Subgroups — #34288637

Order 20224.

608 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23771828	C₂	1	1	yes	maximal, characteristic
2	#23763546	C₂	1	2	yes	maximal, complemented
3	#23763522	C₂	1	2	yes	maximal, complemented
4	#23763504	C₂	1	1	yes	maximal, characteristic
5	#23763448	C₂	1	1	yes	maximal, characteristic
6	#23763444	C₂	1	1	yes	maximal, characteristic
7	#23763283	C₂	1	2	yes	maximal, complemented
8	#23763203	C₂	1	1	yes	maximal, characteristic
9	#23763202	C₂	1	1	yes	maximal, characteristic
10	#23763154	C₂	1	2	yes	maximal, complemented
11	C₇₉×#1827	C₂	1	1	yes	maximal, characteristic
12	C₂×#14579168	C₂²	1	1	yes	characteristic
13	C₂×#14579168	C₂²	1	1	yes	characteristic
14	C₂×#14579096	C₂²	1	2	yes	—
15	#19098599	C₂²	1	2	yes	—
16	#19098547	C₂²	1	1	yes	characteristic
17	#19098537	C₂²	1	1	yes	characteristic
18	#19098521	C₂²	1	2	yes	—
19	#19098519	C₄	1	2	yes	complemented
20	#19098519	C₄	1	2	yes	complemented
21	#19098519	C₂²	1	2	yes	—
22	#19098519	C₂²	1	1	yes	characteristic
23	#19098519	C₂²	1	1	yes	characteristic
24	C₄×#11929608	C₄	1	2	yes	complemented
25	C₄×#11929608	C₂²	1	1	yes	characteristic
26	C₂×#14579072	C₄	1	2	yes	complemented
27	C₂×#14579072	C₂²	1	1	yes	characteristic
28	#19098335	C₂²	1	2	yes	—
29	#19098333	C₂²	1	2	yes	—
30	#19098323	C₂²	1	2	yes	—
31	C₇₉⋊#338	C₂²	1	2	yes	—
32	#19098319	C₂²	1	2	yes	—
33	#19098319	C₂²	1	2	yes	—
34	C₄×#41×C₇₉	C₂²	1	1	yes	characteristic
35	#413×C₇₉	C₂²	1	1	yes	characteristic
36	#410×C₇₉	C₂²	1	1	yes	characteristic
37	#339×C₇₉	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₇₉	C₂²	1	2	yes	—
39	#14579168	—	2	2	no	—
40	#14579168	—	2	2	no	—
41	#14579168	—	2	2	no	—
42	#14579168	—	2	2	no	—
43	C₂×#11929608	C₂×C₄	1	2	yes	—
44	C₂×#11929608	C₂×C₄	1	2	yes	—
45	C₂×#11929608	C₂³	1	2	yes	—
46	C₂×#11929608	C₂³	1	1	yes	characteristic
47	C₂×#11929608	C₂³	1	1	yes	characteristic
48	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
49	C₂²×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
51	C₂²×C₇₉⋊C₈	C₂³	1	1	yes	characteristic
52	#14579120	C₂³	1	2	yes	—
53	#14579096	—	2	2	no	—
54	#14579096	—	2	2	no	—
55	#14579094	—	2	2	no	—
56	#14579092	—	2	2	no	—
57	#14579080	—	2	1	no	—
58	#14579078	—	2	1	no	—
59	#14579076	—	2	2	no	—
60	#14579072	—	2	2	no	—
61	#14579072	C₂×C₄	1	1	yes	characteristic
62	#14579072	C₂×C₄	1	2	yes	—
63	#14579072	C₂×C₄	1	2	yes	—
64	#14579072	C₂×C₄	1	2	yes	—
65	#14579072	C₂×C₄	1	1	yes	characteristic
66	#14579072	C₂×C₄	1	1	yes	characteristic
67	#14579072	C₂×C₄	1	1	yes	characteristic
68	#14579071	—	2	2	no	—
69	C₄×C₇₉⋊C₈	—	2	2	no	—
70	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
71	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
72	C₄×C₇₉⋊C₈	C₂×C₄	1	2	yes	—
73	C₈×Dic₇₉	—	2	2	no	—
74	C₂×#41×C₇₉	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
