Subgroups — #34307972

Order 20224.

710 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14579183	C₂	1	1	yes	maximal, characteristic
2	#23772065	C₂	1	2	yes	maximal, complemented
3	#23764293	C₂	1	2	yes	maximal
4	#23764049	C₂	1	4	yes	maximal, complemented
5	#23763572	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763421	C₂	1	2	yes	maximal, complemented
7	#23763284	C₂	1	2	yes	maximal, complemented
8	C₇₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14579183	C₂²	1	1	yes	characteristic
10	C₄×#11929624	C₂²	1	2	yes	—
11	C₄×Dic₃₁₆	C₄	1	1	yes	characteristic, complemented
12	C₄×Dic₃₁₆	C₄	1	1	yes	characteristic, complemented
13	C₄×#11929613	C₄	1	2	yes	complemented
14	C₄×#11929613	C₄	1	2	yes	complemented
15	C₄×D₆₃₂	C₄	1	1	yes	characteristic, complemented
16	C₄×D₆₃₂	C₄	1	1	yes	characteristic, complemented
17	#19098623	—	2	4	no	—
18	#19098619	—	2	2	no	—
19	#19098615	—	2	2	no	—
20	C₂×#14579097	C₂²	1	2	yes	—
21	C₂×#14579097	C₂²	1	2	yes	—
22	#19098601	C₂²	1	1	yes	characteristic
23	#19098587	C₂²	1	2	yes	—
24	C₂×#14579089	C₂²	1	4	yes	—
25	#19098560	C₂²	1	2	yes	—
26	#19098529	C₂²	1	4	yes	—
27	#19098518	C₂²	1	4	yes	—
28	#19098323	C₂²	1	4	yes	—
29	C₂×C₄⋊C₈×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#97×C₇₉	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₇₉	C₂²	1	1	yes	characteristic
33	#336×C₇₉	C₂²	1	2	yes	—
34	C₂×#11929624	C₂³	1	2	yes	—
35	#14579183	—	2	2	no	—
36	#14579183	—	2	2	no	—
37	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×#11929613	C₂×C₄	1	2	yes	—
39	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic
40	C₂×#11929608	C₂³	1	4	yes	—
41	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₅₈	C₂×C₄	1	2	yes	—
43	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
44	C₄×D₃₁₆	C₂×C₄	1	2	yes	—
45	C₄×C₂²⋊D₇₉	—	2	2	no	—
46	C₄×C₂²⋊D₇₉	—	2	2	no	—
47	#14579120	C₂³	1	2	yes	—
48	#14579118	—	2	2	no	—
49	C₄²×D₇₉	—	2	2	no	—
50	#14579097	—	2	4	no	—
51	#14579097	—	4	2	no	—
52	#14579095	C₂×C₄	1	1	yes	characteristic
53	#14579095	C₂×C₄	1	1	yes	characteristic
54	#14579093	C₂×C₄	1	2	yes	—
55	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₇₉⋊C₈)	—	2	2	no	—
57	#14579089	—	4	4	no	—
58	#14579086	C₂×C₄	1	2	yes	—
59	#14579086	C₂×C₄	1	2	yes	—
60	#14579083	C₂×C₄	1	2	yes	—
61	#14579083	C₂×C₄	1	2	yes	—
62	(C₂×D₇₉)⋊C₈	—	2	4	no	—
63	(C₂×D₇₉)⋊C₈	—	2	4	no	—
64	#14579071	—	2	4	no	—
65	C₈×Dic₇₉	—	2	4	no	—
66	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
68	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
70	C₂×C₄²×C₇₉	C₂³	1	2	yes	—
71	C₂×C₄²×C₇₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
73	C₄⋊C₈×C₇₉	D₄	1	2	yes	—
74	#97×C₇₉	—	2	4	no	—
75	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
79	#95×C₇₉	—	2	1	no	—
80	#11929624	—	2	2	no	—
81	#11929624	—	2	2	no	—
82	#11929624	—	2	2	no	—
83	#11929624	—	2	2	no	—
84	C₂×Dic₁₅₈	C₂²×C₄	1	2	yes	—
85	C₂×D₃₁₆	C₂²×C₄	1	2	yes	—
