Subgroups — C₂×#23761391 #34198423

Order 20224.

480 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19098345	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇₉⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
3	#23761391	C₂	1	4	yes	maximal, direct product factor
4	C₂×#469×C₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929608	—	2	1	no	—
6	C₂×#14579101	—	2	1	no	—
7	#19098345	—	2	1	no	—
8	#19098345	C₂²	1	2	yes	complemented
9	C₇₉⋊#338	—	2	1	no	—
10	C₇₉⋊#338	C₂²	1	2	yes	complemented
11	C₂²×D₈×C₇₉	C₄	1	1	yes	characteristic
12	C₂²×M₁₆×C₇₉	C₂²	1	1	yes	characteristic
13	C₂×#121×C₇₉	—	2	1	no	—
14	#469×C₇₉	—	2	1	no	—
15	#469×C₇₉	—	2	1	no	—
16	#469×C₇₉	C₄	1	2	yes	—
17	#469×C₇₉	C₂²	1	2	yes	—
18	C₂×#106×C₇₉	C₄	1	1	yes	characteristic
19	C₂×#102×C₇₉	—	2	1	no	—
20	C₂×#11929608	—	2	2	no	—
21	C₂×#11929608	—	2	2	no	—
22	C₂×#11929608	—	4	1	no	—
23	C₂×#11929608	—	4	1	no	—
24	C₂×#11929608	—	4	1	no	—
25	C₂×#11929608	—	4	1	no	—
26	C₂²×C₇₉⋊C₈	—	2	2	no	—
27	#14579101	—	2	2	no	—
28	#14579101	—	4	1	no	—
29	C₂×C₇₉⋊C₁₆	—	4	1	no	—
30	C₂²×D₄×C₇₉	—	2	1	no	—
31	C₂²×D₄×C₇₉	—	4	1	no	—
32	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
33	C₂×D₈×C₇₉	—	2	1	no	—
34	C₂×D₈×C₇₉	—	4	1	no	—
35	C₂×D₈×C₇₉	—	4	1	no	—
36	C₂×D₈×C₇₉	C₂×C₄	1	2	yes	—
37	C₂×M₁₆×C₇₉	—	4	1	no	—
38	C₂×M₁₆×C₇₉	—	4	1	no	—
39	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₇₉	D₄	1	1	yes	characteristic
42	C₂×M₁₆×C₇₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
45	#121×C₇₉	—	2	2	no	—
46	#121×C₇₉	—	2	2	no	—
47	#121×C₇₉	—	4	1	no	—
48	#121×C₇₉	—	4	1	no	—
49	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
51	#106×C₇₉	—	2	1	no	—
52	#106×C₇₉	C₂×C₄	1	2	yes	—
53	#102×C₇₉	—	2	2	no	—
54	#102×C₇₉	—	2	2	no	—
55	#11929608	—	4	2	no	—
56	#11929608	—	4	2	no	—
57	#11929608	—	4	2	no	—
58	#11929608	—	4	2	no	—
59	C₂×C₇₉⋊C₈	—	2	4	no	—
60	C₂×C₇₉⋊C₈	—	4	2	no	—
61	C₂×C₇₉⋊C₈	—	4	2	no	—
62	C₇₉⋊C₁₆	—	4	2	no	—
63	C₂⁴×C₇₉	—	4	1	no	—
64	C₂⁴×C₇₉	SD₁₆	1	1	yes	characteristic
65	C₂×D₄×C₇₉	—	2	2	no	—
66	C₂×D₄×C₇₉	—	4	2	no	—
67	C₂×D₄×C₇₉	—	4	2	no	—
68	C₂×D₄×C₇₉	—	4	1	no	—
69	C₂×D₄×C₇₉	—	4	1	no	—
70	C₂×D₄×C₇₉	—	4	1	no	—
71	C₂×D₄×C₇₉	—	4	1	no	—
72	C₂×D₄×C₇₉	—	4	1	no	—
73	C₂×D₄×C₇₉	—	8	4	no	—
74	C₂²×C₄×C₇₉	—	2	1	no	—
75	C₂²×C₄×C₇₉	—	2	1	no	—
76	C₂²×C₄×C₇₉	—	2	1	no	—
77	C₂²×C₄×C₇₉	—	2	1	no	—
78	C₂²×C₄×C₇₉	—	4	1	no	—
79	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₇₉	SD₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
