Subgroups — #34443671

Order 20224.

834 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098995	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19098993	C₂	1	2	yes	maximal, complemented
3	#23773824	C₂	1	2	yes	maximal, complemented
4	#23765662	C₂	1	1	yes	maximal, characteristic, complemented
5	#23765660	C₂	1	2	yes	maximal, complemented
6	#23765658	C₂	1	1	yes	maximal, characteristic, complemented
7	#23765654	C₂	1	2	yes	maximal, complemented
8	#23765651	C₂	1	1	yes	maximal, characteristic, complemented
9	#23765648	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₆×Dic₇₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#508×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14579183	C₂²	1	2	yes	—
13	C₂×#14579181	—	2	2	no	—
14	#19099109	—	2	2	no	—
15	#19098995	—	2	4	no	—
16	#19098993	—	2	8	no	—
17	C₂×Dic₆₃₂	C₂²	1	1	yes	characteristic
18	C₂×#14579107	C₂²	1	2	yes	—
19	C₂×#14579106	C₂²	1	2	yes	—
20	C₂×#14579105	C₂²	1	2	yes	—
21	C₂×#14579104	C₂²	1	2	yes	—
22	C₂×D₁₂₆₄	C₂²	1	1	yes	characteristic
23	C₂×#14579102	C₂²	1	2	yes	—
24	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
25	#19098948	C₂²	1	2	yes	—
26	#19098946	C₂²	1	2	yes	—
27	#19098944	—	2	2	no	—
28	#19098941	C₂²	1	2	yes	—
29	#19098938	C₂²	1	2	yes	—
30	#19098934	—	2	2	no	—
31	C₂×C₈×Dic₇₉	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇₉	C₂²	1	1	yes	characteristic
36	C₂×#135×C₇₉	C₂²	1	2	yes	—
37	#508×C₇₉	—	2	4	no	—
38	C₂×Q₃₂×C₇₉	C₂²	1	1	yes	characteristic
39	C₂×SD₃₂×C₇₉	C₂²	1	2	yes	—
40	C₂×D₁₆×C₇₉	C₂²	1	1	yes	characteristic
41	C₂²×C₁₆×C₇₉	C₂²	1	1	yes	characteristic
42	C₂×#11929624	—	2	2	no	—
43	#14579183	—	2	4	no	—
44	#14579183	—	2	4	no	—
45	#14579181	—	4	8	no	—
46	C₂×Dic₃₁₆	C₂³	1	2	yes	—
47	C₂×#11929614	—	2	2	no	—
48	C₂×#11929613	—	2	2	no	—
49	C₂×#11929612	—	2	2	no	—
50	C₂×#11929611	—	2	2	no	—
51	C₂×D₆₃₂	C₂³	1	2	yes	—
52	C₂×#11929609	—	2	2	no	—
53	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
54	#14579159	—	2	2	no	—
55	#14579157	—	2	2	no	—
56	#14579150	—	2	2	no	—
57	#14579148	—	2	2	no	—
58	#14579133	—	4	4	no	—
59	C₄×C₂²⋊D₇₉	—	4	4	no	—
60	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
61	Dic₆₃₂	—	2	2	no	—
62	#14579107	—	2	4	no	—
63	#14579106	—	2	4	no	—
64	#14579105	—	2	4	no	—
65	#14579104	—	2	4	no	—
66	D₁₂₆₄	—	2	2	no	—
67	#14579102	—	2	4	no	—
68	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	complemented
69	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₇₉⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₈×Dic₇₉	D₄	1	1	yes	characteristic
72	C₈×Dic₇₉	D₄	1	1	yes	characteristic
73	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
74	C₈×Dic₇₉	C₂³	1	1	yes	characteristic
75	C₂×#41×C₇₉	—	2	2	no	—
76	#135×C₇₉	—	2	4	no	—
77	#135×C₇₉	—	2	4	no	—
78	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
79	C₂×SD₁₆×C₇₉	—	2	2	no	—
80	C₂×D₈×C₇₉	C₂³	1	2	yes	—
