Subgroups — #34258951

Order 20224.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773968	C₂	1	1	yes	maximal, characteristic
2	#23772550	C₂	1	2	yes	maximal, complemented
3	#23762299	C₂	1	2	yes	maximal, complemented
4	#23762296	C₂	1	1	yes	maximal, characteristic, complemented
5	#23762285	C₂	1	1	yes	maximal, characteristic, complemented
6	#23762212	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#23762157	C₂	1	1	yes	maximal, characteristic, complemented
9	#23761862	C₂	1	2	yes	maximal, complemented
10	#23761682	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099095	C₂²	1	2	yes	—
13	#19098964	C₄	1	1	yes	characteristic
14	#19098946	C₄	1	1	yes	characteristic
15	#19098916	C₄	1	2	yes	—
16	#19098805	C₄	1	1	yes	characteristic
17	#19098796	C₄	1	1	yes	characteristic
18	#19098791	C₄	1	2	yes	—
19	#19098690	C₂²	1	2	yes	—
20	C₈×Dic₁₅₈	C₂²	1	2	yes	—
21	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
22	C₄×#11929612	C₂²	1	2	yes	—
23	#19098641	C₂²	1	2	yes	—
24	#19098638	—	2	2	no	—
25	#19098634	—	2	2	no	—
26	#19098621	—	2	2	no	—
27	#19098614	C₂²	1	1	yes	characteristic
28	#19098614	C₂²	1	1	yes	characteristic
29	#19098589	—	2	2	no	—
30	#19098585	C₂²	1	2	yes	—
31	C₄×C₈×D₇₉	C₂²	1	1	yes	characteristic
32	#19098316	C₂²	1	1	yes	characteristic
33	#19098315	C₂²	1	2	yes	—
34	#19098295	C₂²	1	2	yes	—
35	#19098293	C₂²	1	2	yes	—
36	D₄⋊(C₇₉⋊C₈)	C₂²	1	2	yes	—
37	#19098285	C₂²	1	1	yes	characteristic
38	#19098285	C₂²	1	1	yes	characteristic
39	C₈×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×D₈×C₇₉	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₇₉	C₂²	1	2	yes	—
42	#335×C₇₉	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₇₉	C₂²	1	2	yes	—
44	C₈²×C₇₉	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×#11929612	C₂×C₄	1	2	yes	—
47	C₂×#11929608	—	2	2	no	—
48	C₂×#11929607	—	2	2	no	—
49	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₇₉	C₂×C₄	1	1	yes	characteristic
51	#14579154	—	2	2	no	—
52	#14579152	C₂×C₄	1	2	yes	—
53	#14579148	C₂×C₄	1	2	yes	—
54	#14579143	—	2	2	no	—
55	#14579141	C₂×C₄	1	2	yes	—
56	#14579132	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₅₈	C₂³	1	2	yes	—
58	C₄×C₂²⋊D₇₉	—	2	2	no	—
59	#14579120	—	2	2	no	—
60	C₄²×D₇₉	C₂³	1	1	yes	characteristic
61	#14579095	C₂×C₄	1	1	yes	characteristic
62	#14579092	C₂×C₄	1	2	yes	—
63	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
64	Dic₇₉⋊C₈	—	2	2	no	—
65	Dic₇₉⋊C₈	—	2	2	no	—
66	Dic₇₉⋊C₈	—	2	2	no	—
67	Dic₇₉⋊C₈	—	2	1	no	—
68	Dic₇₉⋊C₈	D₄	1	1	yes	characteristic
69	Dic₇₉⋊C₈	D₄	1	1	yes	characteristic
70	#14579086	C₂×C₄	1	2	yes	—
71	#14579084	C₂×C₄	1	2	yes	—
72	#14579082	C₂×C₄	1	2	yes	—
73	C₂²⋊(C₇₉⋊C₈)	—	2	2	no	—
74	(C₂×D₇₉)⋊C₈	—	2	1	no	—
75	(C₂×D₇₉)⋊C₈	D₄	1	1	yes	characteristic
76	(C₂×D₇₉)⋊C₈	D₄	1	1	yes	characteristic
77	#14579071	—	2	2	no	—
78	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₇₉	—	2	2	no	—
