Subgroups — #34333406

Order 20224.

620 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098839	C₂	1	1	yes	maximal, characteristic
2	#23772269	C₂	1	1	yes	maximal, characteristic, complemented
3	#23764865	C₂	1	2	yes	maximal, complemented
4	#23764682	C₂	1	2	yes	maximal
5	#23764313	C₂	1	1	yes	maximal, characteristic, complemented
6	#23763893	C₂	1	2	yes	maximal
7	#23763525	C₂	1	1	yes	maximal, characteristic, complemented
8	#23763513	C₂	1	2	yes	maximal, complemented
9	#23763455	C₂	1	1	yes	maximal, characteristic
10	#23763201	C₂	1	1	yes	maximal, characteristic
11	C₇₉×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579194	C₂²	1	2	yes	—
13	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
14	C₂×#14579167	C₂²	1	1	yes	characteristic
15	#19099040	C₂²	1	2	yes	—
16	#19098839	—	2	2	no	—
17	#19098839	—	2	2	no	—
18	#19098671	—	2	2	no	—
19	C₄×Dic₃₁₆	—	2	2	no	—
20	#19098601	C₂²	1	1	yes	characteristic
21	#19098579	C₂²	1	2	yes	—
22	#19098573	C₂²	1	2	yes	—
23	#19098573	C₂²	1	2	yes	—
24	#19098545	C₂²	1	1	yes	characteristic
25	C₂×#14579079	C₂²	1	2	yes	—
26	C₂×#14579079	C₂²	1	2	yes	—
27	#19098536	C₂²	1	1	yes	characteristic
28	#19098518	C₂²	1	1	yes	characteristic
29	#19098335	C₂²	1	1	yes	characteristic
30	#19098335	C₂²	1	2	yes	—
31	#19098334	C₂²	1	2	yes	—
32	#19098333	C₂²	1	1	yes	characteristic
33	#19098320	C₂²	1	2	yes	—
34	#430×C₇₉	C₂²	1	2	yes	—
35	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
36	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
37	#405×C₇₉	C₂²	1	1	yes	characteristic
38	#344×C₇₉	C₂²	1	2	yes	—
39	C₂×#11929624	C₂³	1	1	yes	characteristic
40	C₂²×Dic₁₅₈	C₂³	1	2	yes	—
41	#14579194	—	2	4	no	—
42	#14579194	—	4	2	no	—
43	C₂×Dic₃₁₆	—	2	1	no	—
44	C₂×Dic₃₁₆	—	2	1	no	—
45	C₂×Dic₃₁₆	—	2	1	no	—
46	C₂×Dic₃₁₆	—	2	1	no	—
47	C₂×Dic₃₁₆	—	2	2	no	—
48	C₂×Dic₃₁₆	D₄	1	2	yes	—
49	C₂×Dic₃₁₆	D₄	1	2	yes	—
50	C₂×#11929613	—	2	2	no	—
51	#14579167	—	2	1	no	—
52	#14579167	—	2	1	no	—
53	#14579167	—	2	1	no	—
54	#14579167	—	2	1	no	—
55	C₂×#11929608	C₂³	1	1	yes	characteristic
56	C₂×#11929608	C₂³	1	1	yes	characteristic
57	C₂×#11929607	—	2	1	no	—
58	C₂×C₈×D₇₉	—	2	1	no	—
59	C₄×Dic₁₅₈	—	2	4	no	—
60	#14579126	—	4	2	no	—
61	#14579120	C₂³	1	1	yes	characteristic
62	#14579120	C₂³	1	2	yes	—
63	#14579117	—	2	4	no	—
64	#14579095	—	2	1	no	—
65	#14579093	—	2	1	no	—
66	#14579087	—	4	2	no	—
67	#14579086	—	2	4	no	—
68	#14579086	—	4	2	no	—
69	#14579079	—	2	4	no	—
70	#14579079	—	2	1	no	—
