Subgroups — C₂×#23765489 #34442691

Order 20224.

702 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#19098804	C₂	1	1	yes	maximal, characteristic
4	#23765489	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19098388	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19098367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19098366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#467×C₇₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₁₆	C₂²	1	1	yes	characteristic
10	C₂×#14579140	—	2	1	no	—
11	C₂×#14579107	—	2	4	no	—
12	C₂×#14579107	C₂²	1	4	yes	complemented
13	C₂²×C₇₉⋊C₁₆	—	2	4	no	—
14	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₇₉⋊C₁₆	C₂²	1	4	yes	—
16	C₂²×C₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
17	#19098804	—	2	2	no	—
18	#19098804	C₂²	1	4	yes	—
19	C₂×#14579095	C₂²	1	1	yes	characteristic
20	C₂×#14579086	—	2	1	no	—
21	#19098388	C₂²	1	4	yes	complemented
22	#19098367	C₂²	1	4	yes	complemented
23	#19098366	C₂²	1	4	yes	complemented
24	C₂²×Q₁₆×C₇₉	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₇₉	C₂²	1	1	yes	characteristic
26	C₂×C₂²⋊Q₈×C₇₉	—	2	1	no	—
27	#467×C₇₉	—	2	2	no	—
28	#467×C₇₉	C₂²	1	4	yes	—
29	C₂×#107×C₇₉	C₂²	1	1	yes	characteristic
30	C₂×#103×C₇₉	—	2	1	no	—
31	C₂²×Dic₁₅₈	—	2	1	no	—
32	C₂×Dic₃₁₆	—	2	2	no	—
33	C₂×Dic₃₁₆	—	2	2	no	—
34	C₂×Dic₃₁₆	—	2	1	no	—
35	C₂×Dic₃₁₆	C₂³	1	2	yes	—
36	#14579140	—	2	4	no	—
37	#14579140	—	4	2	no	—
38	C₂×C₄⋊Dic₇₉	—	2	1	no	—
39	C₂×Dic₇₉⋊C₄	—	4	1	no	—
40	C₂×C₂²⋊Dic₇₉	—	4	1	no	—
41	#14579107	—	2	8	no	—
42	C₂×C₇₉⋊C₁₆	—	2	8	no	—
43	C₂×C₇₉⋊C₁₆	—	2	2	no	—
44	C₂×C₇₉⋊C₁₆	D₄	1	4	yes	—
45	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
46	C₂×C₇₉⋊C₁₆	C₂³	1	2	yes	—
47	#14579095	—	2	1	no	—
48	#14579095	C₂³	1	2	yes	—
49	#14579086	—	2	4	no	—
50	C₂²×Q₈×C₇₉	—	2	1	no	—
51	C₂³×C₄×C₇₉	D₄	1	1	yes	characteristic
52	C₂×Q₁₆×C₇₉	—	2	2	no	—
53	C₂×Q₁₆×C₇₉	—	2	1	no	—
54	C₂×Q₁₆×C₇₉	—	2	2	no	—
55	C₂×Q₁₆×C₇₉	C₂³	1	2	yes	—
56	C₂²×C₈×C₇₉	—	2	2	no	—
57	C₂²×C₈×C₇₉	—	2	2	no	—
58	C₂²×C₈×C₇₉	D₄	1	2	yes	—
59	C₂²×C₈×C₇₉	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₇₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
62	C₂²⋊Q₈×C₇₉	—	2	4	no	—
63	C₂²⋊Q₈×C₇₉	—	4	2	no	—
64	C₂×C₄⋊C₄×C₇₉	—	2	1	no	—
65	C₂×C₄⋊C₄×C₇₉	—	4	1	no	—
66	C₂×C₂²⋊C₄×C₇₉	—	4	1	no	—
67	#107×C₇₉	—	2	1	no	—
68	#107×C₇₉	C₂³	1	2	yes	—
69	#103×C₇₉	—	2	4	no	—
70	C₂×Dic₁₅₈	—	2	2	no	—
71	C₂×Dic₁₅₈	—	4	2	no	—
72	C₂×Dic₁₅₈	—	4	1	no	—
73	C₂×Dic₁₅₈	—	4	2	no	—
74	C₂²×Dic₇₉	—	4	1	no	—
75	C₂²×Dic₇₉	—	4	1	no	—
76	Dic₃₁₆	—	2	2	no	—
77	Dic₃₁₆	—	2	2	no	—
78	Dic₃₁₆	—	4	2	no	—
79	C₄⋊Dic₇₉	—	2	2	no	—
80	C₄⋊Dic₇₉	—	4	1	no	—
81	Dic₇₉⋊C₄	—	4	4	no	—
82	C₂²⋊Dic₇₉	—	4	4	no	—