76	C₄×Q₈×C₇₉	—	2	1	no	—
77	C₄×D₄×C₇₉	—	2	2	no	—
78	C₄×D₄×C₇₉	—	2	1	no	—
79	#117×C₇₉	C₂×C₄	1	2	yes	—
80	#117×C₇₉	C₂³	1	2	yes	—
81	#117×C₇₉	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₇₉	C₂×C₄	1	2	yes	—
83	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
84	#101×C₇₉	—	2	1	no	—
85	#100×C₇₉	—	2	1	no	—
86	#99×C₇₉	—	2	2	no	—
87	#11929608	—	2	4	no	—
88	#11929608	—	2	4	no	—
89	#11929608	—	2	2	no	—
90	#11929608	—	2	2	no	—
91	#11929608	—	2	2	no	—
92	#11929608	—	2	2	no	—
93	C₂×C₇₉⋊C₈	—	2	2	no	—
94	C₂×C₇₉⋊C₈	—	2	2	no	—
95	C₂×C₇₉⋊C₈	—	2	2	no	—
96	C₂×C₇₉⋊C₈	—	2	2	no	—
97	C₂×C₇₉⋊C₈	—	2	2	no	—
98	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
99	C₂×C₇₉⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₇₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₇₉⋊C₈	#41	1	2	yes	—
105	C₂×C₇₉⋊C₈	#41	1	2	yes	—
106	C₂×C₇₉⋊C₈	#41	1	2	yes	—
107	C₂×C₇₉⋊C₈	#41	1	2	yes	—
108	C₄⋊Dic₇₉	—	2	2	no	—
109	C₂²⋊Dic₇₉	—	2	2	no	—
110	C₄×Dic₇₉	—	2	2	no	—
111	#41×C₇₉	—	2	2	no	—
112	#41×C₇₉	—	2	2	no	—
113	C₂×Q₈×C₇₉	#41	1	1	yes	characteristic
114	C₂×D₄×C₇₉	#41	1	2	yes	—
115	C₂×D₄×C₇₉	#41	1	1	yes	characteristic
116	C₂²×C₄×C₇₉	—	2	2	no	—
117	C₂²×C₄×C₇₉	—	2	1	no	—
118	C₂²×C₄×C₇₉	C₂²×C₄	1	2	yes	—
119	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
120	M₁₆×C₇₉	—	2	2	no	—
121	M₁₆×C₇₉	—	2	2	no	—
122	C₂×C₈×C₇₉	—	2	2	no	—
123	C₄⋊C₄×C₇₉	—	2	2	no	—
124	C₄⋊C₄×C₇₉	—	2	1	no	—
125	C₄⋊C₄×C₇₉	—	2	2	no	—
126	C₂²⋊C₄×C₇₉	—	2	2	no	—
127	C₂²⋊C₄×C₇₉	—	2	1	no	—
128	C₂²⋊C₄×C₇₉	—	2	2	no	—
129	C₄²×C₇₉	—	2	2	no	—
130	C₄²×C₇₉	—	2	1	no	—
131	C₄²×C₇₉	—	2	2	no	—
132	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₇₉	C₂²×C₄	1	2	yes	—
134	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
135	C₂³×C₇₉	#117	1	2	yes	—
136	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
137	Q₈×C₇₉	—	2	2	no	—
138	D₄×C₇₉	—	2	4	no	—
139	D₄×C₇₉	—	2	2	no	—
140	C₂×C₄×C₇₉	—	2	2	no	—
141	C₂×C₄×C₇₉	—	2	2	no	—
142	C₂×C₄×C₇₉	—	2	1	no	—
143	C₂×C₄×C₇₉	—	2	2	no	—
144	C₂×C₄×C₇₉	—	2	1	no	—
145	C₂×C₄×C₇₉	—	4	2	no	—
146	C₂×C₄×C₇₉	—	8	1	no	—
147	C₂×C₄×C₇₉	—	8	2	no	—
148	C₂×C₄×C₇₉	#117	1	2	yes	—
149	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
151	C₂×C₄×C₇₉	C₂×#41	1	2	yes	—
152	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
153	C₂×Dic₇₉	—	4	2	no	—
154	C₈×C₇₉	—	4	2	no	—
155	C₇₉⋊C₈	—	2	2	no	—
156	C₇₉⋊C₈	—	2	2	no	—
157	C₇₉⋊C₈	—	4	2	no	—
158	C₇₉⋊C₈	#117	1	1	yes	characteristic
159	C₇₉⋊C₈	#117	1	1	yes	characteristic
160	C₇₉⋊C₈	#117	1	1	yes	characteristic
161	C₇₉⋊C₈	#117	1	1	yes	characteristic
162	C₇₉⋊C₈	#117	1	2	yes	—
163	C₇₉⋊C₈	#117	1	2	yes	—
164	C₂²×C₇₉	—	2	2	no	—
165	C₂²×C₇₉	—	4	1	no	—
166	C₂²×C₇₉	—	4	2	no	—
167	C₂²×C₇₉	C₂×#117	1	1	yes	characteristic
168	C₂²×C₇₉	#529	1	2	yes	—
169	C₄×C₇₉	—	2	2	no	—
170	C₄×C₇₉	—	4	2	no	—
171	C₄×C₇₉	—	4	1	no	—