82	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₇₉	C₂×C₄	1	1	yes	characteristic
84	C₂×D₈×C₇₉	C₂×C₄	1	1	yes	characteristic
85	C₂²×C₈×C₇₉	—	2	2	no	—
86	C₄×D₄×C₇₉	C₂³	1	2	yes	—
87	#107×C₇₉	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₇₉	C₂³	1	2	yes	—
89	#102×C₇₉	C₂×C₄	1	2	yes	—
90	C₂²⋊C₈×C₇₉	—	2	2	no	—
91	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
95	C₂×C₂²⋊D₇₉	—	2	2	no	—
96	C₂×C₄×D₇₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×D₇₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₇₉	#41	1	1	yes	characteristic
99	Dic₃₁₆	—	2	2	no	—
100	#11929612	—	2	4	no	—
101	#11929608	—	8	2	no	—
102	#11929607	—	8	2	no	—
103	C₂×C₇₉⋊C₈	—	2	2	no	—
104	C₂×C₇₉⋊C₈	—	2	1	no	—
105	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
109	C₈×D₇₉	—	2	2	no	—
110	C₈×D₇₉	—	2	2	no	—
111	C₄⋊Dic₇₉	—	2	2	no	—
112	C₄⋊Dic₇₉	—	2	2	no	—
113	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
114	Dic₇₉⋊C₄	—	2	2	no	—
115	Dic₇₉⋊C₄	—	2	2	no	—
116	C₂²⋊Dic₇₉	—	2	2	no	—
117	C₂²⋊Dic₇₉	—	2	2	no	—
118	C₂²⋊Dic₇₉	—	2	2	no	—
119	(C₂×D₇₉)⋊C₄	—	2	2	no	—
120	C₄×Dic₇₉	—	2	2	no	—
121	C₄×Dic₇₉	C₂²×C₄	1	1	yes	characteristic
122	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
123	C₄×Dic₇₉	#41	1	1	yes	characteristic
124	C₂×D₄×C₇₉	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₇₉	—	2	2	no	—
126	D₈×C₇₉	—	2	2	no	—
127	C₂×C₈×C₇₉	—	2	2	no	—
128	C₂×C₈×C₇₉	—	2	1	no	—
129	C₂×C₈×C₇₉	—	8	2	no	—
130	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₇₉	C₂²×C₄	1	2	yes	—
135	C₂²⋊C₄×C₇₉	—	2	2	no	—
136	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
137	C₂³×C₇₉	—	2	2	no	—
138	C₂²×D₇₉	C₄×D₄	1	1	yes	characteristic
139	D₄×C₇₉	—	2	2	no	—
140	D₄×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	—	4	2	no	—
144	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₇₉	—	8	2	no	—
148	C₂×Dic₇₉	—	2	2	no	—
149	C₂×Dic₇₉	—	2	2	no	—
150	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
151	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₇₉	C₄×D₄	1	1	yes	characteristic
153	C₄×D₇₉	—	2	1	no	—
154	C₄×D₇₉	—	2	1	no	—
155	C₄×D₇₉	—	2	1	no	—
156	C₄×D₇₉	—	2	1	no	—
157	C₄×D₇₉	—	4	1	no	—
158	Dic₁₅₈	—	2	2	no	—
159	Dic₁₅₈	—	2	2	no	—
160	C₈×C₇₉	—	2	1	no	—
161	C₈×C₇₉	—	2	1	no	—
162	C₈×C₇₉	—	2	1	no	—
163	C₈×C₇₉	—	4	1	no	—
164	C₈×C₇₉	—	4	2	no	—
165	C₈×C₇₉	#131	1	2	yes	—
166	C₇₉⋊C₈	—	2	1	no	—
167	C₇₉⋊C₈	—	2	1	no	—
168	C₇₉⋊C₈	—	2	1	no	—
169	C₇₉⋊C₈	—	4	1	no	—
170	C₇₉⋊C₈	—	4	2	no	—
171	C₇₉⋊C₈	#131	1	2	yes	—
172	C₂²×C₇₉	—	4	2	no	—
173	C₂²×C₇₉	—	4	2	no	—
174	C₂²×C₇₉	—	4	2	no	—
175	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₇₉	—	2	1	no	—
177	C₂×D₇₉	—	2	1	no	—
178	C₂×D₇₉	—	2	1	no	—
179	C₄×C₇₉	—	2	1	no	—
180	C₄×C₇₉	—	4	2	no	—
181	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
182	C₄×C₇₉	C₂×#131	1	1	yes	characteristic
183	C₄×C₇₉	#576	1	1	yes	characteristic
184	C₄×C₇₉	#578	1	1	yes	characteristic
185	Dic₇₉	—	2	1	no	—
186	Dic₇₉	—	2	1	no	—
187	Dic₇₉	—	2	1	no	—
