Subgroups — #34441727

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23776536	C₂	1	1	yes	maximal, characteristic
2	#23772316	C₂	1	1	yes	maximal, characteristic, complemented
3	#23772283	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765297	C₂	1	4	yes	maximal, complemented
5	#23765290	C₂	1	4	yes	maximal, complemented
6	#23765147	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098372	C₂	1	1	yes	maximal, characteristic, complemented
8	#23765004	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#365×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14579183	C₂²	1	1	yes	characteristic
11	C₂×#14579183	C₂²	1	1	yes	characteristic
12	#19099329	—	2	1	no	—
13	#19099044	—	2	1	no	—
14	#19099006	C₄	1	2	yes	complemented
15	#19099006	C₂²	1	2	yes	—
16	#19099001	C₄	1	2	yes	complemented
17	#19099001	C₂²	1	2	yes	—
18	#19098995	C₄	1	4	yes	complemented
19	C₂×#14579101	C₂²	1	1	yes	characteristic
20	#19098684	C₂²	1	4	yes	—
21	#19098679	—	2	2	no	—
22	#19098674	C₂²	1	2	yes	complemented
23	#19098669	C₂²	1	2	yes	complemented
24	#19098650	—	2	2	no	—
25	#19098579	—	2	1	no	—
26	#19098562	—	2	1	no	—
27	#19098518	C₂²	1	1	yes	characteristic
28	#19098378	C₂²	1	2	yes	complemented
29	#19098375	C₂²	1	2	yes	complemented
30	#19098372	C₂²	1	4	yes	—
31	#19098340	C₂²	1	4	yes	complemented
32	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
33	C₂×#108×C₇₉	C₂²	1	1	yes	characteristic
34	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
35	#365×C₇₉	C₂²	1	4	yes	complemented
36	C₂×#11929624	—	2	1	no	—
37	C₂×#11929624	—	2	1	no	—
38	#14579183	—	2	1	no	—
39	#14579183	—	2	1	no	—
40	#14579183	—	2	1	no	—
41	#14579183	—	2	1	no	—
42	#14579183	C₂×C₄	1	2	yes	—
43	#14579183	C₂×C₄	1	2	yes	—
44	#14579183	C₂×C₄	1	2	yes	complemented
45	#14579183	C₂³	1	2	yes	—
46	C₂×Dic₃₁₆	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
48	C₂×#11929613	—	2	1	no	—
49	C₂×#11929613	—	2	1	no	—
50	C₂×D₆₃₂	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
52	#14579169	—	2	4	no	—
53	#14579167	—	4	2	no	—
54	C₂×#11929608	D₄	1	1	yes	characteristic
55	M₁₆×D₇₉	—	4	2	no	—
56	#14579126	—	2	2	no	—
57	#14579123	—	2	2	no	—
58	C₄×C₂²⋊D₇₉	—	4	2	no	—
59	#14579120	D₄	1	1	yes	characteristic
60	#14579119	—	4	1	no	—
61	C₄⋊C₄×D₇₉	—	4	1	no	—
62	Dic₆₃₂	C₂×C₄	1	4	yes	complemented
63	#14579106	C₂×C₄	1	4	yes	complemented
64	#14579106	C₂×C₄	1	4	yes	complemented
65	D₁₂₆₄	C₂×C₄	1	4	yes	complemented
66	#14579101	—	2	1	no	—
67	#14579101	C₂³	1	2	yes	—
68	C₂×C₇₉⋊C₁₆	—	2	1	no	—
69	#14579087	—	2	4	no	—
