Subgroups — #34272122

Order 20224.

602 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23773986	C₂	1	1	yes	maximal, characteristic
2	#23773207	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₅₈	C₂	1	1	yes	maximal, characteristic
4	#23762859	C₂	1	4	yes	maximal, complemented
5	#23762805	C₂	1	2	yes	maximal, complemented
6	#23762771	C₂	1	2	yes	maximal, complemented
7	#23762091	C₂	1	2	yes	maximal
8	C₇₉×#1567	C₂	1	1	yes	maximal, characteristic
9	#19099120	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₅₈	C₂²	1	1	yes	characteristic
11	#19098968	C₂²	1	1	yes	characteristic
12	#19098943	C₂²	1	1	yes	characteristic
13	#19098914	C₂²	1	2	yes	—
14	#19098871	—	2	4	no	—
15	#19098859	C₂²	1	4	yes	—
16	#19098849	C₂²	1	4	yes	—
17	#19098804	—	2	1	no	—
18	#19098804	—	2	1	no	—
19	C₈×Dic₁₅₈	—	2	1	no	—
20	C₈×Dic₁₅₈	—	2	1	no	—
21	C₈×Dic₁₅₈	C₂²	1	2	yes	—
22	C₈×Dic₁₅₈	C₂²	1	2	yes	—
23	C₄×Dic₃₁₆	—	2	1	no	—
24	C₄×Dic₃₁₆	—	2	1	no	—
25	#19098605	—	2	2	no	—
26	C₂×C₄⋊(C₇₉⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×Dic₇₉⋊C₈	—	2	1	no	—
28	#19098573	—	2	2	no	—
29	C₂×C₈×Dic₇₉	—	2	1	no	—
30	#19098311	C₂²	1	4	yes	—
31	#19098308	C₂²	1	4	yes	—
32	#423×C₇₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
34	#335×C₇₉	C₂²	1	2	yes	—
35	#335×C₇₉	C₂²	1	2	yes	—
36	C₂²×Dic₁₅₈	D₄	1	1	yes	characteristic
37	C₂×Dic₃₁₆	—	2	1	no	—
38	C₂×Dic₃₁₆	—	2	1	no	—
39	C₂²×C₇₉⋊C₈	—	2	1	no	—
40	#14579161	C₂³	1	2	yes	—
41	#14579161	C₂³	1	2	yes	—
42	#14579153	—	2	2	no	—
43	#14579140	—	2	2	no	—
44	#14579140	—	2	2	no	—
45	C₄×Dic₁₅₈	—	2	1	no	—
46	C₄×Dic₁₅₈	—	2	1	no	—
47	C₄×Dic₁₅₈	—	2	2	no	—
48	C₄×Dic₁₅₈	—	2	2	no	—
49	C₄×Dic₁₅₈	D₄	1	2	yes	—
50	C₄×Dic₁₅₈	C₂³	1	2	yes	—
51	#14579120	—	2	2	no	—
52	C₂×C₄⋊Dic₇₉	D₄	1	1	yes	characteristic
53	C₂×Dic₇₉⋊C₄	—	2	1	no	—
54	C₂×C₄×Dic₇₉	—	2	1	no	—
55	#14579095	—	2	1	no	—
56	#14579095	—	2	1	no	—
57	#14579095	—	2	4	no	—
58	#14579093	—	2	4	no	—
59	C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₇₉⋊C₈)	C₂³	1	2	yes	—
61	Dic₇₉⋊C₈	—	2	2	no	—
62	Dic₇₉⋊C₈	—	2	2	no	—
63	#14579086	—	2	2	no	—
64	#14579086	—	2	2	no	—
65	#14579086	—	2	4	no	—
66	#14579086	—	2	4	no	—
67	C₈×Dic₇₉	—	2	2	no	—
68	C₈×Dic₇₉	—	2	2	no	—
69	C₂×M₁₆×C₇₉	—	2	2	no	—
70	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
71	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₇₉	—	2	2	no	—
74	C₄⋊C₈×C₇₉	C₂³	1	4	yes	—
75	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
76	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₅₈	—	2	1	no	—
