Subgroups — #34187274

Order 20224.

640 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23763253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14579072	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19098303	C₂	1	4	yes	maximal, complemented
4	C₇₉×#2858	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₇₉⋊C₈)	—	2	1	no	—
6	C₂×#14579072	—	2	1	no	—
7	C₂×#14579072	C₂²	1	2	yes	—
8	C₂×#14579072	C₂²	1	2	yes	—
9	C₂×C₄×C₇₉⋊C₈	—	2	1	no	—
10	#19098318	—	2	1	no	—
11	#19098303	—	2	8	no	—
12	C₂×C₄⋊Q₈×C₇₉	C₄	1	1	yes	characteristic
13	C₂×#127×C₇₉	C₄	1	1	yes	characteristic
14	C₂×#124×C₇₉	C₄	1	2	yes	—
15	C₂×#124×C₇₉	C₂²	1	2	yes	—
16	#390×C₇₉	—	2	1	no	—
17	#390×C₇₉	—	2	1	no	—
18	#390×C₇₉	—	2	1	no	—
19	#390×C₇₉	—	2	1	no	—
20	C₄³×C₇₉	C₂²	1	1	yes	characteristic
21	C₂²×C₇₉⋊C₈	—	2	2	no	—
22	C₂²×C₇₉⋊C₈	—	2	2	no	—
23	C₄⋊(C₇₉⋊C₈)	—	2	4	no	—
24	#14579072	—	2	4	no	—
25	#14579072	—	2	2	no	—
26	#14579072	—	2	4	no	—
27	#14579072	D₄	1	4	yes	complemented
28	C₄×C₇₉⋊C₈	—	2	4	no	—
29	C₂²×Q₈×C₇₉	C₂×C₄	1	2	yes	—
30	C₂²×D₄×C₇₉	—	2	1	no	—
31	C₂²×D₄×C₇₉	—	2	1	no	—
32	C₂²×D₄×C₇₉	C₂×C₄	1	2	yes	—
33	C₄⋊Q₈×C₇₉	—	4	2	no	—
34	#127×C₇₉	—	4	2	no	—
35	#124×C₇₉	—	2	4	no	—
36	#124×C₇₉	—	2	4	no	—
37	#124×C₇₉	—	4	4	no	—
38	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
40	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₇₉	—	2	2	no	—
44	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₇₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
51	C₂×C₇₉⋊C₈	—	2	4	no	—
52	C₂×C₇₉⋊C₈	—	2	4	no	—
53	C₂×C₇₉⋊C₈	—	2	4	no	—
54	C₂×C₇₉⋊C₈	—	2	4	no	—
55	C₂×C₇₉⋊C₈	—	4	2	no	—
56	C₂×C₇₉⋊C₈	—	4	2	no	—
57	C₂⁴×C₇₉	—	2	2	no	—
58	C₂×Q₈×C₇₉	—	2	4	no	—
59	C₂×Q₈×C₇₉	—	4	2	no	—
60	C₂×Q₈×C₇₉	—	4	2	no	—
61	C₂×D₄×C₇₉	—	2	4	no	—
62	C₂×D₄×C₇₉	—	4	2	no	—
63	C₂×D₄×C₇₉	—	4	2	no	—
64	C₂×D₄×C₇₉	—	4	2	no	—
65	C₂×D₄×C₇₉	—	4	2	no	—
66	C₂×D₄×C₇₉	—	4	2	no	—
67	C₂×D₄×C₇₉	—	4	2	no	—
68	C₂×D₄×C₇₉	—	4	2	no	—
69	C₂×D₄×C₇₉	—	4	2	no	—
70	C₂²×C₄×C₇₉	—	2	2	no	—
71	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₇₉	—	4	2	no	—
79	C₄⋊C₄×C₇₉	—	4	2	no	—
80	C₂²⋊C₄×C₇₉	—	4	4	no	—
81	C₂²⋊C₄×C₇₉	—	4	4	no	—
82	C₂²⋊C₄×C₇₉	—	4	4	no	—
83	C₂²⋊C₄×C₇₉	—	4	2	no	—
84	C₂²⋊C₄×C₇₉	—	4	2	no	—
85	C₄²×C₇₉	—	2	2	no	—
86	C₄²×C₇₉	—	2	2	no	—
87	C₄²×C₇₉	—	2	2	no	—
88	C₄²×C₇₉	—	2	2	no	—
89	C₄²×C₇₉	—	2	2	no	—
90	C₄²×C₇₉	—	4	1	no	—
91	C₄²×C₇₉	C₂×D₄	1	2	yes	—
92	C₄²×C₇₉	#41	1	2	yes	—
93	C₂³×C₇₉	—	4	2	no	—
94	C₂³×C₇₉	—	4	2	no	—
95	C₂³×C₇₉	—	4	2	no	—
96	C₂³×C₇₉	—	4	2	no	—
97	C₂³×C₇₉	—	4	2	no	—
98	C₂³×C₇₉	—	4	2	no	—
99	C₂³×C₇₉	—	4	2	no	—
100	C₂³×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	Q₈×C₇₉	—	8	4	no	—
102	D₄×C₇₉	—	8	4	no	—
103	D₄×C₇₉	—	8	4	no	—
104	D₄×C₇₉	—	8	4	no	—
105	C₂×C₄×C₇₉	—	2	1	no	—
106	C₂×C₄×C₇₉	—	2	1	no	—
107	C₂×C₄×C₇₉	—	2	1	no	—
108	C₂×C₄×C₇₉	—	2	1	no	—
109	C₂×C₄×C₇₉	—	2	1	no	—
110	C₂×C₄×C₇₉	—	2	1	no	—
111	C₂×C₄×C₇₉	—	2	1	no	—
112	C₂×C₄×C₇₉	—	2	1	no	—