86	C₂×C₂²⋊D₇₉	—	2	2	no	—
87	C₂×C₄×D₇₉	—	2	2	no	—
88	C₂×C₄×D₇₉	—	2	2	no	—
89	C₂×C₄×D₇₉	—	2	2	no	—
90	Dic₃₁₆	—	2	2	no	—
91	#11929613	—	2	4	no	—
92	D₆₃₂	—	2	2	no	—
93	#11929608	—	2	4	no	—
94	#11929608	—	2	4	no	—
95	C₂×C₇₉⋊C₈	—	2	4	no	—
96	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
97	C₄⋊Dic₇₉	C₂²×C₄	1	2	yes	—
98	Dic₇₉⋊C₄	—	2	2	no	—
99	Dic₇₉⋊C₄	—	2	2	no	—
100	C₂²⋊Dic₇₉	—	2	2	no	—
101	(C₂×D₇₉)⋊C₄	—	2	2	no	—
102	(C₂×D₇₉)⋊C₄	—	2	2	no	—
103	C₄×Dic₇₉	—	2	2	no	—
104	C₄×Dic₇₉	—	2	2	no	—
105	C₄×Dic₇₉	—	2	2	no	—
106	C₂²×C₄×C₇₉	—	2	2	no	—
107	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
108	C₂²×C₄×C₇₉	#41	1	1	yes	characteristic
109	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₇₉	—	2	2	no	—
111	C₂×C₈×C₇₉	—	2	1	no	—
112	C₂×C₈×C₇₉	—	2	1	no	—
113	C₂×C₈×C₇₉	—	2	2	no	—
114	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	C₂×D₄	1	4	yes	—
119	C₂×C₈×C₇₉	#41	1	2	yes	—
120	C₂×C₈×C₇₉	#41	1	2	yes	—
121	C₄⋊C₄×C₇₉	—	4	2	no	—
122	C₄²×C₇₉	—	2	2	no	—
123	C₄²×C₇₉	—	2	2	no	—
124	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₇₉	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
129	C₂²×D₇₉	—	2	2	no	—
130	C₂×C₄×C₇₉	—	2	2	no	—
131	C₂×C₄×C₇₉	—	4	1	no	—
132	C₂×C₄×C₇₉	—	4	2	no	—
133	C₂×C₄×C₇₉	—	4	2	no	—
134	C₂×C₄×C₇₉	—	4	2	no	—
135	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
144	C₂×C₄×C₇₉	2₋¹⁺⁴	1	1	yes	characteristic
145	C₂²⋊D₇₉	—	4	2	no	—
146	C₂²⋊D₇₉	—	4	2	no	—
147	C₂×Dic₇₉	—	2	2	no	—
148	C₂×Dic₇₉	—	2	2	no	—
149	C₂×Dic₇₉	—	2	2	no	—
150	D₃₁₆	—	2	2	no	—
151	D₃₁₆	—	2	2	no	—
152	C₄×D₇₉	—	4	2	no	—
153	C₄×D₇₉	—	4	2	no	—
154	C₄×D₇₉	—	4	4	no	—
155	C₄×D₇₉	—	8	2	no	—
156	Dic₁₅₈	—	2	2	no	—
157	Dic₁₅₈	—	2	2	no	—
158	C₈×C₇₉	—	2	1	no	—
159	C₈×C₇₉	—	2	1	no	—
160	C₈×C₇₉	—	2	4	no	—
161	C₈×C₇₉	#117	1	2	yes	—
162	C₈×C₇₉	#117	1	2	yes	—
163	C₇₉⋊C₈	—	4	4	no	—
164	C₂²×C₇₉	—	2	1	no	—
165	C₂²×C₇₉	—	2	1	no	—
166	C₂²×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₇₉	#555	1	1	yes	characteristic
168	C₂²×C₇₉	#556	1	1	yes	characteristic
169	C₂×D₇₉	—	4	2	no	—
170	C₂×D₇₉	—	4	2	no	—
171	C₂×D₇₉	—	4	2	no	—
172	C₄×C₇₉	—	2	1	no	—
173	C₄×C₇₉	—	2	1	no	—
174	C₄×C₇₉	—	2	1	no	—
175	C₄×C₇₉	—	8	2	no	—
176	C₄×C₇₉	#444	1	2	yes	—
177	C₄×C₇₉	C₂×#117	1	1	yes	characteristic
178	C₄×C₇₉	#520	1	1	yes	characteristic
179	C₄×C₇₉	#547	1	1	yes	characteristic
180	C₄×C₇₉	#550	1	1	yes	characteristic
181	Dic₇₉	—	4	2	no	—
182	Dic₇₉	—	4	2	no	—
183	Dic₇₉	—	4	2	no	—
184	#22803	—	79	1	no	maximal, Sylow
185	C₂×C₇₉	—	2	1	no	—
186	C₂×C₇₉	—	2	1	no	—
187	C₂×C₇₉	#2354	1	1	yes	characteristic
188	C₂×C₇₉	C₂×#444	1	1	yes	characteristic
189	C₂×C₇₉	#2957	1	1	yes	characteristic
190	D₇₉	—	8	2	no	—
191	C₄×#135	—	79	1	no	—
192	#2925	—	79	2	no	—
193	#1975	—	79	2	no	—
194	#1929	—	79	4	no	—
195	#1848	—	79	1	no	—
196	#1806	—	79	2	no	—
197	#1773	—	79	2	no	—
198	#1770	D₇₉	1	1	yes	characteristic, complemented
199	C₇₉	#22803	1	1	yes	characteristic, complemented, Sylow
