Subgroups — #34306414

Order 20224.

982 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098601	C₂	1	2	yes	maximal, complemented
2	#23766349	C₂	1	2	yes	maximal
3	#23764164	C₂	1	2	yes	maximal, complemented
4	#23763601	C₂	1	4	yes	maximal, complemented
5	#23763560	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763429	C₂	1	2	yes	maximal, complemented
7	C₄×#14579095	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#14579120	C₂²	1	4	yes	—
10	#19098601	—	2	8	no	—
11	C₂×#14579095	C₂²	1	2	yes	—
12	C₂×#14579095	C₂²	1	1	yes	characteristic
13	C₂×#14579095	C₂²	1	2	yes	—
14	C₂×#14579095	C₂²	1	1	yes	characteristic
15	C₂×#14579093	C₂²	1	4	yes	—
16	#19098466	—	2	2	no	—
17	#19098456	—	2	4	no	—
18	#19098433	C₂²	1	2	yes	—
19	#19098423	C₂²	1	4	yes	—
20	C₄⋊C₄×Dic₇₉	—	2	2	no	—
21	C₄×C₄⋊Dic₇₉	C₂²	1	2	yes	—
22	#19098326	C₂²	1	2	yes	—
23	#19098326	C₂²	1	4	yes	—
24	C₂³×C₈×C₇₉	C₂²	1	2	yes	—
25	C₂×C₄×D₄×C₇₉	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₇₉	C₄	1	8	yes	complemented
27	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₈×C₇₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
30	#14579120	—	2	8	no	—
31	#14579120	—	4	4	no	—
32	C₂×C₄⋊Dic₇₉	—	2	2	no	—
33	C₂×C₄⋊Dic₇₉	—	2	2	no	—
34	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
35	C₂×C₄⋊Dic₇₉	C₂³	1	4	yes	—
36	C₂×C₄⋊Dic₇₉	C₂³	1	2	yes	—
37	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
38	C₂×C₂²⋊Dic₇₉	—	2	4	no	—
39	C₂×C₄×Dic₇₉	—	2	2	no	—
40	C₂×C₄×Dic₇₉	—	2	4	no	—
41	#14579095	—	2	4	no	—
42	#14579095	—	2	2	no	—
43	#14579095	—	4	2	no	—
44	#14579095	D₄	1	4	yes	complemented
45	#14579093	—	2	8	no	—
46	C₇₉⋊#95	—	2	2	no	—
47	C₇₉⋊#95	—	2	4	no	—
48	C₂²×D₄×C₇₉	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₇₉	C₂³	1	2	yes	—
50	C₂²×C₈×C₇₉	—	2	4	no	—
51	C₂²×C₈×C₇₉	—	2	4	no	—
52	C₂²×C₈×C₇₉	C₂×C₄	1	8	yes	—
53	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
54	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
56	C₄×D₄×C₇₉	C₂×C₄	1	4	yes	—
57	C₄×D₄×C₇₉	Q₈	1	4	yes	—
58	C₂×C₄⋊C₄×C₇₉	D₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₇₉	D₄	1	2	yes	—
60	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₉	C₂×C₄	1	4	yes	—
62	C₂²⋊C₈×C₇₉	C₂×C₄	1	8	yes	—
63	C₄×C₈×C₇₉	C₂×C₄	1	4	yes	—
64	C₂²×Dic₇₉	—	2	2	no	—
65	C₂²×Dic₇₉	—	2	2	no	—
66	C₂²×Dic₇₉	—	2	4	no	—
67	C₄⋊Dic₇₉	—	2	4	no	—
68	C₄⋊Dic₇₉	—	2	2	no	—
69	C₄⋊Dic₇₉	—	2	2	no	—
70	C₄⋊Dic₇₉	—	2	4	no	—
71	C₄⋊Dic₇₉	—	2	2	no	—
72	C₄⋊Dic₇₉	—	4	4	no	—
73	C₄⋊Dic₇₉	—	8	2	no	—
74	C₄⋊Dic₇₉	C₂×D₄	1	4	yes	—
75	C₂²⋊Dic₇₉	—	4	8	no	—
76	C₂²⋊Dic₇₉	—	8	4	no	—
77	C₄×Dic₇₉	—	4	8	no	—
78	C₄×Dic₇₉	—	4	4	no	—
79	C₂⁴×C₇₉	C₂×D₄	1	2	yes	—
80	C₂×D₄×C₇₉	—	2	2	no	—
81	C₂×D₄×C₇₉	—	2	2	no	—
82	C₂×D₄×C₇₉	C₄⋊C₄	1	4	yes	—
83	C₂²×C₄×C₇₉	—	2	2	no	—
84	C₂²×C₄×C₇₉	—	2	2	no	—
85	C₂²×C₄×C₇₉	—	2	2	no	—
86	C₂²×C₄×C₇₉	—	2	2	no	—