83	C₇₉⋊C₁₆	—	2	2	no	—
84	C₇₉⋊C₁₆	C₂×D₄	1	4	yes	—
85	C₂⁴×C₇₉	D₈	1	1	yes	characteristic
86	C₂×Q₈×C₇₉	—	2	2	no	—
87	C₂×Q₈×C₇₉	—	4	1	no	—
88	C₂×Q₈×C₇₉	—	4	2	no	—
89	C₂×Q₈×C₇₉	—	4	2	no	—
90	C₂²×C₄×C₇₉	—	2	2	no	—
91	C₂²×C₄×C₇₉	—	2	2	no	—
92	C₂²×C₄×C₇₉	—	4	1	no	—
93	C₂²×C₄×C₇₉	—	4	1	no	—
94	C₂²×C₄×C₇₉	D₈	1	1	yes	characteristic
95	C₂²×C₄×C₇₉	D₈	1	2	yes	—
96	C₂²×C₄×C₇₉	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₇₉	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₇₉	—	2	2	no	—
99	Q₁₆×C₇₉	—	2	2	no	—
100	Q₁₆×C₇₉	—	4	2	no	—
101	C₂×C₈×C₇₉	—	2	4	no	—
102	C₂×C₈×C₇₉	—	2	2	no	—
103	C₂×C₈×C₇₉	—	2	2	no	—
104	C₂×C₈×C₇₉	—	2	2	no	—
105	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₇₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₉	#41	1	2	yes	—
109	C₂×C₈×C₇₉	C₂⁴	1	1	yes	characteristic
110	C₄⋊C₄×C₇₉	—	2	2	no	—
111	C₄⋊C₄×C₇₉	—	4	1	no	—
112	C₄⋊C₄×C₇₉	—	4	4	no	—
113	C₂²⋊C₄×C₇₉	—	4	4	no	—
114	C₂³×C₇₉	—	2	2	no	—
115	C₂³×C₇₉	—	2	2	no	—
116	C₂³×C₇₉	Q₃₂	1	4	yes	complemented
117	C₂³×C₇₉	C₂×D₈	1	2	yes	—
118	C₂³×C₇₉	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₇₉	—	4	2	no	—
120	Q₈×C₇₉	—	4	2	no	—
121	Q₈×C₇₉	—	8	2	no	—
122	C₂×C₄×C₇₉	—	2	2	no	—
123	C₂×C₄×C₇₉	—	2	2	no	—
124	C₂×C₄×C₇₉	—	2	2	no	—
125	C₂×C₄×C₇₉	—	2	4	no	—
126	C₂×C₄×C₇₉	—	4	2	no	—
127	C₂×C₄×C₇₉	—	4	2	no	—
128	C₂×C₄×C₇₉	—	8	2	no	—
129	C₂×C₄×C₇₉	—	8	2	no	—
130	C₂×C₄×C₇₉	#121	1	2	yes	—
131	C₂×C₄×C₇₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₇₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₉	#135	1	2	yes	—
134	C₂×C₄×C₇₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₇₉	—	4	2	no	—
136	C₂×Dic₇₉	—	4	2	no	—
137	C₂×Dic₇₉	—	8	2	no	—
138	C₂×Dic₇₉	—	8	2	no	—
139	Dic₁₅₈	—	4	2	no	—
140	Dic₁₅₈	—	4	2	no	—
141	Dic₁₅₈	—	8	2	no	—
142	C₈×C₇₉	—	2	2	no	—
143	C₈×C₇₉	#121	1	2	yes	—
144	C₈×C₇₉	C₂²×D₄	1	1	yes	characteristic
145	C₈×C₇₉	C₂×#41	1	1	yes	characteristic
146	C₂²×C₇₉	—	2	2	no	—
147	C₂²×C₇₉	—	2	8	no	—
148	C₂²×C₇₉	—	2	2	no	—
149	C₂²×C₇₉	#466	1	2	yes	—
150	C₂²×C₇₉	C₂×Q₃₂	1	2	yes	—
151	C₂²×C₇₉	C₂×Q₃₂	1	4	yes	complemented
152	C₂²×C₇₉	#508	1	2	yes	—
153	C₂²×C₇₉	C₂²×D₈	1	1	yes	characteristic
154	C₄×C₇₉	—	2	2	no	—
155	C₄×C₇₉	—	8	2	no	—
156	C₄×C₇₉	—	8	2	no	—
157	C₄×C₇₉	#466	1	2	yes	—
158	C₄×C₇₉	C₂×#121	1	1	yes	characteristic
159	C₄×C₇₉	C₂×#135	1	1	yes	characteristic
160	Dic₇₉	—	8	2	no	—
161	Dic₇₉	—	8	2	no	—
162	C₂×#2216	—	79	1	no	maximal, Sylow
163	C₂×C₇₉	—	2	4	no	—
164	C₂×C₇₉	#2216	1	4	yes	complemented
165	C₂×C₇₉	C₂×#466	1	1	yes	characteristic
166	C₂×C₇₉	C₂²×Q₃₂	1	1	yes	characteristic
167	C₂×C₇₉	C₂×#508	1	1	yes	characteristic
168	C₂²×Q₃₂	—	79	1	no	—
169	C₂³×C₁₆	—	79	1	no	—
170	C₂×#467	—	79	1	no	—
