Subgroups — #34441616

Order 20224.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099004	C₂	1	1	yes	maximal, characteristic
2	#23772274	C₂	1	1	yes	maximal, characteristic
3	#23772268	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765268	C₂	1	2	yes	maximal, complemented
5	#23765267	C₂	1	2	yes	maximal, complemented
6	#23765256	C₂	1	2	yes	maximal, complemented
7	#23765249	C₂	1	2	yes	maximal, complemented
8	#23765186	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765108	C₂	1	1	yes	maximal, characteristic
10	#23764996	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#358×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	1	yes	characteristic
13	#19099044	—	2	1	no	—
14	#19099041	—	2	1	no	—
15	#19099004	C₄	1	8	yes	complemented
16	C₂×#14579107	C₂²	1	2	yes	—
17	C₂×#14579104	C₂²	1	2	yes	—
18	C₂×#14579101	C₂²	1	1	yes	characteristic
19	#19098670	—	2	2	no	—
20	#19098662	—	2	2	no	—
21	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
22	Q₁₆×Dic₇₉	C₂²	1	2	yes	—
23	#19098657	C₂²	1	2	yes	—
24	#19098645	C₂²	1	2	yes	—
25	#19098601	C₂²	1	1	yes	characteristic
26	#19098572	—	2	1	no	—
27	#19098562	—	2	1	no	—
28	#19098518	C₂²	1	1	yes	characteristic
29	#19098390	C₂²	1	2	yes	—
30	#19098388	C₂²	1	2	yes	—
31	#19098368	C₂²	1	1	yes	characteristic
32	#19098368	C₂²	1	1	yes	characteristic
33	#19098365	C₂²	1	1	yes	characteristic
34	#19098365	C₂²	1	1	yes	characteristic
35	#19098338	C₂²	1	2	yes	—
36	C₁₆×Dic₇₉	C₂²	1	2	yes	—
37	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
40	#358×C₇₉	C₂²	1	2	yes	—
41	#358×C₇₉	C₂²	1	2	yes	—
42	C₂×#11929624	—	2	1	no	—
43	#14579183	—	2	2	no	—
44	#14579183	C₂×C₄	1	4	yes	—
45	C₂×Dic₃₁₆	C₂³	1	1	yes	characteristic
46	C₂×#11929613	—	2	1	no	—
47	C₂×D₆₃₂	C₂³	1	1	yes	characteristic
48	C₂×#11929608	D₄	1	1	yes	characteristic
49	Q₈×Dic₇₉	—	2	2	no	—
50	Q₈×Dic₇₉	—	2	2	no	—
51	Q₈×Dic₇₉	—	4	1	no	—
52	Q₈×Dic₇₉	—	4	1	no	—
53	#14579126	—	2	2	no	—
54	#14579123	—	2	2	no	—
55	C₄×C₂²⋊D₇₉	—	4	2	no	—
56	#14579120	—	2	1	no	—
57	#14579120	—	4	2	no	—
58	#14579120	D₄	1	1	yes	characteristic
59	#14579119	—	4	1	no	—
60	C₄⋊C₄×D₇₉	—	4	1	no	—
61	#14579107	C₂×C₄	1	8	yes	complemented
62	#14579104	C₂×C₄	1	8	yes	complemented
63	#14579101	C₂×C₄	1	4	yes	complemented
64	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
65	#14579095	C₂³	1	1	yes	characteristic
66	#14579095	C₂³	1	1	yes	characteristic
67	#14579093	—	2	1	no	—
68	#14579085	—	2	2	no	—
69	#14579085	—	2	2	no	—
70	#14579084	—	2	2	no	—
71	#14579081	—	2	2	no	—
72	#14579071	D₄	1	2	yes	—
73	C₈×Dic₇₉	C₂³	1	2	yes	—
74	C₂²×Q₈×C₇₉	—	2	1	no	—
75	C₂×Q₁₆×C₇₉	—	2	1	no	—
76	C₂×Q₁₆×C₇₉	—	2	2	no	—
77	C₂×Q₁₆×C₇₉	C₂×C₄	1	4	yes	complemented
78	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
79	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
81	C₂×C₁₆×C₇₉	—	2	2	no	—
82	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
83	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
