91	C₂×C₇₉⋊C₈	—	2	1	no	—
92	C₂×C₇₉⋊C₈	—	2	1	no	—
93	C₂×C₇₉⋊C₈	—	2	2	no	—
94	C₂×C₇₉⋊C₈	—	8	1	no	—
95	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
97	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
98	#41×C₇₉	—	2	4	no	—
99	C₂×Q₈×C₇₉	C₄⋊C₄	1	1	yes	characteristic
100	C₂×D₄×C₇₉	C₄⋊C₄	1	2	yes	—
101	C₂×D₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₇₉	C₂×Q₈	1	1	yes	characteristic
108	C₂×C₈×C₇₉	—	2	1	no	—
109	C₂×C₈×C₇₉	—	2	1	no	—
110	C₂×C₈×C₇₉	—	2	2	no	—
111	C₂×C₈×C₇₉	—	8	1	no	—
112	C₂×C₈×C₇₉	—	8	2	no	—
113	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
114	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
115	C₄⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
116	C₄⋊C₄×C₇₉	C₄⋊C₄	1	1	yes	characteristic
117	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
118	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
119	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	2	yes	—
120	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
122	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₉	C₂×D₄	1	2	yes	—
124	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
125	C₄²×C₇₉	C₂×Q₈	1	1	yes	characteristic
126	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂³×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	Q₈×C₇₉	—	2	2	no	—
130	D₄×C₇₉	—	2	4	no	—
131	D₄×C₇₉	—	2	2	no	—
132	C₂×C₄×C₇₉	—	2	4	no	—
133	C₂×C₄×C₇₉	—	2	1	no	—
134	C₂×C₄×C₇₉	—	2	2	no	—
135	C₂×C₄×C₇₉	—	2	1	no	—
136	C₂×C₄×C₇₉	—	2	2	no	—
137	C₂×C₄×C₇₉	—	4	1	no	—
138	C₂×C₄×C₇₉	—	4	2	no	—
139	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
143	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
144	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	#131	1	4	yes	—
146	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
149	C₈×C₇₉	—	4	1	no	—
150	C₈×C₇₉	—	4	1	no	—
151	C₈×C₇₉	—	4	2	no	—
152	C₇₉⋊C₈	—	2	2	no	—
153	C₇₉⋊C₈	—	2	2	no	—
154	C₇₉⋊C₈	—	2	4	no	—
155	C₇₉⋊C₈	—	4	1	no	—
156	C₇₉⋊C₈	—	4	2	no	—
157	C₇₉⋊C₈	—	4	1	no	—
158	C₂²×C₇₉	—	2	1	no	—
159	C₂²×C₇₉	—	2	1	no	—
160	C₂²×C₇₉	—	2	2	no	—
161	C₂²×C₇₉	—	2	2	no	—
162	C₂²×C₇₉	—	2	2	no	—
163	C₂²×C₇₉	#424	1	2	yes	—
164	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₇₉	—	2	2	no	—
166	C₄×C₇₉	—	2	1	no	—
167	C₄×C₇₉	—	2	2	no	—
168	C₄×C₇₉	—	2	1	no	—
169	C₄×C₇₉	—	2	1	no	—
170	C₄×C₇₉	—	2	2	no	—
171	C₄×C₇₉	#424	1	2	yes	—
172	C₄×C₇₉	C₂×#131	1	2	yes	—
173	C₄×C₇₉	#577	1	1	yes	characteristic
174	C₄×C₇₉	#577	1	1	yes	characteristic
175	#17811	—	79	1	no	maximal, Sylow
176	C₂×C₇₉	—	2	2	no	—
177	C₂×C₇₉	—	4	2	no	—
178	C₂×C₇₉	C₂×#424	1	1	yes	characteristic
179	C₂×C₇₉	#2912	1	1	yes	characteristic
180	C₂×C₇₉	#2916	1	1	yes	characteristic
181	#2989	—	79	1	no	—
182	#2988	D₇₉	1	1	yes	characteristic
