Subgroups — C₂×#23765288 #34441699

Order 20224.

646 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099003	C₂	1	1	yes	maximal, characteristic
2	C₂×#19098685	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098669	C₂	1	1	yes	maximal, characteristic, complemented
4	#23765288	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19098385	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19098380	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#360×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14579168	—	2	1	no	—
11	C₂×#14579126	—	2	1	no	—
12	#19099003	C₄	1	4	yes	complemented
13	#19099003	C₂²	1	4	yes	—
14	C₂×#14579107	C₄	1	2	yes	complemented
15	C₂×#14579105	C₄	1	2	yes	complemented
16	C₂×#14579101	C₂²	1	1	yes	characteristic
17	#19098685	—	2	2	no	—
18	#19098685	C₂²	1	4	yes	—
19	#19098669	—	2	2	no	—
20	#19098669	C₂²	1	4	yes	complemented
21	C₂×#14579097	C₂²	1	1	yes	characteristic
22	C₂×#14579088	—	2	1	no	—
23	C₂×#14579084	—	2	1	no	—
24	C₂×#14579071	C₂²	1	1	yes	characteristic
25	#19098385	C₂²	1	4	yes	complemented
26	#19098380	C₂²	1	4	yes	complemented
27	#19098340	C₂²	1	4	yes	complemented
28	C₂×#135×C₇₉	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#106×C₇₉	C₂²	1	1	yes	characteristic
31	#360×C₇₉	C₂²	1	4	yes	—
32	C₂²×Dic₁₅₈	—	2	1	no	—
33	C₂×Dic₃₁₆	—	2	1	no	—
34	C₂×Dic₃₁₆	—	2	1	no	—
35	C₂×Dic₃₁₆	—	2	1	no	—
36	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
37	C₂×Dic₃₁₆	C₂×C₄	1	2	yes	—
38	C₂×Dic₃₁₆	C₂³	1	2	yes	—
39	#14579168	—	2	4	no	—
40	#14579168	—	4	2	no	—
41	C₂×#11929608	—	2	1	no	—
42	C₂×#11929608	—	4	1	no	—
43	C₂²×C₇₉⋊C₈	—	4	1	no	—
44	C₂²×C₇₉⋊C₈	D₄	1	1	yes	characteristic
45	#14579126	—	2	4	no	—
46	#14579126	—	4	2	no	—
47	C₂×Dic₇₉⋊C₄	—	4	1	no	—
48	C₂×C₄×Dic₇₉	—	4	1	no	—
49	C₂×C₄×Dic₇₉	D₄	1	1	yes	characteristic
50	#14579107	C₂×C₄	1	8	yes	complemented
51	#14579105	C₂×C₄	1	8	yes	complemented
52	#14579101	C₂×C₄	1	2	yes	—
53	#14579101	C₂³	1	2	yes	—
54	C₂×C₇₉⋊C₁₆	C₂×C₄	1	2	yes	—
55	#14579097	—	2	1	no	—
56	#14579097	C₂³	1	2	yes	—
57	#14579088	—	2	4	no	—
58	#14579084	—	2	4	no	—
59	#14579071	D₄	1	2	yes	—
60	#14579071	C₂³	1	2	yes	—
61	C₂×#41×C₇₉	—	2	1	no	—
62	#135×C₇₉	—	2	1	no	—
63	#135×C₇₉	—	2	1	no	—
64	#135×C₇₉	C₂×C₄	1	2	yes	—
65	#135×C₇₉	C₂³	1	2	yes	—
66	C₂×Q₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
67	C₂×SD₁₆×C₇₉	—	2	1	no	—
68	C₂×D₈×C₇₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
71	M₃₂×C₇₉	—	2	1	no	—
72	M₃₂×C₇₉	C₂³	1	2	yes	—
73	C₂×C₁₆×C₇₉	—	2	1	no	—
74	#106×C₇₉	—	2	1	no	—
75	#106×C₇₉	C₂³	1	2	yes	—
76	C₂×Dic₁₅₈	—	2	2	no	—
77	C₂×Dic₁₅₈	—	2	2	no	—
78	C₂×Dic₁₅₈	—	2	2	no	—
79	C₂×Dic₁₅₈	—	4	1	no	—
80	C₂×Dic₁₅₈	—	4	1	no	—
81	C₂×Dic₁₅₈	—	4	1	no	—
82	C₂²×Dic₇₉	—	2	1	no	—
83	C₂²×Dic₇₉	—	4	1	no	—
84	Dic₃₁₆	—	2	2	no	—
85	Dic₃₁₆	—	2	2	no	—
86	Dic₃₁₆	—	2	2	no	—
87	Dic₃₁₆	C₂²×C₄	1	4	yes	—
88	#11929608	—	2	2	no	—