84	#107×C₇₉	—	2	1	no	—
85	#107×C₇₉	C₂³	1	2	yes	—
86	#11929624	—	2	4	no	—
87	#11929624	—	4	2	no	—
88	C₂×Dic₁₅₈	—	2	1	no	—
89	C₂×D₃₁₆	—	2	1	no	—
90	C₂×C₂²⋊D₇₉	—	4	1	no	—
91	C₂×C₄×D₇₉	—	4	2	no	—
92	C₂×C₄×D₇₉	—	4	1	no	—
93	Dic₃₁₆	—	2	1	no	—
94	Dic₃₁₆	C₂²×C₄	1	2	yes	—
95	#11929613	—	2	4	no	—
96	D₆₃₂	—	2	1	no	—
97	D₆₃₂	C₂²×C₄	1	2	yes	—
98	#11929608	—	2	2	no	—
99	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
100	C₄⋊Dic₇₉	—	2	1	no	—
101	C₄⋊Dic₇₉	—	2	1	no	—
102	C₄⋊Dic₇₉	—	4	2	no	—
103	C₄⋊Dic₇₉	—	4	2	no	—
104	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
105	C₄⋊Dic₇₉	D₈	1	1	yes	characteristic
106	Dic₇₉⋊C₄	—	4	2	no	—
107	C₂²⋊Dic₇₉	—	4	1	no	—
108	C₂²⋊Dic₇₉	D₈	1	2	yes	—
109	(C₂×D₇₉)⋊C₄	—	4	2	no	—
110	C₄×Dic₇₉	—	4	2	no	—
111	C₄×Dic₇₉	—	4	2	no	—
112	C₄×Dic₇₉	—	4	2	no	—
113	C₄×Dic₇₉	—	4	1	no	—
114	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
115	C₇₉⋊C₁₆	C₂²×C₄	1	4	yes	—
116	C₂×Q₈×C₇₉	—	2	4	no	—
117	C₂×Q₈×C₇₉	—	2	2	no	—
118	C₂×Q₈×C₇₉	—	4	1	no	—
119	C₂×Q₈×C₇₉	—	4	2	no	—
120	C₂²×C₄×C₇₉	—	4	1	no	—
121	C₂²×C₄×C₇₉	—	4	1	no	—
122	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₇₉	—	2	4	no	—
124	Q₁₆×C₇₉	—	2	2	no	—
125	Q₁₆×C₇₉	C₂²×C₄	1	4	yes	—
126	C₂×C₈×C₇₉	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₉	#41	1	2	yes	—
129	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
130	C₄⋊C₄×C₇₉	—	2	2	no	—
131	C₄⋊C₄×C₇₉	—	4	1	no	—
132	C₁₆×C₇₉	—	2	1	no	—
133	C₁₆×C₇₉	—	2	1	no	—
134	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₇₉	—	4	1	no	—
136	Q₈×C₇₉	—	2	4	no	—
137	Q₈×C₇₉	—	4	2	no	—
138	Q₈×C₇₉	—	4	4	no	—
139	C₂×C₄×C₇₉	—	4	2	no	—
140	C₂×C₄×C₇₉	—	4	4	no	—
141	C₂×C₄×C₇₉	—	4	2	no	—
142	C₂×C₄×C₇₉	—	16	1	no	—
143	C₂×C₄×C₇₉	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₇₉	#135	1	2	yes	—
146	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₇₉	—	4	4	no	—
148	C₂×Dic₇₉	—	4	1	no	—
149	C₂×Dic₇₉	—	4	1	no	—
150	C₂×Dic₇₉	—	4	1	no	—
151	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
152	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
153	D₃₁₆	—	2	2	no	—
154	D₃₁₆	—	4	1	no	—
155	C₄×D₇₉	—	4	4	no	—
156	C₄×D₇₉	—	16	2	no	—
157	Dic₁₅₈	—	2	2	no	—
158	Dic₁₅₈	—	4	1	no	—
159	C₈×C₇₉	C₄×D₄	1	2	yes	—
160	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
161	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
162	C₇₉⋊C₈	—	2	2	no	—
163	C₂²×C₇₉	—	2	1	no	—
164	C₂²×C₇₉	—	2	1	no	—
165	C₂²×C₇₉	C₄×D₈	1	2	yes	—
166	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
167	C₂×D₇₉	—	4	2	no	—
168	C₂×D₇₉	—	8	1	no	—
169	C₂×D₇₉	—	8	1	no	—
170	C₄×C₇₉	—	4	4	no	—
171	C₄×C₇₉	—	8	2	no	—
172	C₄×C₇₉	C₄×D₈	1	2	yes	—
173	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
175	Dic₇₉	—	2	2	no	—
176	Dic₇₉	—	2	2	no	—
177	Dic₇₉	—	4	2	no	—
178	Dic₇₉	—	8	1	no	—
179	Dic₇₉	—	8	1	no	—
180	#33455	—	79	1	no	maximal, Sylow
181	C₂×C₇₉	—	2	2	no	—
182	C₂×C₇₉	C₂×C₄×D₈	1	1	yes	characteristic
183	C₂×C₇₉	#3416	1	1	yes	characteristic