172	C₄×C₇₉	—	4	1	no	—
173	C₄×C₇₉	—	8	2	no	—
174	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
175	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
176	C₄×C₇₉	#529	1	2	yes	—
177	Dic₇₉	—	8	2	no	—
178	#20788	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	2	no	—
180	C₂×C₇₉	—	2	1	no	—
181	C₂×C₇₉	—	4	2	no	—
182	C₂×C₇₉	#2283	1	1	yes	characteristic
183	#2875	—	79	1	no	—
184	#1841	—	79	2	no	—
185	#1833	—	79	2	no	—
186	#1827	—	79	1	no	—
187	#1827	D₇₉	1	1	yes	characteristic
188	#1813	—	79	1	no	—
189	#1812	—	79	1	no	—
190	#1773	—	79	2	no	—
191	#1757	—	79	1	no	—
192	#1757	—	79	1	no	—
193	#1747	—	79	2	no	—
194	C₇₉	#20788	1	1	yes	characteristic, complemented, Sylow
195	C₂×#131	—	79	1	no	—
196	C₂×#131	—	79	1	no	—
197	C₄×#41	C₂×D₇₉	1	1	yes	characteristic
198	C₂×#108	—	79	2	no	—
199	#427	—	79	2	no	—
200	#413	—	79	1	no	—
201	#413	C₂×D₇₉	1	1	yes	characteristic
202	#410	—	79	1	no	—
203	#410	C₂×D₇₉	1	1	yes	characteristic
204	#405	—	79	1	no	—
205	#405	—	79	1	no	—
206	#405	—	79	2	no	—
207	#405	—	79	2	no	—
208	#405	—	79	2	no	—
209	#405	—	79	2	no	—
210	C₄×M₁₆	—	79	1	no	—
211	C₄×M₁₆	—	79	2	no	—
212	C₂×#97	—	79	1	no	—
213	C₂×#97	—	79	2	no	—
214	#344	—	79	2	no	—
215	#344	—	79	2	no	—
216	#339	—	79	2	no	—
217	#339	C₂×D₇₉	1	2	yes	—
218	#338	—	79	2	no	—
219	H(3,ℤ/4)	—	79	2	no	—
220	H(3,ℤ/4)	—	79	2	no	—
221	H(3,ℤ/4)	C₂×D₇₉	1	2	yes	—
222	C₂×#41	C₂²×D₇₉	1	1	yes	characteristic
223	#131	—	158	2	no	—
224	#131	—	158	2	no	—
225	#131	—	158	2	no	—
226	#131	—	158	2	no	—
227	C₂×M₁₆	—	79	2	no	—
228	C₂×M₁₆	—	79	2	no	—
229	C₂×M₁₆	—	79	1	no	—
230	C₂×M₁₆	—	79	1	no	—
231	C₂×M₁₆	—	79	2	no	—
232	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
233	C₂²×C₈	—	79	1	no	—
234	C₂²×C₈	—	79	1	no	—
235	C₂²×C₈	—	79	2	no	—
236	C₂²×C₈	—	79	2	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	2	2	no	—
240	#117	—	79	2	no	—
241	#117	C₄×D₇₉	1	2	yes	—
242	#117	C₂²×D₇₉	1	2	yes	—
243	#117	C₂²×D₇₉	1	1	yes	characteristic
244	C₂×C₄²	C₄×D₇₉	1	2	yes	—
245	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
246	#108	—	158	2	no	—
247	#108	—	158	2	no	—
248	#107	—	158	2	no	—
249	#106	—	158	2	no	—
250	#101	—	2	1	no	—
251	#101	—	158	1	no	—
252	#100	—	2	1	no	—
253	#100	—	158	1	no	—
254	#99	—	2	2	no	—
255	#99	—	158	2	no	—
256	#97	—	79	1	no	—
257	#97	—	79	1	no	—
258	#97	—	79	2	no	—
259	#97	—	79	1	no	—
260	#97	—	79	1	no	—
261	#97	—	79	2	no	—
262	#97	—	79	2	no	—
263	#97	—	158	2	no	—
264	#97	—	158	2	no	—
265	C₄×C₈	—	79	2	no	—
266	C₄×C₈	—	79	2	no	—
267	C₄×C₈	—	79	2	no	—
268	C₄×C₈	—	158	2	no	—
269	C₄×C₈	—	158	2	no	—
270	#41	—	2	2	no	—
271	#41	—	2	2	no	—
272	C₂×Q₈	#11929623	1	1	yes	characteristic
273	C₂×D₄	#11929623	1	1	yes	characteristic
274	C₂×D₄	#11929623	1	2	yes	—
275	C₂²×C₄	—	2	1	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	C₂×C₄×D₇₉	1	2	yes	—