89	#11929608	—	4	4	no	—
90	#11929608	—	4	1	no	—
91	C₂×C₇₉⋊C₈	—	2	1	no	—
92	C₂×C₇₉⋊C₈	—	2	1	no	—
93	C₂×C₇₉⋊C₈	—	4	4	no	—
94	C₂×C₇₉⋊C₈	—	4	1	no	—
95	C₂×C₇₉⋊C₈	C₂×D₄	1	2	yes	—
96	Dic₇₉⋊C₄	—	4	4	no	—
97	C₄×Dic₇₉	—	2	1	no	—
98	C₄×Dic₇₉	—	4	4	no	—
99	C₄×Dic₇₉	C₂×D₄	1	2	yes	—
100	C₇₉⋊C₁₆	C₂²×C₄	1	4	yes	—
101	#41×C₇₉	—	2	2	no	—
102	#41×C₇₉	—	2	2	no	—
103	#41×C₇₉	—	4	1	no	—
104	#41×C₇₉	—	4	1	no	—
105	C₂×Q₈×C₇₉	—	2	1	no	—
106	C₂×D₄×C₇₉	—	2	1	no	—
107	C₂×D₄×C₇₉	—	4	1	no	—
108	C₂²×C₄×C₇₉	—	4	1	no	—
109	C₂²×C₄×C₇₉	—	4	1	no	—
110	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
111	Q₁₆×C₇₉	—	2	1	no	—
112	Q₁₆×C₇₉	C₂²×C₄	1	2	yes	—
113	SD₁₆×C₇₉	—	2	2	no	—
114	SD₁₆×C₇₉	—	2	2	no	—
115	D₈×C₇₉	—	2	1	no	—
116	D₈×C₇₉	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₇₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
121	C₂×C₈×C₇₉	#41	1	1	yes	characteristic
122	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
123	C₄⋊C₄×C₇₉	—	2	2	no	—
124	C₄⋊C₄×C₇₉	—	4	1	no	—
125	C₁₆×C₇₉	—	2	2	no	—
126	C₂³×C₇₉	—	4	1	no	—
127	C₂³×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	Q₈×C₇₉	—	2	2	no	—
129	Q₈×C₇₉	—	4	1	no	—
130	D₄×C₇₉	—	2	2	no	—
131	D₄×C₇₉	—	4	2	no	—
132	D₄×C₇₉	—	4	2	no	—
133	D₄×C₇₉	—	4	1	no	—
134	C₂×C₄×C₇₉	—	4	1	no	—
135	C₂×C₄×C₇₉	—	4	2	no	—
136	C₂×C₄×C₇₉	—	4	2	no	—
137	C₂×C₄×C₇₉	—	4	2	no	—
138	C₂×C₄×C₇₉	—	8	2	no	—
139	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₉	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×Dic₇₉	—	2	2	no	—
146	C₂×Dic₇₉	—	4	2	no	—
147	C₂×Dic₇₉	—	4	2	no	—
148	C₂×Dic₇₉	—	4	2	no	—
149	C₂×Dic₇₉	—	4	2	no	—
150	Dic₁₅₈	—	2	4	no	—
151	Dic₁₅₈	—	4	2	no	—
152	Dic₁₅₈	—	4	2	no	—
153	Dic₁₅₈	—	4	2	no	—
154	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
155	C₈×C₇₉	C₄×D₄	1	1	yes	characteristic
156	C₈×C₇₉	C₂³×C₄	1	1	yes	characteristic
157	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
158	C₇₉⋊C₈	—	2	2	no	—
159	C₇₉⋊C₈	—	4	2	no	—
160	C₂²×C₇₉	—	2	2	no	—
161	C₂²×C₇₉	—	4	2	no	—
162	C₂²×C₇₉	—	8	2	no	—
163	C₂²×C₇₉	#442	1	1	yes	characteristic
164	C₂²×C₇₉	#442	1	1	yes	characteristic
165	C₂²×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	C₄×C₇₉	—	4	2	no	—
167	C₄×C₇₉	—	8	2	no	—
168	C₄×C₇₉	#442	1	1	yes	characteristic
169	C₄×C₇₉	#442	1	1	yes	characteristic
170	C₄×C₇₉	C₂×C₄×D₄	1	1	yes	characteristic
171	C₄×C₇₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
172	Dic₇₉	—	4	2	no	—
173	Dic₇₉	—	4	4	no	—
174	C₂×#2180	—	79	1	no	maximal, Sylow
175	C₂×C₇₉	—	2	1	no	—
176	C₂×C₇₉	—	2	1	no	—
177	C₂×C₇₉	—	8	2	no	—
178	C₂×C₇₉	#2180	1	2	yes	complemented
179	C₂×C₇₉	C₂×#442	1	1	yes	characteristic
180	C₂×#510	—	79	1	no	—
181	C₂×#444	—	79	1	no	—
182	C₂×#441	—	79	1	no	—
183	#2180	—	79	8	no	—
184	C₂×#365	—	79	1	no	—
185	C₂×#362	—	79	1	no	—
186	C₂×#360	D₇₉	1	1	yes	characteristic
187	C₂×#347	—	79	1	no	—