71	#14579079	—	4	2	no	—
72	#14579077	—	2	1	no	—
73	#14579075	—	2	2	no	—
74	#14579071	—	2	1	no	—
75	C₈×Dic₇₉	—	2	1	no	—
76	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
77	C₂×M₁₆×C₇₉	C₂³	1	2	yes	—
78	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
80	#117×C₇₉	C₂³	1	1	yes	characteristic
81	#108×C₇₉	—	2	1	no	—
82	#108×C₇₉	—	2	1	no	—
83	#108×C₇₉	—	2	1	no	—
84	#108×C₇₉	—	2	1	no	—
85	#108×C₇₉	—	4	2	no	—
86	#97×C₇₉	—	2	1	no	—
87	C₄×C₈×C₇₉	—	2	1	no	—
88	#11929624	—	2	1	no	—
89	#11929624	—	2	1	no	—
90	#11929624	—	2	1	no	—
91	#11929624	—	2	1	no	—
92	C₂×Dic₁₅₈	—	2	2	no	—
93	C₂×Dic₁₅₈	—	2	2	no	—
94	C₂×Dic₁₅₈	—	2	2	no	—
95	C₂×Dic₁₅₈	—	2	2	no	—
96	C₂×Dic₁₅₈	—	2	2	no	—
97	C₂×Dic₁₅₈	C₂×D₄	1	4	yes	—
98	C₂×Dic₁₅₈	#41	1	1	yes	characteristic
99	C₂×D₃₁₆	#41	1	1	yes	characteristic
100	C₂×C₂²⋊D₇₉	—	2	1	no	—
101	C₂²×Dic₇₉	—	2	2	no	—
102	C₂×C₄×D₇₉	—	2	1	no	—
103	Dic₃₁₆	—	2	2	no	—
104	Dic₃₁₆	—	2	2	no	—
105	Dic₃₁₆	—	2	2	no	—
106	Dic₃₁₆	—	2	2	no	—
107	Dic₃₁₆	—	4	4	no	—
108	#11929613	—	4	4	no	—
109	#11929608	—	2	1	no	—
110	#11929608	—	2	1	no	—
111	#11929608	—	2	1	no	—
112	#11929608	—	2	1	no	—
113	#11929607	—	4	1	no	—
114	#11929607	—	4	1	no	—
115	C₂×C₇₉⋊C₈	—	2	1	no	—
116	C₂×C₇₉⋊C₈	—	2	1	no	—
117	C₈×D₇₉	—	4	1	no	—
118	C₈×D₇₉	—	4	1	no	—
119	C₄⋊Dic₇₉	—	2	2	no	—
120	C₄⋊Dic₇₉	—	2	1	no	—
121	Dic₇₉⋊C₄	—	4	4	no	—
122	C₂²⋊Dic₇₉	—	2	1	no	—
123	C₂²⋊Dic₇₉	—	2	2	no	—
124	C₄×Dic₇₉	—	2	1	no	—
125	C₄×Dic₇₉	—	2	2	no	—
126	C₄×Dic₇₉	—	4	4	no	—
127	C₂²×C₄×C₇₉	C₂⁴	1	1	yes	characteristic
128	M₁₆×C₇₉	—	2	2	no	—
129	M₁₆×C₇₉	—	2	2	no	—
130	M₁₆×C₇₉	—	2	1	no	—
131	M₁₆×C₇₉	—	2	1	no	—
132	M₁₆×C₇₉	—	2	2	no	—
133	M₁₆×C₇₉	—	2	2	no	—
134	C₂×C₈×C₇₉	—	2	1	no	—
135	C₂×C₈×C₇₉	—	2	2	no	—
136	C₂×C₈×C₇₉	—	2	2	no	—
137	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
140	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
141	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
142	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
143	C₄⋊C₄×C₇₉	—	2	1	no	—
144	C₂²⋊C₄×C₇₉	#41	1	2	yes	—
145	C₄²×C₇₉	—	2	1	no	—
146	C₂³×C₇₉	C₂×#41	1	1	yes	characteristic
147	C₂²×D₇₉	—	2	1	no	—
148	C₂×C₄×C₇₉	—	2	2	no	—
149	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₇₉	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂×C₄×C₇₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²⋊D₇₉	—	4	1	no	—
156	C₂²⋊D₇₉	—	4	1	no	—
157	C₂×Dic₇₉	—	2	1	no	—
158	C₂×Dic₇₉	—	2	4	no	—
159	C₂×Dic₇₉	—	4	2	no	—
160	C₂×Dic₇₉	—	4	2	no	—
