Subgroups — #34445918

Order 20224.

506 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776404	C₂	1	1	yes	maximal, characteristic, complemented
2	#23776252	C₂	1	1	yes	maximal, characteristic
3	#23765626	C₂	1	2	yes	maximal, complemented
4	#23765540	C₂	1	2	yes	maximal, complemented
5	#23765539	C₂	1	1	yes	maximal, characteristic, complemented
6	#23765444	C₂	1	2	yes	maximal
7	#23765442	C₂	1	1	yes	maximal, characteristic
8	C₁₆×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14579106	C₂	1	1	yes	maximal, characteristic
10	#23765249	C₂	1	2	yes	maximal
11	C₇₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	#19099288	—	2	1	no	—
13	#19099228	—	2	1	no	—
14	C₂×#14579106	C₂²	1	1	yes	characteristic
15	C₂×#14579104	C₂²	1	2	yes	—
16	#19098967	C₂²	1	2	yes	—
17	#19098941	C₂²	1	2	yes	—
18	#19098842	—	2	2	no	—
19	#19098840	—	2	1	no	—
20	#19098771	—	2	2	no	—
21	#19098770	—	2	1	no	—
22	C₈×Dic₁₅₈	C₂²	1	1	yes	characteristic
23	C₄×Dic₃₁₆	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
25	C₄×D₆₃₂	C₂²	1	1	yes	characteristic
26	#19098645	C₂²	1	2	yes	—
27	#19098391	C₂²	1	1	yes	characteristic
28	#19098390	C₂²	1	2	yes	—
29	C₄⋊(C₇₉⋊C₁₆)	C₂²	1	1	yes	characteristic
30	Dic₇₉⋊C₁₆	C₂²	1	2	yes	—
31	#19098368	C₂²	1	1	yes	characteristic
32	#19098367	C₂²	1	2	yes	—
33	#19098366	C₂²	1	2	yes	—
34	#19098365	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₉	C₂²	1	2	yes	—
36	#500×C₇₉	C₂²	1	1	yes	characteristic
37	#494×C₇₉	C₂²	1	1	yes	characteristic
38	#358×C₇₉	C₂²	1	2	yes	—
39	#358×C₇₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂×Q₈×D₇₉	—	4	1	no	—
42	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
43	C₂×#11929614	—	2	2	no	—
44	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
45	#14579160	—	2	2	no	—
46	#14579159	—	2	2	no	—
47	#14579152	—	4	1	no	—
48	#14579151	—	4	2	no	—
49	#14579139	—	4	2	no	—
50	#14579137	—	4	1	no	—
51	Q₈×Dic₇₉	—	2	2	no	—
52	C₄×Dic₁₅₈	—	2	1	no	—
53	C₄×Dic₁₅₈	D₄	1	1	yes	characteristic
54	#14579126	—	4	1	no	—
55	C₄×D₃₁₆	—	2	1	no	—
56	#14579123	—	2	2	no	—
57	#14579106	—	2	2	no	—
58	#14579104	—	2	4	no	—
59	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
60	#14579095	C₂³	1	1	yes	characteristic
61	#14579095	C₂³	1	1	yes	characteristic
62	#14579094	—	2	2	no	—
63	C₄⋊(C₇₉⋊C₈)	D₄	1	1	yes	characteristic
64	Dic₇₉⋊C₈	D₄	1	2	yes	—
65	#14579086	—	2	1	no	—
66	#14579085	—	2	2	no	—
67	#14579083	—	2	1	no	—
68	#14579081	—	2	2	no	—
