Subgroups — C₂×#22620442 #33955586

Order 18688.

1942 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	C₂×#17989142	C₂	1	2	yes	maximal
2	#22620442	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17988992	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#17988988	C₂	1	4	yes	maximal, complemented
5	C₂×#17988983	C₂	1	2	yes	maximal, complemented
6	C₂×#17988624	C₂	1	4	yes	maximal
7	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#500×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₇₃	C₂²	1	2	yes	—
10	#17989142	—	2	8	no	—
11	#17989142	C₂²	1	16	yes	—
12	C₂×#14556591	C₂²	1	2	yes	—
13	C₂×#14556582	C₂²	1	4	yes	—
14	C₂×#14556571	—	2	2	no	—
15	C₂×#14556567	—	2	2	no	—
16	C₂×Q₈×Dic₇₃	—	2	2	no	—
17	C₂×#14556556	—	2	2	no	—
18	C₂×#14556549	C₂²	1	4	yes	—
19	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
20	#17988992	C₂²	1	8	yes	complemented
21	#17988988	C₂²	1	32	yes	complemented
22	#17988983	C₂²	1	16	yes	complemented
23	#17988624	C₂²	1	32	yes	—
24	C₂×#14556525	C₂²	1	4	yes	—
25	C₂×#14556522	C₂²	1	8	yes	—
26	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
28	C₄×C₈×D₇₃	C₂²	1	8	yes	—
29	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
30	#500×C₇₃	C₂²	1	8	yes	complemented
31	C₂×#107×C₇₃	C₂²	1	4	yes	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂²×Dic₁₄₆	—	2	2	no	—
34	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
36	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
37	C₂²×#10596831	C₂³	1	2	yes	—
38	C₂×C₈×D₇₃	Q₈	1	16	yes	complemented
39	C₂×C₈×D₇₃	C₂³	1	8	yes	—
40	#14556591	—	2	4	no	—
41	#14556591	C₂³	1	8	yes	—
42	#14556582	—	2	8	no	—
43	#14556582	C₂³	1	16	yes	—
44	#14556571	—	2	16	no	—
45	#14556567	—	2	16	no	—
46	Q₈×Dic₇₃	—	2	16	no	—
47	#14556556	—	2	16	no	—
48	#14556549	—	2	8	no	—
49	#14556549	C₂³	1	16	yes	—
50	C₂×C₄⋊Dic₇₃	—	2	4	no	—
51	C₂×C₄⋊Dic₇₃	—	2	2	no	—
52	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
53	C₂×#11924007	—	2	4	no	—
54	C₂×#11924005	—	2	4	no	—
55	C₂×C₄×Dic₇₃	—	2	4	no	—
56	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
57	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
58	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
59	C₄²×D₇₃	D₄	1	4	yes	—
60	C₄²×D₇₃	C₂³	1	4	yes	—
61	#14556525	C₂³	1	16	yes	—
62	#14556522	C₂³	1	32	yes	—
63	C₄×#10596831	C₂³	1	4	yes	—
64	C₈×Dic₇₃	C₂³	1	8	yes	—
65	C₂²×Q₈×C₇₃	—	2	2	no	—
66	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₇₃	—	2	4	no	—
68	C₄⋊Q₈×C₇₃	C₂³	1	8	yes	—
69	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
70	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
71	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
72	#107×C₇₃	C₂³	1	16	yes	—
73	C₄×C₈×C₇₃	C₂³	1	4	yes	—
74	C₂³×D₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₁₄₆	—	2	8	no	—
76	C₂×Dic₁₄₆	—	4	8	no	—
