Subgroups — C₄×#19098675 #34286752

Order 20224.

1326 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098675	C₂	1	2	yes	maximal
2	C₄×D₄×Dic₇₉	C₂	1	2	yes	maximal, complemented
3	#23763299	C₂	1	2	yes	maximal, complemented
4	C₄×#14579093	C₂	1	1	yes	maximal, characteristic, complemented
5	#23763227	C₂	1	4	yes	maximal, complemented
6	C₄×#14579082	C₂	1	2	yes	maximal, complemented
7	C₄×#14579071	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₈×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×Dic₇₉	C₂²	1	4	yes	—
10	C₂²×C₄×Dic₇₉	—	2	2	no	—
11	#19098675	C₄	1	16	yes	direct product factor
12	C₂×#14579093	C₂²	1	2	yes	—
13	C₂×#14579082	C₂²	1	4	yes	—
14	C₂×#14579071	C₂²	1	2	yes	—
15	C₂×#14579071	C₂²	1	2	yes	—
16	C₂×C₄×C₇₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×Dic₇₉	C₂²	1	4	yes	—
18	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
19	C₂²⋊C₄×Dic₇₉	—	2	4	no	—
20	C₄×C₂²⋊Dic₇₉	—	2	2	no	—
21	C₄²×Dic₇₉	C₂²	1	1	yes	characteristic
22	#19098326	C₂²	1	2	yes	—
23	#19098325	C₂²	1	4	yes	—
24	C₂²×D₈×C₇₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₇₉	C₂²	1	2	yes	—
26	C₄×D₈×C₇₉	C₄	1	8	yes	complemented
27	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×#102×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂³×Dic₇₉	—	2	4	no	—
31	C₂²×C₇₉⋊C₈	C₂³	1	2	yes	—
32	D₄×Dic₇₉	—	2	8	no	—
33	D₄×Dic₇₉	C₂×C₄	1	16	yes	complemented
34	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
35	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
36	C₂×C₄×Dic₇₉	—	2	2	no	—
37	C₂×C₄×Dic₇₉	—	2	4	no	—
38	C₂×C₄×Dic₇₉	—	4	8	no	—
39	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
40	C₂×C₄×Dic₇₉	D₄	1	2	yes	—
41	C₂×C₄×Dic₇₉	C₂³	1	2	yes	—
42	#14579093	C₂×C₄	1	8	yes	complemented
43	#14579082	C₂×C₄	1	16	yes	complemented
44	#14579071	—	2	4	no	—
45	#14579071	C₂×C₄	1	8	yes	complemented
46	C₄×C₇₉⋊C₈	—	2	2	no	—
47	C₇₉⋊#95	—	2	2	no	—
48	C₇₉⋊#95	—	2	4	no	—
49	C₂²×D₄×C₇₉	C₂³	1	2	yes	—
50	C₂³×C₄×C₇₉	—	2	2	no	—
51	C₂×D₈×C₇₉	C₂×C₄	1	8	yes	—
52	C₂×D₈×C₇₉	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
55	C₄×D₄×C₇₉	—	2	4	no	—
56	C₄×D₄×C₇₉	C₂×C₄	1	8	yes	complemented
57	C₂×C₄⋊C₄×C₇₉	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
59	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
60	#107×C₇₉	C₂×C₄	1	4	yes	complemented
61	#102×C₇₉	C₂×C₄	1	8	yes	complemented
62	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	complemented
63	C₂²×Dic₇₉	—	2	16	no	—
64	C₂²×Dic₇₉	—	2	4	no	—
65	C₂²×Dic₇₉	—	4	8	no	—
66	C₂²×Dic₇₉	—	4	8	no	—
67	C₂²×Dic₇₉	C₂×D₄	1	4	yes	—
68	C₂×C₇₉⋊C₈	—	2	2	no	—
69	C₂×C₇₉⋊C₈	—	2	2	no	—
70	C₂×C₇₉⋊C₈	C₂²×C₄	1	4	yes	—
71	C₄⋊Dic₇₉	—	2	4	no	—
72	C₄⋊Dic₇₉	C₂²×C₄	1	8	yes	—
73	C₂²⋊Dic₇₉	—	2	16	no	—
74	C₄×Dic₇₉	—	2	4	no	—
75	C₄×Dic₇₉	—	4	2	no	—