183	#2922	—	79	1	no	—
184	#1574	—	79	2	no	—
185	#1573	—	79	2	no	—
186	#1572	—	79	2	no	—
187	#1570	—	79	2	no	—
188	#1565	—	79	1	no	—
189	#1565	—	79	1	no	—
190	#1564	—	79	1	no	—
191	C₂×#334	—	79	1	no	—
192	C₇₉	#17811	1	1	yes	characteristic, complemented, Sylow
193	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
194	#446	—	158	1	no	—
195	C₈×Q₈	—	2	1	no	—
196	#436	—	158	1	no	—
197	#435	—	158	2	no	—
198	C₈×D₄	—	2	2	no	—
199	C₈×D₄	—	2	1	no	—
200	#433	—	79	4	no	—
201	#432	—	79	2	no	—
202	#431	—	79	2	no	—
203	#431	—	79	2	no	—
204	#431	—	79	1	no	—
205	#431	C₂×D₇₉	1	1	yes	characteristic
206	#431	C₂×D₇₉	1	2	yes	—
207	#424	—	79	2	no	—
208	#423	—	79	2	no	—
209	C₂×C₄⋊C₈	—	79	1	no	—
210	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
211	C₂×C₄⋊C₈	C₂×D₇₉	1	2	yes	—
212	#408	—	79	2	no	—
213	C₄×M₁₆	—	79	1	no	—
214	C₂×C₄×C₈	—	79	1	no	—
215	#335	—	79	1	no	—
216	#335	—	79	1	no	—
217	#335	—	79	2	no	—
218	#335	—	79	2	no	—
219	#334	—	79	2	no	—
220	#334	—	79	2	no	—
221	#334	—	79	1	no	—
222	#334	—	79	1	no	—
223	#334	—	79	2	no	—
224	#334	—	79	2	no	—
225	C₂×#41	C₄×D₇₉	1	1	yes	characteristic
226	C₂×M₁₆	—	79	2	no	—
227	C₂×M₁₆	—	158	2	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	—	79	2	no	—
231	C₂²×C₈	—	158	1	no	—
232	C₄×Q₈	Dic₁₅₈	1	1	yes	characteristic
233	C₄×Q₈	C₂²⋊D₇₉	1	1	yes	characteristic
234	C₄×D₄	Dic₁₅₈	1	2	yes	—
235	C₄×D₄	Dic₁₅₈	1	1	yes	characteristic
236	C₄×D₄	C₂²⋊D₇₉	1	2	yes	—
237	C₄×D₄	C₂²⋊D₇₉	1	1	yes	characteristic
238	#117	C₄×D₇₉	1	1	yes	characteristic
239	#117	C₄×D₇₉	1	2	yes	—
240	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	2	no	—
244	C₄⋊C₈	—	79	4	no	—
245	C₄⋊C₈	—	79	4	no	—
246	C₄⋊C₈	—	79	2	no	—
247	C₄⋊C₈	—	79	1	no	—
248	C₄⋊C₈	—	79	1	no	—
249	C₄⋊C₈	—	158	1	no	—
250	C₄⋊C₈	—	158	2	no	—
251	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
252	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
253	C₄⋊C₈	C₂²×D₇₉	1	1	yes	characteristic
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	2	2	no	—
256	C₂²⋊C₈	—	79	4	no	—
257	C₂²⋊C₈	—	79	4	no	—
258	C₂²⋊C₈	—	158	1	no	—
259	C₂²⋊C₈	—	158	2	no	—
260	#97	—	79	2	no	—
261	#97	—	158	2	no	—
262	C₄×C₈	—	2	1	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	79	2	no	—
265	C₄×C₈	—	79	1	no	—
266	C₄×C₈	—	79	1	no	—
267	C₄×C₈	—	79	2	no	—
268	C₄×C₈	—	158	1	no	—
269	#41	—	2	4	no	—
270	C₂×Q₈	Dic₇₉⋊C₄	1	1	yes	characteristic
271	C₂×D₄	Dic₇₉⋊C₄	1	2	yes	—
272	C₂×D₄	Dic₇₉⋊C₄	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
274	C₂²×C₄	C₂×C₄×D₇₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
277	C₂²×C₄	C₂×Dic₁₅₈	1	2	yes	—
278	C₂²×C₄	C₂×Dic₁₅₈	1	1	yes	characteristic
279	M₁₆	—	632	2	no	—
280	M₁₆	—	158	4	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	8	1	no	—
285	C₂×C₈	—	8	2	no	—