278	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	158	2	no	—
282	M₁₆	—	158	2	no	—
283	M₁₆	—	158	4	no	—
284	M₁₆	—	158	4	no	—
285	M₁₆	—	158	2	no	—
286	M₁₆	—	158	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	79	2	no	—
289	C₂×C₈	—	79	2	no	—
290	C₂×C₈	—	79	2	no	—
291	C₂×C₈	—	79	2	no	—
292	C₂×C₈	—	79	1	no	—
293	C₂×C₈	—	79	1	no	—
294	C₂×C₈	—	79	1	no	—
295	C₂×C₈	—	79	1	no	—
296	C₂×C₈	—	79	2	no	—
297	C₂×C₈	—	79	2	no	—
298	C₂×C₈	—	158	2	no	—
299	C₂×C₈	—	158	2	no	—
300	C₂×C₈	—	158	2	no	—
301	C₂×C₈	—	158	2	no	—
302	C₂×C₈	—	158	2	no	—
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	158	2	no	—
307	C₂²⋊C₄	—	2	1	no	—
308	C₂²⋊C₄	—	2	2	no	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	158	2	no	—
311	C₄²	—	2	1	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	158	2	no	—
315	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₇₉	1	2	yes	—
317	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
318	C₂³	#14579117	1	2	yes	—
319	C₂³	C₂×#11929623	1	1	yes	characteristic
320	Q₈	—	2	2	no	—
321	D₄	—	2	4	no	—
322	D₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	316	2	no	—
332	C₂×C₄	#14579117	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
334	C₂×C₄	#41×D₇₉	1	2	yes	—
335	C₂×C₄	#41×D₇₉	1	2	yes	—
336	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
337	C₈	—	4	2	no	—
338	C₈	—	316	2	no	—
339	C₈	—	79	1	no	—
340	C₈	—	79	1	no	—
341	C₈	—	79	2	no	—
342	C₈	—	79	1	no	—
343	C₈	—	79	1	no	—
344	C₈	—	79	2	no	—
345	C₈	—	158	2	no	—
346	C₈	—	158	2	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	2	no	—
350	C₂²	C₂×#14579117	1	1	yes	characteristic
351	C₂²	#19099090	1	2	yes	—
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	632	2	no	—
358	C₄	#117×D₇₉	1	1	yes	characteristic
359	C₄	#117×D₇₉	1	1	yes	characteristic
360	C₄	#19099090	1	2	yes	—
361	C₂	—	2	1	no	—
362	C₂	—	2	2	no	—
363	C₂	—	4	2	no	—
364	C₂	#23765791	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₇₉
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1827	C₂
socle	C₂×C₇₉	#2283
prime radical	C₂×#41×C₇₉	C₂³
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1827	D₇₉
hypocenter	C₇₉	#20788
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₄ ◁ #117 ◁ #1827

Complements

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

N	H ≅ G/N	type
2. #23763546	C₂	semidirect
3. #23763522	C₂	semidirect
7. #23763283	C₂	semidirect
10. #23763154	C₂	semidirect
19. #19098519	C₄	semidirect
20. #19098519	C₄	semidirect
24. C₄×#11929608	C₄	semidirect
26. C₂×#14579072	C₄	semidirect
194. C₇₉	#20788	semidirect

Maximal subgroups

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

subgroup	index
#23771828	2
#23763546	2
#23763522	2
#23763504	2
#23763448	2
#23763444	2
#23763283	2
#23763203	2
#23763202	2
#23763154	2
C₇₉×#1827	2
#20788	79