Subgroups — C₄×#19098610 #34286373

Order 20224.

1034 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098610	C₂	1	2	yes	maximal
2	C₂×C₄×#11929607	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×D₇₉	C₂	1	2	yes	maximal
4	#23763308	C₂	1	2	yes	maximal, complemented
5	#23763305	C₂	1	2	yes	maximal, complemented
6	C₄×#14579096	C₂	1	2	yes	maximal, complemented
7	C₄×#14579097	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763125	C₂	1	2	yes	maximal
9	C₄×#108×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#11929607	C₂²	1	1	yes	characteristic
11	C₂²×#11929607	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×D₇₉	C₂²	1	4	yes	—
13	C₂×C₄²×D₇₉	C₂²	1	1	yes	characteristic
14	#19098610	C₄	1	16	yes	direct product factor
15	C₂×#14579097	C₂²	1	2	yes	—
16	C₂×#14579096	C₂²	1	4	yes	—
17	C₄×#11929608	C₂²	1	2	yes	—
18	#19098515	—	2	2	no	—
19	M₁₆×Dic₇₉	C₂²	1	4	yes	—
20	C₄×#11929607	—	2	2	no	—
21	C₄×#11929607	C₄	1	8	yes	—
22	C₂×#14579071	C₂²	1	1	yes	characteristic
23	C₂×#14579071	C₂²	1	1	yes	characteristic
24	#97×D₇₉	—	2	2	no	—
25	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₇₉	—	2	2	no	—
27	#19098328	C₂²	1	2	yes	—
28	#19098327	C₂²	1	4	yes	—
29	C₂×#108×C₇₉	C₂²	1	2	yes	—
30	C₄×M₁₆×C₇₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	#341×C₇₉	C₂²	1	2	yes	—
33	C₂²×C₄×D₇₉	Q₈	1	2	yes	—
34	C₂²×C₄×D₇₉	C₂³	1	1	yes	characteristic
35	C₂×#11929608	C₂³	1	4	yes	—
36	C₂×#11929607	—	2	4	no	—
37	C₂×#11929607	—	2	4	no	—
38	C₂×#11929607	C₂×C₄	1	4	yes	—
39	C₂×#11929607	C₂×C₄	1	4	yes	—
40	C₂×#11929607	C₂×C₄	1	8	yes	—
41	M₁₆×D₇₉	—	2	8	no	—
42	M₁₆×D₇₉	C₂×C₄	1	16	yes	—
43	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₈×D₇₉	—	2	4	no	—
46	C₂×C₄×Dic₇₉	Q₈	1	2	yes	—
47	C₂×C₄×Dic₇₉	C₂³	1	1	yes	characteristic
48	C₄²×D₇₉	C₂×C₄	1	4	yes	—
49	C₄²×D₇₉	D₄	1	2	yes	—
50	C₄²×D₇₉	D₄	1	2	yes	—
51	#14579097	C₂×C₄	1	8	yes	complemented
52	#14579096	C₂×C₄	1	16	yes	complemented
53	#14579072	—	2	2	no	—
54	#14579071	—	2	4	no	—
55	#14579071	C₂×C₄	1	4	yes	—
56	#14579071	C₂×C₄	1	4	yes	—
57	C₄×C₇₉⋊C₈	—	2	2	no	—
58	C₄×C₇₉⋊C₈	C₂×C₄	1	4	yes	—
59	C₈×Dic₇₉	—	2	4	no	—
60	C₂×M₁₆×C₇₉	C₂³	1	4	yes	—
61	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
64	#108×C₇₉	C₂×C₄	1	8	yes	complemented
65	#97×C₇₉	—	2	2	no	—
66	C₄×C₈×C₇₉	—	2	2	no	—
67	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
68	C₂³×D₇₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×Dic₇₉	C₂×Q₈	1	2	yes	—