69	C₈×Dic₇₉	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₇₉	—	2	1	no	—
71	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₇₉	—	2	1	no	—
73	C₄⋊Q₈×C₇₉	—	2	1	no	—
74	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
76	#107×C₇₉	—	2	1	no	—
77	#107×C₇₉	C₂³	1	2	yes	—
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅₈	—	2	1	no	—
80	C₂×Dic₁₅₈	—	4	2	no	—
81	C₂×Dic₁₅₈	D₈	1	1	yes	characteristic
82	Q₈×D₇₉	—	8	4	no	—
83	C₂×D₃₁₆	—	2	1	no	—
84	C₂×C₄×D₇₉	—	4	2	no	—
85	C₂×C₄×D₇₉	—	4	1	no	—
86	Dic₃₁₆	—	2	1	no	—
87	Dic₃₁₆	—	2	1	no	—
88	#11929614	—	8	2	no	—
89	D₆₃₂	—	2	1	no	—
90	D₆₃₂	#41	1	2	yes	—
91	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
92	C₄⋊Dic₇₉	—	2	1	no	—
93	C₄⋊Dic₇₉	—	2	1	no	—
94	C₄⋊Dic₇₉	—	4	2	no	—
95	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
96	Dic₇₉⋊C₄	—	4	2	no	—
97	Dic₇₉⋊C₄	—	4	2	no	—
98	Dic₇₉⋊C₄	—	4	2	no	—
99	Dic₇₉⋊C₄	—	4	1	no	—
100	Dic₇₉⋊C₄	D₈	1	2	yes	—
101	(C₂×D₇₉)⋊C₄	—	4	2	no	—
102	(C₂×D₇₉)⋊C₄	—	4	1	no	—
103	C₄×Dic₇₉	—	4	2	no	—
104	C₄×Dic₇₉	—	4	1	no	—
105	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
106	C₇₉⋊C₁₆	—	2	2	no	—
107	C₂×Q₈×C₇₉	—	2	2	no	—
108	C₂×Q₈×C₇₉	—	4	1	no	—
109	Q₁₆×C₇₉	—	2	2	no	—
110	Q₁₆×C₇₉	—	2	2	no	—
111	Q₁₆×C₇₉	—	8	1	no	—
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₇₉	—	2	2	no	—
115	C₄⋊C₄×C₇₉	—	4	1	no	—
116	C₄⋊C₄×C₇₉	—	4	1	no	—
117	C₄⋊C₄×C₇₉	—	4	1	no	—
118	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₇₉	—	2	1	no	—
120	C₁₆×C₇₉	—	2	1	no	—
121	C₂²×D₇₉	—	4	1	no	—
122	Q₈×C₇₉	—	4	2	no	—
123	Q₈×C₇₉	—	4	2	no	—
124	Q₈×C₇₉	—	8	2	no	—
125	C₂×C₄×C₇₉	—	4	2	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₇₉	—	4	1	no	—
131	C₂×Dic₇₉	—	4	1	no	—
132	C₂×Dic₇₉	—	4	1	no	—
133	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
134	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
135	D₃₁₆	—	2	2	no	—
136	D₃₁₆	—	4	1	no	—
137	C₄×D₇₉	—	8	2	no	—
138	C₄×D₇₉	—	8	4	no	—
139	Dic₁₅₈	—	4	1	no	—
140	Dic₁₅₈	—	4	1	no	—
141	Dic₁₅₈	—	8	4	no	—
142	Dic₁₅₈	SD₃₂	1	4	yes	complemented
143	C₈×C₇₉	—	2	1	no	—
144	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
146	C₇₉⋊C₈	—	2	2	no	—
147	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₇₉	—	4	2	no	—
149	C₂×D₇₉	—	8	1	no	—
150	C₂×D₇₉	—	8	1	no	—
151	C₄×C₇₉	—	2	1	no	—
152	C₄×C₇₉	—	8	2	no	—
153	C₄×C₇₉	—	8	2	no	—
154	C₄×C₇₉	C₂×SD₃₂	1	2	yes	—
155	C₄×C₇₉	#548	1	1	yes	characteristic
156	C₄×C₇₉	#577	1	1	yes	characteristic
157	Dic₇₉	—	2	2	no	—
158	Dic₇₉	—	8	1	no	—