200	C₂×#135	—	79	1	no	—
201	C₄×#41	—	79	2	no	—
202	C₄×Q₁₆	—	79	1	no	—
203	C₄×Q₁₆	—	79	1	no	—
204	C₄×SD₁₆	—	79	2	no	—
205	C₄×SD₁₆	—	79	2	no	—
206	C₄×D₈	—	79	1	no	—
207	C₄×D₈	—	79	1	no	—
208	#433	—	158	4	no	—
209	#432	—	158	2	no	—
210	#431	—	158	2	no	—
211	C₂×#108	—	79	2	no	—
212	C₂×#108	—	79	2	no	—
213	#427	—	79	1	no	—
214	#423	—	79	2	no	—
215	C₂×C₄⋊C₈	C₂×D₇₉	1	1	yes	characteristic
216	C₂×#104	—	79	4	no	—
217	#416	—	79	2	no	—
218	#408	—	79	4	no	—
219	#405	—	79	4	no	—
220	C₂×#97	C₂×D₇₉	1	2	yes	—
221	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
222	C₄×C₄⋊C₄	C₂×D₇₉	1	1	yes	characteristic
223	#339	—	79	4	no	—
224	#336	C₂×D₇₉	1	2	yes	—
225	C₂×#41	—	79	2	no	—
226	#135	—	158	2	no	—
227	#135	—	158	2	no	—
228	C₂×Q₁₆	—	79	1	no	—
229	C₂×SD₁₆	—	79	2	no	—
230	C₂×D₈	—	79	1	no	—
231	C₂×M₁₆	—	79	4	no	—
232	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
234	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
235	C₄×Q₈	—	79	2	no	—
236	C₄×Q₈	—	79	2	no	—
237	C₄×D₄	—	79	2	no	—
238	C₄×D₄	—	79	2	no	—
239	C₄×D₄	—	158	2	no	—
240	C₄×D₄	—	158	2	no	—
241	#117	—	79	2	no	—
242	#117	—	158	2	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄²	—	158	2	no	—
245	C₂×C₄²	C₂²×D₇₉	1	2	yes	—
246	C₂×C₄²	C₂²×D₇₉	1	1	yes	characteristic
247	#108	—	316	2	no	—
248	#108	—	158	4	no	—
249	#107	—	79	1	no	—
250	#107	—	79	1	no	—
251	#106	—	79	2	no	—
252	C₄⋊C₈	—	158	2	no	—
253	C₄⋊C₈	—	158	2	no	—
254	C₄⋊C₈	D₃₁₆	1	2	yes	—
255	C₄⋊C₈	D₃₁₆	1	2	yes	—
256	#104	—	316	4	no	—
257	#103	—	79	2	no	—
258	#103	—	79	2	no	—
259	#102	—	79	2	no	—
260	#102	—	79	2	no	—
261	C₂²⋊C₈	—	158	4	no	—
262	C₂²⋊C₈	—	158	4	no	—
263	#97	—	2	4	no	—
264	#97	—	158	4	no	—
265	C₄×C₈	—	158	4	no	—
266	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
267	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
268	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
269	C₄×C₈	C₄×D₇₉	1	1	yes	characteristic
270	#95	—	2	1	no	—
271	#41	—	158	2	no	—
272	#41	—	158	2	no	—
273	#41	—	158	2	no	—
274	#41	—	158	2	no	—
275	C₂×Q₈	—	79	2	no	—
276	C₂×D₄	—	79	2	no	—
277	C₂×D₄	—	158	2	no	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	158	2	no	—
280	C₂²×C₄	—	158	2	no	—
281	C₂²×C₄	—	158	2	no	—
282	C₂²×C₄	#11929624	1	1	yes	characteristic
283	C₂²×C₄	#11929624	1	1	yes	characteristic
284	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
285	Q₁₆	—	158	2	no	—
286	SD₁₆	—	158	4	no	—
287	D₈	—	158	2	no	—
288	M₁₆	—	158	4	no	—
289	M₁₆	—	158	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	158	4	no	—
295	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
299	C₂×C₈	C₂×D₃₁₆	1	4	yes	—
300	C₂×C₈	#11929623	1	2	yes	—
301	C₂×C₈	#11929625	1	2	yes	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	79	2	no	—
304	C₄⋊C₄	—	79	2	no	—
305	C₄⋊C₄	—	158	2	no	—
306	C₄⋊C₄	—	158	2	no	—
307	C₂²⋊C₄	—	158	2	no	—
308	C₂²⋊C₄	—	158	2	no	—
309	C₂²⋊C₄	—	158	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	158	2	no	—