85	D₈×C₇₉	—	4	2	no	—
86	D₈×C₇₉	—	4	2	no	—
87	M₁₆×C₇₉	—	4	1	no	—
88	M₁₆×C₇₉	—	4	1	no	—
89	M₁₆×C₇₉	—	4	1	no	—
90	M₁₆×C₇₉	—	4	1	no	—
91	C₂×C₈×C₇₉	—	2	1	no	—
92	C₂×C₈×C₇₉	—	2	1	no	—
93	C₂×C₈×C₇₉	—	2	1	no	—
94	C₂×C₈×C₇₉	—	2	1	no	—
95	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₇₉	—	2	2	no	—
100	C₄⋊C₄×C₇₉	—	4	1	no	—
101	C₂²⋊C₄×C₇₉	—	4	2	no	—
102	C₂²⋊C₄×C₇₉	—	4	2	no	—
103	C₂³×C₇₉	—	2	1	no	—
104	C₂³×C₇₉	—	2	1	no	—
105	C₂³×C₇₉	—	4	2	no	—
106	C₂³×C₇₉	—	4	2	no	—
107	C₂³×C₇₉	—	8	2	no	—
108	C₂³×C₇₉	—	8	2	no	—
109	C₂³×C₇₉	—	8	1	no	—
110	C₂³×C₇₉	—	8	1	no	—
111	C₂³×C₇₉	#102	1	1	yes	characteristic
112	C₂³×C₇₉	#102	1	1	yes	characteristic
113	C₂³×C₇₉	C₂×SD₁₆	1	1	yes	characteristic
114	D₄×C₇₉	—	4	2	no	—
115	D₄×C₇₉	—	4	2	no	—
116	D₄×C₇₉	—	8	4	no	—
117	D₄×C₇₉	—	8	4	no	—
118	D₄×C₇₉	—	8	1	no	—
119	D₄×C₇₉	—	8	1	no	—
120	C₂×C₄×C₇₉	—	2	1	no	—
121	C₂×C₄×C₇₉	—	2	1	no	—
122	C₂×C₄×C₇₉	—	2	1	no	—
123	C₂×C₄×C₇₉	—	2	1	no	—
124	C₂×C₄×C₇₉	—	2	1	no	—
125	C₂×C₄×C₇₉	—	2	1	no	—
126	C₂×C₄×C₇₉	—	2	1	no	—
127	C₂×C₄×C₇₉	—	2	1	no	—
128	C₂×C₄×C₇₉	—	2	1	no	—
129	C₂×C₄×C₇₉	—	2	1	no	—
130	C₂×C₄×C₇₉	—	4	2	no	—
131	C₂×C₄×C₇₉	—	8	2	no	—
132	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#99	1	1	yes	characteristic
134	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
135	C₂×C₄×C₇₉	#102	1	1	yes	characteristic
136	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
137	C₂×C₄×C₇₉	#104	1	1	yes	characteristic
138	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
140	C₈×C₇₉	—	2	1	no	—
141	C₈×C₇₉	—	2	1	no	—
142	C₈×C₇₉	—	2	1	no	—
143	C₈×C₇₉	—	2	1	no	—
144	C₇₉⋊C₈	—	4	4	no	—
145	C₂²×C₇₉	—	2	2	no	—
146	C₂²×C₇₉	—	2	1	no	—
147	C₂²×C₇₉	—	2	1	no	—
148	C₂²×C₇₉	—	4	1	no	—
149	C₂²×C₇₉	—	4	1	no	—
150	C₂²×C₇₉	—	4	2	no	—
151	C₂²×C₇₉	—	8	1	no	—
152	C₂²×C₇₉	—	8	2	no	—
153	C₂²×C₇₉	—	8	2	no	—
154	C₂²×C₇₉	—	8	2	no	—
155	C₂²×C₇₉	—	16	2	no	—
156	C₂²×C₇₉	—	16	2	no	—
157	C₂²×C₇₉	#327	1	1	yes	characteristic
158	C₂²×C₇₉	#327	1	1	yes	characteristic
159	C₂²×C₇₉	C₂×#102	1	1	yes	characteristic
160	C₄×C₇₉	—	2	1	no	—
161	C₄×C₇₉	—	2	1	no	—
162	C₄×C₇₉	—	8	2	no	—
163	C₄×C₇₉	#327	1	1	yes	characteristic
164	C₄×C₇₉	#327	1	1	yes	characteristic
165	C₄×C₇₉	C₂×#99	1	1	yes	characteristic
166	C₄×C₇₉	C₂×#104	1	1	yes	characteristic
167	C₂×#1341	—	79	1	no	maximal, Sylow
168	C₂×C₇₉	—	2	1	no	—
169	C₂×C₇₉	—	2	1	no	—
170	C₂×C₇₉	—	4	1	no	—
171	C₂×C₇₉	—	4	1	no	—
172	C₂×C₇₉	—	16	2	no	—
173	C₂×C₇₉	#1341	1	2	yes	complemented