70	#14579084	—	2	2	no	—
71	#14579081	—	2	2	no	—
72	#14579071	C₂³	1	2	yes	—
73	C₈×Dic₇₉	D₄	1	2	yes	—
74	C₂×M₁₆×C₇₉	—	2	1	no	—
75	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
77	M₃₂×C₇₉	C₂×C₄	1	2	yes	complemented
78	M₃₂×C₇₉	C₂³	1	2	yes	—
79	C₂×C₁₆×C₇₉	C₂×C₄	1	2	yes	complemented
80	#108×C₇₉	—	2	1	no	—
81	#108×C₇₉	C₂³	1	2	yes	—
82	#106×C₇₉	—	2	1	no	—
83	#106×C₇₉	C₂³	1	2	yes	—
84	#11929624	—	2	2	no	—
85	#11929624	—	2	2	no	—
86	#11929624	—	2	2	no	—
87	#11929624	—	2	2	no	—
88	#11929624	—	4	1	no	—
89	#11929624	—	4	1	no	—
90	#11929624	—	4	1	no	—
91	#11929624	—	4	1	no	—
92	C₂×Dic₁₅₈	—	2	1	no	—
93	C₂×Dic₁₅₈	—	2	1	no	—
94	C₂×D₃₁₆	—	2	1	no	—
95	C₂×D₃₁₆	—	2	1	no	—
96	C₂×C₂²⋊D₇₉	—	4	1	no	—
97	C₂×C₂²⋊D₇₉	—	4	1	no	—
98	C₂×C₄×D₇₉	—	4	2	no	—
99	C₂×C₄×D₇₉	—	4	1	no	—
100	C₂×C₄×D₇₉	—	4	1	no	—
101	Dic₃₁₆	—	2	1	no	—
102	Dic₃₁₆	—	2	1	no	—
103	Dic₃₁₆	C₂²×C₄	1	2	yes	—
104	Dic₃₁₆	C₂²×C₄	1	2	yes	—
105	#11929613	—	2	2	no	—
106	#11929613	—	2	2	no	—
107	#11929613	—	2	2	no	—
108	#11929613	—	2	2	no	—
109	D₆₃₂	—	2	1	no	—
110	D₆₃₂	—	2	1	no	—
111	D₆₃₂	C₂²×C₄	1	2	yes	—
112	D₆₃₂	C₂²×C₄	1	2	yes	—
113	#11929608	—	2	1	no	—
114	#11929608	—	2	1	no	—
115	#11929607	—	4	4	no	—
116	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
117	C₈×D₇₉	—	4	4	no	—
118	C₄⋊Dic₇₉	—	2	1	no	—
119	Dic₇₉⋊C₄	—	4	2	no	—
120	C₂²⋊Dic₇₉	—	2	1	no	—
121	(C₂×D₇₉)⋊C₄	—	4	2	no	—
122	C₄×Dic₇₉	—	4	2	no	—
123	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
124	C₇₉⋊C₁₆	—	2	2	no	—
125	C₂²×C₄×C₇₉	—	4	1	no	—
126	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
127	M₁₆×C₇₉	—	2	2	no	—
128	M₁₆×C₇₉	—	4	1	no	—
129	C₂×C₈×C₇₉	—	4	1	no	—
130	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
134	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
135	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
136	C₄⋊C₄×C₇₉	—	2	2	no	—
137	C₄⋊C₄×C₇₉	—	4	1	no	—
138	C₁₆×C₇₉	C₂²×C₄	1	4	yes	—
139	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂²×D₇₉	—	4	1	no	—
141	C₂²×D₇₉	—	4	1	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	—	16	1	no	—
144	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₇₉	—	4	2	no	—
151	C₂²⋊D₇₉	—	4	2	no	—
152	C₂²⋊D₇₉	—	4	2	no	—
153	C₂²⋊D₇₉	—	4	2	no	—
154	C₂×Dic₇₉	—	2	2	no	—
155	C₂×Dic₇₉	—	4	1	no	—
156	C₂×Dic₇₉	—	4	1	no	—
157	D₃₁₆	—	2	2	no	—
158	D₃₁₆	—	2	2	no	—
159	D₃₁₆	—	4	1	no	—
160	D₃₁₆	—	4	1	no	—
161	C₄×D₇₉	—	4	2	no	—
162	C₄×D₇₉	—	4	2	no	—
163	C₄×D₇₉	—	4	2	no	—
164	C₄×D₇₉	—	4	2	no	—
165	C₄×D₇₉	—	16	2	no	—
166	Dic₁₅₈	—	2	2	no	—
167	Dic₁₅₈	—	2	2	no	—
168	Dic₁₅₈	—	4	1	no	—
169	Dic₁₅₈	—	4	1	no	—