77	C₂²×Dic₇₃	—	2	4	no	—
78	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×D₇₃	—	2	4	no	—
82	C₂×C₄×D₇₃	—	2	4	no	—
83	C₂×C₄×D₇₃	C₂×D₄	1	4	yes	—
84	C₂×C₄×D₇₃	C₂×D₄	1	4	yes	—
85	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
86	C₂×C₄×D₇₃	C₂×Q₈	1	8	yes	—
87	C₂×C₄×D₇₃	C₂⁴	1	2	yes	—
88	C₂×#10596831	C₂×Q₈	1	8	yes	—
89	C₂×#10596831	C₂⁴	1	4	yes	—
90	C₈×D₇₃	C₂×Q₈	1	32	yes	complemented
91	C₄⋊Dic₇₃	—	2	8	no	—
92	C₄⋊Dic₇₃	—	2	4	no	—
93	C₄⋊Dic₇₃	—	2	16	no	—
94	C₄⋊Dic₇₃	C₂⁴	1	8	yes	—
95	#11924007	—	2	16	no	—
96	#11924005	—	2	16	no	—
97	C₄×Dic₇₃	—	2	16	no	—
98	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
99	C₄×Dic₇₃	C₂×D₄	1	4	yes	—
100	C₄×Dic₇₃	C₂×D₄	1	4	yes	—
101	C₄×Dic₇₃	C₂⁴	1	2	yes	—
102	C₂×Q₈×C₇₃	—	2	8	no	—
103	C₂×Q₈×C₇₃	—	4	8	no	—
104	C₂²×C₄×C₇₃	—	2	4	no	—
105	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂×Q₈	1	8	yes	—
109	C₂×C₈×C₇₃	C₂⁴	1	4	yes	—
110	C₄⋊C₄×C₇₃	—	2	8	no	—
111	C₄⋊C₄×C₇₃	—	2	4	no	—
112	C₄⋊C₄×C₇₃	C₂⁴	1	8	yes	—
113	C₄²×C₇₃	C₂×D₄	1	2	yes	—
114	C₄²×C₇₃	C₂⁴	1	2	yes	—
115	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
116	C₂²×D₇₃	—	2	2	no	—
117	C₂²×D₇₃	—	2	2	no	—
118	C₂²×D₇₃	C₄⋊Q₈	1	4	yes	—
119	C₂²×D₇₃	C₂²×D₄	1	2	yes	—
120	Q₈×C₇₃	—	4	16	no	—
121	C₂×C₄×C₇₃	—	2	8	no	—
122	C₂×C₄×C₇₃	—	4	8	no	—
123	C₂×C₄×C₇₃	#135	1	4	yes	—
124	C₂×C₄×C₇₃	#135	1	4	yes	—
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
128	C₂×C₄×C₇₃	C₂²×Q₈	1	4	yes	—
129	C₂×C₄×C₇₃	C₂⁵	1	1	yes	characteristic
130	C₂×Dic₇₃	—	2	2	no	—
131	C₂×Dic₇₃	—	2	2	no	—
132	C₂×Dic₇₃	—	2	8	no	—
133	C₂×Dic₇₃	—	4	8	no	—
134	C₂×Dic₇₃	C₄⋊Q₈	1	4	yes	—
135	C₂×Dic₇₃	C₂²×D₄	1	2	yes	—
136	C₂×Dic₇₃	C₂²×D₄	1	2	yes	—
137	C₂×Dic₇₃	C₂²×D₄	1	2	yes	—
138	C₄×D₇₃	—	2	8	no	—
139	C₄×D₇₃	—	2	8	no	—
140	C₄×D₇₃	C₄⋊Q₈	1	8	yes	—
141	C₄×D₇₃	C₂²×Q₈	1	8	yes	—
142	Dic₁₄₆	—	4	16	no	—
143	C₈×C₇₃	C₂²×Q₈	1	8	yes	—
144	#10596831	C₂²×Q₈	1	8	yes	—
145	C₂²×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
146	C₂²×C₇₃	C₂×#135	1	2	yes	—
147	C₂²×C₇₃	C₂×#135	1	2	yes	—
148	C₂²×C₇₃	C₂³×D₄	1	1	yes	characteristic
149	C₂×D₇₃	—	2	8	no	—
150	C₂×D₇₃	—	2	2	no	—
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	C₂×C₄⋊Q₈	1	4	yes	—
153	C₄×C₇₃	—	4	8	no	—
154	C₄×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
155	C₄×C₇₃	C₂×#135	1	4	yes	—
156	C₄×C₇₃	C₂×#135	1	4	yes	—
157	C₄×C₇₃	C₂³×Q₈	1	2	yes	—
158	Dic₇₃	—	2	2	no	—
159	Dic₇₃	—	2	2	no	—
160	Dic₇₃	—	4	8	no	—
161	Dic₇₃	C₂×C₄⋊Q₈	1	4	yes	—
162	C₂×#3160	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	#3160	1	4	yes	complemented
164	C₂×C₇₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
165	C₂×C₇₃	C₂²×#135	1	1	yes	characteristic
166	C₂×C₇₃	C₂²×#135	1	1	yes	characteristic
167	D₇₃	—	2	4	no	—
168	C₂×#533	—	73	2	no	—
169	#3160	—	73	16	no	—
170	C₂×#500	—	73	1	no	—
171	C₂×#500	D₇₃	1	1	yes	characteristic, complemented