188	Dic₇₉	—	4	2	no	—
189	Dic₇₉	—	4	2	no	—
190	#17973	—	79	1	no	maximal, Sylow
191	C₂×C₇₉	—	8	2	no	—
192	C₂×C₇₉	#2931	1	1	yes	characteristic
193	C₂×C₇₉	#2951	1	1	yes	characteristic
194	C₂×C₇₉	#2957	1	1	yes	characteristic
195	D₇₉	—	4	1	no	—
196	#3168	—	79	1	no	—
197	#2993	—	79	2	no	—
198	#1592	—	79	2	no	—
199	#1591	—	79	1	no	—
200	#1589	—	79	1	no	—
201	SD₁₆⋊C₈	—	79	2	no	—
202	C₈×Q₁₆	—	79	1	no	—
203	C₈×D₈	D₇₉	1	1	yes	characteristic, complemented
204	#1567	—	79	1	no	—
205	#1495	—	79	2	no	—
206	#1450	—	79	1	no	—
207	C₇₉	#17973	1	1	yes	characteristic, complemented, Sylow
208	#529	—	79	2	no	—
209	#499	—	79	1	no	—
210	#495	—	79	1	no	—
211	#488	—	79	2	no	—
212	#467	—	79	1	no	—
213	#466	—	79	1	no	—
214	#465	—	79	2	no	—
215	#446	—	79	2	no	—
216	C₈×Q₈	—	79	2	no	—
217	C₄×Q₁₆	—	79	1	no	—
218	C₄×SD₁₆	—	79	2	no	—
219	C₄×D₈	C₂×D₇₉	1	1	yes	characteristic
220	#436	—	79	2	no	—
221	#436	—	158	2	no	—
222	#435	—	158	2	no	—
223	C₈×D₄	C₂×D₇₉	1	2	yes	—
224	#433	—	158	2	no	—
225	#431	—	79	1	no	—
226	#431	—	79	1	no	—
227	#424	—	158	2	no	—
228	#423	—	79	2	no	—
229	C₂×C₄×C₈	—	79	1	no	—
230	#335	—	79	2	no	—
231	#335	—	79	1	no	—
232	#335	C₂×D₇₉	1	1	yes	characteristic
233	Q₈⋊C₈	—	79	2	no	—
234	Q₈⋊C₈	—	79	2	no	—
235	D₄⋊C₈	—	79	2	no	—
236	D₄⋊C₈	C₂×D₇₉	1	2	yes	—
237	#322	—	79	1	no	—
238	#322	—	79	1	no	—
239	C₈²	—	79	1	no	—
240	C₈²	C₂×D₇₉	1	1	yes	characteristic
241	C₂×Q₁₆	—	79	1	no	—
242	C₂×SD₁₆	—	79	2	no	—
243	C₂×D₈	C₄×D₇₉	1	1	yes	characteristic
244	C₂×M₁₆	—	158	2	no	—
245	C₂×M₁₆	—	158	2	no	—
246	C₂²×C₈	—	2	2	no	—
247	C₂²×C₈	—	79	1	no	—
248	C₂²×C₈	—	79	1	no	—
249	#126	—	158	2	no	—
250	#125	—	79	2	no	—
251	#124	—	79	2	no	—
252	#123	—	158	2	no	—
253	C₂²⋊Q₈	—	79	2	no	—
254	#121	—	79	2	no	—
255	C₄×Q₈	—	79	2	no	—
256	C₄×D₄	—	158	2	no	—
257	C₄×D₄	C₂²×D₇₉	1	2	yes	—
258	#117	—	158	2	no	—
259	C₂×C₄²	—	79	1	no	—
260	#107	—	79	1	no	—
261	#107	C₄×D₇₉	1	1	yes	characteristic
262	#106	—	79	2	no	—
263	C₄⋊C₈	—	79	1	no	—
264	C₄⋊C₈	—	79	1	no	—
265	C₄⋊C₈	—	79	2	no	—
266	C₄⋊C₈	—	158	2	no	—
267	C₄⋊C₈	—	158	1	no	—
268	C₄⋊C₈	—	158	2	no	—
269	C₄⋊C₈	—	158	2	no	—
270	C₄⋊C₈	C₂²×D₇₉	1	2	yes	—
271	#103	—	79	2	no	—
272	#103	—	79	2	no	—
273	#102	—	79	2	no	—
274	#102	C₄×D₇₉	1	2	yes	—
275	C₂²⋊C₈	—	2	2	no	—
276	C₂²⋊C₈	—	79	1	no	—
277	C₂²⋊C₈	—	79	1	no	—
278	C₂²⋊C₈	—	158	1	no	—
279	C₂²⋊C₈	—	158	2	no	—
280	#97	—	158	2	no	—
281	C₄×C₈	—	79	1	no	—
282	C₄×C₈	—	79	1	no	—
283	C₄×C₈	—	79	1	no	—
284	C₄×C₈	—	79	1	no	—
285	C₄×C₈	—	79	1	no	—
286	C₄×C₈	—	158	2	no	—
287	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
290	C₂×Q₈	—	79	2	no	—
291	C₂×D₄	—	158	2	no	—
292	C₂×D₄	C₂×C₄×D₇₉	1	2	yes	—
293	C₂²×C₄	—	2	2	no	—
294	C₂²×C₄	—	79	1	no	—
295	C₂²×C₄	—	79	1	no	—
296	C₂²×C₄	—	79	1	no	—
297	Q₁₆	—	158	2	no	—
298	SD₁₆	—	158	4	no	—
299	D₈	—	2	2	no	—
300	M₁₆	—	632	2	no	—
301	M₁₆	—	632	2	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	8	2	no	—
305	C₂×C₈	—	79	1	no	—
306	C₂×C₈	—	79	1	no	—
