Subgroups — C₂×#22611758 #33239142

Order 18688.

754 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×#14556530	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
4	#22611758	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988366	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₂₉₂	C₄	1	1	yes	characteristic
12	C₂×#14556530	—	2	4	no	—
13	C₂×#14556530	C₂²	1	4	yes	complemented
14	C₂²×#11924003	C₂²	1	1	yes	characteristic
15	C₂²×#11924003	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	—	2	4	no	—
17	C₂×C₁₆×D₇₃	C₂²	1	4	yes	complemented
18	C₈×D₂₉₂	C₄	1	4	yes	—
19	C₈×D₂₉₂	C₂²	1	4	yes	—
20	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
21	C₂×#14556503	C₄	1	1	yes	characteristic
22	C₂×#14556503	C₄	1	1	yes	characteristic
23	C₄⋊#11924003	C₂²	1	4	yes	complemented
24	#17988366	C₂²	1	4	yes	complemented
25	#17988366	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	#346×C₇₃	C₂²	1	4	yes	complemented
30	C₂²×D₂₉₂	C₈	1	1	yes	characteristic
31	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
34	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	—	2	2	no	—
36	C₂×C₈×D₇₃	—	2	2	no	—
37	C₂×C₈×D₇₃	—	2	2	no	—
38	C₂×C₈×D₇₃	—	2	2	no	—
39	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂³	1	2	yes	—
42	C₂×C₈×D₇₃	C₂³	1	2	yes	—
43	C₄×D₂₉₂	C₈	1	4	yes	—
44	C₄×D₂₉₂	C₂×C₄	1	4	yes	—
45	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
46	C₂×#11924005	C₈	1	1	yes	characteristic
47	C₂×#11924005	C₈	1	1	yes	characteristic
48	#14556530	—	2	8	no	—
49	C₂×#11924003	—	2	2	no	—
50	C₂×#11924003	—	2	2	no	—
51	C₂×#11924003	C₂³	1	2	yes	—
52	C₂×#11924003	C₂³	1	2	yes	—
53	C₁₆×D₇₃	—	2	8	no	—
54	C₄⋊#10596831	C₂×C₄	1	4	yes	—
55	#14556503	C₂×C₄	1	4	yes	—
56	#14556503	C₂×C₄	1	4	yes	—
57	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₁₆×C₇₃	—	2	2	no	—
61	C₂×C₁₆×C₇₃	D₄	1	4	yes	complemented
62	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
63	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
64	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₇₃	C₂³	1	2	yes	—
66	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
67	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
68	C₂×D₂₉₂	—	2	4	no	—
69	C₂×D₂₉₂	C₂×C₈	1	4	yes	—
70	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
71	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₄×D₇₃	—	2	2	no	—
73	C₂×C₄×D₇₃	—	2	2	no	—
74	C₂×C₄×D₇₃	—	2	2	no	—
75	C₂×C₄×D₇₃	—	2	2	no	—
76	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
80	C₂×#10596831	—	2	2	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	C₂²×C₄	1	2	yes	—
83	C₂×#10596831	C₂²×C₄	1	2	yes	—
84	C₈×D₇₃	—	2	4	no	—
85	C₈×D₇₃	—	2	4	no	—
86	C₈×D₇₃	—	2	2	no	—
87	C₈×D₇₃	—	2	2	no	—
88	C₈×D₇₃	—	2	2	no	—
89	C₈×D₇₃	—	2	2	no	—
90	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
91	#11924005	C₂×C₈	1	4	yes	—
92	#11924005	C₂×C₈	1	4	yes	—
93	#11924003	—	2	2	no	—
94	#11924003	—	2	2	no	—
95	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	—	2	2	no	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₇₃	#41	1	2	yes	—
103	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
104	C₄²×C₇₃	C₂×C₈	1	2	yes	—
105	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
106	C₁₆×C₇₃	—	2	2	no	—
107	C₁₆×C₇₃	C₂×D₄	1	4	yes	complemented
108	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₇₃	—	2	2	no	—
110	C₂²×D₇₃	—	2	2	no	—
111	C₂²×D₇₃	—	2	2	no	—
112	C₂²×D₇₃	—	2	2	no	—
113	C₂²×D₇₃	—	2	2	no	—
114	C₂²×D₇₃	M₃₂	1	4	yes	complemented
115	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
116	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₇₃	—	2	2	no	—
118	C₂×C₄×C₇₃	—	2	2	no	—
119	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
121	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