170	C₈×C₇₉	—	4	2	no	—
171	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
172	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
173	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
174	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
175	C₇₉⋊C₈	—	2	2	no	—
176	C₂²×C₇₉	—	4	1	no	—
177	C₂²×C₇₉	#442	1	1	yes	characteristic
178	C₂²×C₇₉	#442	1	1	yes	characteristic
179	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₂×D₇₉	—	4	2	no	—
181	C₂×D₇₉	—	4	2	no	—
182	C₂×D₇₉	—	8	1	no	—
183	C₂×D₇₉	—	8	1	no	—
184	C₂×D₇₉	—	16	1	no	—
185	C₄×C₇₉	—	8	2	no	—
186	C₄×C₇₉	#442	1	1	yes	characteristic
187	C₄×C₇₉	#442	1	1	yes	characteristic
188	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	Dic₇₉	—	4	2	no	—
191	Dic₇₉	—	4	2	no	—
192	Dic₇₉	—	4	2	no	—
193	#33469	—	79	1	no	maximal, Sylow
194	C₂×C₇₉	—	2	1	no	—
195	C₂×C₇₉	—	2	1	no	—
196	C₂×C₇₉	—	2	1	no	—
197	C₂×C₇₉	C₂×#442	1	1	yes	characteristic
198	D₇₉	—	8	2	no	—
199	D₇₉	—	8	2	no	—
200	#3414	—	79	1	no	—
201	#2957	—	79	1	no	—
202	#2952	—	79	1	no	—
203	AGL(1,ℤ/16)	—	79	4	no	—
204	#2180	—	79	4	no	—
205	C₂×#365	D₇₉	1	1	yes	characteristic, complemented
206	#2148	—	79	1	no	—
207	C₂×#360	—	79	1	no	—
208	#2110	—	79	1	no	—
209	C₇₉	#33469	1	1	yes	characteristic, complemented, Sylow
210	C₂×#135	—	79	1	no	—
211	C₂×#135	—	79	1	no	—
212	#568	—	158	1	no	—
213	#520	—	158	1	no	—
214	#510	—	79	2	no	—
215	#510	—	79	2	no	—
216	#509	—	79	2	no	—
217	#509	—	79	2	no	—
218	#508	—	79	4	no	—
219	C₂×M₃₂	—	79	1	no	—
220	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
221	#444	—	79	4	no	—
222	#443	—	158	2	no	—
223	#442	—	79	2	no	—
224	#441	—	79	2	no	—
225	C₄×SD₁₆	—	158	2	no	—
226	C₂×#108	C₂×D₇₉	1	1	yes	characteristic
227	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
228	#421	—	158	1	no	—
229	#417	—	158	1	no	—
230	#405	—	79	1	no	—
231	#365	C₂×D₇₉	1	4	yes	complemented
232	#362	—	79	2	no	—
233	#361	—	79	2	no	—
234	#360	—	79	4	no	—
235	#347	—	79	4	no	—
236	C₂×#41	—	158	1	no	—
237	C₂×#41	—	158	1	no	—
238	#135	—	79	2	no	—
239	#135	—	79	2	no	—
240	#135	—	79	2	no	—
241	#135	—	79	2	no	—
242	#135	—	158	1	no	—
243	#135	—	158	1	no	—
244	#135	—	158	1	no	—
245	#135	—	158	1	no	—
246	C₂×Q₁₆	—	79	1	no	—
247	C₂×Q₁₆	—	79	1	no	—
248	C₂×SD₁₆	—	158	1	no	—
249	C₂×SD₁₆	—	158	1	no	—
250	C₂×D₈	—	79	1	no	—
251	C₂×D₈	—	79	1	no	—
252	#131	—	316	2	no	—
253	#131	—	158	4	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	316	2	no	—
256	C₂×M₁₆	—	79	1	no	—
257	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
258	C₄×Q₈	—	158	2	no	—