159	Dic₇₉	—	8	1	no	—
160	Dic₇₉	—	8	1	no	—
161	Dic₇₉	C₂×SD₃₂	1	4	yes	—
162	#33926	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	#3358	1	1	yes	characteristic
164	C₂×C₇₉	C₂²×SD₃₂	1	1	yes	characteristic
165	C₂×C₇₉	#3417	1	1	yes	characteristic
166	D₇₉	—	8	2	no	—
167	#3395	—	79	1	no	—
168	Q₈×D₈	—	79	1	no	—
169	#2241	—	79	2	no	—
170	#2241	D₇₉	1	1	yes	characteristic, complemented
171	#2225	—	79	1	no	—
172	#2225	—	79	2	no	—
173	#2208	—	79	1	no	—
174	#2208	—	79	2	no	—
175	Q₈×C₁₆	—	79	1	no	—
176	C₄×SD₃₂	—	79	2	no	—
177	C₄×SD₃₂	—	79	1	no	—
178	C₇₉	#33926	1	1	yes	characteristic, complemented, Sylow
179	#557	—	158	1	no	—
180	D₄×Q₈	—	158	1	no	—
181	C₂×SD₃₂	—	79	2	no	—
182	C₂×SD₃₂	—	79	1	no	—
183	#500	—	79	2	no	—
184	#500	C₂×D₇₉	1	1	yes	characteristic
185	#494	—	79	2	no	—
186	#494	C₂×D₇₉	1	1	yes	characteristic
187	#474	—	158	2	no	—
188	#474	—	158	1	no	—
189	#462	—	158	1	no	—
190	#462	—	158	2	no	—
191	C₈×Q₈	—	79	1	no	—
192	C₄×Q₁₆	—	79	2	no	—
193	C₄×Q₁₆	—	79	1	no	—
194	C₄×D₈	—	79	2	no	—
195	C₄×D₈	—	79	1	no	—
196	#367	—	79	1	no	—
197	#367	—	79	2	no	—
198	C₄⋊C₁₆	—	79	2	no	—
199	C₄⋊C₁₆	—	79	1	no	—
200	#358	—	79	2	no	—
201	#358	—	79	1	no	—
202	#358	—	79	2	no	—
203	#358	C₂×D₇₉	1	2	yes	—
204	#358	C₂×D₇₉	1	2	yes	—
205	#357	—	79	1	no	—
206	C₄×C₁₆	—	79	2	no	—
207	C₄×C₁₆	C₂×D₇₉	1	1	yes	characteristic
208	C₂²×Q₈	—	316	1	no	—
209	C₂×Q₁₆	—	2	1	no	—
210	C₂×Q₁₆	—	79	1	no	—
211	C₂×Q₁₆	—	158	2	no	—
212	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
213	C₂×D₈	—	79	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	158	2	no	—
217	C₄⋊Q₈	—	158	2	no	—
218	#125	—	316	2	no	—
219	#125	—	316	1	no	—
220	C₂²⋊Q₈	—	316	2	no	—
221	C₂²⋊Q₈	—	316	1	no	—
222	C₄×Q₈	—	316	1	no	—
223	C₄×Q₈	—	79	1	no	—
224	C₄×Q₈	—	158	2	no	—
225	C₄×Q₈	—	158	1	no	—
226	C₄×D₄	—	158	2	no	—
227	C₄×D₄	—	158	1	no	—
228	SD₃₂	—	158	2	no	—
229	SD₃₂	—	158	4	no	—
230	C₂×C₁₆	—	79	2	no	—
231	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
233	#107	—	2	1	no	—
234	#107	—	79	1	no	—
235	#107	—	79	1	no	—
236	#107	—	158	2	no	—
237	#107	C₂²×D₇₉	1	2	yes	—
238	C₄⋊C₈	—	79	2	no	—
239	C₄⋊C₈	—	79	1	no	—
240	#103	—	158	2	no	—
241	#103	—	158	1	no	—
242	#102	—	158	2	no	—
243	#102	—	158	1	no	—
244	C₄×C₈	—	79	2	no	—
245	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	316	2	no	—
249	C₂×Q₈	—	79	1	no	—
250	C₂×Q₈	—	632	4	no	—
251	C₂×Q₈	—	158	1	no	—
252	C₂×D₄	—	158	1	no	—
253	C₂²×C₄	—	316	2	no	—