286	C₂×C₈	—	79	2	no	—
287	C₂×C₈	—	79	2	no	—
288	C₂×C₈	—	79	4	no	—
289	C₂×C₈	—	632	1	no	—
290	C₂×C₈	—	158	1	no	—
291	C₂×C₈	—	158	1	no	—
292	C₂×C₈	—	158	2	no	—
293	C₂×C₈	—	158	4	no	—
294	C₄⋊C₄	Dic₇₉⋊C₄	1	2	yes	—
295	C₄⋊C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
296	C₄⋊C₄	Dic₇₉⋊C₄	1	2	yes	—
297	C₄⋊C₄	Dic₇₉⋊C₄	1	1	yes	characteristic
298	C₂²⋊C₄	Dic₇₉⋊C₄	1	2	yes	—
299	C₂²⋊C₄	Dic₇₉⋊C₄	1	2	yes	—
300	C₂²⋊C₄	Dic₇₉⋊C₄	1	2	yes	—
301	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₇₉	1	2	yes	—
303	C₄²	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
304	C₄²	C₂×C₂²⋊D₇₉	1	2	yes	—
305	C₄²	C₂×Dic₁₅₈	1	2	yes	—
306	C₄²	C₂×Dic₁₅₈	1	1	yes	characteristic
307	C₄²	C₂³×D₇₉	1	1	yes	characteristic
308	C₂³	C₂×Dic₇₉⋊C₄	1	2	yes	—
309	C₂³	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
310	Q₈	—	2	2	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	2	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
322	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
323	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
324	C₂×C₄	C₂×Dic₇₉⋊C₄	1	2	yes	—
325	C₂×C₄	C₂×Dic₇₉⋊C₄	1	1	yes	characteristic
326	C₂×C₄	#14579167	1	4	yes	—
327	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
329	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
330	C₈	—	4	1	no	—
331	C₈	—	4	1	no	—
332	C₈	—	4	2	no	—
333	C₈	—	316	2	no	—
334	C₈	—	316	1	no	—
335	C₈	—	316	1	no	—
336	C₈	—	158	2	no	—
337	C₈	—	158	2	no	—
338	C₈	—	158	4	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	#19098589	1	2	yes	—
345	C₂²	C₂²×Dic₇₉⋊C₄	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	#19098589	1	2	yes	—
353	C₄	C₂×#14579167	1	2	yes	—
354	C₄	#19099371	1	1	yes	characteristic
355	C₄	#19099374	1	1	yes	characteristic
356	C₂	—	2	2	no	—
357	C₂	—	4	2	no	—
358	C₂	C₂×#19098589	1	1	yes	characteristic
359	C₂	#23771993	1	1	yes	characteristic
360	C₂	#23772009	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#2988	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₇₉	C₂×C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	#2988	D₇₉
hypocenter	C₇₉	#17811
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2988

Complements

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

N	H ≅ G/N	type
3. #23762188	C₂	semidirect
4. #23762183	C₂	semidirect
5. #23762178	C₂	semidirect
6. #23762170	C₂	semidirect
7. #23762153	C₂	semidirect
8. #23762150	C₂	semidirect
9. #23762146	C₂	semidirect
10. C₂×#19098313	C₂	semidirect
192. C₇₉	#17811	semidirect

Maximal subgroups

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

subgroup	index
#23772517	2
#23772044	2
#23762188	2
#23762183	2
#23762178	2
#23762170	2
#23762153	2
#23762150	2
#23762146	2
C₂×#19098313	2
C₇₉×#2988	2
#17811	79