171	C₂×#467	D₇₉	1	1	yes	characteristic
172	#2216	—	79	8	no	—
173	C₂×#366	—	79	1	no	—
174	C₂×#358	—	79	1	no	—
175	C₂×#358	—	79	1	no	—
176	C₇₉	C₂×#2216	1	1	yes	characteristic, complemented, Sylow
177	C₂²×Q₁₆	—	79	1	no	—
178	C₂²×Q₁₆	C₂×D₇₉	1	1	yes	characteristic
179	C₂³×C₈	C₂×D₇₉	1	1	yes	characteristic
180	C₂×C₂²⋊Q₈	—	2	1	no	—
181	C₂×C₂²⋊Q₈	—	158	1	no	—
182	C₂×Q₃₂	—	79	4	no	—
183	C₂×Q₃₂	—	158	4	no	—
184	C₂²×C₁₆	—	79	1	no	—
185	C₂²×C₁₆	—	79	1	no	—
186	C₂²×C₁₆	—	79	4	no	—
187	C₂²×C₁₆	—	158	4	no	—
188	#467	—	2	2	no	—
189	#467	—	79	4	no	—
190	#467	—	158	2	no	—
191	#467	C₂×D₇₉	1	4	yes	—
192	C₂×#107	—	79	1	no	—
193	C₂×#107	C₂×D₇₉	1	1	yes	characteristic
194	C₂×#103	—	2	1	no	—
195	C₂×#103	—	158	1	no	—
196	#366	—	79	4	no	—
197	#358	—	79	4	no	—
198	#358	—	79	4	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×Q₈	—	158	1	no	—
201	C₂³×C₄	C₂²⋊D₇₉	1	1	yes	characteristic
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	—	2	2	no	—
205	C₂×Q₁₆	—	79	2	no	—
206	C₂×Q₁₆	—	158	2	no	—
207	C₂×Q₁₆	—	158	1	no	—
208	C₂×Q₁₆	—	158	2	no	—
209	C₂×Q₁₆	C₂²×D₇₉	1	2	yes	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	C₂²⋊D₇₉	1	1	yes	characteristic
213	C₂²×C₈	C₂²⋊D₇₉	1	2	yes	—
214	C₂²×C₈	C₂²×D₇₉	1	2	yes	—
215	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
216	C₂²⋊Q₈	—	2	4	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₂²⋊Q₈	—	316	2	no	—
219	C₂²⋊Q₈	—	158	4	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	4	1	no	—
222	C₂×C₄⋊C₄	—	316	1	no	—
223	C₂×C₄⋊C₄	—	158	1	no	—
224	C₂×C₂²⋊C₄	—	4	1	no	—
225	C₂×C₂²⋊C₄	—	316	1	no	—
226	Q₃₂	—	158	8	no	—
227	C₂×C₁₆	—	79	4	no	—
228	C₂×C₁₆	—	79	2	no	—
229	C₂×C₁₆	—	79	2	no	—
230	C₂×C₁₆	—	158	8	no	—
231	C₂×C₁₆	—	158	2	no	—
232	#107	—	2	1	no	—
233	#107	—	79	2	no	—
234	#107	—	158	1	no	—
235	#107	C₂²×D₇₉	1	2	yes	—
236	#103	—	2	4	no	—
237	#103	—	158	4	no	—
238	C₂⁴	#11929609	1	1	yes	characteristic
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	—	4	2	no	—
243	C₂×Q₈	—	316	1	no	—
244	C₂×Q₈	—	316	2	no	—
245	C₂×Q₈	—	316	2	no	—
246	C₂×Q₈	—	158	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	316	1	no	—
252	C₂²×C₄	—	316	1	no	—
253	C₂²×C₄	#11929609	1	2	yes	—
254	C₂²×C₄	#11929609	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₇₉	1	2	yes	—
256	C₂²×C₄	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	—	4	2	no	—
260	Q₁₆	—	316	2	no	—
261	Q₁₆	—	158	2	no	—
262	Q₁₆	—	158	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	D₄×D₇₉	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₇₉	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
270	C₂×C₈	#11929623	1	2	yes	—
271	C₂×C₈	C₂³×D₇₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	4	no	—