174	C₂×C₇₉	C₂×#327	1	1	yes	characteristic
175	C₂×#469	D₇₉	1	1	yes	characteristic
176	C₂×#349	—	79	1	no	—
177	C₂×#338	—	79	1	no	—
178	#1341	—	79	4	no	—
179	C₇₉	C₂×#1341	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₈	Dic₇₉	1	1	yes	characteristic
181	C₂²×M₁₆	—	158	1	no	—
182	C₂²×M₁₆	C₂×D₇₉	1	1	yes	characteristic
183	C₂×#121	—	2	1	no	—
184	C₂×M₃₂	—	158	1	no	—
185	#469	—	2	1	no	—
186	#469	—	2	1	no	—
187	#469	Dic₇₉	1	2	yes	—
188	#469	C₂×D₇₉	1	2	yes	—
189	C₂×#106	Dic₇₉	1	1	yes	characteristic
190	C₂×#102	—	2	1	no	—
191	#349	—	79	2	no	—
192	#349	—	158	1	no	—
193	#338	—	79	2	no	—
194	#338	—	158	1	no	—
195	C₂²×D₄	—	2	1	no	—
196	C₂²×D₄	—	4	1	no	—
197	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
198	C₂×D₈	—	2	1	no	—
199	C₂×D₈	—	4	1	no	—
200	C₂×D₈	—	4	1	no	—
201	C₂×D₈	C₂×Dic₇₉	1	2	yes	—
202	C₂×M₁₆	—	4	1	no	—
203	C₂×M₁₆	—	4	1	no	—
204	C₂×M₁₆	—	316	1	no	—
205	C₂×M₁₆	—	316	1	no	—
206	C₂×M₁₆	—	316	1	no	—
207	C₂×M₁₆	—	316	1	no	—
208	C₂×M₁₆	—	158	2	no	—
209	C₂×M₁₆	—	158	2	no	—
210	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
211	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
212	C₂×M₁₆	C₂²⋊D₇₉	1	1	yes	characteristic
213	C₂×M₁₆	C₂²×D₇₉	1	1	yes	characteristic
214	C₂²×C₈	—	158	2	no	—
215	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
216	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
217	#121	—	2	2	no	—
218	#121	—	2	2	no	—
219	#121	—	4	1	no	—
220	#121	—	4	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	4	1	no	—
223	M₃₂	—	316	1	no	—
224	M₃₂	—	158	2	no	—
225	C₂×C₁₆	—	316	1	no	—
226	#106	—	2	1	no	—
227	#106	C₂×Dic₇₉	1	2	yes	—
228	#102	—	2	2	no	—
229	#102	—	2	2	no	—
230	C₂⁴	—	4	1	no	—
231	C₂⁴	#11929612	1	1	yes	characteristic
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂×D₄	—	4	1	no	—
240	C₂×D₄	—	8	4	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²⋊Dic₇₉	1	1	yes	characteristic
248	C₂²×C₄	#11929609	1	1	yes	characteristic
249	C₂²×C₄	#11929609	1	1	yes	characteristic
250	C₂²×C₄	#11929612	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
252	D₈	—	4	2	no	—
253	D₈	—	4	2	no	—
254	M₁₆	—	4	1	no	—
255	M₁₆	—	4	1	no	—
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	316	2	no	—
259	M₁₆	—	316	2	no	—
260	M₁₆	—	316	2	no	—
261	M₁₆	—	316	2	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	316	2	no	—
267	C₂×C₈	—	316	2	no	—
268	C₂×C₈	—	158	4	no	—
