Subgroups — #34445152

Order 20224.

548 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23775792	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×D₃₁₆	C₂	1	1	yes	maximal, characteristic
3	#23765629	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765604	C₂	1	2	yes	maximal, complemented
5	#23765492	C₂	1	2	yes	maximal, complemented
6	#23765451	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765335	C₂	1	2	yes	maximal, complemented
8	C₄×#14579107	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765219	C₂	1	1	yes	maximal, characteristic
10	#358×D₇₉	C₂	1	2	yes	maximal, complemented
11	C₇₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₉	C₂²	1	2	yes	—
13	#19099274	—	2	1	no	—
14	Q₈×D₃₁₆	—	2	1	no	—
15	C₂×Dic₆₃₂	C₂²	1	2	yes	—
16	C₂×#14579107	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₇₉	C₂²	1	2	yes	—
18	#19098943	C₂²	1	1	yes	characteristic
19	#19098900	—	2	2	no	—
20	#19098855	—	2	1	no	—
21	#19098805	C₂²	1	2	yes	—
22	#19098805	C₂²	1	2	yes	—
23	#19098773	—	2	1	no	—
24	#19098720	—	2	2	no	—
25	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
26	C₈×D₃₁₆	C₂²	1	1	yes	characteristic
27	#107×D₇₉	C₂²	1	2	yes	—
28	#19098388	C₂²	1	1	yes	characteristic
29	#19098368	C₂²	1	2	yes	—
30	#19098367	C₂²	1	1	yes	characteristic
31	#19098366	C₂²	1	1	yes	characteristic
32	#19098366	C₂²	1	2	yes	—
33	#19098366	C₂²	1	2	yes	—
34	(C₂×D₇₉)⋊C₁₆	C₂²	1	2	yes	—
35	C₄×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
36	#500×C₇₉	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	#366×C₇₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₇₉	C₂²	1	1	yes	characteristic
40	#358×C₇₉	C₂²	1	2	yes	—
41	C₂×Q₈×D₇₉	—	2	2	no	—
42	C₂×Dic₃₁₆	—	2	1	no	—
43	C₂×Dic₃₁₆	C₂³	1	2	yes	—
44	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
45	C₂×#11929614	—	2	2	no	—
46	Q₁₆×D₇₉	—	2	4	no	—
47	Q₁₆×D₇₉	—	4	2	no	—
48	C₂×C₈×D₇₉	C₂³	1	2	yes	—
49	#14579161	—	2	1	no	—
50	#14579161	—	4	1	no	—
51	#14579152	—	4	1	no	—
52	#14579141	—	2	2	no	—
53	#14579141	—	2	2	no	—
54	#14579141	—	4	2	no	—
55	#14579137	—	4	2	no	—
56	C₄×Dic₁₅₈	—	2	1	no	—
57	C₄×D₃₁₆	—	4	1	no	—
58	C₄×D₃₁₆	D₄	1	1	yes	characteristic
59	#14579123	—	4	1	no	—
60	C₄⋊C₄×D₇₉	—	2	2	no	—
61	Dic₆₃₂	—	4	2	no	—
62	#14579107	—	2	2	no	—
63	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
65	C₁₆×D₇₉	—	4	2	no	—
66	#14579095	C₂³	1	1	yes	characteristic
67	#14579094	—	2	2	no	—
68	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
69	#14579086	—	2	1	no	—
70	#14579084	—	2	2	no	—
71	#14579084	—	2	2	no	—
72	(C₂×D₇₉)⋊C₈	D₄	1	2	yes	—