113	C₂×C₄×C₇₉	—	2	1	no	—
114	C₂×C₄×C₇₉	—	2	1	no	—
115	C₂×C₄×C₇₉	—	2	1	no	—
116	C₂×C₄×C₇₉	—	2	1	no	—
117	C₂×C₄×C₇₉	—	2	1	no	—
118	C₂×C₄×C₇₉	—	4	2	no	—
119	C₂×C₄×C₇₉	—	4	2	no	—
120	C₂×C₄×C₇₉	—	4	2	no	—
121	C₂×C₄×C₇₉	#100	1	2	yes	—
122	C₂×C₄×C₇₉	#100	1	2	yes	—
123	C₂×C₄×C₇₉	#104	1	2	yes	—
124	C₂×C₄×C₇₉	#104	1	2	yes	—
125	C₂×C₄×C₇₉	#104	1	2	yes	—
126	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₇₉	#121	1	2	yes	—
128	C₂×C₄×C₇₉	#124	1	1	yes	characteristic
129	C₂×C₄×C₇₉	#127	1	1	yes	characteristic
130	C₇₉⋊C₈	—	2	8	no	—
131	C₇₉⋊C₈	—	4	4	no	—
132	C₂²×C₇₉	—	8	2	no	—
133	C₂²×C₇₉	—	8	2	no	—
134	C₂²×C₇₉	—	8	2	no	—
135	C₂²×C₇₉	—	8	2	no	—
136	C₂²×C₇₉	—	8	2	no	—
137	C₂²×C₇₉	—	8	2	no	—
138	C₂²×C₇₉	—	8	2	no	—
139	C₂²×C₇₉	#390	1	1	yes	characteristic
140	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
141	C₂²×C₇₉	C₂×#100	1	1	yes	characteristic
142	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
143	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
144	C₂²×C₇₉	C₂×#104	1	1	yes	characteristic
145	C₂²×C₇₉	#421	1	1	yes	characteristic
146	C₄×C₇₉	—	2	2	no	—
147	C₄×C₇₉	—	2	2	no	—
148	C₄×C₇₉	—	2	1	no	—
149	C₄×C₇₉	—	2	1	no	—
150	C₄×C₇₉	—	2	2	no	—
151	C₄×C₇₉	—	2	2	no	—
152	C₄×C₇₉	—	4	1	no	—
153	C₄×C₇₉	—	8	2	no	—
154	C₄×C₇₉	#329	1	4	yes	complemented
155	#9685	—	79	1	no	maximal, Sylow
156	C₂×C₇₉	—	16	2	no	—
157	C₂×C₇₉	C₂×#329	1	1	yes	characteristic
158	C₂×C₇₉	#1525	1	1	yes	characteristic
159	C₂×C₇₉	#1527	1	1	yes	characteristic
160	C₂×C₇₉	#1530	1	1	yes	characteristic
161	C₂×C₇₉	#1962	1	1	yes	characteristic
162	C₂×C₇₉	#1965	1	1	yes	characteristic
163	C₂×C₇₉	#1980	1	1	yes	characteristic
164	#2858	D₇₉	1	1	yes	characteristic
165	#1767	—	79	1	no	—
166	C₄×#97	—	79	1	no	—
167	C₂×#329	—	79	4	no	—
168	C₇₉	#9685	1	1	yes	characteristic, complemented, Sylow
169	C₂×C₄⋊Q₈	Dic₇₉	1	1	yes	characteristic
170	C₂×#127	Dic₇₉	1	1	yes	characteristic
171	C₂×#124	Dic₇₉	1	2	yes	—
172	C₂×#124	C₂×D₇₉	1	2	yes	—
173	C₂×C₄⋊C₈	—	158	1	no	—
174	C₂×#97	—	79	2	no	—
175	C₂×#97	—	79	2	no	—
176	C₂×#97	—	158	1	no	—
177	C₂×C₄×C₈	—	158	1	no	—
178	#390	—	2	1	no	—
179	#390	—	2	1	no	—
180	#390	—	2	1	no	—
181	#390	—	2	1	no	—
182	C₄³	C₂×D₇₉	1	1	yes	characteristic
183	#336	—	158	1	no	—
184	#329	—	158	8	no	—
185	C₂²×Q₈	C₂×Dic₇₉	1	2	yes	—
186	C₂²×D₄	—	2	1	no	—
187	C₂²×D₄	—	2	1	no	—
188	C₂²×D₄	C₂×Dic₇₉	1	2	yes	—
189	C₂²×C₈	—	158	2	no	—
190	C₂²×C₈	—	158	2	no	—
191	C₄⋊Q₈	—	4	2	no	—
192	#127	—	4	2	no	—
193	#124	—	2	4	no	—
194	#124	—	2	4	no	—
195	#124	—	4	4	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
203	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
205	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₇₉	1	1	yes	characteristic
208	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
209	C₄⋊C₈	—	158	4	no	—
210	#97	—	79	4	no	—
211	#97	—	158	4	no	—
212	#97	—	158	4	no	—
213	#97	—	158	2	no	—
214	C₄×C₈	—	158	4	no	—