87	C₂²×C₄×C₇₉	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₉	C₂×Q₈	1	4	yes	—
92	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
93	C₂×C₈×C₇₉	—	2	16	no	—
94	C₂×C₈×C₇₉	—	2	2	no	—
95	C₂×C₈×C₇₉	—	2	2	no	—
96	C₂×C₈×C₇₉	—	2	1	no	—
97	C₂×C₈×C₇₉	—	2	1	no	—
98	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₇₉	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₇₉	#41	1	2	yes	—
103	C₄⋊C₄×C₇₉	C₄⋊C₄	1	4	yes	—
104	C₂²⋊C₄×C₇₉	C₄⋊C₄	1	8	yes	—
105	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
106	C₄²×C₇₉	C₂×Q₈	1	2	yes	—
107	C₂³×C₇₉	—	2	2	no	—
108	C₂³×C₇₉	—	2	2	no	—
109	C₂³×C₇₉	—	2	2	no	—
110	C₂³×C₇₉	—	2	2	no	—
111	C₂³×C₇₉	—	2	2	no	—
112	C₂³×C₇₉	—	2	2	no	—
113	C₂³×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
114	C₂³×C₇₉	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₇₉	—	2	8	no	—
116	C₂×C₄×C₇₉	—	2	2	no	—
117	C₂×C₄×C₇₉	—	2	2	no	—
118	C₂×C₄×C₇₉	—	2	1	no	—
119	C₂×C₄×C₇₉	—	2	1	no	—
120	C₂×C₄×C₇₉	—	2	4	no	—
121	C₂×C₄×C₇₉	—	2	4	no	—
122	C₂×C₄×C₇₉	—	2	4	no	—
123	C₂×C₄×C₇₉	—	2	4	no	—
124	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×C₄×C₇₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₇₉	#135	1	2	yes	—
128	C₂×C₄×C₇₉	#135	1	2	yes	—
129	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₉	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
135	C₂×Dic₇₉	—	2	4	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	4	2	no	—
138	C₂×Dic₇₉	—	4	4	no	—
139	C₂×Dic₇₉	—	4	2	no	—
140	C₂×Dic₇₉	—	4	2	no	—
141	C₂×Dic₇₉	—	4	4	no	—
142	C₂×Dic₇₉	—	4	2	no	—
143	C₂×Dic₇₉	—	4	4	no	—
144	C₈×C₇₉	—	2	4	no	—
145	C₈×C₇₉	—	2	2	no	—
146	C₈×C₇₉	C₄×D₄	1	4	yes	—
147	C₂²×C₇₉	—	2	4	no	—
148	C₂²×C₇₉	—	2	4	no	—
149	C₂²×C₇₉	—	2	4	no	—
150	C₂²×C₇₉	—	4	2	no	—
151	C₂²×C₇₉	—	4	2	no	—
152	C₂²×C₇₉	—	4	2	no	—
153	C₂²×C₇₉	—	4	2	no	—
154	C₂²×C₇₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂²×C₇₉	D₄²	1	1	yes	characteristic
156	C₂²×C₇₉	#547	1	1	yes	characteristic
157	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
158	C₂²×C₇₉	C₂×#135	1	1	yes	characteristic
159	C₂²×C₇₉	#576	1	1	yes	characteristic
160	C₂²×C₇₉	#578	1	1	yes	characteristic
161	C₄×C₇₉	—	2	4	no	—
162	C₄×C₇₉	—	2	2	no	—
163	C₄×C₇₉	#427	1	4	yes	complemented
164	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
165	C₄×C₇₉	#520	1	1	yes	characteristic
166	C₄×C₇₉	D₄×Q₈	1	1	yes	characteristic
167	C₄×C₇₉	#554	1	1	yes	characteristic
168	Dic₇₉	—	4	4	no	—
169	Dic₇₉	—	8	2	no	—
170	Dic₇₉	—	8	4	no	—
171	#22658	—	79	1	no	maximal, Sylow
172	C₂×C₇₉	—	4	4	no	—
173	C₂×C₇₉	D₄×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₇₉	C₂×#427	1	1	yes	characteristic
175	C₂×C₇₉	#2913	1	1	yes	characteristic
176	C₂×C₇₉	#3283	1	1	yes	characteristic
177	C₂×C₇₉	#3287	1	1	yes	characteristic
178	C₂×C₇₉	#3341	1	1	yes	characteristic
179	C₂×C₇₉	#3342	1	1	yes	characteristic
180	C₂×C₈×D₄	D₇₉	1	1	yes	characteristic