161	C₂×Dic₇₉	—	4	2	no	—
162	C₂×Dic₇₉	—	4	1	no	—
163	D₃₁₆	—	2	1	no	—
164	D₃₁₆	—	2	1	no	—
165	C₄×D₇₉	—	4	1	no	—
166	C₄×D₇₉	—	4	1	no	—
167	Dic₁₅₈	—	2	4	no	—
168	Dic₁₅₈	—	2	4	no	—
169	Dic₁₅₈	—	2	1	no	—
170	Dic₁₅₈	—	2	1	no	—
171	Dic₁₅₈	—	4	2	no	—
172	Dic₁₅₈	—	4	2	no	—
173	C₈×C₇₉	—	4	2	no	—
174	C₈×C₇₉	—	4	2	no	—
175	C₈×C₇₉	—	4	1	no	—
176	C₈×C₇₉	#123	1	1	yes	characteristic
177	C₈×C₇₉	#123	1	1	yes	characteristic
178	C₈×C₇₉	#123	1	1	yes	characteristic
179	C₈×C₇₉	#123	1	1	yes	characteristic
180	C₇₉⋊C₈	—	4	1	no	—
181	C₇₉⋊C₈	—	4	1	no	—
182	C₂²×C₇₉	—	4	1	no	—
183	C₂²×C₇₉	#529	1	1	yes	characteristic
184	C₂²×C₇₉	#546	1	1	yes	characteristic
185	C₂²×C₇₉	#555	1	1	yes	characteristic
186	C₂×D₇₉	—	4	1	no	—
187	C₂×D₇₉	—	8	1	no	—
188	C₄×C₇₉	—	4	2	no	—
189	C₄×C₇₉	C₂×#123	1	1	yes	characteristic
190	C₄×C₇₉	#540	1	1	yes	characteristic
191	C₄×C₇₉	#546	1	1	yes	characteristic
192	C₄×C₇₉	#553	1	1	yes	characteristic
193	Dic₇₉	—	4	1	no	—
194	Dic₇₉	—	4	4	no	—
195	Dic₇₉	—	8	2	no	—
196	Dic₇₉	—	8	1	no	—
197	#24989	—	79	1	no	maximal, Sylow
198	C₂×C₇₉	—	2	1	no	—
199	C₂×C₇₉	—	2	1	no	—
200	C₂×C₇₉	—	2	1	no	—
201	C₂×C₇₉	#2420	1	1	yes	characteristic
202	D₇₉	—	8	1	no	—
203	C₂×#473	—	79	1	no	—
204	#2950	—	79	1	no	—
205	#2082	—	79	2	no	—
206	#2049	—	79	2	no	—
207	#1979	—	79	1	no	—
208	#1953	D₇₉	1	1	yes	characteristic, complemented
209	#1905	—	79	2	no	—
210	#1833	—	79	1	no	—
211	#1830	—	79	2	no	—
212	#1814	—	79	1	no	—
213	#1757	—	79	1	no	—
214	C₇₉	#24989	1	1	yes	characteristic, complemented, Sylow
215	C₂×#137	—	79	2	no	—
216	C₂²×Q₁₆	—	79	1	no	—
217	C₂×#131	—	79	1	no	—
218	#519	—	79	2	no	—
219	#473	—	158	2	no	—
220	#473	—	158	2	no	—
221	#441	—	158	2	no	—
222	C₄×Q₁₆	—	158	2	no	—
223	#430	C₂×D₇₉	1	2	yes	—
224	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
225	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
226	#427	—	79	1	no	—
227	#421	—	79	2	no	—
228	#419	—	79	2	no	—
229	#419	—	79	2	no	—
230	#413	—	79	1	no	—
231	C₂×#101	—	79	2	no	—
232	C₂×#101	—	79	2	no	—
233	#410	—	79	1	no	—
234	#405	—	79	1	no	—
235	#405	C₂×D₇₉	1	1	yes	characteristic
236	#344	—	79	2	no	—
237	#344	—	79	2	no	—
238	#344	—	79	1	no	—
239	#344	—	79	1	no	—
240	#344	C₂×D₇₉	1	2	yes	—
241	H(3,ℤ/4)	—	79	2	no	—
242	C₂×#41	—	79	1	no	—
243	C₂²×Q₈	—	79	2	no	—
244	#137	—	316	2	no	—
245	#137	—	158	4	no	—
246	C₂×Q₁₆	—	79	2	no	—
247	C₂×Q₁₆	—	79	2	no	—
248	C₂×Q₁₆	—	158	2	no	—
249	C₂×Q₁₆	—	158	1	no	—
250	C₂×Q₁₆	—	158	1	no	—
