Subgroups — #34204351

Order 20224.

410 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₃₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098380	C₂	1	2	yes	maximal
3	#23763572	C₂	1	1	yes	maximal, characteristic
4	#23761509	C₂	1	2	yes	maximal
5	C₇₉×#2150	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₁₅₈	—	2	1	no	—
8	C₄×Dic₃₁₆	—	2	1	no	—
9	C₄×Dic₃₁₆	—	2	1	no	—
10	C₄×Dic₃₁₆	C₄	1	2	yes	—
11	C₄×Dic₃₁₆	C₄	1	2	yes	—
12	C₂×#14579097	—	2	1	no	—
13	C₂×#14579097	C₂²	1	2	yes	—
14	C₂×#14579095	C₂²	1	1	yes	characteristic
15	#19098587	—	2	1	no	—
16	C₂×#14579086	—	2	1	no	—
17	#19098380	—	2	4	no	—
18	C₂×M₃₂×C₇₉	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₁₆×C₇₉	—	2	1	no	—
21	C₄⋊C₁₆×C₇₉	—	2	1	no	—
22	C₂²×Dic₁₅₈	—	2	1	no	—
23	C₂×Dic₃₁₆	—	2	1	no	—
24	C₂×Dic₃₁₆	—	2	1	no	—
25	C₂×Dic₃₁₆	—	2	1	no	—
26	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
27	C₂×Dic₃₁₆	D₄	1	2	yes	—
28	C₂×Dic₃₁₆	Q₈	1	2	yes	—
29	C₂×#11929608	—	2	2	no	—
30	C₄×Dic₁₅₈	—	2	2	no	—
31	C₄×Dic₁₅₈	—	2	2	no	—
32	C₄×Dic₁₅₈	—	4	1	no	—
33	C₄×Dic₁₅₈	—	4	1	no	—
34	C₂×C₄⋊Dic₇₉	—	2	1	no	—
35	C₂×Dic₇₉⋊C₄	—	4	1	no	—
36	C₂×C₄×Dic₇₉	—	4	1	no	—
37	#14579097	—	2	2	no	—
38	#14579097	—	2	2	no	—
39	#14579097	—	4	2	no	—
40	#14579095	—	2	1	no	—
41	#14579095	C₂×C₄	1	2	yes	—
42	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
43	C₄⋊(C₇₉⋊C₈)	—	4	1	no	—
44	#14579086	—	2	2	no	—
45	#14579086	—	2	2	no	—
46	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
49	M₃₂×C₇₉	—	2	2	no	—
50	M₃₂×C₇₉	—	2	2	no	—
51	C₂×C₁₆×C₇₉	—	2	2	no	—
52	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
55	C₄×C₈×C₇₉	D₄	1	1	yes	characteristic
56	C₂×Dic₁₅₈	—	2	2	no	—
57	C₂×Dic₁₅₈	—	2	2	no	—
58	C₂×Dic₁₅₈	—	2	2	no	—
59	C₂×Dic₁₅₈	—	4	1	no	—
60	C₂×Dic₁₅₈	—	4	1	no	—
61	C₂×Dic₁₅₈	—	4	1	no	—
62	C₂²×Dic₇₉	—	4	1	no	—
63	C₂²×Dic₇₉	—	4	1	no	—
64	Dic₃₁₆	—	2	2	no	—
65	Dic₃₁₆	—	2	2	no	—
66	Dic₃₁₆	—	2	2	no	—
67	Dic₃₁₆	C₄⋊C₄	1	4	yes	—
68	#11929608	—	4	2	no	—
69	#11929608	—	4	2	no	—
70	C₂×C₇₉⋊C₈	—	4	2	no	—
71	C₄⋊Dic₇₉	—	2	2	no	—
72	C₄⋊Dic₇₉	—	4	1	no	—
73	Dic₇₉⋊C₄	—	4	2	no	—
74	Dic₇₉⋊C₄	—	4	2	no	—
75	C₄×Dic₇₉	—	4	2	no	—
76	C₄×Dic₇₉	—	4	2	no	—
77	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
78	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
79	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₇₉	—	2	1	no	—
81	C₂×C₈×C₇₉	—	2	1	no	—
82	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₇₉	C₂×Q₈	1	1	yes	characteristic
88	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
89	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
90	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₄²×C₇₉	D₈	1	1	yes	characteristic
93	C₄²×C₇₉	D₈	1	1	yes	characteristic
94	C₁₆×C₇₉	—	4	2	no	—
95	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₇₉	—	2	2	no	—
97	C₂×C₄×C₇₉	—	4	1	no	—
98	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
99	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
100	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
101	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
