Subgroups — C₂×#22611695 #33238982

Order 18688.

698 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₇₃	C₂	1	2	yes	maximal
2	C₂×#17988541	C₂	1	1	yes	maximal, characteristic
3	#22611695	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988408	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988407	C₂	1	2	yes	maximal, complemented
6	C₂×#364×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11924011	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₂×#14556548	C₄	1	1	yes	characteristic
10	C₂×#14556531	C₂²	1	2	yes	—
11	C₂×#14556530	—	2	2	no	—
12	M₃₂×D₇₃	—	2	4	no	—
13	M₃₂×D₇₃	C₂²	1	8	yes	—
14	C₂×C₁₆×D₇₃	—	2	2	no	—
15	#17988541	C₄	1	4	yes	—
16	#17988541	C₂²	1	4	yes	—
17	C₂×#14556502	C₄	1	1	yes	characteristic
18	C₂×#14556501	C₄	1	1	yes	characteristic
19	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
20	#17988408	C₂²	1	4	yes	complemented
21	#17988407	C₂²	1	8	yes	complemented
22	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#364×C₇₃	C₂²	1	4	yes	complemented
25	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×#11924011	C₈	1	4	yes	—
27	C₂×#11924011	C₈	1	4	yes	—
28	C₂×#11924011	C₂×C₄	1	4	yes	—
29	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₇₃	D₄	1	2	yes	—
33	C₂×C₈×D₇₃	D₄	1	2	yes	—
34	C₂×C₈×D₇₃	Q₈	1	2	yes	—
35	C₂×C₈×D₇₃	Q₈	1	2	yes	—
36	C₂×C₈×D₇₃	C₂³	1	2	yes	—
37	#14556548	—	2	2	no	—
38	#14556548	C₂×C₄	1	4	yes	—
39	C₂×#11924007	—	2	1	no	—
40	C₂×#11924005	—	2	1	no	—
41	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
42	#14556531	—	2	2	no	—
43	#14556531	C₂³	1	4	yes	—
44	#14556530	—	2	8	no	—
45	C₂×#11924003	—	2	2	no	—
46	C₁₆×D₇₃	—	2	8	no	—
47	#14556502	C₂×C₄	1	4	yes	—
48	#14556501	C₂×C₄	1	4	yes	—
49	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
50	C₈×Dic₇₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
54	M₃₂×C₇₃	—	2	2	no	—
55	M₃₂×C₇₃	C₂³	1	4	yes	—
56	C₂×C₁₆×C₇₃	—	2	2	no	—
57	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₇₃	C₂³	1	2	yes	—
59	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
60	C₂²×Dic₇₃	—	2	1	no	—
61	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
62	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
66	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
67	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
68	#11924011	C₂×C₈	1	16	yes	—
69	C₂×#10596831	C₄⋊C₄	1	2	yes	—
70	C₂×#10596831	C₄⋊C₄	1	2	yes	—
71	C₂×#10596831	C₂×C₈	1	2	yes	—
72	C₂×#10596831	C₂×C₈	1	2	yes	—
73	C₂×#10596831	C₂²×C₄	1	2	yes	—
74	C₂×#10596831	C₂²×C₄	1	2	yes	—
75	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
76	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
77	C₈×D₇₃	C₂×D₄	1	4	yes	—
78	C₈×D₇₃	C₂×Q₈	1	4	yes	—
79	#11924007	—	2	4	no	—
80	#11924005	—	2	4	no	—
81	C₄×Dic₇₃	—	2	1	no	—
82	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
83	#11924003	—	2	4	no	—
84	C₂²×C₄×C₇₃	—	2	1	no	—
85	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
87	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
95	C₄²×C₇₃	—	2	1	no	—
96	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
97	C₁₆×C₇₃	—	2	4	no	—
98	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂²×D₇₃	—	4	2	no	—
100	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
101	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