73	C₂×Q₁₆×C₇₉	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₇₉	—	2	1	no	—
75	C₄×Q₈×C₇₉	—	2	1	no	—
76	C₂×C₁₆×C₇₉	C₂³	1	2	yes	—
77	#107×C₇₉	—	2	1	no	—
78	#107×C₇₉	C₂³	1	1	yes	characteristic
79	#107×C₇₉	C₂³	1	2	yes	—
80	#103×C₇₉	—	2	1	no	—
81	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₅₈	—	2	2	no	—
83	C₂×Dic₁₅₈	—	2	1	no	—
84	C₂×Dic₁₅₈	—	4	2	no	—
85	Q₈×D₇₉	—	4	4	no	—
86	Q₈×D₇₉	—	8	2	no	—
87	C₂×D₃₁₆	D₈	1	1	yes	characteristic
88	C₂×C₄×D₇₉	—	4	2	no	—
89	C₂×C₄×D₇₉	—	4	2	no	—
90	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
91	Dic₃₁₆	—	2	1	no	—
92	Dic₃₁₆	—	2	2	no	—
93	Dic₃₁₆	—	2	1	no	—
94	Dic₃₁₆	—	2	2	no	—
95	Dic₃₁₆	—	8	1	no	—
96	#11929614	—	4	4	no	—
97	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
98	C₈×D₇₉	—	2	2	no	—
99	C₈×D₇₉	—	2	2	no	—
100	C₄⋊Dic₇₉	—	2	1	no	—
101	C₄⋊Dic₇₉	—	4	2	no	—
102	C₄⋊Dic₇₉	—	4	2	no	—
103	C₄⋊Dic₇₉	—	4	1	no	—
104	C₄⋊Dic₇₉	—	4	1	no	—
105	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
106	Dic₇₉⋊C₄	—	4	2	no	—
107	Dic₇₉⋊C₄	—	4	1	no	—
108	(C₂×D₇₉)⋊C₄	—	4	2	no	—
109	(C₂×D₇₉)⋊C₄	—	4	2	no	—
110	(C₂×D₇₉)⋊C₄	D₈	1	2	yes	—
111	C₄×Dic₇₉	—	4	1	no	—
112	C₇₉⋊C₁₆	—	2	1	no	—
113	C₇₉⋊C₁₆	—	2	1	no	—
114	C₂×Q₈×C₇₉	—	2	1	no	—
115	C₂×Q₈×C₇₉	—	4	1	no	—
116	Q₁₆×C₇₉	—	2	1	no	—
117	Q₁₆×C₇₉	#41	1	2	yes	—
118	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
120	C₄⋊C₄×C₇₉	—	2	1	no	—
121	C₄⋊C₄×C₇₉	—	2	2	no	—
122	C₄⋊C₄×C₇₉	—	4	1	no	—
123	C₄⋊C₄×C₇₉	—	4	1	no	—
124	C₄²×C₇₉	—	4	1	no	—
125	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₇₉	—	2	2	no	—
127	C₂²×D₇₉	C₂×D₈	1	2	yes	—
128	Q₈×C₇₉	—	2	2	no	—
129	Q₈×C₇₉	—	4	1	no	—
130	Q₈×C₇₉	—	16	1	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	1	no	—
133	C₂×C₄×C₇₉	—	4	2	no	—
134	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₇₉	—	4	1	no	—
138	C₂×Dic₇₉	—	4	1	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
141	D₃₁₆	Q₃₂	1	4	yes	complemented
142	C₄×D₇₉	—	2	2	no	—
143	C₄×D₇₉	—	2	2	no	—
144	C₄×D₇₉	—	8	4	no	—
145	C₄×D₇₉	—	16	2	no	—
146	Dic₁₅₈	—	4	2	no	—
147	Dic₁₅₈	—	4	1	no	—
148	Dic₁₅₈	—	4	2	no	—
149	Dic₁₅₈	—	4	1	no	—
150	Dic₁₅₈	—	8	4	no	—
151	C₈×C₇₉	—	2	1	no	—
152	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
153	C₈×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
154	C₇₉⋊C₈	—	2	2	no	—
155	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₇₉	—	2	2	no	—
157	C₂×D₇₉	—	2	2	no	—
158	C₂×D₇₉	C₂×Q₃₂	1	4	yes	—
159	C₄×C₇₉	—	2	1	no	—
160	C₄×C₇₉	—	4	2	no	—
161	C₄×C₇₉	—	8	1	no	—
162	C₄×C₇₉	—	8	2	no	—
163	C₄×C₇₉	C₂×Q₃₂	1	2	yes	—
164	C₄×C₇₉	#546	1	1	yes	characteristic
165	C₄×C₇₉	#577	1	1	yes	characteristic
166	Dic₇₉	—	2	2	no	—