102	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₄×C₇₉	#137	1	1	yes	characteristic
108	C₂×Dic₇₉	—	4	2	no	—
109	C₂×Dic₇₉	—	4	2	no	—
110	C₂×Dic₇₉	—	4	2	no	—
111	C₂×Dic₇₉	—	4	2	no	—
112	C₂×Dic₇₉	—	8	2	no	—
113	Dic₁₅₈	—	2	4	no	—
114	Dic₁₅₈	—	4	2	no	—
115	Dic₁₅₈	—	4	2	no	—
116	Dic₁₅₈	—	4	2	no	—
117	C₈×C₇₉	—	2	1	no	—
118	C₈×C₇₉	—	2	1	no	—
119	C₈×C₇₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₈×C₇₉	#117	1	1	yes	characteristic
121	C₈×C₇₉	#121	1	1	yes	characteristic
122	C₈×C₇₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₇₉⋊C₈	—	8	2	no	—
124	C₂²×C₇₉	—	2	1	no	—
125	C₂²×C₇₉	—	2	1	no	—
126	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
127	C₂²×C₇₉	#461	1	1	yes	characteristic
128	C₂²×C₇₉	#476	1	1	yes	characteristic
129	C₄×C₇₉	—	2	1	no	—
130	C₄×C₇₉	—	2	1	no	—
131	C₄×C₇₉	—	2	1	no	—
132	C₄×C₇₉	#362	1	2	yes	complemented
133	C₄×C₇₉	#380	1	1	yes	characteristic
134	C₄×C₇₉	#417	1	1	yes	characteristic
135	C₄×C₇₉	#459	1	1	yes	characteristic
136	C₄×C₇₉	#474	1	1	yes	characteristic
137	Dic₇₉	—	4	4	no	—
138	Dic₇₉	—	8	2	no	—
139	#12819	—	79	1	no	maximal, Sylow
140	C₂×C₇₉	—	2	1	no	—
141	C₂×C₇₉	—	2	1	no	—
142	C₂×C₇₉	#1802	1	1	yes	characteristic
143	C₂×C₇₉	C₂×#362	1	1	yes	characteristic
144	C₂×C₇₉	#2148	1	1	yes	characteristic
145	C₂×C₄×Q₁₆	—	79	1	no	—
146	#2150	D₇₉	1	1	yes	characteristic
147	C₂×#362	—	79	2	no	—
148	#1848	—	79	1	no	—
149	#1387	—	79	2	no	—
150	C₇₉	#12819	1	1	yes	characteristic, complemented, Sylow
151	C₂²×Q₁₆	—	79	1	no	—
152	C₂×C₄×Q₈	—	158	1	no	—
153	C₂×M₃₂	C₂×D₇₉	1	2	yes	—
154	C₄×Q₁₆	—	79	2	no	—
155	C₄×Q₁₆	—	79	2	no	—
156	C₄×Q₁₆	—	158	1	no	—
157	C₄×Q₁₆	—	158	1	no	—
158	C₂×#108	—	79	2	no	—
159	C₂×#108	—	158	1	no	—
160	C₂×#107	—	79	1	no	—
161	#423	—	158	1	no	—
162	C₂×#103	—	158	1	no	—
163	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
164	C₄⋊C₁₆	—	2	1	no	—
165	C₄⋊C₁₆	—	2	1	no	—
166	#362	—	158	4	no	—
167	C₂²×Q₈	—	158	1	no	—
168	C₂×Q₁₆	—	79	2	no	—
169	C₂×Q₁₆	—	79	2	no	—
170	C₂×Q₁₆	—	79	2	no	—
171	C₂×Q₁₆	—	158	1	no	—
172	C₂×Q₁₆	—	158	1	no	—
173	C₂×Q₁₆	—	158	1	no	—
174	C₂×M₁₆	—	158	2	no	—
175	C₂²×C₈	D₃₁₆	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
177	C₄×Q₈	—	316	1	no	—
178	C₄×Q₈	—	316	1	no	—
179	C₄×Q₈	—	158	2	no	—
180	C₄×Q₈	—	158	2	no	—
181	C₂×C₄⋊C₄	—	316	1	no	—
182	C₂×C₄⋊C₄	—	158	1	no	—
183	C₂×C₄²	—	316	1	no	—
184	C₂×C₄²	D₃₁₆	1	1	yes	characteristic
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	#108	—	316	2	no	—
189	#108	—	158	2	no	—
190	#108	—	158	2	no	—
191	#107	—	79	2	no	—
192	#107	—	158	1	no	—
193	C₄⋊C₈	—	316	1	no	—
194	C₄⋊C₈	—	316	1	no	—
195	#103	—	158	2	no	—
196	#103	—	158	2	no	—
197	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
198	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
199	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
201	C₂×Q₈	—	316	1	no	—
202	C₂×Q₈	—	316	1	no	—
203	C₂×Q₈	—	316	1	no	—
204	C₂×Q₈	—	158	2	no	—
205	C₂×Q₈	—	158	2	no	—
206	C₂×Q₈	—	158	2	no	—
207	C₂²×C₄	—	316	1	no	—
208	C₂²×C₄	—	316	1	no	—