81	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
82	Q₃₂×C₇₉	—	2	2	no	—
83	SD₃₂×C₇₉	—	2	4	no	—
84	D₁₆×C₇₉	—	2	2	no	—
85	C₂×C₁₆×C₇₉	D₄	1	4	yes	complemented
86	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
87	C₂×C₁₆×C₇₉	C₂³	1	1	yes	characteristic
88	#11929624	—	4	4	no	—
89	#11929624	—	4	4	no	—
90	C₂×Dic₁₅₈	—	2	2	no	—
91	C₂×D₃₁₆	—	2	2	no	—
92	C₂×C₂²⋊D₇₉	—	4	4	no	—
93	C₂×C₂²⋊D₇₉	—	4	2	no	—
94	C₂²×Dic₇₉	D₈	1	2	yes	—
95	C₂×C₄×D₇₉	—	4	2	no	—
96	Dic₃₁₆	—	2	2	no	—
97	Dic₃₁₆	—	2	2	no	—
98	#11929614	—	4	4	no	—
99	#11929613	—	4	4	no	—
100	#11929612	—	4	4	no	—
101	#11929611	—	4	4	no	—
102	D₆₃₂	—	2	2	no	—
103	D₆₃₂	—	2	2	no	—
104	#11929609	—	4	4	no	—
105	C₂×C₇₉⋊C₈	D₈	1	4	yes	—
106	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
107	C₂×C₇₉⋊C₈	C₂×D₄	1	1	yes	characteristic
108	Dic₇₉⋊C₄	—	4	4	no	—
109	C₂²⋊Dic₇₉	—	4	4	no	—
110	(C₂×D₇₉)⋊C₄	—	4	4	no	—
111	C₄×Dic₇₉	D₈	1	2	yes	—
112	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₇₉	C₂×D₄	1	1	yes	characteristic
114	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
115	C₇₉⋊C₁₆	C₂×D₄	1	2	yes	—
116	#41×C₇₉	—	4	4	no	—
117	#41×C₇₉	—	4	4	no	—
118	C₂×Q₈×C₇₉	—	2	2	no	—
119	C₂×D₄×C₇₉	—	2	2	no	—
120	C₂×D₄×C₇₉	—	4	2	no	—
121	C₂²×C₄×C₇₉	—	4	2	no	—
122	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₇₉	—	2	2	no	—
124	Q₁₆×C₇₉	—	2	2	no	—
125	SD₁₆×C₇₉	—	4	4	no	—
126	D₈×C₇₉	—	2	2	no	—
127	D₈×C₇₉	—	2	2	no	—
128	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
132	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
133	C₁₆×C₇₉	C₂×D₄	1	2	yes	—
134	C₂³×C₇₉	—	4	2	no	—
135	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
136	C₂²×D₇₉	—	4	2	no	—
137	Q₈×C₇₉	—	4	2	no	—
138	Q₈×C₇₉	—	4	2	no	—
139	D₄×C₇₉	—	4	2	no	—
140	D₄×C₇₉	—	4	2	no	—
141	D₄×C₇₉	—	8	4	no	—
142	C₂×C₄×C₇₉	—	4	2	no	—
143	C₂×C₄×C₇₉	—	8	4	no	—
144	C₂×C₄×C₇₉	#127	1	2	yes	—
145	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₇₉	—	8	4	no	—
149	C₂²⋊D₇₉	—	8	8	no	—
150	C₂×Dic₇₉	—	2	4	no	—
151	C₂×Dic₇₉	—	4	2	no	—
152	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
153	C₂×Dic₇₉	C₂×D₈	1	2	yes	—
154	D₃₁₆	—	4	2	no	—
155	D₃₁₆	—	4	2	no	—
156	C₄×D₇₉	—	8	4	no	—
157	Dic₁₅₈	—	4	2	no	—
158	Dic₁₅₈	—	4	2	no	—
159	C₈×C₇₉	#127	1	2	yes	—
160	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
162	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
163	C₇₉⋊C₈	C₂×D₈	1	2	yes	—
164	C₂²×C₇₉	—	2	2	no	—
165	C₂²×C₇₉	—	8	2	no	—
166	C₂²×C₇₉	—	8	4	no	—
167	C₂²×C₇₉	#493	1	2	yes	—
168	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
169	C₂×D₇₉	—	8	2	no	—
170	C₂×D₇₉	—	8	4	no	—
171	C₄×C₇₉	—	8	2	no	—
172	C₄×C₇₉	#493	1	2	yes	—
173	C₄×C₇₉	C₂×#127	1	1	yes	characteristic
174	C₄×C₇₉	C₂²×D₈	1	1	yes	characteristic
175	Dic₇₉	—	2	2	no	—
176	Dic₇₉	—	8	2	no	—
177	Dic₇₉	#508	1	4	yes	complemented
178	#33713	—	79	1	no	maximal, Sylow
179	C₂×C₇₉	—	2	2	no	—
180	C₂×C₇₉	—	16	2	no	—