167	Dic₇₉	—	8	1	no	—
168	Dic₇₉	—	8	2	no	—
169	Dic₇₉	—	8	1	no	—
170	#33866	—	79	1	no	maximal, Sylow
171	C₂×C₇₉	#3294	1	1	yes	characteristic
172	C₂×C₇₉	C₂²×Q₃₂	1	1	yes	characteristic
173	C₂×C₇₉	#3416	1	1	yes	characteristic
174	D₇₉	—	2	4	no	—
175	#3338	—	79	1	no	—
176	D₄×Q₁₆	—	79	1	no	—
177	#2241	—	79	1	no	—
178	#2236	—	79	2	no	—
179	#2227	D₇₉	1	1	yes	characteristic, complemented
180	#2216	—	79	2	no	—
181	#2209	—	79	1	no	—
182	#2188	—	79	2	no	—
183	C₄×Q₃₂	—	79	1	no	—
184	D₄×C₁₆	—	79	1	no	—
185	C₂×#358	—	79	2	no	—
186	C₇₉	#33866	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₁₆	—	79	2	no	—
188	#554	—	158	1	no	—
189	D₄×Q₈	—	158	1	no	—
190	C₂×Q₃₂	—	79	1	no	—
191	C₂×Q₃₂	—	79	2	no	—
192	C₂²×C₁₆	—	79	2	no	—
193	#500	C₂×D₇₉	1	1	yes	characteristic
194	#494	—	79	1	no	—
195	#484	—	158	2	no	—
196	#477	—	158	1	no	—
197	#467	—	79	2	no	—
198	#467	—	79	2	no	—
199	#462	—	158	1	no	—
200	#451	—	158	2	no	—
201	C₄×Q₁₆	—	79	1	no	—
202	C₄×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
203	C₈×D₄	—	79	1	no	—
204	C₂×#107	—	79	2	no	—
205	#366	—	79	1	no	—
206	#366	C₂×D₇₉	1	2	yes	—
207	C₄⋊C₁₆	C₂×D₇₉	1	1	yes	characteristic
208	#358	—	79	1	no	—
209	#358	—	79	2	no	—
210	#358	—	79	2	no	—
211	#358	—	79	1	no	—
212	#358	—	79	2	no	—
213	#358	C₂×D₇₉	1	2	yes	—
214	C₂²⋊C₁₆	—	79	2	no	—
215	C₄×C₁₆	—	79	1	no	—
216	C₂²×Q₈	—	158	2	no	—
217	C₂×Q₁₆	—	316	2	no	—
218	C₂×Q₁₆	—	79	2	no	—
219	C₂×Q₁₆	—	79	1	no	—
220	C₂×Q₁₆	—	158	2	no	—
221	C₂×Q₁₆	—	158	4	no	—
222	C₂×Q₁₆	—	158	1	no	—
223	C₂×Q₁₆	C₂²×D₇₉	1	1	yes	characteristic
224	C₂²×C₈	—	79	2	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	316	1	no	—
227	C₄⋊Q₈	—	158	1	no	—
228	#125	—	316	1	no	—
229	C₂²⋊Q₈	—	316	2	no	—
230	C₂²⋊Q₈	—	316	2	no	—
231	C₂²⋊Q₈	—	158	2	no	—
232	C₂²⋊Q₈	—	158	2	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	158	1	no	—
235	C₄×D₄	—	316	1	no	—
236	C₄×D₄	—	316	1	no	—
237	C₄×D₄	—	79	1	no	—
238	C₂×C₄⋊C₄	—	158	2	no	—
239	Q₃₂	—	316	2	no	—
240	Q₃₂	—	158	2	no	—
241	C₂×C₁₆	—	316	2	no	—
242	C₂×C₁₆	—	79	1	no	—
243	C₂×C₁₆	—	79	1	no	—
244	C₂×C₁₆	C₂²×D₇₉	1	2	yes	—
245	#107	—	2	1	no	—
246	#107	—	79	1	no	—
247	#107	—	158	2	no	—
248	#107	C₂²×D₇₉	1	2	yes	—
249	#107	C₂²×D₇₉	1	1	yes	characteristic
250	C₄⋊C₈	—	79	1	no	—
251	#103	—	2	1	no	—
252	#103	—	158	2	no	—
253	#103	—	158	2	no	—
254	#103	—	158	1	no	—
255	C₂²⋊C₈	—	79	2	no	—
256	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	316	2	no	—
260	C₂×Q₈	—	316	4	no	—
261	C₂×Q₈	—	632	2	no	—
262	C₂×Q₈	—	158	1	no	—
263	C₂×Q₈	—	158	2	no	—
264	C₂×D₄	—	79	1	no	—
265	C₂²×C₄	—	316	2	no	—
266	C₂²×C₄	—	316	2	no	—