122	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	#131	1	2	yes	—
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×Dic₇₃	—	2	2	no	—
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
128	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
129	D₂₉₂	—	2	8	no	—
130	C₄×D₇₃	—	2	2	no	—
131	C₄×D₇₃	—	2	2	no	—
132	C₄×D₇₃	—	2	2	no	—
133	C₄×D₇₃	—	2	2	no	—
134	C₄×D₇₃	—	2	4	no	—
135	C₄×D₇₃	—	2	4	no	—
136	C₈×C₇₃	—	2	2	no	—
137	C₈×C₇₃	C₄×D₄	1	2	yes	—
138	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
140	#10596831	—	2	2	no	—
141	#10596831	—	2	2	no	—
142	C₂²×C₇₃	C₈×D₄	1	2	yes	—
143	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
144	C₂²×C₇₃	#504	1	2	yes	—
145	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
146	C₂×D₇₃	—	2	2	no	—
147	C₂×D₇₃	—	2	2	no	—
148	C₂×D₇₃	—	2	8	no	—
149	C₂×D₇₃	—	2	2	no	—
150	C₂×D₇₃	—	2	2	no	—
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	—	4	4	no	—
153	C₂×D₇₃	C₂×M₃₂	1	4	yes	complemented
154	C₄×C₇₃	—	2	2	no	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	C₈×D₄	1	2	yes	—
157	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
158	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
159	Dic₇₃	—	2	2	no	—
160	Dic₇₃	—	2	2	no	—
161	C₂×#2168	—	73	1	no	maximal, Sylow
162	C₂×C₇₃	#2168	1	4	yes	complemented
163	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
164	C₂×C₇₃	C₂²×M₃₂	1	1	yes	characteristic
165	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
166	D₇₃	—	2	4	no	—
167	D₇₃	—	4	2	no	—
168	C₂²×M₃₂	—	73	1	no	—
169	C₂³×C₁₆	—	73	1	no	—
170	C₂×C₈×D₄	—	73	1	no	—
171	#2168	—	73	8	no	—
172	C₂×C₄⋊C₁₆	—	73	1	no	—
173	C₂×C₂²⋊C₁₆	—	73	1	no	—
174	C₂×C₂²⋊C₁₆	—	73	1	no	—
175	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
176	C₇₃	C₂×#2168	1	1	yes	characteristic, complemented, Sylow
177	C₂³×C₈	—	73	1	no	—
178	C₂³×C₈	—	73	1	no	—
179	C₂×C₄×D₄	—	73	1	no	—
180	C₂×M₃₂	—	73	4	no	—
181	C₂×M₃₂	—	146	4	no	—
182	C₂²×C₁₆	—	73	1	no	—
183	C₂²×C₁₆	—	73	1	no	—
184	C₂²×C₁₆	—	73	4	no	—
185	C₂²×C₁₆	—	146	4	no	—
186	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
187	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
188	C₈×D₄	—	73	4	no	—
189	C₈×D₄	—	73	4	no	—
190	C₂×C₄⋊C₈	—	73	1	no	—
191	C₂×C₂²⋊C₈	—	73	1	no	—
192	C₂×C₂²⋊C₈	—	73	1	no	—
193	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
194	C₄⋊C₁₆	—	73	4	no	—
195	C₂²⋊C₁₆	—	73	4	no	—
196	C₂²⋊C₁₆	—	73	4	no	—
197	#346	C₂×D₇₃	1	4	yes	complemented
198	C₂²×D₄	—	73	1	no	—
199	C₂³×C₄	—	73	1	no	—
200	C₂³×C₄	—	73	1	no	—
201	C₂²×C₈	—	73	1	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	—	73	2	no	—
204	C₂²×C₈	—	73	2	no	—
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₂²×C₈	—	146	2	no	—
208	C₂²×C₈	—	146	2	no	—
209	C₂²×C₈	—	146	2	no	—
210	C₂²×C₈	—	146	2	no	—
211	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
212	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
213	C₄×D₄	—	73	4	no	—
214	C₄×D₄	—	73	4	no	—
215	C₂×C₄⋊C₄	—	73	1	no	—
216	C₂×C₂²⋊C₄	—	73	1	no	—
217	C₂×C₂²⋊C₄	—	73	1	no	—
218	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
219	M₃₂	—	146	8	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	73	2	no	—
222	C₂×C₁₆	—	73	2	no	—
223	C₂×C₁₆	—	146	2	no	—
224	C₂×C₁₆	—	146	2	no	—
225	C₂×C₁₆	—	146	8	no	—
226	C₂×C₁₆	D₂₉₂	1	4	yes	complemented
227	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
228	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
229	C₄⋊C₈	—	73	4	no	—
230	C₂²⋊C₈	—	73	4	no	—
231	C₂²⋊C₈	—	73	4	no	—
232	C₄×C₈	C₄×D₇₃	1	2	yes	—
233	C₄×C₈	C₂²×D₇₃	1	2	yes	—
234	C₂⁴	—	73	1	no	—
235	C₂⁴	—	73	1	no	—
236	C₂×D₄	—	73	4	no	—
237	C₂×D₄	—	146	4	no	—
238	C₂²×C₄	—	73	1	no	—
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	—	73	2	no	—
241	C₂²×C₄	—	73	2	no	—
242	C₂²×C₄	—	73	2	no	—
243	C₂²×C₄	—	73	2	no	—
244	C₂²×C₄	—	146	2	no	—