80	C₂×Dic₁₅₈	—	2	2	no	—
81	C₂×Dic₁₅₈	—	2	2	no	—
82	C₂×Dic₁₅₈	—	2	2	no	—
83	C₂×Dic₁₅₈	Q₁₆	1	4	yes	complemented
84	C₂×Dic₁₅₈	C₂×D₄	1	2	yes	—
85	C₂²×Dic₇₉	—	2	1	no	—
86	C₂²×Dic₇₉	—	2	1	no	—
87	Dic₃₁₆	—	4	2	no	—
88	Dic₃₁₆	—	4	2	no	—
89	C₂×C₇₉⋊C₈	—	2	2	no	—
90	C₂×C₇₉⋊C₈	—	4	2	no	—
91	C₄⋊Dic₇₉	—	2	1	no	—
92	C₄⋊Dic₇₉	—	2	2	no	—
93	C₄⋊Dic₇₉	—	2	2	no	—
94	C₄⋊Dic₇₉	—	2	2	no	—
95	C₄⋊Dic₇₉	—	2	2	no	—
96	C₄⋊Dic₇₉	C₂×D₄	1	2	yes	—
97	Dic₇₉⋊C₄	—	2	2	no	—
98	Dic₇₉⋊C₄	—	2	2	no	—
99	Dic₇₉⋊C₄	—	4	2	no	—
100	Dic₇₉⋊C₄	—	4	2	no	—
101	C₂²⋊Dic₇₉	—	4	2	no	—
102	C₄×Dic₇₉	—	2	2	no	—
103	C₄×Dic₇₉	—	2	2	no	—
104	C₄×Dic₇₉	—	4	2	no	—
105	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
108	M₁₆×C₇₉	—	4	4	no	—
109	C₂×C₈×C₇₉	—	2	2	no	—
110	C₂×C₈×C₇₉	—	2	4	no	—
111	C₂×C₈×C₇₉	—	4	1	no	—
112	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₇₉	#41	1	1	yes	characteristic
119	C₄²×C₇₉	C₂⁴	1	1	yes	characteristic
120	C₂³×C₇₉	#121	1	1	yes	characteristic
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	4	1	no	—
124	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₇₉	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₇₉	#135	1	2	yes	—
129	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
132	C₂×Dic₇₉	—	2	2	no	—
133	C₂×Dic₇₉	—	2	4	no	—
134	C₂×Dic₇₉	—	2	2	no	—
135	C₂×Dic₇₉	—	4	2	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	Dic₁₅₈	—	2	2	no	—
139	Dic₁₅₈	—	2	4	no	—
140	Dic₁₅₈	—	4	2	no	—
141	Dic₁₅₈	—	4	2	no	—
142	Dic₁₅₈	—	8	2	no	—
143	Dic₁₅₈	C₂×Q₁₆	1	4	yes	—
144	C₈×C₇₉	—	2	1	no	—
145	C₈×C₇₉	—	2	1	no	—
146	C₈×C₇₉	—	2	1	no	—
147	C₈×C₇₉	—	4	4	no	—
148	C₈×C₇₉	#137	1	2	yes	—
149	C₇₉⋊C₈	—	4	2	no	—
150	C₂²×C₇₉	—	2	1	no	—
151	C₂²×C₇₉	—	2	1	no	—
152	C₂²×C₇₉	#467	1	2	yes	—
153	C₂²×C₇₉	C₂×#121	1	1	yes	characteristic
154	C₄×C₇₉	—	2	1	no	—
155	C₄×C₇₉	—	2	1	no	—
156	C₄×C₇₉	—	2	1	no	—
157	C₄×C₇₉	#467	1	2	yes	—
158	C₄×C₇₉	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
160	C₄×C₇₉	C₂×#137	1	1	yes	characteristic
161	C₄×C₇₉	#575	1	1	yes	characteristic
162	Dic₇₉	—	2	4	no	—
163	Dic₇₉	—	4	2	no	—
164	Dic₇₉	—	8	2	no	—
165	Dic₇₉	—	8	2	no	—
166	#19262	—	79	1	no	maximal, Sylow
167	C₂×C₇₉	—	2	2	no	—
168	C₂×C₇₉	#3037	1	1	yes	characteristic
169	C₂×C₇₉	C₂×#467	1	1	yes	characteristic
170	C₂×C₇₉	#3067	1	1	yes	characteristic
171	#3170	—	79	1	no	—
172	#3078	—	79	2	no	—
173	C₂×C₈×Q₈	—	79	1	no	—
174	#1684	—	79	4	no	—
175	#1676	—	79	2	no	—
176	#1670	—	79	2	no	—
177	#1567	D₇₉	1	1	yes	characteristic
178	#1550	—	79	2	no	—