188	C₇₉	C₂×#2180	1	1	yes	characteristic, complemented, Sylow
189	C₂×#135	C₂×D₇₉	1	1	yes	characteristic
190	C₂²×Q₁₆	—	79	1	no	—
191	C₂×#131	—	158	1	no	—
192	C₂×C₄×Q₈	—	158	1	no	—
193	#510	—	79	4	no	—
194	#510	—	79	4	no	—
195	C₂×Q₃₂	—	79	2	no	—
196	C₂×SD₃₂	—	79	2	no	—
197	C₂×M₃₂	—	79	1	no	—
198	C₂×M₃₂	C₂×D₇₉	1	1	yes	characteristic
199	#444	—	79	4	no	—
200	#444	—	158	2	no	—
201	#441	—	79	4	no	—
202	#441	—	158	2	no	—
203	C₂×#108	—	79	1	no	—
204	C₂×#106	C₂×D₇₉	1	1	yes	characteristic
205	C₂×#104	—	158	1	no	—
206	C₂×#103	—	158	1	no	—
207	C₂×#97	—	79	1	no	—
208	#365	—	79	4	no	—
209	#362	—	79	4	no	—
210	#360	C₂×D₇₉	1	4	yes	—
211	#347	—	79	4	no	—
212	C₂×#41	—	2	1	no	—
213	C₂²×Q₈	—	158	1	no	—
214	#135	—	2	1	no	—
215	#135	—	2	1	no	—
216	#135	C₄×D₇₉	1	2	yes	—
217	#135	C₂²×D₇₉	1	2	yes	—
218	C₂×Q₁₆	—	79	2	no	—
219	C₂×Q₁₆	—	79	2	no	—
220	C₂×Q₁₆	—	79	2	no	—
221	C₂×Q₁₆	—	158	1	no	—
222	C₂×Q₁₆	—	158	1	no	—
223	C₂×Q₁₆	—	158	1	no	—
224	C₂×Q₁₆	C₄×D₇₉	1	1	yes	characteristic
225	C₂×SD₁₆	—	2	1	no	—
226	C₂×D₈	C₄×D₇₉	1	1	yes	characteristic
227	#131	—	316	2	no	—
228	#131	—	158	4	no	—
229	C₂×M₁₆	—	316	1	no	—
230	C₂×M₁₆	—	158	1	no	—
231	C₂²×C₈	—	316	1	no	—
232	C₂²×C₈	—	79	1	no	—
233	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
234	C₄×Q₈	—	316	2	no	—
235	C₄×Q₈	—	158	4	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	316	1	no	—
238	C₂×C₄²	—	316	1	no	—
239	C₂×C₄²	—	79	1	no	—
240	Q₃₂	—	79	8	no	—
241	SD₃₂	—	79	8	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	—	79	2	no	—
244	M₃₂	—	79	2	no	—
245	M₃₂	C₂²×D₇₉	1	2	yes	—
246	C₂×C₁₆	—	2	1	no	—
247	C₂×C₁₆	—	79	2	no	—
248	#108	—	79	2	no	—
249	#108	—	158	1	no	—
250	#106	—	2	1	no	—
251	#106	C₂²×D₇₉	1	2	yes	—
252	#104	—	158	4	no	—
253	#103	—	158	4	no	—
254	#97	—	79	2	no	—
255	#97	—	79	2	no	—
256	#41	—	2	2	no	—
257	#41	—	2	2	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	316	1	no	—
262	C₂×Q₈	—	316	1	no	—
263	C₂×Q₈	—	316	1	no	—
264	C₂×Q₈	—	158	2	no	—
265	C₂×Q₈	—	158	2	no	—
266	C₂×Q₈	—	158	2	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	4	1	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	316	1	no	—
272	C₂²×C₄	—	158	1	no	—
273	C₂²×C₄	D₄×D₇₉	1	1	yes	characteristic
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	79	4	no	—
276	Q₁₆	—	158	2	no	—
277	Q₁₆	—	158	2	no	—
278	Q₁₆	—	158	2	no	—
279	Q₁₆	C₂×C₄×D₇₉	1	2	yes	—
280	SD₁₆	—	2	2	no	—
281	SD₁₆	—	2	2	no	—
282	D₈	—	2	1	no	—
283	D₈	C₂×C₄×D₇₉	1	2	yes	—
284	M₁₆	—	316	1	no	—
285	M₁₆	—	316	4	no	—
286	M₁₆	—	158	2	no	—
287	C₂×C₈	—	316	1	no	—
288	C₂×C₈	—	316	4	no	—
289	C₂×C₈	—	79	2	no	—
290	C₂×C₈	—	158	1	no	—
291	C₂×C₈	—	158	1	no	—
292	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₇₉	1	1	yes	characteristic
294	C₂×C₈	D₄×D₇₉	1	1	yes	characteristic
295	C₂×C₈	#11929623	1	1	yes	characteristic
296	C₂×C₈	#11929623	1	1	yes	characteristic
297	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	316	4	no	—