70	C₂²×Dic₇₉	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₄×D₇₉	—	4	4	no	—
72	C₂×C₄×D₇₉	C₄⋊C₄	1	4	yes	—
73	C₂×C₄×D₇₉	C₄⋊C₄	1	4	yes	—
74	C₂×C₄×D₇₉	C₂²×C₄	1	2	yes	—
75	C₂×C₄×D₇₉	C₂²×C₄	1	4	yes	—
76	C₂×C₄×D₇₉	C₂×D₄	1	2	yes	—
77	C₂×C₄×D₇₉	#41	1	4	yes	—
78	#11929608	—	2	4	no	—
79	#11929608	C₂²×C₄	1	8	yes	—
80	#11929607	—	2	8	no	—
81	#11929607	—	2	16	no	—
82	#11929607	C₄²	1	16	yes	—
83	C₂×C₇₉⋊C₈	—	2	4	no	—
84	C₂×C₇₉⋊C₈	—	2	2	no	—
85	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
87	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
88	C₈×D₇₉	—	2	16	no	—
89	C₄×Dic₇₉	C₄⋊C₄	1	4	yes	—
90	C₄×Dic₇₉	C₄⋊C₄	1	4	yes	—
91	C₄×Dic₇₉	C₂²×C₄	1	2	yes	—
92	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₇₉	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
95	M₁₆×C₇₉	—	2	4	no	—
96	M₁₆×C₇₉	C₂²×C₄	1	8	yes	—
97	C₂×C₈×C₇₉	—	2	4	no	—
98	C₂×C₈×C₇₉	—	2	2	no	—
99	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
102	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
103	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
105	C₂³×C₇₉	C₂²×Q₈	1	1	yes	characteristic
106	C₂²×D₇₉	—	4	2	no	—
107	C₂²×D₇₉	C₂×C₄⋊C₄	1	2	yes	—
108	C₂²×D₇₉	C₄×Q₈	1	4	yes	—
109	C₂×C₄×C₇₉	—	2	4	no	—
110	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
114	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
116	C₂×Dic₇₉	—	4	2	no	—
117	C₂×Dic₇₉	C₂×C₄⋊C₄	1	4	yes	—
118	C₂×Dic₇₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×Dic₇₉	C₄×Q₈	1	4	yes	—
120	C₄×D₇₉	—	2	8	no	—
121	C₄×D₇₉	—	4	4	no	—
122	C₄×D₇₉	C₂×C₄²	1	4	yes	—
123	C₄×D₇₉	#117	1	4	yes	—
124	C₄×D₇₉	C₄×D₄	1	8	yes	—
125	C₈×C₇₉	—	2	2	no	—
126	C₈×C₇₉	—	2	8	no	—
127	C₈×C₇₉	C₂×C₄²	1	4	yes	—
128	C₇₉⋊C₈	—	2	2	no	—
129	C₇₉⋊C₈	—	2	8	no	—
130	C₇₉⋊C₈	C₂×C₄²	1	4	yes	—
131	C₂²×C₇₉	—	2	2	no	—
132	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₇₉	C₂×C₄×Q₈	1	2	yes	—
134	C₂×D₇₉	—	2	4	no	—
135	C₂×D₇₉	—	4	4	no	—
136	C₂×D₇₉	C₄×C₄⋊C₄	1	4	yes	—
137	C₄×C₇₉	—	2	2	no	—
138	C₄×C₇₉	#430	1	4	yes	complemented
139	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
140	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
141	C₄×C₇₉	C₂×C₄×D₄	1	2	yes	—
142	Dic₇₉	—	2	4	no	—
143	Dic₇₉	C₄×C₄⋊C₄	1	4	yes	—
144	C₄×#430	—	79	1	no	maximal, Sylow
145	C₂×C₇₉	—	2	2	no	—
146	C₂×C₇₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₇₉	C₂×#430	1	2	yes	—
148	D₇₉	—	2	4	no	—
149	C₂×#430	—	79	2	no	—
150	C₂×C₄×M₁₆	—	79	1	no	—
151	C₂×C₄×M₁₆	—	79	2	no	—
152	#1778	—	79	2	no	—
153	#1778	—	79	2	no	—