76	C₄×Dic₇₉	—	4	8	no	—
77	C₄×Dic₇₉	—	4	4	no	—
78	C₄×Dic₇₉	C₂²×C₄	1	4	yes	—
79	C₄×Dic₇₉	#41	1	4	yes	—
80	C₂⁴×C₇₉	—	2	2	no	—
81	C₂×D₄×C₇₉	—	2	2	no	—
82	C₂×D₄×C₇₉	—	2	4	no	—
83	C₂×D₄×C₇₉	C₂²×C₄	1	8	yes	—
84	C₂×D₄×C₇₉	C₂²×C₄	1	4	yes	—
85	C₂²×C₄×C₇₉	—	2	2	no	—
86	C₂²×C₄×C₇₉	—	2	8	no	—
87	C₂²×C₄×C₇₉	—	4	4	no	—
88	C₂²×C₄×C₇₉	—	4	4	no	—
89	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
92	D₈×C₇₉	C₄²	1	16	yes	complemented
93	C₂×C₈×C₇₉	—	2	2	no	—
94	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
97	C₄⋊C₄×C₇₉	—	2	2	no	—
98	C₄⋊C₄×C₇₉	C₂²×C₄	1	4	yes	—
99	C₂²⋊C₄×C₇₉	—	2	8	no	—
100	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
101	C₄²×C₇₉	#41	1	2	yes	—
102	C₂³×C₇₉	—	2	8	no	—
103	C₂³×C₇₉	—	2	4	no	—
104	C₂³×C₇₉	—	4	2	no	—
105	C₂³×C₇₉	—	4	2	no	—
106	C₂³×C₇₉	—	4	4	no	—
107	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
108	D₄×C₇₉	—	2	8	no	—
109	D₄×C₇₉	—	2	4	no	—
110	D₄×C₇₉	C₂×C₄²	1	8	yes	complemented
111	C₂×C₄×C₇₉	—	2	2	no	—
112	C₂×C₄×C₇₉	—	2	4	no	—
113	C₂×C₄×C₇₉	—	4	2	no	—
114	C₂×C₄×C₇₉	—	4	2	no	—
115	C₂×C₄×C₇₉	—	4	16	no	—
116	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	4	yes	—
119	C₂×C₄×C₇₉	C₂³×C₄	1	2	yes	—
120	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
122	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
123	C₂×Dic₇₉	—	2	4	no	—
124	C₂×Dic₇₉	—	2	4	no	—
125	C₂×Dic₇₉	—	2	8	no	—
126	C₂×Dic₇₉	—	4	32	no	—
127	C₂×Dic₇₉	—	4	4	no	—
128	C₂×Dic₇₉	—	4	4	no	—
129	C₂×Dic₇₉	C₄×D₄	1	8	yes	—
130	C₈×C₇₉	—	2	2	no	—
131	C₈×C₇₉	C₂×C₄²	1	4	yes	complemented
132	C₇₉⋊C₈	—	2	4	no	—
133	C₂²×C₇₉	—	2	8	no	—
134	C₂²×C₇₉	—	4	4	no	—
135	C₂²×C₇₉	—	4	4	no	—
136	C₂²×C₇₉	—	4	8	no	—
137	C₂²×C₇₉	—	4	8	no	—
138	C₂²×C₇₉	C₂×C₄×D₄	1	2	yes	—
139	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
141	C₂²×C₇₉	#575	1	2	yes	—
142	C₄×C₇₉	—	2	2	no	—
143	C₄×C₇₉	—	4	4	no	—
144	C₄×C₇₉	#442	1	4	yes	complemented
145	C₄×C₇₉	C₂²×C₄²	1	1	yes	characteristic
146	C₄×C₇₉	C₄×#41	1	2	yes	—
147	C₄×C₇₉	C₄×#41	1	1	yes	characteristic
148	Dic₇₉	—	2	8	no	—
149	Dic₇₉	—	4	8	no	—
150	C₄×#442	—	79	1	no	maximal, Sylow
151	C₂×C₇₉	—	4	8	no	—
152	C₂×C₇₉	C₄²×D₄	1	1	yes	characteristic
153	C₂×C₇₉	C₂×#442	1	1	yes	characteristic
154	C₂×C₇₉	#2943	1	1	yes	characteristic
155	C₂×C₇₉	C₄×AGL(1,ℤ/8)	1	4	yes	—
156	C₂×#442	—	79	2	no	—
157	C₂×C₄×D₈	D₇₉	1	1	yes	characteristic
158	C₄²×D₄	—	79	2	no	—
159	#1776	—	79	2	no	—
160	C₄×#106	—	79	1	no	—
161	#1762	—	79	4	no	—
162	C₄×#102	—	79	2	no	—
163	C₄×#97	—	79	1	no	—
164	C₇₉	C₄×#442	1	1	yes	characteristic, complemented, Sylow
165	C₂²×D₈	C₂×D₇₉	1	1	yes	characteristic
166	C₂×C₄×D₄	—	79	4	no	—
167	C₂×C₄×D₄	C₂×D₇₉	1	2	yes	—
168	C₂²×C₄²	—	158	2	no	—
169	#442	—	79	16	no	—
170	C₄×D₈	Dic₇₉	1	8	yes	complemented