301	C₄²	—	316	4	no	—
302	C₄²	—	79	2	no	—
303	C₄²	—	158	1	no	—
304	C₁₆	—	2	2	no	—
305	C₁₆	—	79	4	no	—
306	C₂³	—	4	1	no	—
307	C₂³	#14579185	1	1	yes	characteristic
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	316	2	no	—
311	Q₈	—	316	2	no	—
312	Q₈	—	316	2	no	—
313	Q₈	—	158	4	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	316	2	no	—
324	C₂×C₄	—	316	2	no	—
325	C₂×C₄	—	316	2	no	—
326	C₂×C₄	—	316	2	no	—
327	C₂×C₄	—	158	2	no	—
328	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
329	C₂×C₄	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
330	C₂×C₄	#14579185	1	1	yes	characteristic
331	C₂×C₄	#14579188	1	1	yes	characteristic
332	C₂×C₄	#14579188	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₇₉	1	1	yes	characteristic
334	C₈	—	316	2	no	—
335	C₈	—	158	2	no	—
336	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
337	C₈	(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
338	C₈	C₂²×C₄×D₇₉	1	1	yes	characteristic
339	C₈	C₂×#11929623	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	4	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	#19098672	1	1	yes	characteristic
344	C₂²	#19098672	1	1	yes	characteristic
345	C₂²	C₂×#14579185	1	1	yes	characteristic
346	C₄	—	4	2	no	—
347	C₄	—	8	2	no	—
348	C₄	—	316	2	no	—
349	C₄	—	316	4	no	—
350	C₄	#19098672	1	1	yes	characteristic
351	C₄	#19098672	1	1	yes	characteristic
352	C₄	C₂×(C₂²⋊D₇₉)⋊C₄	1	1	yes	characteristic
353	C₄	C₂×#14579188	1	1	yes	characteristic
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	—	8	2	no	—
357	C₂	#23765288	1	2	yes	direct product factor
358	C₂	C₂×#19098672	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19098672
commutator	C₈×C₇₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#360×C₇₉	C₂
socle	C₂²×C₇₉	#442
prime radical	C₂×#135×C₇₉	C₂²
cosocle	C₂×C₈×C₇₉	C₂⁴
hypercenter	C₂×#360	D₇₉
hypocenter	C₇₉	C₂×#2180
LCS	G ▷ C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#360

Complements

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

N	H ≅ G/N	type
3. C₂×#19098669	C₂	semidirect
4. #23765288	C₂	direct (N × H)
5. C₂×#19098385	C₂	semidirect
6. C₂×#19098380	C₂	semidirect
7. C₂×#19098340	C₂	semidirect
12. #19099003	C₄	semidirect
14. C₂×#14579107	C₄	semidirect
15. C₂×#14579105	C₄	semidirect
20. #19098669	C₂²	semidirect
25. #19098385	C₂²	semidirect
26. #19098380	C₂²	semidirect
27. #19098340	C₂²	semidirect
50. #14579107	C₂×C₄	semidirect
51. #14579105	C₂×C₄	semidirect
178. C₂×C₇₉	#2180	semidirect
188. C₇₉	C₂×#2180	semidirect
357. C₂	#23765288	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×#19099003	2
C₂×#19098685	2
C₂×#19098669	2
#23765288	2
C₂×#19098385	2
C₂×#19098380	2
C₂×#19098340	2
C₂×#360×C₇₉	2
C₂×#2180	79

Subgroups — C2×#23765288 #34441699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765288 #34441699