259	C₄×D₄	—	316	2	no	—
260	C₄×D₄	—	158	2	no	—
261	#117	—	316	1	no	—
262	#117	—	79	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	316	1	no	—
265	Q₃₂	—	79	4	no	—
266	SD₃₂	—	79	4	no	—
267	SD₃₂	—	79	4	no	—
268	D₁₆	—	79	4	no	—
269	M₃₂	—	79	2	no	—
270	M₃₂	—	158	1	no	—
271	M₃₂	C₄×D₇₉	1	2	yes	complemented
272	M₃₂	C₂²×D₇₉	1	2	yes	—
273	C₂×C₁₆	—	158	1	no	—
274	C₂×C₁₆	C₄×D₇₉	1	2	yes	complemented
275	#108	—	2	1	no	—
276	#108	C₂²×D₇₉	1	2	yes	—
277	#106	—	2	1	no	—
278	#106	C₂²×D₇₉	1	2	yes	—
279	#104	—	158	4	no	—
280	#103	—	158	2	no	—
281	#102	—	158	2	no	—
282	#97	—	79	2	no	—
283	C₄×C₈	—	79	2	no	—
284	#41	—	316	1	no	—
285	#41	—	316	1	no	—
286	#41	—	316	1	no	—
287	#41	—	316	1	no	—
288	#41	—	158	2	no	—
289	#41	—	158	2	no	—
290	#41	—	158	2	no	—
291	#41	—	158	2	no	—
292	C₂×Q₈	—	158	1	no	—
293	C₂×Q₈	—	158	1	no	—
294	C₂×D₄	—	316	1	no	—
295	C₂×D₄	—	316	1	no	—
296	C₂×D₄	—	158	1	no	—
297	C₂×D₄	—	158	1	no	—
298	C₂²×C₄	—	4	1	no	—
299	C₂²×C₄	—	316	2	no	—
300	C₂²×C₄	—	316	1	no	—
301	C₂²×C₄	—	316	1	no	—
302	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
303	Q₁₆	—	79	2	no	—
304	Q₁₆	—	79	2	no	—
305	Q₁₆	—	158	1	no	—
306	Q₁₆	—	158	1	no	—
307	SD₁₆	—	158	2	no	—
308	SD₁₆	—	158	2	no	—
309	SD₁₆	—	158	2	no	—
310	SD₁₆	—	158	2	no	—
311	D₈	—	79	2	no	—
312	D₈	—	79	2	no	—
313	D₈	—	158	1	no	—
314	D₈	—	158	1	no	—
315	M₁₆	—	2	2	no	—
316	M₁₆	—	4	1	no	—
317	M₁₆	—	316	4	no	—
318	M₁₆	—	158	1	no	—
319	M₁₆	—	158	1	no	—
320	C₂×C₈	—	4	1	no	—
321	C₂×C₈	—	316	4	no	—
322	C₂×C₈	—	79	2	no	—
323	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
324	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
325	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
326	C₂×C₈	#11929625	1	1	yes	characteristic
327	C₂×C₈	#11929625	1	1	yes	characteristic
328	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	316	2	no	—
332	C₄⋊C₄	—	158	1	no	—
333	C₂²⋊C₄	—	316	2	no	—
334	C₂²⋊C₄	—	158	1	no	—
335	C₄²	—	316	2	no	—
336	C₄²	—	79	2	no	—
337	C₁₆	—	158	2	no	—
338	C₁₆	C₂×C₄×D₇₉	1	4	yes	—
339	C₂³	—	316	1	no	—
340	C₂³	—	316	1	no	—
341	C₂³	#14579188	1	1	yes	characteristic
342	Q₈	—	316	1	no	—
343	Q₈	—	316	1	no	—
344	Q₈	—	158	2	no	—
345	Q₈	—	158	2	no	—
346	D₄	—	316	2	no	—
347	D₄	—	316	2	no	—
348	D₄	—	316	2	no	—
349	D₄	—	316	2	no	—
350	D₄	—	316	1	no	—
351	D₄	—	316	1	no	—
352	D₄	—	158	2	no	—
353	D₄	—	158	2	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	16	1	no	—
356	C₂×C₄	—	316	1	no	—
357	C₂×C₄	—	316	1	no	—
358	C₂×C₄	—	316	2	no	—
359	C₂×C₄	—	316	2	no	—
360	C₂×C₄	—	316	2	no	—
361	C₂×C₄	—	316	2	no	—