181	C₂×#427	—	79	2	no	—
182	#2354	—	79	2	no	—
183	#1948	—	79	2	no	—
184	#1854	—	79	4	no	—
185	#1845	—	79	1	no	—
186	#1808	—	79	2	no	—
187	C₄×#107	—	79	1	no	—
188	C₇₉	#22658	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₈	C₂×D₇₉	1	2	yes	—
190	C₂×C₄×D₄	C₂×D₇₉	1	1	yes	characteristic
191	C₂×#117	—	79	4	no	—
192	C₈×D₄	Dic₇₉	1	8	yes	complemented
193	#427	—	158	8	no	—
194	C₂×#107	—	79	2	no	—
195	C₂×#107	—	79	1	no	—
196	C₂×#107	—	79	1	no	—
197	C₂×#107	—	79	2	no	—
198	C₂×#106	—	79	4	no	—
199	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
200	C₂×C₂²⋊C₈	C₂×D₇₉	1	2	yes	—
201	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
202	#390	—	158	2	no	—
203	#388	—	158	4	no	—
204	#384	—	79	2	no	—
205	#380	—	79	4	no	—
206	C₄×C₄⋊C₄	—	79	2	no	—
207	C₄×C₄⋊C₄	—	158	2	no	—
208	#340	—	79	4	no	—
209	#340	—	79	2	no	—
210	C₂²×D₄	D₃₁₆	1	1	yes	characteristic
211	C₂³×C₄	C₂²×D₇₉	1	2	yes	—
212	C₂²×C₈	—	2	4	no	—
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	C₂×Dic₇₉	1	8	yes	—
215	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
216	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
217	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
218	C₄×D₄	Dic₁₅₈	1	4	yes	—
219	C₄×D₄	C₂×Dic₇₉	1	4	yes	—
220	#117	—	316	4	no	—
221	#117	—	158	8	no	—
222	C₂×C₄⋊C₄	—	79	2	no	—
223	C₂×C₄⋊C₄	—	79	4	no	—
224	C₂×C₄⋊C₄	—	79	2	no	—
225	C₂×C₄⋊C₄	—	158	2	no	—
226	C₂×C₄⋊C₄	—	158	2	no	—
227	C₂×C₄⋊C₄	D₃₁₆	1	1	yes	characteristic
228	C₂×C₂²⋊C₄	—	158	4	no	—
229	C₂×C₂²⋊C₄	—	158	4	no	—
230	C₂×C₂²⋊C₄	D₃₁₆	1	2	yes	—
231	C₂×C₄²	—	158	4	no	—
232	C₂×C₄²	—	158	2	no	—
233	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
234	#107	—	316	2	no	—
235	#107	—	79	4	no	—
236	#107	—	158	4	no	—
237	#107	—	158	2	no	—
238	#106	—	158	8	no	—
239	C₄⋊C₈	C₂×Dic₇₉	1	4	yes	—
240	C₂²⋊C₈	C₂×Dic₇₉	1	8	yes	—
241	C₄×C₈	C₂×Dic₇₉	1	4	yes	—
242	#95	—	158	2	no	—
243	#95	—	158	4	no	—
244	C₂⁴	C₂×D₃₁₆	1	2	yes	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	2	2	no	—
247	C₂×D₄	C₄⋊Dic₇₉	1	4	yes	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	158	2	no	—
253	C₂²×C₄	—	158	4	no	—
254	C₂²×C₄	—	158	2	no	—
255	C₂²×C₄	C₂²×Dic₇₉	1	4	yes	—
256	C₂²×C₄	C₂×D₃₁₆	1	2	yes	—
257	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₃₁₆	1	1	yes	characteristic
259	C₂²×C₄	C₂×Dic₁₅₈	1	4	yes	—
260	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	16	no	—
266	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
267	C₂×C₈	C₂²×Dic₇₉	1	4	yes	—
268	C₂×C₈	C₂²×Dic₇₉	1	2	yes	—
269	C₂×C₈	D₄×D₇₉	1	2	yes	—
270	C₂×C₈	#11929623	1	2	yes	—
271	C₄⋊C₄	—	316	4	no	—
272	C₄⋊C₄	—	79	4	no	—
273	C₄⋊C₄	—	632	2	no	—
274	C₄⋊C₄	—	158	2	no	—
275	C₄⋊C₄	—	158	4	no	—
276	C₄⋊C₄	—	158	2	no	—
277	C₄⋊C₄	—	158	4	no	—
278	C₄⋊C₄	—	158	2	no	—
279	C₄⋊C₄	C₄⋊Dic₇₉	1	4	yes	—
280	C₂²⋊C₄	—	316	8	no	—
281	C₂²⋊C₄	—	632	4	no	—