184	C₂×C₇₉	#3417	1	1	yes	characteristic
185	D₇₉	—	8	2	no	—
186	C₂×#510	—	79	1	no	—
187	#2951	—	79	1	no	—
188	#2950	—	79	1	no	—
189	#2176	—	79	2	no	—
190	#2175	—	79	2	no	—
191	C₄×Q₃₂	—	79	2	no	—
192	C₄×SD₃₂	—	79	2	no	—
193	#2159	—	79	1	no	—
194	#2138	—	79	1	no	—
195	C₂×#358	D₇₉	1	1	yes	characteristic, complemented
196	#2108	—	79	1	no	—
197	C₇₉	#33455	1	1	yes	characteristic, complemented, Sylow
198	C₂×#135	—	79	1	no	—
199	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
200	#520	—	158	1	no	—
201	#519	—	158	1	no	—
202	#510	—	79	8	no	—
203	C₂×Q₃₂	—	79	2	no	—
204	C₂×SD₃₂	—	79	2	no	—
205	C₂×M₃₂	—	79	1	no	—
206	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
207	#441	—	158	2	no	—
208	#440	—	158	2	no	—
209	C₄×Q₁₆	—	79	2	no	—
210	C₄×Q₁₆	—	79	2	no	—
211	C₄×Q₁₆	—	79	2	no	—
212	C₄×D₈	—	79	2	no	—
213	#427	—	79	1	no	—
214	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
215	#419	—	158	1	no	—
216	#417	—	158	1	no	—
217	#405	—	79	1	no	—
218	#367	—	79	2	no	—
219	#366	—	79	2	no	—
220	#358	—	79	1	no	—
221	#358	—	79	1	no	—
222	#358	C₂×D₇₉	1	2	yes	—
223	#358	C₂×D₇₉	1	2	yes	—
224	#357	—	79	1	no	—
225	#357	—	79	1	no	—
226	#346	—	79	2	no	—
227	C₄×C₁₆	—	79	2	no	—
228	C₂×#41	—	158	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#135	—	79	4	no	—
231	#135	—	158	2	no	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×Q₁₆	—	79	1	no	—
235	C₂×Q₁₆	C₄×D₇₉	1	4	yes	complemented
236	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
237	C₂×SD₁₆	—	158	1	no	—
238	C₂×D₈	—	79	1	no	—
239	C₂×M₁₆	—	79	1	no	—
240	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
241	C₄×Q₈	—	316	1	no	—
242	C₄×Q₈	—	316	1	no	—
243	C₄×Q₈	—	158	2	no	—
244	C₄×Q₈	—	158	2	no	—
245	C₄×Q₈	—	158	2	no	—
246	C₄×D₄	—	316	2	no	—
247	C₄×D₄	—	158	2	no	—
248	#117	—	316	1	no	—
249	#117	—	316	2	no	—
250	#117	—	79	1	no	—
251	#117	—	158	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	316	1	no	—
254	Q₃₂	—	79	8	no	—
255	SD₃₂	—	79	8	no	—
256	M₃₂	—	79	4	no	—
257	C₂×C₁₆	—	2	2	no	—
258	C₂×C₁₆	—	79	2	no	—
259	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
261	#107	—	2	1	no	—
262	#107	—	79	1	no	—
263	#107	—	79	1	no	—
264	#107	C₂²×D₇₉	1	2	yes	—
265	#106	—	158	1	no	—
266	#103	—	158	2	no	—
267	#103	—	158	2	no	—
268	#103	—	158	2	no	—
269	#102	—	158	2	no	—
270	#97	—	79	2	no	—
271	C₄×C₈	—	79	2	no	—
272	#41	—	316	2	no	—
273	#41	—	158	4	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	1	no	—
277	C₂×Q₈	—	4	2	no	—
278	C₂×Q₈	—	158	1	no	—
279	C₂×D₄	—	316	1	no	—
280	C₂×D₄	—	158	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	316	2	no	—
284	C₂²×C₄	—	316	1	no	—
285	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
286	Q₁₆	—	2	2	no	—
287	Q₁₆	—	2	4	no	—
288	Q₁₆	—	79	2	no	—
289	Q₁₆	—	158	1	no	—
290	Q₁₆	C₂×C₄×D₇₉	1	4	yes	—
291	SD₁₆	—	158	4	no	—
292	D₈	—	79	2	no	—
293	D₈	—	158	1	no	—
294	M₁₆	—	158	2	no	—
295	C₂×C₈	—	79	2	no	—
296	C₂×C₈	C₂×C₄×D₇₉	1	2	yes	—