154	C₄×#108	—	79	2	no	—
155	C₄×#108	—	79	1	no	—
156	C₄×#108	D₇₉	1	1	yes	characteristic, complemented
157	#1738	—	79	2	no	—
158	C₇₉	C₄×#430	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	—	79	1	no	—
160	C₂²×M₁₆	—	79	1	no	—
161	C₂²×M₁₆	—	79	4	no	—
162	C₂²×C₄²	—	79	1	no	—
163	#430	—	79	16	no	—
164	C₂×#108	—	79	4	no	—
165	C₂×#108	—	79	2	no	—
166	C₂×#108	C₂×D₇₉	1	2	yes	—
167	C₄×M₁₆	—	79	8	no	—
168	C₄×M₁₆	—	79	4	no	—
169	C₄×M₁₆	—	79	2	no	—
170	C₄×M₁₆	—	158	2	no	—
171	C₄×M₁₆	—	158	2	no	—
172	C₄×M₁₆	C₂×D₇₉	1	2	yes	—
173	C₂×#97	—	79	1	no	—
174	C₂×#97	—	79	1	no	—
175	C₂×#97	—	158	2	no	—
176	C₂×C₄×C₈	—	79	1	no	—
177	C₂×C₄×C₈	—	158	2	no	—
178	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
179	#341	—	79	4	no	—
180	#341	—	79	2	no	—
181	#341	C₂×D₇₉	1	2	yes	—
182	C₂³×C₄	—	79	2	no	—
183	C₂³×C₄	—	79	1	no	—
184	C₂×M₁₆	—	79	4	no	—
185	C₂×M₁₆	—	79	8	no	—
186	C₂×M₁₆	—	79	4	no	—
187	C₂×M₁₆	—	79	4	no	—
188	C₂×M₁₆	—	79	16	no	—
189	C₂×M₁₆	—	158	4	no	—
190	C₂×M₁₆	—	158	4	no	—
191	C₂×M₁₆	—	158	8	no	—
192	C₂×M₁₆	C₂²×D₇₉	1	4	yes	—
193	C₂²×C₈	—	79	1	no	—
194	C₂²×C₈	—	79	1	no	—
195	C₂²×C₈	—	158	4	no	—
196	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
198	C₂×C₄²	—	79	2	no	—
199	C₂×C₄²	—	79	1	no	—
200	C₂×C₄²	—	79	4	no	—
201	C₂×C₄²	—	79	2	no	—
202	C₂×C₄²	—	79	2	no	—
203	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
204	#108	—	79	16	no	—
205	#108	—	79	8	no	—
206	#108	C₄×D₇₉	1	8	yes	complemented
207	#97	—	2	2	no	—
208	#97	—	79	4	no	—
209	#97	—	79	4	no	—
210	#97	—	158	4	no	—
211	#97	—	158	2	no	—
212	C₄×C₈	—	2	2	no	—
213	C₄×C₈	—	79	4	no	—
214	C₄×C₈	—	158	4	no	—
215	C₄×C₈	—	158	2	no	—
216	C₄×C₈	C₄×D₇₉	1	4	yes	—
217	C₂⁴	—	79	1	no	—
218	C₂²×C₄	—	316	4	no	—
219	C₂²×C₄	—	79	2	no	—
220	C₂²×C₄	—	79	4	no	—
221	C₂²×C₄	—	79	1	no	—
222	C₂²×C₄	—	79	4	no	—
223	C₂²×C₄	—	79	2	no	—
224	C₂²×C₄	—	79	4	no	—
225	C₂²×C₄	—	79	2	no	—
226	C₂²×C₄	—	79	4	no	—
227	C₂²×C₄	Q₈×D₇₉	1	2	yes	—
228	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
229	M₁₆	—	2	4	no	—
230	M₁₆	—	79	16	no	—
231	M₁₆	—	79	8	no	—
232	M₁₆	—	158	8	no	—
233	M₁₆	—	158	16	no	—
234	M₁₆	—	158	4	no	—
235	M₁₆	C₂×C₄×D₇₉	1	8	yes	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	79	2	no	—
239	C₂×C₈	—	79	4	no	—
240	C₂×C₈	—	79	2	no	—
241	C₂×C₈	—	158	16	no	—
242	C₂×C₈	—	158	2	no	—
243	C₂×C₈	—	158	4	no	—
244	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
245	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