209	C₂²×C₄	#11929613	1	1	yes	characteristic
210	C₂²×C₄	#11929613	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
212	Q₁₆	—	79	4	no	—
213	Q₁₆	—	158	2	no	—
214	Q₁₆	—	158	2	no	—
215	Q₁₆	—	158	2	no	—
216	M₁₆	—	316	2	no	—
217	M₁₆	—	316	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	316	2	no	—
221	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
222	C₂×C₈	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
223	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
224	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
226	C₂×C₈	Q₈×D₇₉	1	1	yes	characteristic
227	C₂×C₈	#11929623	1	1	yes	characteristic
228	C₂×C₈	#11929625	1	1	yes	characteristic
229	C₄⋊C₄	—	316	2	no	—
230	C₄⋊C₄	—	316	2	no	—
231	C₄⋊C₄	—	316	1	no	—
232	C₄⋊C₄	—	158	2	no	—
233	C₄²	—	316	2	no	—
234	C₄²	—	316	2	no	—
235	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
236	C₄²	(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
237	C₄²	D₆₃₂	1	1	yes	characteristic
238	C₄²	D₆₃₂	1	1	yes	characteristic
239	C₁₆	—	4	2	no	—
240	C₂³	C₂×#11929613	1	1	yes	characteristic
241	Q₈	—	316	2	no	—
242	Q₈	—	316	2	no	—
243	Q₈	—	316	2	no	—
244	Q₈	—	158	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	316	2	no	—
248	C₂×C₄	—	316	2	no	—
249	C₂×C₄	—	316	2	no	—
250	C₂×C₄	—	316	2	no	—
251	C₂×C₄	—	632	2	no	—
252	C₂×C₄	#14579083	1	1	yes	characteristic
253	C₂×C₄	#14579083	1	1	yes	characteristic
254	C₂×C₄	#14579083	1	1	yes	characteristic
255	C₂×C₄	#14579083	1	1	yes	characteristic
256	C₂×C₄	C₂×(C₂×D₇₉)⋊C₄	1	1	yes	characteristic
257	C₂×C₄	#14579136	1	1	yes	characteristic
258	C₂×C₄	#14579141	1	1	yes	characteristic
259	C₂×C₄	C₂×D₆₃₂	1	1	yes	characteristic
260	C₂×C₄	#14579189	1	1	yes	characteristic
261	C₂×C₄	#14579194	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	632	2	no	—
265	C₈	C₄⋊C₄×D₇₉	1	1	yes	characteristic
266	C₈	#14579119	1	1	yes	characteristic
267	C₈	#14579132	1	1	yes	characteristic
268	C₈	#14579137	1	1	yes	characteristic
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×#14579083	1	1	yes	characteristic
272	C₂²	#19098769	1	1	yes	characteristic
273	C₂²	#19098854	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	316	4	no	—
278	C₄	—	632	2	no	—
279	C₄	#19098380	1	2	yes	complemented
280	C₄	#19098420	1	1	yes	characteristic
281	C₄	#19098565	1	1	yes	characteristic
282	C₄	#19098759	1	1	yes	characteristic
283	C₄	#19098844	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	#23763401	1	1	yes	characteristic
287	C₂	C₂×#19098380	1	1	yes	characteristic
288	C₂	#23765150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14579083
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2150	C₂
socle	C₂²×C₇₉	C₂×#102
prime radical	C₂³×C₇₉	C₂×SD₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2150	D₇₉
hypocenter	C₇₉	#12819
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
132. C₄×C₇₉	#362	semidirect
150. C₇₉	#12819	semidirect
279. C₄	#19098380	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₃₁₆	2
C₂×#19098380	2
#23763572	2
#23761509	2
C₇₉×#2150	2
#12819	79