313	C₄²	—	158	2	no	—
314	C₄²	—	158	2	no	—
315	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₇₉	1	1	yes	characteristic
317	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
318	C₄²	C₂×D₃₁₆	1	1	yes	characteristic
319	C₂³	—	158	2	no	—
320	C₂³	C₂×#11929624	1	1	yes	characteristic
321	Q₈	—	158	2	no	—
322	Q₈	—	158	2	no	—
323	D₄	—	316	2	no	—
324	D₄	—	316	2	no	—
325	D₄	—	158	2	no	—
326	D₄	—	158	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	316	4	no	—
333	C₂×C₄	—	316	2	no	—
334	C₂×C₄	—	316	2	no	—
335	C₂×C₄	—	632	2	no	—
336	C₂×C₄	—	158	2	no	—
337	C₂×C₄	—	158	2	no	—
338	C₂×C₄	—	158	2	no	—
339	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
340	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
341	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
342	C₂×C₄	C₄×D₃₁₆	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₄×D₇₉	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₃₁₆	1	1	yes	characteristic
345	C₂×C₄	C₂×#11929623	1	1	yes	characteristic
346	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
347	C₂×C₄	#14579207	1	1	yes	characteristic
348	C₂×C₄	#14579208	1	1	yes	characteristic
349	C₈	—	2	4	no	—
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	316	4	no	—
353	C₈	#14579119	1	2	yes	—
354	C₈	#14579119	1	2	yes	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	316	2	no	—
358	C₂²	—	316	2	no	—
359	C₂²	—	316	2	no	—
360	C₂²	C₂×C₄×D₃₁₆	1	1	yes	characteristic
361	C₂²	#19099278	1	1	yes	characteristic
362	C₂²	#19099286	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	316	2	no	—
368	C₄	—	316	2	no	—
369	C₄	—	316	2	no	—
370	C₄	#19098686	1	2	yes	—
371	C₄	C₂×#14579119	1	1	yes	characteristic
372	C₄	#19099043	1	1	yes	characteristic
373	C₄	#19099217	1	1	yes	characteristic
374	C₄	#19099235	1	1	yes	characteristic
375	C₂	—	2	1	no	—
376	C₂	—	2	1	no	—
377	C₂	—	632	2	no	—
378	C₂	#23766347	1	1	yes	characteristic
379	C₂	C₂×#19098686	1	1	yes	characteristic
380	C₂	#23772318	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₆
commutator	C₂×C₄×C₇₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1770	C₂
socle	C₂²×C₇₉	C₂×C₄×D₄
prime radical	C₂×#14579183	C₂²
cosocle	C₂²×C₄×C₇₉	C₂⁴
hypercenter	#1770	D₇₉
hypocenter	C₇₉	#22803
LCS	G ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄⋊C₈ ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #23772065	C₂	semidirect
4. #23764049	C₂	semidirect
5. #23763572	C₂	semidirect
6. #23763421	C₂	semidirect
7. #23763284	C₂	semidirect
8. C₇₉×#1770	C₂	semidirect
11. C₄×Dic₃₁₆	C₄	semidirect
12. C₄×Dic₃₁₆	C₄	semidirect
13. C₄×#11929613	C₄	semidirect
14. C₄×#11929613	C₄	semidirect
15. C₄×D₆₃₂	C₄	semidirect
16. C₄×D₆₃₂	C₄	semidirect
198. #1770	D₇₉	semidirect
199. C₇₉	#22803	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14579183	2
#23772065	2
#23764293	2
#23764049	2
#23763572	2
#23763421	2
#23763284	2
C₇₉×#1770	2
#22803	79