246	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
247	C₄²	—	79	4	no	—
248	C₄²	—	79	4	no	—
249	C₄²	—	79	2	no	—
250	C₄²	—	79	2	no	—
251	C₄²	C₂×C₄×D₇₉	1	2	yes	—
252	C₄²	D₄×D₇₉	1	1	yes	characteristic
253	C₄²	D₄×D₇₉	1	1	yes	characteristic
254	C₂³	—	316	2	no	—
255	C₂³	—	79	4	no	—
256	C₂³	—	79	2	no	—
257	C₂³	C₂×Q₈×D₇₉	1	1	yes	characteristic
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	316	4	no	—
260	C₂×C₄	—	316	2	no	—
261	C₂×C₄	—	79	2	no	—
262	C₂×C₄	—	79	4	no	—
263	C₂×C₄	—	79	4	no	—
264	C₂×C₄	—	79	4	no	—
265	C₂×C₄	—	79	8	no	—
266	C₂×C₄	—	79	4	no	—
267	C₂×C₄	—	158	8	no	—
268	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
269	C₂×C₄	C₄⋊C₄×D₇₉	1	2	yes	—
270	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
271	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
273	C₂×C₄	#41×D₇₉	1	2	yes	—
274	C₈	—	2	2	no	—
275	C₈	—	2	8	no	—
276	C₈	—	79	4	no	—
277	C₈	—	158	2	no	—
278	C₈	—	158	8	no	—
279	C₈	C₄²×D₇₉	1	4	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	316	4	no	—
282	C₂²	—	79	4	no	—
283	C₂²	—	158	4	no	—
284	C₂²	C₂×C₄⋊C₄×D₇₉	1	1	yes	characteristic
285	C₂²	C₄×Q₈×D₇₉	1	2	yes	—
286	C₄	—	2	2	no	—
287	C₄	—	79	4	no	—
288	C₄	—	158	4	no	—
289	C₄	#19098610	1	4	yes	direct product factor
290	C₄	C₂×C₄²×D₇₉	1	1	yes	characteristic
291	C₄	#117×D₇₉	1	1	yes	characteristic
292	C₄	C₄×D₄×D₇₉	1	2	yes	—
293	C₂	—	2	2	no	—
294	C₂	—	158	4	no	—
295	C₂	C₄×C₄⋊C₄×D₇₉	1	1	yes	characteristic
296	C₂	C₂×#19098610	1	2	yes	—

	subgroup	G/H
center	C₄²	D₄×D₇₉
commutator	C₄×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₄×#108×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₉	C₂×Q₈
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₄×#108	D₇₉
hypocenter	C₇₉	C₄×#430
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
4. #23763308	C₂	semidirect
5. #23763305	C₂	semidirect
6. C₄×#14579096	C₂	semidirect
7. C₄×#14579097	C₂	semidirect
9. C₄×#108×C₇₉	C₂	semidirect
14. #19098610	C₄	direct (N × H)
51. #14579097	C₂×C₄	semidirect
52. #14579096	C₂×C₄	semidirect
64. #108×C₇₉	C₂×C₄	semidirect
138. C₄×C₇₉	#430	semidirect
156. C₄×#108	D₇₉	semidirect
158. C₇₉	C₄×#430	semidirect
206. #108	C₄×D₇₉	semidirect
289. C₄	#19098610	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098610	2
C₂×C₄×#11929607	2
C₄×M₁₆×D₇₉	2
#23763308	2
#23763305	2
C₄×#14579096	2
C₄×#14579097	2
#23763125	2
C₄×#108×C₇₉	2
C₄×#430	79

Subgroups — C4×#19098610 #34286373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098610 #34286373