254	C₂²×C₄	—	316	1	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	—	8	1	no	—
258	Q₁₆	—	632	2	no	—
259	Q₁₆	—	158	1	no	—
260	Q₁₆	—	158	1	no	—
261	D₈	—	79	2	no	—
262	D₈	—	158	1	no	—
263	C₂×C₈	—	79	2	no	—
264	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	316	2	no	—
271	C₄⋊C₄	—	316	2	no	—
272	C₄⋊C₄	—	316	2	no	—
273	C₄⋊C₄	—	316	2	no	—
274	C₄⋊C₄	—	316	1	no	—
275	C₄⋊C₄	—	79	2	no	—
276	C₄⋊C₄	—	79	1	no	—
277	C₄⋊C₄	—	158	1	no	—
278	C₄⋊C₄	—	158	1	no	—
279	C₂²⋊C₄	—	316	2	no	—
280	C₂²⋊C₄	—	316	1	no	—
281	C₄²	—	316	2	no	—
282	C₄²	—	316	1	no	—
283	C₄²	—	79	2	no	—
284	C₄²	D₄×D₇₉	1	1	yes	characteristic
285	C₁₆	—	2	1	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	—	158	2	no	—
288	C₂³	—	316	1	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	8	2	no	—
292	Q₈	—	316	1	no	—
293	Q₈	—	316	1	no	—
294	Q₈	—	79	4	no	—
295	Q₈	—	632	4	no	—
296	D₄	—	316	1	no	—
297	D₄	—	158	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	316	1	no	—
301	C₂×C₄	—	316	1	no	—
302	C₂×C₄	—	316	1	no	—
303	C₂×C₄	—	79	2	no	—
304	C₂×C₄	—	79	2	no	—
305	C₂×C₄	—	632	4	no	—
306	C₂×C₄	—	632	2	no	—
307	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
308	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	158	2	no	—
312	C₈	#41×D₇₉	1	1	yes	characteristic
313	C₈	#14579208	1	1	yes	characteristic
314	C₂²	—	316	2	no	—
315	C₂²	—	632	1	no	—
316	C₂²	—	632	1	no	—
317	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	79	4	no	—
322	C₄	—	632	1	no	—
323	C₄	—	632	1	no	—
324	C₄	—	632	1	no	—
325	C₄	—	158	2	no	—
326	C₄	SD₃₂×D₇₉	1	2	yes	—
327	C₄	#19099221	1	1	yes	characteristic
328	C₄	#19099369	1	1	yes	characteristic
329	C₂	—	632	2	no	—
330	C₂	#23776032	1	1	yes	characteristic
331	C₂	C₂×SD₃₂×D₇₉	1	1	yes	characteristic
332	C₂	#23776549	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×#2241	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×D₆₃₂	C₂³
hypercenter	#2241	D₇₉
hypocenter	C₇₉	#33926
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #23776404	C₂	semidirect
3. #23765626	C₂	semidirect
4. #23765540	C₂	semidirect
5. #23765539	C₂	semidirect
8. C₁₆×Dic₁₅₈	C₂	semidirect
11. C₇₉×#2241	C₂	semidirect
142. Dic₁₅₈	SD₃₂	semidirect
170. #2241	D₇₉	semidirect
178. C₇₉	#33926	semidirect

Maximal subgroups

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

subgroup	index
#23776404	2
#23776252	2
#23765626	2
#23765540	2
#23765539	2
#23765444	2
#23765442	2
C₁₆×Dic₁₅₈	2
C₄×#14579106	2
#23765249	2
C₇₉×#2241	2
#33926	79