245	C₂²×C₄	—	146	2	no	—
246	C₂²×C₄	—	146	2	no	—
247	C₂²×C₄	—	146	2	no	—
248	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
249	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	73	2	no	—
253	C₂×C₈	—	73	2	no	—
254	C₂×C₈	—	146	2	no	—
255	C₂×C₈	—	146	2	no	—
256	C₂×C₈	—	146	4	no	—
257	C₂×C₈	—	146	4	no	—
258	C₂×C₈	—	146	2	no	—
259	C₂×C₈	—	146	2	no	—
260	C₂×C₈	—	146	2	no	—
261	C₂×C₈	—	146	2	no	—
262	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
263	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
264	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
265	C₂×C₈	#11924028	1	2	yes	—
266	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
267	C₄⋊C₄	—	73	4	no	—
268	C₂²⋊C₄	—	73	4	no	—
269	C₂²⋊C₄	—	73	4	no	—
270	C₄²	C₈×D₇₃	1	2	yes	—
271	C₄²	C₂×C₄×D₇₃	1	2	yes	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	146	2	no	—
274	C₁₆	—	146	2	no	—
275	C₁₆	C₂×D₂₉₂	1	4	yes	complemented
276	C₂³	—	73	4	no	—
277	C₂³	—	73	2	no	—
278	C₂³	—	73	2	no	—
279	C₂³	—	146	2	no	—
280	C₂³	—	146	2	no	—
281	C₂³	—	146	2	no	—
282	C₂³	—	146	2	no	—
283	C₂³	—	146	2	no	—
284	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
285	D₄	—	146	8	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	73	2	no	—
290	C₂×C₄	—	146	2	no	—
291	C₂×C₄	—	146	2	no	—
292	C₂×C₄	—	146	4	no	—
293	C₂×C₄	—	146	4	no	—
294	C₂×C₄	—	146	2	no	—
295	C₂×C₄	—	146	2	no	—
296	C₂×C₄	—	146	2	no	—
297	C₂×C₄	—	146	2	no	—
298	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
299	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
300	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
301	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
302	C₂×C₄	#14556597	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	146	2	no	—
306	C₈	—	146	2	no	—
307	C₈	C₄×D₂₉₂	1	2	yes	—
308	C₈	C₂²×D₂₉₂	1	1	yes	characteristic
309	C₈	C₂×#11924028	1	1	yes	characteristic
310	C₂²	—	292	4	no	—
311	C₂²	—	73	4	no	—
312	C₂²	—	146	2	no	—
313	C₂²	—	146	2	no	—
314	C₂²	—	146	2	no	—
315	C₂²	—	146	8	no	—
316	C₂²	—	146	2	no	—
317	C₂²	—	146	2	no	—
318	C₂²	C₈×D₂₉₂	1	2	yes	—
319	C₂²	M₃₂×D₇₃	1	2	yes	—
320	C₂²	#17989005	1	2	yes	—
321	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	146	2	no	—
325	C₄	—	146	2	no	—
326	C₄	C₈×D₂₉₂	1	2	yes	—
327	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
328	C₄	C₂×#14556597	1	1	yes	characteristic
329	C₂	—	292	2	no	—
330	C₂	—	146	4	no	—
331	C₂	#22611758	1	4	yes	direct product factor
332	C₂	C₂×C₈×D₂₉₂	1	1	yes	characteristic
333	C₂	C₂×M₃₂×D₇₃	1	1	yes	characteristic
334	C₂	C₂×#17989005	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#346×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂²×D₂₉₂	C₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	C₂×#2168
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂²×#14556530	C₂	semidirect
2. C₂²×C₁₆×D₇₃	C₂	semidirect
4. #22611758	C₂	direct (N × H)
5. C₂×C₄⋊#11924003	C₂	semidirect
6. C₂×#17988366	C₂	semidirect
7. C₂×#17988366	C₂	semidirect
8. C₂×#346×C₇₃	C₂	semidirect
13. C₂×#14556530	C₂²	semidirect
17. C₂×C₁₆×D₇₃	C₂²	semidirect
23. C₄⋊#11924003	C₂²	semidirect
24. #17988366	C₂²	semidirect
25. #17988366	C₂²	semidirect
29. #346×C₇₃	C₂²	semidirect
61. C₂×C₁₆×C₇₃	D₄	semidirect
107. C₁₆×C₇₃	C₂×D₄	semidirect
114. C₂²×D₇₃	M₃₂	semidirect
153. C₂×D₇₃	C₂×M₃₂	semidirect
162. C₂×C₇₃	#2168	semidirect
175. C₂×#346	D₇₃	semidirect
176. C₇₃	C₂×#2168	semidirect
197. #346	C₂×D₇₃	semidirect
226. C₂×C₁₆	D₂₉₂	semidirect
275. C₁₆	C₂×D₂₉₂	semidirect
331. C₂	#22611758	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×#14556530	2
C₂²×C₁₆×D₇₃	2
C₂×C₈×D₂₉₂	2
#22611758	2
C₂×C₄⋊#11924003	2
C₂×#17988366	2
C₂×#346×C₇₃	2
C₂×#2168	73

Subgroups — C2×#22611758 #33239142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611758 #33239142