171	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
172	C₂×#106	—	79	2	no	—
173	C₂×#102	—	79	4	no	—
174	C₂×#102	C₂×D₇₉	1	2	yes	—
175	C₂×#97	—	79	2	no	—
176	C₂×#97	—	79	2	no	—
177	C₂×C₄×C₈	—	79	1	no	—
178	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
179	C₄×C₄⋊C₄	—	79	4	no	—
180	C₄×C₄⋊C₄	—	79	2	no	—
181	C₄×C₂²⋊C₄	—	158	4	no	—
182	C₄×C₂²⋊C₄	—	158	2	no	—
183	C₄³	—	79	1	no	—
184	#340	—	79	2	no	—
185	#340	—	79	4	no	—
186	C₂²×D₄	C₂²×D₇₉	1	2	yes	—
187	C₂³×C₄	—	2	2	no	—
188	C₂³×C₄	—	158	4	no	—
189	C₂×D₈	C₄×D₇₉	1	8	yes	—
190	C₂×D₈	C₂×Dic₇₉	1	4	yes	—
191	C₂²×C₈	—	79	2	no	—
192	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
193	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
194	C₄×D₄	—	2	4	no	—
195	C₄×D₄	—	79	16	no	—
196	C₄×D₄	—	158	8	no	—
197	C₄×D₄	C₂×Dic₇₉	1	8	yes	complemented
198	C₂×C₄⋊C₄	—	79	4	no	—
199	C₂×C₄⋊C₄	C₂²×D₇₉	1	2	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	158	4	no	—
202	C₂×C₄²	—	316	8	no	—
203	C₂×C₄²	—	79	2	no	—
204	C₂×C₄²	—	79	2	no	—
205	C₂×C₄²	—	79	2	no	—
206	C₂×C₄²	—	158	4	no	—
207	C₂×C₄²	—	158	2	no	—
208	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
209	#107	C₂×Dic₇₉	1	4	yes	complemented
210	#106	—	79	8	no	—
211	#102	—	79	16	no	—
212	#102	C₂×Dic₇₉	1	8	yes	complemented
213	#97	—	79	8	no	—
214	#97	—	158	4	no	—
215	C₄×C₈	—	158	2	no	—
216	C₄×C₈	C₂×Dic₇₉	1	4	yes	complemented
217	#95	—	158	4	no	—
218	#95	—	158	2	no	—
219	C₂⁴	—	2	2	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	C₂×C₄×D₇₉	1	8	yes	—
223	C₂×D₄	C₂²×Dic₇₉	1	4	yes	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	4	4	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	—	316	8	no	—
229	C₂²×C₄	—	316	8	no	—
230	C₂²×C₄	—	79	4	no	—
231	C₂²×C₄	—	158	16	no	—
232	C₂²×C₄	—	158	4	no	—
233	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
234	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
235	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
236	D₈	C₄×Dic₇₉	1	16	yes	complemented
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	79	4	no	—
239	C₂×C₈	—	158	2	no	—
240	C₂×C₈	—	158	2	no	—
241	C₂×C₈	C₂×C₄×D₇₉	1	4	yes	—
242	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
243	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	79	8	no	—
246	C₄⋊C₄	—	158	4	no	—
247	C₄⋊C₄	C₂²×Dic₇₉	1	4	yes	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₂²⋊C₄	—	158	16	no	—
250	C₄²	—	316	2	no	—
251	C₄²	—	316	8	no	—
252	C₄²	—	316	4	no	—
253	C₄²	—	79	4	no	—
254	C₄²	—	79	4	no	—
255	C₄²	—	158	4	no	—
256	C₄²	C₂²×Dic₇₉	1	2	yes	—
257	C₄²	#11929623	1	2	yes	—
258	C₂³	—	2	8	no	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	C₂×D₄×D₇₉	1	1	yes	characteristic
264	D₄	—	2	8	no	—
265	D₄	—	2	4	no	—
266	D₄	C₂×C₄×Dic₇₉	1	8	yes	complemented