307	C₂×C₈	—	79	1	no	—
308	C₂×C₈	—	79	1	no	—
309	C₂×C₈	—	158	2	no	—
310	C₂×C₈	—	158	1	no	—
311	C₂×C₈	—	158	2	no	—
312	C₂×C₈	—	158	2	no	—
313	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
315	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
316	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
317	C₄⋊C₄	—	79	2	no	—
318	C₄⋊C₄	—	158	2	no	—
319	C₄⋊C₄	—	158	2	no	—
320	C₄⋊C₄	—	158	2	no	—
321	C₄⋊C₄	—	158	2	no	—
322	C₄⋊C₄	C₂×C₄×D₇₉	1	2	yes	—
323	C₂²⋊C₄	—	2	2	no	—
324	C₂²⋊C₄	—	158	2	no	—
325	C₂²⋊C₄	—	158	2	no	—
326	C₂²⋊C₄	—	158	2	no	—
327	C₂²⋊C₄	—	158	2	no	—
328	C₄²	—	79	1	no	—
329	C₄²	—	79	1	no	—
330	C₄²	—	79	1	no	—
331	C₄²	—	158	2	no	—
332	C₄²	C₂³×D₇₉	1	1	yes	characteristic
333	C₂³	—	2	2	no	—
334	C₂³	—	79	1	no	—
335	Q₈	—	158	2	no	—
336	Q₈	—	158	2	no	—
337	D₄	—	2	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	632	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	316	1	no	—
344	C₂×C₄	—	79	1	no	—
345	C₂×C₄	—	79	1	no	—
346	C₂×C₄	—	79	1	no	—
347	C₂×C₄	—	158	2	no	—
348	C₂×C₄	—	158	2	no	—
349	C₂×C₄	—	158	1	no	—
350	C₂×C₄	—	158	1	no	—
351	C₂×C₄	—	158	1	no	—
352	C₂×C₄	—	158	1	no	—
353	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
355	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	316	1	no	—
362	C₈	—	316	2	no	—
363	C₈	—	79	2	no	—
364	C₈	—	158	1	no	—
365	C₈	—	158	1	no	—
366	C₈	—	158	1	no	—
367	C₈	#14579167	1	2	yes	—
368	C₂²	—	4	2	no	—
369	C₂²	—	4	2	no	—
370	C₂²	—	4	2	no	—
371	C₂²	—	158	1	no	—
372	C₂²	—	158	1	no	—
373	C₂²	—	158	1	no	—
374	C₂²	C₄×D₄×D₇₉	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	316	2	no	—
378	C₄	—	316	2	no	—
379	C₄	—	158	1	no	—
380	C₄	—	158	1	no	—
381	C₄	—	158	1	no	—
382	C₄	#131×D₇₉	1	1	yes	characteristic
383	C₄	C₂×#14579167	1	1	yes	characteristic
384	C₄	#19099364	1	1	yes	characteristic
385	C₄	#19099377	1	1	yes	characteristic
386	C₂	—	8	2	no	—
387	C₂	—	316	1	no	—
388	C₂	#23772107	1	1	yes	characteristic
389	C₂	#23772270	1	1	yes	characteristic
390	C₂	#23772312	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₄²	C₂³×D₇₉
Fitting	C₈×D₈×C₇₉	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	#19098796	C₄
cosocle	C₄²×C₇₉	C₂⁴
hypercenter	C₈×D₈	D₇₉
hypocenter	C₇₉	#17973
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×D₈

Complements

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

N	H ≅ G/N	type
2. #23772550	C₂	semidirect
3. #23762299	C₂	semidirect
4. #23762296	C₂	semidirect
5. #23762285	C₂	semidirect
6. #23762212	C₂	semidirect
7. C₈×Dic₃₁₆	C₂	semidirect
8. #23762157	C₂	semidirect
9. #23761862	C₂	semidirect
10. #23761682	C₂	semidirect
11. C₈×D₈×C₇₉	C₂	semidirect
203. C₈×D₈	D₇₉	semidirect
207. C₇₉	#17973	semidirect

Maximal subgroups

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

subgroup	index
#23773968	2
#23772550	2
#23762299	2
#23762296	2
#23762285	2
#23762212	2
C₈×Dic₃₁₆	2
#23762157	2
#23761862	2
#23761682	2
C₈×D₈×C₇₉	2
#17973	79