172	C₂×#499	—	73	4	no	—
173	C₂×#498	—	73	2	no	—
174	C₂×#427	—	73	4	no	—
175	C₂²×C₄×C₈	—	73	1	no	—
176	C₇₃	C₂×#3160	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₈	—	73	2	no	—
178	#533	—	73	16	no	—
179	#533	—	146	8	no	—
180	C₂×C₄⋊Q₈	—	73	2	no	—
181	C₂×C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
182	C₂×#125	—	73	4	no	—
183	C₂×C₂²⋊Q₈	—	146	2	no	—
184	C₂×C₂²⋊Q₈	—	146	2	no	—
185	C₂×C₄×Q₈	—	146	2	no	—
186	C₂×C₄×Q₈	—	146	2	no	—
187	C₂×#117	—	73	4	no	—
188	C₂²×C₄²	—	73	1	no	—
189	#500	—	73	8	no	—
190	#500	C₂×D₇₃	1	8	yes	complemented
191	#499	—	73	32	no	—
192	#498	—	73	16	no	—
193	#427	—	73	32	no	—
194	C₂×#107	—	73	4	no	—
195	C₂×#107	C₂×D₇₃	1	4	yes	—
196	C₂×#106	—	73	8	no	—
197	C₂×C₄×C₈	—	73	2	no	—
198	C₂×C₄×C₈	—	73	1	no	—
199	C₂×C₄×C₈	—	73	8	no	—
200	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
201	C₂²×Q₈	—	2	2	no	—
202	C₂²×Q₈	—	146	2	no	—
203	C₂³×C₄	—	73	1	no	—
204	C₂³×C₄	—	73	1	no	—
205	C₂³×C₄	—	73	1	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₂²×C₈	—	73	16	no	—
208	C₂²×C₈	—	73	8	no	—
209	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	73	8	no	—
212	C₄⋊Q₈	—	146	4	no	—
213	C₄⋊Q₈	C₂²×D₇₃	1	8	yes	—
214	#125	—	73	16	no	—
215	#125	—	146	8	no	—
216	C₂²⋊Q₈	—	146	16	no	—
217	C₂²⋊Q₈	—	146	16	no	—
218	C₄×Q₈	—	146	16	no	—
219	C₄×Q₈	—	146	16	no	—
220	#117	—	73	16	no	—
221	#117	—	146	8	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	73	4	no	—
224	C₂×C₄⋊C₄	—	146	4	no	—
225	C₂×C₄⋊C₄	—	146	4	no	—
226	C₂×C₄⋊C₄	—	146	2	no	—
227	C₂×C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
228	C₂×C₂²⋊C₄	—	146	4	no	—
229	C₂×C₄²	—	73	1	no	—
230	C₂×C₄²	—	73	1	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₄²	—	73	4	no	—
233	C₂×C₄²	—	73	4	no	—
234	C₂×C₄²	—	146	4	no	—
235	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
236	#107	—	73	16	no	—
237	#107	C₂²×D₇₃	1	16	yes	—
238	#106	—	73	32	no	—
239	C₄×C₈	—	73	8	no	—
240	C₄×C₈	—	73	4	no	—
241	C₄×C₈	C₂²×D₇₃	1	4	yes	—
242	C₂⁴	—	73	1	no	—
243	C₂×Q₈	—	2	8	no	—
244	C₂×Q₈	—	4	8	no	—
245	C₂×Q₈	—	292	8	no	—
246	C₂×Q₈	—	146	8	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	73	1	no	—
249	C₂²×C₄	—	73	2	no	—
250	C₂²×C₄	—	73	8	no	—
251	C₂²×C₄	—	73	2	no	—
252	C₂²×C₄	—	73	1	no	—
253	C₂²×C₄	—	73	1	no	—
254	C₂²×C₄	—	73	4	no	—
255	C₂²×C₄	—	73	4	no	—
256	C₂²×C₄	—	146	4	no	—
257	C₂²×C₄	—	146	4	no	—
258	C₂²×C₄	—	146	4	no	—
259	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
260	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
261	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
262	C₂×C₈	—	73	8	no	—
263	C₂×C₈	—	73	4	no	—
264	C₂×C₈	—	73	32	no	—
265	C₂×C₈	Q₈×D₇₃	1	8	yes	—
266	C₂×C₈	C₂³×D₇₃	1	4	yes	—
267	C₄⋊C₄	—	2	8	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	73	8	no	—
270	C₄⋊C₄	—	146	16	no	—
271	C₄⋊C₄	—	146	16	no	—