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	16	no	—
272	C₂×C₄	—	316	4	no	—
273	C₂×C₄	—	316	4	no	—
274	C₂×C₄	—	316	32	no	—
275	C₂×C₄	—	79	8	no	—
276	C₂×C₄	—	158	8	no	—
277	C₂×C₄	—	158	4	no	—
278	C₂×C₄	—	158	4	no	—
279	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
280	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
281	C₂×C₄	#14579185	1	4	yes	—
282	C₂×C₄	C₂²×C₄×D₇₉	1	2	yes	—
283	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
284	C₂×C₄	#41×D₇₉	1	2	yes	—
285	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	158	4	no	—
288	C₈	C₂×C₄×Dic₇₉	1	4	yes	complemented
289	C₂²	—	2	8	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	8	no	—
293	C₂²	—	4	8	no	—
294	C₂²	C₄×D₄×D₇₉	1	2	yes	—
295	C₂²	C₂×D₄×Dic₇₉	1	1	yes	characteristic
296	C₂²	C₂×#14579185	1	2	yes	—
297	C₂²	#19099360	1	2	yes	—
298	C₄	—	2	2	no	—
299	C₄	—	4	4	no	—
300	C₄	—	316	8	no	—
301	C₄	—	158	8	no	—
302	C₄	#19098675	1	4	yes	direct product factor
303	C₄	C₂²×C₄×Dic₇₉	1	1	yes	characteristic
304	C₄	C₄×#11929623	1	2	yes	—
305	C₄	#41×Dic₇₉	1	1	yes	characteristic
306	C₂	—	4	8	no	—
307	C₂	C₄×D₄×Dic₇₉	1	1	yes	characteristic
308	C₂	C₂×#19098675	1	1	yes	characteristic
309	C₂	#23772201	1	1	yes	characteristic
310	C₂	C₄×#14579185	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₉
commutator	C₄×C₇₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₄×D₈×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂²×D₈×C₇₉	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₄×D₈	D₇₉
hypocenter	C₇₉	C₄×#442
LCS	G ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₄×Dic₇₉	C₂	semidirect
3. #23763299	C₂	semidirect
4. C₄×#14579093	C₂	semidirect
5. #23763227	C₂	semidirect
6. C₄×#14579082	C₂	semidirect
7. C₄×#14579071	C₂	semidirect
11. #19098675	C₄	direct (N × H)
26. C₄×D₈×C₇₉	C₄	semidirect
33. D₄×Dic₇₉	C₂×C₄	semidirect
42. #14579093	C₂×C₄	semidirect
43. #14579082	C₂×C₄	semidirect
45. #14579071	C₂×C₄	semidirect
56. C₄×D₄×C₇₉	C₂×C₄	semidirect
60. #107×C₇₉	C₂×C₄	semidirect
61. #102×C₇₉	C₂×C₄	semidirect
62. C₄×C₈×C₇₉	C₂×C₄	semidirect
92. D₈×C₇₉	C₄²	semidirect
110. D₄×C₇₉	C₂×C₄²	semidirect
131. C₈×C₇₉	C₂×C₄²	semidirect
144. C₄×C₇₉	#442	semidirect
164. C₇₉	C₄×#442	semidirect
170. C₄×D₈	Dic₇₉	semidirect
197. C₄×D₄	C₂×Dic₇₉	semidirect
209. #107	C₂×Dic₇₉	semidirect
212. #102	C₂×Dic₇₉	semidirect
216. C₄×C₈	C₂×Dic₇₉	semidirect
236. D₈	C₄×Dic₇₉	semidirect
266. D₄	C₂×C₄×Dic₇₉	semidirect
288. C₈	C₂×C₄×Dic₇₉	semidirect
302. C₄	#19098675	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×#19098675	2
C₄×D₄×Dic₇₉	2
#23763299	2
C₄×#14579093	2
#23763227	2
C₄×#14579082	2
C₄×#14579071	2
C₂×C₄×D₈×C₇₉	2
C₄×#442	79

Subgroups — C4×#19098675 #34286752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19098675 #34286752