Subgroups — C₂×#22608226 #33051602

Order 18688.

866 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988640	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11924012	C₂	1	1	yes	maximal, characteristic
3	#22608226	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988309	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988308	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#431×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11924012	C₄	1	2	yes	—
10	#17988640	C₄	1	4	yes	—
11	#17988640	C₂²	1	4	yes	—
12	C₂×C₄⋊#10596831	C₄	1	2	yes	—
13	C₂×C₂²⋊#10596831	C₄	1	2	yes	—
14	C₄×#11924012	C₄	1	4	yes	—
15	C₄×#11924012	C₂²	1	4	yes	—
16	C₂×#14556502	C₄	1	2	yes	—
17	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
19	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
20	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
21	#17988309	C₂²	1	4	yes	complemented
22	#17988309	C₂²	1	4	yes	complemented
23	#17988308	C₂²	1	4	yes	complemented
24	C₈×#10596831	C₂²	1	4	yes	complemented
25	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
26	#431×C₇₃	C₄	1	4	yes	—
27	#431×C₇₃	C₂²	1	4	yes	—
28	C₂×C₄⋊C₈×C₇₃	C₄	1	2	yes	—
29	C₂×C₂²⋊C₈×C₇₃	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	C₂×#11924012	—	2	8	no	—
33	C₂×#11924012	C₂×C₄	1	8	yes	—
34	C₂²×#10596831	C₂×C₄	1	2	yes	—
35	C₂²×#10596831	C₂×C₄	1	2	yes	—
36	C₂²×#10596831	C₂×C₄	1	2	yes	—
37	C₂²×#10596831	C₂×C₄	1	2	yes	—
38	C₄⋊#10596831	C₂×C₄	1	8	yes	—
39	C₂²⋊#10596831	C₂×C₄	1	8	yes	—
40	#14556502	C₂×C₄	1	8	yes	—
41	C₄×#10596831	C₂×C₄	1	2	yes	—
42	C₄×#10596831	C₂×C₄	1	2	yes	—
43	C₄×#10596831	C₂×C₄	1	2	yes	—
44	C₄×#10596831	C₂×C₄	1	2	yes	—
45	C₄×#10596831	C₂³	1	2	yes	—
46	C₄×#10596831	C₂³	1	2	yes	—
47	C₄×#10596831	C₂³	1	2	yes	—
48	C₄×#10596831	C₂³	1	2	yes	—
49	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
53	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
63	C₂²⋊C₈×C₇₃	C₂×C₄	1	8	yes	—
64	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₇₃	C₂³	1	2	yes	—
67	C₄×C₈×C₇₃	C₂³	1	2	yes	—
68	#11924012	—	2	16	no	—
69	C₂×#10596831	—	2	4	no	—
70	C₂×#10596831	—	2	4	no	—
71	C₂×#10596831	—	2	4	no	—
72	C₂×#10596831	M₁₆	1	8	yes	—
73	C₂×#10596831	C₂²×C₄	1	4	yes	—
74	C₂×#10596831	C₂²×C₄	1	4	yes	—
75	C₂×#10596831	C₂²×C₄	1	4	yes	—
76	C₂×#10596831	C₂²×C₄	1	4	yes	—
77	C₂⁴×C₇₃	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	—	2	4	no	—
79	C₂²×C₄×C₇₃	—	2	4	no	—
80	C₂²×C₄×C₇₃	—	2	4	no	—
81	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
83	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
86	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₃	—	2	4	no	—
94	C₂×C₈×C₇₃	—	2	4	no	—
95	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
96	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
97	C₄²×C₇₃	C₄²	1	2	yes	—
98	C₄²×C₇₃	C₄²	1	2	yes	—
99	C₄²×C₇₃	C₄²	1	2	yes	—
100	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
107	C₂³×C₇₃	—	2	2	no	—
108	C₂³×C₇₃	—	2	2	no	—
109	C₂³×C₇₃	—	2	2	no	—
110	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
111	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	2	2	no	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	4	no	—
116	C₂×C₄×C₇₃	—	2	4	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	2	4	no	—
119	C₂×C₄×C₇₃	—	2	4	no	—