297	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
298	C₂×C₈	#11929623	1	2	yes	—
299	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	316	2	no	—
303	C₄⋊C₄	—	316	2	no	—
304	C₄⋊C₄	—	316	2	no	—
305	C₄⋊C₄	—	79	1	no	—
306	C₄⋊C₄	—	79	1	no	—
307	C₄⋊C₄	—	158	1	no	—
308	C₄⋊C₄	—	158	1	no	—
309	C₂²⋊C₄	—	316	2	no	—
310	C₂²⋊C₄	—	316	1	no	—
311	C₂²⋊C₄	—	79	2	no	—
312	C₄²	—	316	2	no	—
313	C₄²	—	316	2	no	—
314	C₄²	—	316	2	no	—
315	C₄²	—	316	1	no	—
316	C₄²	—	79	2	no	—
317	C₁₆	—	2	1	no	—
318	C₁₆	—	2	1	no	—
319	C₁₆	—	79	4	no	—
320	C₂³	—	316	1	no	—
321	C₂³	D₈×D₇₉	1	1	yes	characteristic
322	Q₈	—	2	4	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	4	2	no	—
325	Q₈	—	316	1	no	—
326	Q₈	—	158	2	no	—
327	D₄	—	316	4	no	—
328	D₄	—	316	1	no	—
329	D₄	—	158	2	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	16	1	no	—
334	C₂×C₄	—	316	1	no	—
335	C₂×C₄	—	316	4	no	—
336	C₂×C₄	—	316	1	no	—
337	C₂×C₄	—	316	1	no	—
338	C₂×C₄	—	79	2	no	—
339	C₂×C₄	—	79	2	no	—
340	C₂×C₄	—	1264	2	no	—
341	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
342	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
343	C₂×C₄	#14579182	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
345	C₈	—	158	2	no	—
346	C₈	(C₂²⋊D₇₉)⋊C₄	1	2	yes	—
347	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
348	C₈	C₂×#11929623	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	316	2	no	—
352	C₂²	—	632	1	no	—
353	C₂²	—	632	1	no	—
354	C₂²	#19098643	1	2	yes	—
355	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
356	C₄	—	4	4	no	—
357	C₄	—	8	2	no	—
358	C₄	—	316	2	no	—
359	C₄	—	632	1	no	—
360	C₄	—	632	1	no	—
361	C₄	—	158	2	no	—
362	C₄	—	158	2	no	—
363	C₄	#19098643	1	2	yes	—
364	C₄	C₂×(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
365	C₄	C₂×#14579182	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	632	2	no	—
368	C₂	C₂×#19098643	1	1	yes	characteristic
369	C₂	#23776542	1	1	yes	characteristic
370	C₂	#23776549	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₉
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#358×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
prime radical	C₂×#14579183	C₂²
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#358	D₇₉
hypocenter	C₇₉	#33455
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
3. #23772268	C₂	semidirect
4. #23765268	C₂	semidirect
5. #23765267	C₂	semidirect
6. #23765256	C₂	semidirect
7. #23765249	C₂	semidirect
8. #23765186	C₂	semidirect
10. #23764996	C₂	semidirect
11. C₂×#358×C₇₉	C₂	semidirect
15. #19099004	C₄	semidirect
61. #14579107	C₂×C₄	semidirect
62. #14579104	C₂×C₄	semidirect
63. #14579101	C₂×C₄	semidirect
77. C₂×Q₁₆×C₇₉	C₂×C₄	semidirect
195. C₂×#358	D₇₉	semidirect
197. C₇₉	#33455	semidirect
235. C₂×Q₁₆	C₄×D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099004	2
#23772274	2
#23772268	2
#23765268	2
#23765267	2
#23765256	2
#23765249	2
#23765186	2
#23765108	2
#23764996	2
C₂×#358×C₇₉	2
#33455	79