267	C₂²×C₄	—	79	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	316	4	no	—
270	Q₁₆	—	632	1	no	—
271	Q₁₆	—	158	1	no	—
272	Q₁₆	—	158	2	no	—
273	Q₁₆	—	158	1	no	—
274	Q₁₆	—	158	2	no	—
275	Q₁₆	#11929625	1	2	yes	—
276	C₂×C₈	—	79	2	no	—
277	C₂×C₈	—	158	2	no	—
278	C₂×C₈	—	158	2	no	—
279	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	316	2	no	—
286	C₄⋊C₄	—	316	2	no	—
287	C₄⋊C₄	—	316	2	no	—
288	C₄⋊C₄	—	316	1	no	—
289	C₄⋊C₄	—	316	1	no	—
290	C₄⋊C₄	—	316	1	no	—
291	C₄⋊C₄	—	79	1	no	—
292	C₄⋊C₄	—	158	1	no	—
293	C₂²⋊C₄	—	316	2	no	—
294	C₂²⋊C₄	—	316	2	no	—
295	C₂²⋊C₄	—	79	2	no	—
296	C₄²	—	4	1	no	—
297	C₄²	—	316	1	no	—
298	C₄²	D₄×D₇₉	1	1	yes	characteristic
299	C₁₆	—	2	2	no	—
300	C₁₆	—	158	1	no	—
301	C₁₆	—	158	1	no	—
302	C₂³	—	79	2	no	—
303	Q₈	—	2	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	16	1	no	—
306	Q₈	—	316	2	no	—
307	Q₈	—	316	1	no	—
308	Q₈	—	316	2	no	—
309	Q₈	—	316	1	no	—
310	Q₈	—	632	4	no	—
311	D₄	—	79	4	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	316	1	no	—
316	C₂×C₄	—	316	1	no	—
317	C₂×C₄	—	316	2	no	—
318	C₂×C₄	—	79	2	no	—
319	C₂×C₄	—	632	4	no	—
320	C₂×C₄	—	158	2	no	—
321	C₂×C₄	—	158	2	no	—
322	C₂×C₄	—	1264	2	no	—
323	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
324	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	158	2	no	—
328	C₈	C₂×#11929625	1	1	yes	characteristic
329	C₈	#14579207	1	1	yes	characteristic
330	C₂²	—	79	4	no	—
331	C₂²	—	158	2	no	—
332	C₂²	—	158	2	no	—
333	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	632	1	no	—
339	C₄	—	632	2	no	—
340	C₄	—	632	1	no	—
341	C₄	—	158	2	no	—
342	C₄	Q₃₂×D₇₉	1	2	yes	—
343	C₄	#19099204	1	1	yes	characteristic
344	C₄	#19099372	1	1	yes	characteristic
345	C₂	—	158	4	no	—
346	C₂	#23775210	1	1	yes	characteristic
347	C₂	C₂×Q₃₂×D₇₉	1	1	yes	characteristic
348	C₂	#23776546	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₇₉×#2227	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₃₁₆	D₈
hypercenter	#2227	D₇₉
hypocenter	C₇₉	#33866
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
3. #23765629	C₂	semidirect
4. #23765604	C₂	semidirect
5. #23765492	C₂	semidirect
6. #23765451	C₂	semidirect
7. #23765335	C₂	semidirect
8. C₄×#14579107	C₂	semidirect
10. #358×D₇₉	C₂	semidirect
11. C₇₉×#2227	C₂	semidirect
141. D₃₁₆	Q₃₂	semidirect
179. #2227	D₇₉	semidirect
186. C₇₉	#33866	semidirect

Maximal subgroups

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

subgroup	index
#23775792	2
Q₁₆×D₃₁₆	2
#23765629	2
#23765604	2
#23765492	2
#23765451	2
#23765335	2
C₄×#14579107	2
#23765219	2
#358×D₇₉	2
C₇₉×#2227	2
#33866	79