362	C₂×C₄	—	158	2	no	—
363	C₂×C₄	—	1264	2	no	—
364	C₂×C₄	#14579123	1	1	yes	characteristic
365	C₂×C₄	#14579123	1	1	yes	characteristic
366	C₂×C₄	#14579188	1	1	yes	characteristic
367	C₂×C₄	#14579188	1	1	yes	characteristic
368	C₂×C₄	#14579188	1	1	yes	characteristic
369	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
370	C₈	—	4	2	no	—
371	C₈	—	158	2	no	—
372	C₈	#14579123	1	1	yes	characteristic
373	C₈	#14579123	1	1	yes	characteristic
374	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
375	C₈	C₂×#11929625	1	1	yes	characteristic
376	C₂²	—	4	1	no	—
377	C₂²	—	316	2	no	—
378	C₂²	—	316	2	no	—
379	C₂²	—	632	1	no	—
380	C₂²	—	632	1	no	—
381	C₂²	—	1264	1	no	—
382	C₂²	#19098674	1	1	yes	characteristic
383	C₂²	#19098674	1	1	yes	characteristic
384	C₂²	C₂×#14579188	1	1	yes	characteristic
385	C₄	—	8	2	no	—
386	C₄	—	316	2	no	—
387	C₄	—	316	2	no	—
388	C₄	—	316	2	no	—
389	C₄	#19098674	1	1	yes	characteristic
390	C₄	#19098674	1	1	yes	characteristic
391	C₄	C₂×#14579123	1	1	yes	characteristic
392	C₄	C₂×#14579188	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	632	2	no	—
397	C₂	—	632	2	no	—
398	C₂	C₂×#19098674	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19098674
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#365×C₇₉	C₂
socle	C₂×C₇₉	C₂×#442
prime radical	#23776536	C₂
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#365	D₇₉
hypocenter	C₇₉	#33469
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
2. #23772316	C₂	semidirect
3. #23772283	C₂	semidirect
4. #23765297	C₂	semidirect
5. #23765290	C₂	semidirect
6. #23765147	C₂	semidirect
7. C₂×#19098372	C₂	semidirect
8. #23765004	C₂	semidirect
9. C₂×#365×C₇₉	C₂	semidirect
14. #19099006	C₄	semidirect
16. #19099001	C₄	semidirect
18. #19098995	C₄	semidirect
22. #19098674	C₂²	semidirect
23. #19098669	C₂²	semidirect
28. #19098378	C₂²	semidirect
29. #19098375	C₂²	semidirect
31. #19098340	C₂²	semidirect
35. #365×C₇₉	C₂²	semidirect
44. #14579183	C₂×C₄	semidirect
46. C₂×Dic₃₁₆	C₂×C₄	semidirect
50. C₂×D₆₃₂	C₂×C₄	semidirect
62. Dic₆₃₂	C₂×C₄	semidirect
63. #14579106	C₂×C₄	semidirect
64. #14579106	C₂×C₄	semidirect
65. D₁₂₆₄	C₂×C₄	semidirect
77. M₃₂×C₇₉	C₂×C₄	semidirect
79. C₂×C₁₆×C₇₉	C₂×C₄	semidirect
205. C₂×#365	D₇₉	semidirect
209. C₇₉	#33469	semidirect
231. #365	C₂×D₇₉	semidirect
271. M₃₂	C₄×D₇₉	semidirect
274. C₂×C₁₆	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
#23776536	2
#23772316	2
#23772283	2
#23765297	2
#23765290	2
#23765147	2
C₂×#19098372	2
#23765004	2
C₂×#365×C₇₉	2
#33469	79