272	C₄⋊C₄	—	146	4	no	—
273	C₄⋊C₄	—	146	8	no	—
274	C₄⋊C₄	C₂³×D₇₃	1	8	yes	—
275	C₂²⋊C₄	—	146	16	no	—
276	C₄²	—	73	4	no	—
277	C₄²	—	73	4	no	—
278	C₄²	—	73	2	no	—
279	C₄²	—	73	2	no	—
280	C₄²	—	146	16	no	—
281	C₄²	D₄×D₇₃	1	2	yes	—
282	C₄²	C₂³×D₇₃	1	2	yes	—
283	C₂³	—	73	4	no	—
284	C₂³	—	73	2	no	—
285	C₂³	—	146	2	no	—
286	C₂³	—	146	2	no	—
287	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
288	Q₈	—	4	16	no	—
289	Q₈	—	292	16	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	292	8	no	—
293	C₂×C₄	—	73	4	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	73	8	no	—
296	C₂×C₄	—	73	8	no	—
297	C₂×C₄	—	73	2	no	—
298	C₂×C₄	—	73	2	no	—
299	C₂×C₄	—	146	8	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	—	146	2	no	—
302	C₂×C₄	—	146	8	no	—
303	C₂×C₄	—	146	8	no	—
304	C₂×C₄	#14556610	1	4	yes	—
305	C₂×C₄	#14556614	1	4	yes	—
306	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
308	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
309	C₂×C₄	C₂×Q₈×D₇₃	1	4	yes	—
310	C₂×C₄	C₂⁴×D₇₃	1	1	yes	characteristic
311	C₈	—	73	8	no	—
312	C₈	C₂×Q₈×D₇₃	1	8	yes	—
313	C₂²	—	73	4	no	—
314	C₂²	—	146	8	no	—
315	C₂²	—	146	2	no	—
316	C₂²	—	146	2	no	—
317	C₂²	C₄⋊Q₈×D₇₃	1	2	yes	—
318	C₂²	C₂×#14556610	1	2	yes	—
319	C₂²	C₂×#14556614	1	2	yes	—
320	C₂²	C₂²×D₄×D₇₃	1	1	yes	characteristic
321	C₄	—	4	8	no	—
322	C₄	—	292	8	no	—
323	C₄	—	73	4	no	—
324	C₄	—	146	2	no	—
325	C₄	—	146	2	no	—
326	C₄	C₄⋊Q₈×D₇₃	1	2	yes	—
327	C₄	C₂×#14556610	1	4	yes	—
328	C₄	C₂×#14556614	1	4	yes	—
329	C₄	C₂²×Q₈×D₇₃	1	2	yes	—
330	C₂	—	146	4	no	—
331	C₂	#22620442	1	4	yes	direct product factor
332	C₂	C₂×C₄⋊Q₈×D₇₃	1	1	yes	characteristic
333	C₂	C₂²×#14556610	1	1	yes	characteristic
334	C₂	C₂²×#14556614	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₇₃
commutator, cosocle	C₂×C₄×C₇₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₇₃
Fitting	C₂×#500×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×D₄
prime radical	C₂³×D₇₃	C₂×D₄
hypercenter	C₂×#500	D₇₃
hypocenter	C₇₃	C₂×#3160
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. #22620442	C₂	direct (N × H)
3. C₂×#17988992	C₂	semidirect
4. C₂×#17988988	C₂	semidirect
5. C₂×#17988983	C₂	semidirect
8. C₂×#500×C₇₃	C₂	semidirect
20. #17988992	C₂²	semidirect
21. #17988988	C₂²	semidirect
22. #17988983	C₂²	semidirect
30. #500×C₇₃	C₂²	semidirect
38. C₂×C₈×D₇₃	Q₈	semidirect
90. C₈×D₇₃	C₂×Q₈	semidirect
163. C₂×C₇₃	#3160	semidirect
171. C₂×#500	D₇₃	semidirect
176. C₇₃	C₂×#3160	semidirect
190. #500	C₂×D₇₃	semidirect
331. C₂	#22620442	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×#17989142	2
#22620442	2
C₂×#17988992	2
C₂×#17988988	2
C₂×#17988983	2
C₂×#17988624	2
C₂×C₄×C₈×D₇₃	2
C₂×#500×C₇₃	2
C₂×#3160	73

Subgroups — C2×#22620442 #33955586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22620442 #33955586