269	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₇₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₇₉	1	1	yes	characteristic
272	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₁₆	—	316	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	#14579082	1	1	yes	characteristic
287	C₂³	#14579082	1	1	yes	characteristic
288	C₂³	C₂×#11929612	1	1	yes	characteristic
289	D₄	—	4	2	no	—
290	D₄	—	4	2	no	—
291	D₄	—	8	4	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	1	no	—
294	D₄	—	8	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	#14579076	1	1	yes	characteristic
308	C₂×C₄	#14579076	1	1	yes	characteristic
309	C₂×C₄	#14579082	1	1	yes	characteristic
310	C₂×C₄	#14579082	1	1	yes	characteristic
311	C₂×C₄	#14579088	1	1	yes	characteristic
312	C₂×C₄	#14579088	1	1	yes	characteristic
313	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
314	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	316	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	16	2	no	—
331	C₂²	—	16	2	no	—
332	C₂²	#19098297	1	1	yes	characteristic
333	C₂²	#19098297	1	1	yes	characteristic
334	C₂²	C₂×#14579082	1	1	yes	characteristic
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	8	2	no	—
338	C₄	#19098297	1	1	yes	characteristic
339	C₄	#19098297	1	1	yes	characteristic
340	C₄	C₂×#14579076	1	1	yes	characteristic
341	C₄	C₂×#14579088	1	1	yes	characteristic
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	4	1	no	—
345	C₂	—	4	1	no	—
346	C₂	—	16	2	no	—
347	C₂	#23761391	1	2	yes	direct product factor
348	C₂	C₂×#19098297	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098297
commutator	C₂×C₈×C₇₉	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₇₉
Fitting, prime radical	C₂×#469×C₇₉	C₂
socle	C₂²×C₇₉	#327
cosocle	C₂×M₁₆×C₇₉	C₂³
hypercenter	C₂×#469	D₇₉
hypocenter	C₇₉	C₂×#1341
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₂×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#469

Complements

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

N	H ≅ G/N	type
1. C₂×#19098345	C₂	semidirect
2. C₂×C₇₉⋊#338	C₂	semidirect
3. #23761391	C₂	direct (N × H)
8. #19098345	C₂²	semidirect
10. C₇₉⋊#338	C₂²	semidirect
173. C₂×C₇₉	#1341	semidirect
179. C₇₉	C₂×#1341	semidirect
347. C₂	#23761391	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19098345	2
C₂×C₇₉⋊#338	2
#23761391	2
C₂×#469×C₇₉	2
C₂×#1341	79

Subgroups — C2×#23761391 #34198423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23761391 #34198423