179	C₇₉	#19262	1	1	yes	characteristic, complemented, Sylow
180	#533	—	79	2	no	—
181	C₂×C₄×Q₈	—	79	1	no	—
182	#500	—	79	1	no	—
183	#494	—	79	1	no	—
184	#487	—	79	2	no	—
185	#479	—	158	4	no	—
186	#477	—	79	4	no	—
187	#475	—	79	4	no	—
188	#467	—	158	1	no	—
189	#467	—	158	1	no	—
190	C₈×Q₈	—	79	2	no	—
191	C₈×Q₈	—	79	2	no	—
192	C₈×Q₈	—	158	1	no	—
193	C₈×Q₈	—	158	1	no	—
194	C₄×Q₁₆	—	158	1	no	—
195	C₄×Q₁₆	—	158	1	no	—
196	#428	—	158	2	no	—
197	#423	C₂×D₇₉	1	2	yes	—
198	C₂×C₄⋊C₈	—	79	1	no	—
199	C₂×C₄⋊C₈	—	158	1	no	—
200	#419	—	158	2	no	—
201	C₂×C₄×C₈	—	158	1	no	—
202	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
203	#335	C₂×D₇₉	1	2	yes	—
204	#335	C₂×D₇₉	1	2	yes	—
205	#333	—	79	4	no	—
206	#332	—	79	4	no	—
207	C₂²×Q₈	—	79	1	no	—
208	C₂×Q₁₆	—	158	1	no	—
209	C₂×Q₁₆	—	158	1	no	—
210	C₂×M₁₆	—	2	2	no	—
211	C₂²×C₈	—	158	1	no	—
212	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
213	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
214	C₄⋊Q₈	—	79	2	no	—
215	C₄⋊Q₈	—	79	2	no	—
216	#125	—	158	2	no	—
217	C₂²⋊Q₈	—	158	2	no	—
218	C₂²⋊Q₈	—	158	2	no	—
219	C₄×Q₈	—	79	2	no	—
220	C₄×Q₈	—	79	2	no	—
221	C₄×Q₈	—	158	2	no	—
222	C₄×Q₈	—	158	2	no	—
223	C₄×Q₈	—	158	1	no	—
224	C₄×Q₈	—	158	1	no	—
225	#117	—	158	2	no	—
226	C₂×C₄⋊C₄	—	79	1	no	—
227	C₂×C₄⋊C₄	—	158	1	no	—
228	C₂×C₄²	—	158	1	no	—
229	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
230	#107	—	158	4	no	—
231	#107	—	158	1	no	—
232	#107	—	158	1	no	—
233	#106	—	158	4	no	—
234	C₄⋊C₈	—	2	2	no	—
235	C₄⋊C₈	—	79	2	no	—
236	C₄⋊C₈	—	158	2	no	—
237	C₄⋊C₈	—	158	2	no	—
238	C₄⋊C₈	—	158	1	no	—
239	C₄⋊C₈	C₂²×D₇₉	1	4	yes	—
240	#103	—	158	4	no	—
241	#103	—	158	4	no	—
242	#103	—	158	2	no	—
243	#103	—	158	2	no	—
244	C₄×C₈	—	158	2	no	—
245	C₄×C₈	—	158	2	no	—
246	C₄×C₈	D₃₁₆	1	1	yes	characteristic
247	C₄×C₈	D₃₁₆	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
249	C₄×C₈	C₂²×D₇₉	1	1	yes	characteristic
250	C₂×Q₈	—	79	4	no	—
251	C₂×Q₈	—	79	2	no	—
252	C₂×Q₈	—	158	2	no	—
253	C₂×Q₈	—	158	2	no	—
254	C₂×Q₈	—	158	1	no	—
255	C₂×Q₈	—	158	2	no	—
256	C₂²×C₄	—	158	1	no	—
257	C₂²×C₄	—	158	1	no	—
258	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
260	C₂²×C₄	#11929625	1	1	yes	characteristic
261	Q₁₆	—	316	2	no	—
262	Q₁₆	—	316	2	no	—
263	M₁₆	—	4	4	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	316	2	no	—
268	C₂×C₈	—	158	2	no	—
269	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
270	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
271	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