251	C₂×Q₁₆	—	158	1	no	—
252	C₂×Q₁₆	—	158	1	no	—
253	C₂×SD₁₆	—	158	2	no	—
254	#131	—	158	1	no	—
255	#131	—	158	1	no	—
256	#131	—	158	1	no	—
257	#131	—	158	1	no	—
258	C₂×M₁₆	—	79	1	no	—
259	C₂×M₁₆	—	79	1	no	—
260	C₂×M₁₆	—	158	1	no	—
261	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
262	C₂×M₁₆	C₂²×D₇₉	1	2	yes	—
263	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
264	C₂²×C₈	—	158	1	no	—
265	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
266	C₄×Q₈	—	316	2	no	—
267	C₄×Q₈	—	158	4	no	—
268	#117	—	79	2	no	—
269	#117	—	79	1	no	—
270	#117	—	158	4	no	—
271	#117	C₂²×D₇₉	1	1	yes	characteristic
272	#108	—	2	1	no	—
273	#108	—	2	1	no	—
274	#108	—	2	1	no	—
275	#108	—	2	1	no	—
276	#108	—	4	2	no	—
277	#107	—	158	1	no	—
278	#106	—	158	1	no	—
279	#104	—	316	2	no	—
280	#103	—	316	2	no	—
281	#103	—	158	4	no	—
282	#101	—	316	2	no	—
283	#101	—	158	4	no	—
284	#101	—	158	1	no	—
285	#100	—	158	1	no	—
286	#99	—	158	2	no	—
287	#97	—	2	1	no	—
288	#97	—	158	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	158	1	no	—
291	#41	—	158	1	no	—
292	#41	—	158	1	no	—
293	#41	—	158	1	no	—
294	#41	—	158	1	no	—
295	C₂×Q₈	—	79	1	no	—
296	C₂×Q₈	—	79	4	no	—
297	C₂×Q₈	—	158	2	no	—
298	C₂×Q₈	—	158	2	no	—
299	C₂×Q₈	—	158	2	no	—
300	C₂×Q₈	—	158	2	no	—
301	C₂×Q₈	—	158	2	no	—
302	C₂×D₄	—	79	1	no	—
303	C₂×D₄	—	158	1	no	—
304	C₂²×C₄	—	158	2	no	—
305	C₂²×C₄	—	158	1	no	—
306	C₂²×C₄	C₂³×D₇₉	1	1	yes	characteristic
307	Q₁₆	—	316	4	no	—
308	Q₁₆	—	158	2	no	—
309	Q₁₆	—	158	2	no	—
310	Q₁₆	—	158	2	no	—
311	Q₁₆	—	158	2	no	—
312	SD₁₆	—	316	4	no	—
313	M₁₆	—	2	1	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	2	2	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	2	no	—
319	M₁₆	—	316	1	no	—
320	M₁₆	—	316	1	no	—
321	M₁₆	—	158	1	no	—
322	M₁₆	—	158	1	no	—
323	M₁₆	—	158	1	no	—
324	M₁₆	—	158	1	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	316	1	no	—
329	C₂×C₈	—	316	1	no	—
330	C₂×C₈	—	158	1	no	—
331	C₂×C₈	—	158	1	no	—
332	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
333	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
334	C₂×C₈	#11929624	1	1	yes	characteristic
335	C₂×C₈	#11929624	1	1	yes	characteristic
336	C₂×C₈	#11929625	1	1	yes	characteristic
337	C₂×C₈	#11929625	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	316	4	no	—
340	C₄⋊C₄	—	158	1	no	—
341	C₄⋊C₄	—	158	2	no	—
342	C₂²⋊C₄	—	158	1	no	—
343	C₂²⋊C₄	—	158	2	no	—
344	C₂²⋊C₄	#11929625	1	2	yes	—
345	C₄²	—	2	1	no	—