181	C₂×C₇₉	C₂×#493	1	1	yes	characteristic
182	C₂×C₇₉	C₂×#508	1	2	yes	—
183	D₇₉	—	16	2	no	—
184	C₂×#508	—	79	1	no	—
185	C₂×#508	—	79	2	no	—
186	C₂×#508	D₇₉	1	1	yes	characteristic, complemented
187	#3147	—	79	2	no	—
188	#2249	—	79	1	no	—
189	#2249	—	79	2	no	—
190	#2249	—	79	1	no	—
191	#2248	—	79	2	no	—
192	#2247	—	79	1	no	—
193	#2246	—	79	1	no	—
194	C₂×C₄×C₁₆	—	79	1	no	—
195	C₇₉	#33713	1	1	yes	characteristic, complemented, Sylow
196	C₂×#135	—	79	2	no	—
197	C₂×#135	—	158	2	no	—
198	C₂×#135	C₂×D₇₉	1	2	yes	—
199	#532	—	158	2	no	—
200	#508	—	2	4	no	—
201	#508	—	158	8	no	—
202	#508	—	158	4	no	—
203	C₂×Q₃₂	—	79	2	no	—
204	C₂×Q₃₂	—	79	1	no	—
205	C₂×Q₃₂	C₂×D₇₉	1	1	yes	characteristic
206	C₂×SD₃₂	—	79	2	no	—
207	C₂×SD₃₂	—	79	2	no	—
208	C₂×SD₃₂	—	79	2	no	—
209	C₂×SD₃₂	C₂×D₇₉	1	2	yes	—
210	C₂×D₁₆	—	79	2	no	—
211	C₂×D₁₆	—	79	1	no	—
212	C₂×D₁₆	C₂×D₇₉	1	1	yes	characteristic
213	C₂²×C₁₆	—	79	1	no	—
214	C₂²×C₁₆	C₂×D₇₉	1	1	yes	characteristic
215	#495	—	79	2	no	—
216	#495	—	79	2	no	—
217	#495	—	158	2	no	—
218	#494	—	79	2	no	—
219	#493	—	79	2	no	—
220	#492	—	158	2	no	—
221	C₂×C₄×C₈	—	79	1	no	—
222	C₄×C₁₆	—	79	1	no	—
223	C₄×C₁₆	—	79	1	no	—
224	C₄×C₁₆	—	79	1	no	—
225	C₄×C₁₆	—	79	1	no	—
226	C₂×#41	—	2	2	no	—
227	C₂×#41	—	158	2	no	—
228	#135	—	2	4	no	—
229	#135	—	2	4	no	—
230	#135	—	316	8	no	—
231	#135	—	158	4	no	—
232	#135	—	158	4	no	—
233	C₂×Q₁₆	—	79	2	no	—
234	C₂×Q₁₆	—	158	2	no	—
235	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	158	2	no	—
238	C₂×SD₁₆	—	158	2	no	—
239	C₂×SD₁₆	—	158	2	no	—
240	C₂×D₈	—	79	2	no	—
241	C₂×D₈	—	158	2	no	—
242	C₂×D₈	C₂²×D₇₉	1	2	yes	—
243	C₂²×C₈	—	79	1	no	—
244	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
245	C₄⋊Q₈	—	158	2	no	—
246	#127	—	158	2	no	—
247	#124	—	158	2	no	—
248	#124	—	158	2	no	—
249	#121	—	316	4	no	—
250	C₄×D₄	—	316	4	no	—
251	C₂×C₄²	—	79	1	no	—
252	Q₃₂	—	2	2	no	—
253	Q₃₂	—	158	4	no	—
254	Q₃₂	—	158	2	no	—
255	SD₃₂	—	2	4	no	—
256	SD₃₂	—	158	4	no	—
257	SD₃₂	—	158	4	no	—
258	SD₃₂	—	158	4	no	—
259	D₁₆	—	2	2	no	—
260	D₁₆	—	158	4	no	—
261	D₁₆	—	158	2	no	—
262	C₂×C₁₆	—	79	4	no	—
263	C₂×C₁₆	—	79	1	no	—
264	C₂×C₁₆	—	79	1	no	—
265	C₂×C₁₆	C₂²⋊D₇₉	1	4	yes	complemented
266	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
267	C₂×C₁₆	C₂²×D₇₉	1	1	yes	characteristic
268	C₄×C₈	—	79	1	no	—
269	C₄×C₈	—	79	1	no	—
270	C₄×C₈	—	79	1	no	—
271	C₄×C₈	—	79	1	no	—
272	#41	—	4	4	no	—
273	#41	—	4	4	no	—
274	#41	—	316	4	no	—
275	#41	—	316	4	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	158	2	no	—
278	C₂×D₄	—	2	2	no	—
279	C₂×D₄	—	4	2	no	—
280	C₂×D₄	—	316	2	no	—
281	C₂×D₄	—	316	4	no	—
282	C₂×D₄	—	158	2	no	—
283	C₂²×C₄	—	4	2	no	—
284	C₂²×C₄	—	316	2	no	—
285	C₂²×C₄	—	79	2	no	—
286	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	2	no	—
289	Q₁₆	—	316	4	no	—
290	Q₁₆	—	158	2	no	—
291	Q₁₆	—	158	2	no	—
292	SD₁₆	—	4	4	no	—
293	SD₁₆	—	316	4	no	—
294	SD₁₆	—	316	4	no	—