120	C₂×C₄×C₇₃	—	2	4	no	—
121	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
128	C₂×C₄×C₇₃	#131	1	4	yes	—
129	C₂×C₄×C₇₃	#131	1	4	yes	—
130	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
133	C₈×C₇₃	—	2	4	no	—
134	C₈×C₇₃	—	2	4	no	—
135	#10596831	—	2	4	no	—
136	#10596831	—	2	4	no	—
137	#10596831	—	2	4	no	—
138	#10596831	C₂×M₁₆	1	8	yes	—
139	C₂²×C₇₃	—	2	2	no	—
140	C₂²×C₇₃	—	2	2	no	—
141	C₂²×C₇₃	—	2	2	no	—
142	C₂²×C₇₃	—	4	4	no	—
143	C₂²×C₇₃	C₄×M₁₆	1	2	yes	—
144	C₂²×C₇₃	#405	1	2	yes	—
145	C₂²×C₇₃	#405	1	2	yes	—
146	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	C₄×M₁₆	1	2	yes	—
151	C₄×C₇₃	#405	1	2	yes	—
152	C₄×C₇₃	#405	1	2	yes	—
153	C₄×C₇₃	C₂²×M₁₆	1	2	yes	—
154	C₄×C₇₃	C₂×#131	1	2	yes	—
155	C₄×C₇₃	C₂×#131	1	2	yes	—
156	C₂×#1453	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	—	4	2	no	—
158	C₂×C₇₃	#1453	1	4	yes	complemented
159	C₂×C₇₃	C₂×C₄×M₁₆	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#405	1	1	yes	characteristic
161	C₂×C₇₃	C₂×#405	1	1	yes	characteristic
162	C₂×#431	—	73	1	no	—
163	C₂×#431	D₇₃	1	1	yes	characteristic
164	C₂×C₄×M₁₆	—	73	1	no	—
165	#1453	—	73	8	no	—
166	C₂×#322	—	73	1	no	—
167	C₂×#322	—	73	1	no	—
168	C₂×#322	—	73	1	no	—
169	C₂×C₈²	—	73	1	no	—
170	C₇₃	C₂×#1453	1	1	yes	characteristic, complemented, Sylow
171	C₂²×M₁₆	—	73	2	no	—
172	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
173	#431	—	73	4	no	—
174	#431	—	73	4	no	—
175	#431	Dic₇₃	1	4	yes	—
176	#431	C₂×D₇₃	1	4	yes	—
177	C₂×C₄⋊C₈	—	73	2	no	—
178	C₂×C₄⋊C₈	Dic₇₃	1	2	yes	—
179	C₂×C₂²⋊C₈	—	73	2	no	—
180	C₂×C₂²⋊C₈	Dic₇₃	1	2	yes	—
181	C₄×M₁₆	—	73	4	no	—
182	C₄×M₁₆	—	73	4	no	—
183	C₂×#97	—	73	2	no	—
184	C₂×C₄×C₈	—	73	1	no	—
185	C₂×C₄×C₈	—	73	1	no	—
186	C₂×C₄×C₈	—	73	1	no	—
187	C₂×C₄×C₈	—	73	1	no	—
188	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
190	#322	—	73	4	no	—
191	#322	—	73	4	no	—
192	#322	—	73	4	no	—
193	C₈²	—	73	4	no	—
194	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
195	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
196	C₂³×C₄	C₂×Dic₇₃	1	1	yes	characteristic
197	C₂×M₁₆	—	73	8	no	—
198	C₂×M₁₆	—	146	8	no	—
199	C₂²×C₈	—	73	2	no	—
200	C₂²×C₈	—	73	2	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	—	73	2	no	—
203	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
204	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
205	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
206	C₂×C₄²	C₄×D₇₃	1	2	yes	—
207	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
208	C₂×C₄²	C₄×D₇₃	1	2	yes	—
209	C₂×C₄²	C₂×Dic₇₃	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
211	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
212	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
213	C₄⋊C₈	—	73	8	no	—
214	C₄⋊C₈	C₂×Dic₇₃	1	8	yes	—
215	C₂²⋊C₈	—	73	8	no	—
216	C₂²⋊C₈	C₂×Dic₇₃	1	8	yes	—
217	#97	—	73	8	no	—
218	C₄×C₈	—	73	2	no	—
219	C₄×C₈	—	73	2	no	—
220	C₄×C₈	—	73	2	no	—
221	C₄×C₈	—	73	2	no	—
222	C₄×C₈	—	73	2	no	—
223	C₄×C₈	—	73	2	no	—
224	C₄×C₈	—	73	2	no	—
225	C₄×C₈	—	73	2	no	—
226	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
227	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
228	C₄×C₈	C₂²×D₇₃	1	2	yes	—
229	C₄×C₈	C₂²×D₇₃	1	2	yes	—
230	C₂⁴	C₄×Dic₇₃	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