346	C₄²	—	316	4	no	—
347	C₄²	—	158	1	no	—
348	C₄²	—	158	2	no	—
349	C₂³	—	158	1	no	—
350	C₂³	C₂×#11929625	1	1	yes	characteristic
351	Q₈	—	316	2	no	—
352	Q₈	—	316	2	no	—
353	Q₈	—	158	4	no	—
354	Q₈	—	158	1	no	—
355	Q₈	—	158	1	no	—
356	Q₈	—	158	4	no	—
357	D₄	—	316	1	no	—
358	D₄	—	316	1	no	—
359	D₄	—	158	1	no	—
360	D₄	—	158	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	316	2	no	—
363	C₂×C₄	—	316	2	no	—
364	C₂×C₄	—	316	1	no	—
365	C₂×C₄	—	316	1	no	—
366	C₂×C₄	—	316	2	no	—
367	C₂×C₄	—	316	1	no	—
368	C₂×C₄	—	158	4	no	—
369	C₂×C₄	—	158	1	no	—
370	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
371	C₂×C₄	#41×D₇₉	1	1	yes	characteristic
372	C₂×C₄	C₂×#11929624	1	1	yes	characteristic
373	C₂×C₄	C₂×#11929625	1	1	yes	characteristic
374	C₂×C₄	#14579206	1	1	yes	characteristic
375	C₂×C₄	#14579207	1	1	yes	characteristic
376	C₈	—	4	2	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	1	no	—
379	C₈	—	316	1	no	—
380	C₈	—	316	1	no	—
381	C₈	#14579146	1	1	yes	characteristic
382	C₈	#14579146	1	1	yes	characteristic
383	C₈	#14579146	1	1	yes	characteristic
384	C₈	#14579146	1	1	yes	characteristic
385	C₂²	—	4	1	no	—
386	C₂²	—	316	1	no	—
387	C₂²	—	632	1	no	—
388	C₂²	#19099100	1	1	yes	characteristic
389	C₂²	#19099204	1	1	yes	characteristic
390	C₂²	#19099279	1	1	yes	characteristic
391	C₄	—	4	2	no	—
392	C₄	—	316	1	no	—
393	C₄	—	316	4	no	—
394	C₄	—	632	2	no	—
395	C₄	—	632	1	no	—
396	C₄	C₂×#14579146	1	1	yes	characteristic
397	C₄	#19099164	1	1	yes	characteristic
398	C₄	#19099210	1	1	yes	characteristic
399	C₄	#19099266	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	632	1	no	—
404	C₂	#23766893	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23766893
commutator, cosocle	C₂²×C₄×C₇₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₉
Fitting	C₇₉×#1953	C₂
socle	C₂×C₇₉	#2420
prime radical	C₂×#11929624	C₂³
hypercenter	#1953	D₇₉
hypocenter	C₇₉	#24989
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1953

Complements

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

N	H ≅ G/N	type
2. #23772269	C₂	semidirect
3. #23764865	C₂	semidirect
5. #23764313	C₂	semidirect
7. #23763525	C₂	semidirect
8. #23763513	C₂	semidirect
11. C₇₉×#1953	C₂	semidirect
208. #1953	D₇₉	semidirect
214. C₇₉	#24989	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098839	2
#23772269	2
#23764865	2
#23764682	2
#23764313	2
#23763893	2
#23763525	2
#23763513	2
#23763455	2
#23763201	2
C₇₉×#1953	2
#24989	79