282	C₂²⋊C₄	C₄⋊Dic₇₉	1	8	yes	—
283	C₄²	—	316	8	no	—
284	C₄²	—	316	4	no	—
285	C₄²	C₂²×Dic₇₉	1	2	yes	—
286	C₄²	C₂×Dic₁₅₈	1	2	yes	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂×C₄⋊Dic₇₉	1	4	yes	—
294	C₂³	C₂²×D₃₁₆	1	1	yes	characteristic
295	D₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	316	2	no	—
305	C₂×C₄	—	316	2	no	—
306	C₂×C₄	—	316	4	no	—
307	C₂×C₄	—	316	2	no	—
308	C₂×C₄	—	316	2	no	—
309	C₂×C₄	—	316	4	no	—
310	C₂×C₄	—	316	2	no	—
311	C₂×C₄	—	316	4	no	—
312	C₂×C₄	—	158	4	no	—
313	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
314	C₂×C₄	C₂×C₄⋊Dic₇₉	1	4	yes	—
315	C₂×C₄	C₂×C₄⋊Dic₇₉	1	2	yes	—
316	C₂×C₄	#14579183	1	2	yes	—
317	C₂×C₄	#14579183	1	2	yes	—
318	C₂×C₄	C₂³×Dic₇₉	1	1	yes	characteristic
319	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
320	C₂×C₄	C₂²×Dic₁₅₈	1	1	yes	characteristic
321	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
322	C₂×C₄	#14579207	1	1	yes	characteristic
323	C₂×C₄	#14579208	1	1	yes	characteristic
324	C₈	—	2	4	no	—
325	C₈	—	2	2	no	—
326	C₈	D₄×Dic₇₉	1	4	yes	—
327	C₂²	—	2	4	no	—
328	C₂²	—	2	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	4	2	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	C₂²×C₄⋊Dic₇₉	1	1	yes	characteristic
335	C₂²	D₄×D₃₁₆	1	1	yes	characteristic
336	C₂²	#19099217	1	1	yes	characteristic
337	C₂²	C₂×#14579183	1	1	yes	characteristic
338	C₂²	C₂×#14579183	1	1	yes	characteristic
339	C₂²	#19099368	1	1	yes	characteristic
340	C₂²	#19099380	1	1	yes	characteristic
341	C₄	—	2	4	no	—
342	C₄	—	2	2	no	—
343	C₄	—	316	4	no	—
344	C₄	—	632	2	no	—
345	C₄	—	632	4	no	—
346	C₄	#19098601	1	4	yes	complemented
347	C₄	C₂×D₄×Dic₇₉	1	1	yes	characteristic
348	C₄	#19099046	1	1	yes	characteristic
349	C₄	D₄×Dic₁₅₈	1	1	yes	characteristic
350	C₄	#19099270	1	1	yes	characteristic
351	C₂	—	4	4	no	—
352	C₂	D₄×C₄⋊Dic₇₉	1	1	yes	characteristic
353	C₂	C₂×#19098601	1	1	yes	characteristic
354	C₂	#23772000	1	1	yes	characteristic
355	C₂	#23775122	1	1	yes	characteristic
356	C₂	#23775160	1	1	yes	characteristic
357	C₂	#23775823	1	1	yes	characteristic
358	C₂	#23775831	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₂×C₈×D₄×C₇₉	C₂
socle	C₂³×C₇₉	C₂²×D₄
prime radical	C₂²×D₄×C₇₉	D₄
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	C₂×C₈×D₄	D₇₉
hypocenter	C₇₉	#22658
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19098601	C₂	semidirect
3. #23764164	C₂	semidirect
4. #23763601	C₂	semidirect
5. #23763560	C₂	semidirect
6. #23763429	C₂	semidirect
7. C₄×#14579095	C₂	semidirect
26. C₈×D₄×C₇₉	C₄	semidirect
44. #14579095	D₄	semidirect
163. C₄×C₇₉	#427	semidirect
188. C₇₉	#22658	semidirect
192. C₈×D₄	Dic₇₉	semidirect
346. C₄	#19098601	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098601	2
#23766349	2
#23764164	2
#23763601	2
#23763560	2
#23763429	2
C₄×#14579095	2
C₂×C₈×D₄×C₇₉	2
#22658	79