275	C₄⋊C₄	—	316	4	no	—
276	C₄⋊C₄	—	316	1	no	—
277	C₄⋊C₄	—	158	2	no	—
278	C₂²⋊C₄	—	4	4	no	—
279	C₂²⋊C₄	—	316	4	no	—
280	C₁₆	—	79	4	no	—
281	C₁₆	—	158	2	no	—
282	C₂³	—	2	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	#14579107	1	4	yes	complemented
285	C₂³	C₂×#11929609	1	2	yes	—
286	C₂³	C₂×#11929609	1	1	yes	characteristic
287	Q₈	—	4	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	316	2	no	—
291	Q₈	—	316	2	no	—
292	Q₈	—	632	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	2	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	316	2	no	—
302	C₂×C₄	—	316	2	no	—
303	C₂×C₄	—	632	2	no	—
304	C₂×C₄	—	632	2	no	—
305	C₂×C₄	#14579133	1	2	yes	—
306	C₂×C₄	C₂×#11929609	1	2	yes	—
307	C₂×C₄	C₂×#11929609	1	1	yes	characteristic
308	C₂×C₄	#14579181	1	2	yes	—
309	C₂×C₄	C₂²×C₂²⋊D₇₉	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	#14579133	1	2	yes	—
312	C₈	C₂×D₄×D₇₉	1	1	yes	characteristic
313	C₈	C₂×#11929623	1	1	yes	characteristic
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	8	no	—
317	C₂²	#19098797	1	2	yes	—
318	C₂²	C₂×#14579107	1	2	yes	—
319	C₂²	C₂×#14579107	1	4	yes	complemented
320	C₂²	#19098993	1	2	yes	—
321	C₂²	C₂²×#11929609	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	632	2	no	—
326	C₄	—	632	2	no	—
327	C₄	#19098797	1	2	yes	—
328	C₄	C₂×#14579133	1	1	yes	characteristic
329	C₄	C₂×#14579181	1	1	yes	characteristic
330	C₂	—	2	4	no	—
331	C₂	#23765489	1	4	yes	direct product factor
332	C₂	C₂×#19098797	1	1	yes	characteristic
333	C₂	C₂²×#14579107	1	1	yes	characteristic
334	C₂	C₂×#19098993	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11929609
commutator, cosocle	C₂×C₈×C₇₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₉
Fitting	C₂×#467×C₇₉	C₂
socle	C₂³×C₇₉	C₂×D₈
prime radical	C₂⁴×C₇₉	D₈
hypercenter	C₂×#467	D₇₉
hypocenter	C₇₉	C₂×#2216
LCS	G ▷ C₂×C₈×C₇₉ ▷ C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
1. C₂²×#14579107	C₂	semidirect
4. #23765489	C₂	direct (N × H)
5. C₂×#19098388	C₂	semidirect
6. C₂×#19098367	C₂	semidirect
7. C₂×#19098366	C₂	semidirect
12. C₂×#14579107	C₂²	semidirect
21. #19098388	C₂²	semidirect
22. #19098367	C₂²	semidirect
23. #19098366	C₂²	semidirect
116. C₂³×C₇₉	Q₃₂	semidirect
151. C₂²×C₇₉	C₂×Q₃₂	semidirect
164. C₂×C₇₉	#2216	semidirect
176. C₇₉	C₂×#2216	semidirect
284. C₂³	#14579107	semidirect
319. C₂²	C₂×#14579107	semidirect
331. C₂	#23765489	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14579107	2
C₂³×C₇₉⋊C₁₆	2
C₂×#19098804	2
#23765489	2
C₂×#19098388	2
C₂×#19098367	2
C₂×#19098366	2
C₂×#467×C₇₉	2
C₂×#2216	79

Subgroups — C2×#23765489 #34442691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23765489 #34442691