235	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
236	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
237	C₂²×C₄	C₄×Dic₇₃	1	1	yes	characteristic
238	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
239	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
242	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
244	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
245	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
246	M₁₆	—	146	16	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	73	8	no	—
250	C₂×C₈	—	73	4	no	—
251	C₂×C₈	—	73	4	no	—
252	C₂×C₈	—	73	4	no	—
253	C₂×C₈	—	73	4	no	—
254	C₂×C₈	—	146	4	no	—
255	C₂×C₈	—	146	4	no	—
256	C₂×C₈	—	146	4	no	—
257	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
258	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
259	C₄²	C₄×Dic₇₃	1	2	yes	—
260	C₄²	C₄×Dic₇₃	1	2	yes	—
261	C₄²	C₄×Dic₇₃	1	2	yes	—
262	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
263	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
264	C₄²	C₂×C₄×D₇₃	1	2	yes	—
265	C₄²	C₂×C₄×D₇₃	1	2	yes	—
266	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
267	C₄²	C₂²×Dic₇₃	1	2	yes	—
268	C₄²	C₂³×D₇₃	1	1	yes	characteristic
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
273	C₂³	C₂×C₄×Dic₇₃	1	1	yes	characteristic
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
284	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
285	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
286	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
287	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₇₃	1	4	yes	—
290	C₂×C₄	#14556597	1	4	yes	—
291	C₂×C₄	#14556598	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
294	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	73	8	no	—
298	C₈	—	146	4	no	—
299	C₈	—	146	4	no	—
300	C₈	—	146	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	M₁₆×Dic₇₃	1	2	yes	—
306	C₂²	#17988542	1	2	yes	—
307	C₂²	#17988543	1	2	yes	—
308	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	M₁₆×Dic₇₃	1	2	yes	—
313	C₄	#17988542	1	2	yes	—
314	C₄	#17988543	1	2	yes	—
315	C₄	C₂×M₁₆×D₇₃	1	2	yes	—
316	C₄	C₂×#14556597	1	2	yes	—
317	C₄	C₂×#14556598	1	2	yes	—
318	C₂	—	4	2	no	—
319	C₂	#22608226	1	4	yes	direct product factor
320	C₂	C₂×M₁₆×Dic₇₃	1	1	yes	characteristic
321	C₂	C₂×#17988542	1	1	yes	characteristic
322	C₂	C₂×#17988543	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#431×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄²
prime radical	C₂⁴×C₇₃	C₄²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×#431	D₇₃
hypocenter	C₇₃	C₂×#1453
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #22608226	C₂	direct (N × H)
4. C₂×#17988309	C₂	semidirect
5. C₂×#17988309	C₂	semidirect
6. C₂×#17988308	C₂	semidirect
7. C₂×C₈×#10596831	C₂	semidirect
21. #17988309	C₂²	semidirect
22. #17988309	C₂²	semidirect
23. #17988308	C₂²	semidirect
24. C₈×#10596831	C₂²	semidirect
158. C₂×C₇₃	#1453	semidirect
170. C₇₃	C₂×#1453	semidirect
319. C₂	#22608226	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×#17988640	2
C₂×C₄×#11924012	2
#22608226	2
C₂×#17988309	2
C₂×#17988308	2
C₂×C₈×#10596831	2
C₂×#431×C₇₃	2
C₂×#1453	73

Subgroups — C2×#22608226 #33051602

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22608226 #33051602