215	C₂⁴	—	2	2	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	4	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×D₄	—	2	4	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	2	no	—
222	C₂×D₄	—	4	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	2	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
231	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
232	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
233	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
236	C₂×C₈	—	316	2	no	—
237	C₂×C₈	—	316	2	no	—
238	C₂×C₈	—	158	4	no	—
239	C₂×C₈	—	158	4	no	—
240	C₂×C₈	—	158	4	no	—
241	C₂×C₈	—	158	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₂²⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	1	no	—
255	C₄²	D₄×D₇₉	1	2	yes	—
256	C₄²	#11929623	1	2	yes	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
265	Q₈	—	8	4	no	—
266	D₄	—	8	4	no	—
267	D₄	—	8	4	no	—
268	D₄	—	8	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	#14579078	1	2	yes	—
286	C₂×C₄	#14579078	1	2	yes	—
287	C₂×C₄	#14579088	1	2	yes	—
288	C₂×C₄	#14579088	1	2	yes	—
289	C₂×C₄	#14579088	1	2	yes	—
290	C₂×C₄	D₄×Dic₇₉	1	2	yes	—
291	C₂×C₄	#14579132	1	2	yes	—
292	C₂×C₄	#14579148	1	1	yes	characteristic
293	C₂×C₄	#14579157	1	1	yes	characteristic
294	C₈	—	316	4	no	—
295	C₈	—	158	8	no	—
296	C₂²	—	8	2	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	#19098466	1	1	yes	characteristic
304	C₂²	C₂×#14579078	1	1	yes	characteristic
305	C₂²	C₂×#14579078	1	1	yes	characteristic
306	C₂²	C₂×#14579088	1	1	yes	characteristic
307	C₂²	C₂×#14579088	1	1	yes	characteristic
308	C₂²	C₂×#14579088	1	1	yes	characteristic
309	C₂²	#19098580	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	8	2	no	—
318	C₄	#19098303	1	4	yes	complemented
319	C₂	—	16	2	no	—
320	C₂	C₂×#19098303	1	1	yes	characteristic
321	C₂	#23761992	1	1	yes	characteristic
322	C₂	#23761998	1	1	yes	characteristic
323	C₂	#23762011	1	1	yes	characteristic
324	C₂	#23764229	1	1	yes	characteristic
325	C₂	#23764241	1	1	yes	characteristic
326	C₂	#23764320	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₇₉
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₉
Fitting	C₇₉×#2858	C₂
socle	C₂³×C₇₉	C₂×C₂²⋊C₄
prime radical	C₂×#127×C₇₉	C₄
cosocle	C₂×C₄²×C₇₉	C₂³
hypercenter	#2858	D₇₉
hypocenter	C₇₉	#9685
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂²×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2858

Complements

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

N	H ≅ G/N	type
1. #23763253	C₂	semidirect
2. C₄×#14579072	C₂	semidirect
3. C₂×#19098303	C₂	semidirect
27. #14579072	D₄	semidirect
154. C₄×C₇₉	#329	semidirect
168. C₇₉	#9685	semidirect
318. C₄	#19098303	semidirect

Maximal subgroups

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

subgroup	index
#23763253	2
C₄×#14579072	2
C₂×#19098303	2
C₇₉×#2858	2
#9685	79