295	SD₁₆	—	316	4	no	—
296	D₈	—	2	2	no	—
297	D₈	—	2	2	no	—
298	D₈	—	316	4	no	—
299	D₈	—	158	2	no	—
300	D₈	—	158	2	no	—
301	C₂×C₈	—	79	4	no	—
302	C₂×C₈	—	79	1	no	—
303	C₂×C₈	—	79	1	no	—
304	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
305	C₂×C₈	D₄×D₇₉	1	2	yes	—
306	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
307	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
308	C₄⋊C₄	—	316	4	no	—
309	C₂²⋊C₄	—	316	4	no	—
310	C₂²⋊C₄	—	316	4	no	—
311	C₄²	—	79	2	no	—
312	C₄²	—	79	1	no	—
313	C₄²	—	79	1	no	—
314	C₁₆	—	79	2	no	—
315	C₁₆	—	79	2	no	—
316	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
317	C₁₆	C₂×C₂²⋊D₇₉	1	2	yes	—
318	C₂³	—	4	2	no	—
319	C₂³	—	316	2	no	—
320	C₂³	D₈×D₇₉	1	1	yes	characteristic
321	Q₈	—	4	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	316	2	no	—
324	Q₈	—	316	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	8	4	no	—
328	D₄	—	316	2	no	—
329	D₄	—	316	2	no	—
330	D₄	—	632	8	no	—
331	D₄	—	632	4	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	—	316	2	no	—
335	C₂×C₄	—	79	2	no	—
336	C₂×C₄	—	79	2	no	—
337	C₂×C₄	—	632	4	no	—
338	C₂×C₄	—	158	4	no	—
339	C₂×C₄	#14579157	1	2	yes	—
340	C₂×C₄	D₈×D₇₉	1	2	yes	—
341	C₂×C₄	D₈×D₇₉	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
343	C₈	—	79	2	no	—
344	C₈	—	79	2	no	—
345	C₈	#14579157	1	2	yes	—
346	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
347	C₈	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
348	C₂²	—	2	2	no	—
349	C₂²	—	8	4	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	632	4	no	—
352	C₂²	—	632	2	no	—
353	C₂²	#19098938	1	2	yes	—
354	C₂²	C₂×D₈×D₇₉	1	1	yes	characteristic
355	C₄	—	8	2	no	—
356	C₄	—	79	4	no	—
357	C₄	—	632	2	no	—
358	C₄	—	158	2	no	—
359	C₄	#19098938	1	2	yes	—
360	C₄	C₂×#14579157	1	1	yes	characteristic
361	C₄	C₂×D₈×D₇₉	1	1	yes	characteristic
362	C₂	—	2	2	no	—
363	C₂	—	16	2	no	—
364	C₂	—	1264	2	no	—
365	C₂	C₂×#19098938	1	1	yes	characteristic
366	C₂	#508×D₇₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₈×D₇₉
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#508×C₇₉	C₂
socle	C₂²×C₇₉	C₂²×D₈
hypercenter	C₂×#508	D₇₉
hypocenter	C₇₉	#33713
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#19098995	C₂	semidirect
2. C₂×#19098993	C₂	semidirect
3. #23773824	C₂	semidirect
4. #23765662	C₂	semidirect
5. #23765660	C₂	semidirect
6. #23765658	C₂	semidirect
7. #23765654	C₂	semidirect
8. #23765651	C₂	semidirect
9. #23765648	C₂	semidirect
10. C₂×C₁₆×Dic₇₉	C₂	semidirect
11. C₂×#508×C₇₉	C₂	semidirect
68. C₂×C₇₉⋊C₁₆	D₄	semidirect
85. C₂×C₁₆×C₇₉	D₄	semidirect
177. Dic₇₉	#508	semidirect
186. C₂×#508	D₇₉	semidirect
195. C₇₉	#33713	semidirect
265. C₂×C₁₆	C₂²⋊D₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19098995	2
C₂×#19098993	2
#23773824	2
#23765662	2
#23765660	2
#23765658	2
#23765654	2
#23765651	2
#23765648	2
C₂×C₁₆×Dic₇₉	2
C₂×#508×C₇₉	2
#33713	79