272	C₂×C₈	C₂×D₃₁₆	1	1	yes	characteristic
273	C₄⋊C₄	—	316	2	no	—
274	C₄⋊C₄	—	316	2	no	—
275	C₄⋊C₄	—	79	2	no	—
276	C₄⋊C₄	—	158	2	no	—
277	C₄⋊C₄	—	158	2	no	—
278	C₄⋊C₄	—	158	2	no	—
279	C₄⋊C₄	—	158	2	no	—
280	C₄⋊C₄	—	158	2	no	—
281	C₄⋊C₄	—	158	2	no	—
282	C₄⋊C₄	—	158	1	no	—
283	C₂²⋊C₄	—	316	2	no	—
284	C₄²	—	316	2	no	—
285	C₄²	—	158	2	no	—
286	C₄²	—	158	2	no	—
287	C₄²	D₄×D₇₉	1	1	yes	characteristic
288	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
289	C₄²	#11929625	1	1	yes	characteristic
290	C₄²	C₂³×D₇₉	1	1	yes	characteristic
291	C₂³	#14579136	1	1	yes	characteristic
292	Q₈	—	316	2	no	—
293	Q₈	—	316	2	no	—
294	Q₈	—	79	4	no	—
295	Q₈	—	632	2	no	—
296	Q₈	—	158	4	no	—
297	Q₈	—	158	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	316	2	no	—
302	C₂×C₄	—	316	2	no	—
303	C₂×C₄	—	316	2	no	—
304	C₂×C₄	—	158	2	no	—
305	C₂×C₄	—	158	2	no	—
306	C₂×C₄	—	158	4	no	—
307	C₂×C₄	#14579136	1	1	yes	characteristic
308	C₂×C₄	#14579136	1	1	yes	characteristic
309	C₂×C₄	#14579136	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₇₉	1	2	yes	—
311	C₂×C₄	#14579180	1	2	yes	—
312	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
314	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	4	4	no	—
319	C₈	—	316	2	no	—
320	C₈	#14579194	1	2	yes	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	#19098805	1	2	yes	—
324	C₂²	C₂×#14579136	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	316	2	no	—
329	C₄	—	632	2	no	—
330	C₄	—	632	2	no	—
331	C₄	—	158	4	no	—
332	C₄	#19098805	1	2	yes	—
333	C₄	C₂×Q₁₆×D₇₉	1	1	yes	characteristic
334	C₄	C₂×#14579180	1	1	yes	characteristic
335	C₄	C₂×#14579194	1	1	yes	characteristic
336	C₄	#19099363	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	#23772874	1	1	yes	characteristic
339	C₂	C₂×#19098805	1	1	yes	characteristic
340	C₂	#23773121	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579136
commutator, cosocle	C₄²×C₇₉	C₂⁴
Frattini	C₄²	C₂³×D₇₉
Fitting	C₇₉×#1567	C₂
socle	C₂²×C₇₉	C₂×#121
prime radical	C₂³×C₇₉	#121
hypercenter	#1567	D₇₉
hypocenter	C₇₉	#19262
LCS	G ▷ C₄²×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
4. #23762859	C₂	semidirect
5. #23762805	C₂	semidirect
6. #23762771	C₂	semidirect
83. C₂×Dic₁₅₈	Q₁₆	semidirect
179. C₇₉	#19262	semidirect

Maximal subgroups

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

subgroup	index
#23773986	2
#23773207	2
C₂×C₈×Dic₁₅₈	2
#23762859	2
#23762805	2
#23762771	2
#23762091	2
C₇₉×#1567	2
#19262	79