102	C₂²×D₇₃	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	4	2	no	—
105	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×Dic₇₃	—	2	2	no	—
112	C₂×Dic₇₃	—	4	2	no	—
113	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
114	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
115	C₂×Dic₇₃	C₂×M₁₆	1	2	yes	—
116	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
117	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
118	C₄×D₇₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₄×D₇₃	C₂²×C₈	1	4	yes	—
120	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₈×C₇₃	C₂²×C₈	1	4	yes	—
123	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
125	#10596831	C₂×C₄⋊C₄	1	4	yes	—
126	#10596831	C₂²×C₈	1	4	yes	—
127	C₂²×C₇₃	—	2	2	no	—
128	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
129	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
130	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
131	C₂×D₇₃	—	2	4	no	—
132	C₂×D₇₃	—	4	2	no	—
133	C₂×D₇₃	—	4	2	no	—
134	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
135	C₄×C₇₃	—	4	2	no	—
136	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
137	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
139	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
140	Dic₇₃	—	4	2	no	—
141	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
142	C₂×#2153	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	1	no	—
144	C₂×C₇₃	—	2	1	no	—
145	C₂×C₇₃	#2153	1	2	yes	complemented
146	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
147	D₇₃	—	2	4	no	—
148	C₂²×M₃₂	—	73	2	no	—
149	C₂×#405	—	73	1	no	—
150	#2153	—	73	8	no	—
151	C₂×#364	—	73	2	no	—
152	C₂×#364	—	73	1	no	—
153	C₂×#364	D₇₃	1	1	yes	characteristic, complemented
154	C₇₃	C₂×#2153	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	73	1	no	—
156	C₂³×C₈	—	73	1	no	—
157	C₂×#117	—	73	1	no	—
158	C₂×M₃₂	—	73	2	no	—
159	C₂×M₃₂	—	73	8	no	—
160	C₂×M₃₂	—	146	2	no	—
161	C₂×M₃₂	—	146	4	no	—
162	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
163	C₂²×C₁₆	—	146	2	no	—
164	#405	—	73	4	no	—
165	#405	—	73	4	no	—
166	C₂×#97	—	73	1	no	—
167	C₂×#97	—	73	1	no	—
168	C₂×C₄×C₈	—	73	1	no	—
169	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
170	#364	—	73	8	no	—
171	#364	—	73	4	no	—
172	#364	C₂×D₇₃	1	4	yes	complemented
173	C₂³×C₄	—	73	1	no	—
174	C₂×M₁₆	—	73	4	no	—
175	C₂×M₁₆	—	73	4	no	—
176	C₂×M₁₆	—	73	4	no	—
177	C₂²×C₈	—	73	1	no	—
178	C₂²×C₈	—	73	1	no	—
179	C₂²×C₈	—	73	2	no	—
180	C₂²×C₈	—	73	2	no	—
181	C₂²×C₈	—	73	2	no	—
182	C₂²×C₈	—	73	2	no	—
183	C₂²×C₈	—	73	2	no	—
184	C₂²×C₈	—	73	2	no	—
185	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
187	#117	—	73	4	no	—
188	#117	—	146	2	no	—
189	C₂×C₄⋊C₄	—	146	1	no	—
190	C₂×C₂²⋊C₄	—	146	1	no	—
191	C₂×C₄²	—	73	1	no	—
192	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
193	M₃₂	—	2	2	no	—
194	M₃₂	—	73	4	no	—
195	M₃₂	—	146	8	no	—
196	M₃₂	—	146	2	no	—
197	M₃₂	C₂²×D₇₃	1	4	yes	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	146	8	no	—
200	C₂×C₁₆	—	146	2	no	—
201	#97	—	73	4	no	—
202	#97	—	73	4	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	C₄×D₇₃	1	2	yes	—
206	C₄×C₈	C₂²×D₇₃	1	2	yes	—
207	C₂⁴	—	73	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	73	1	no	—
210	C₂²×C₄	—	73	2	no	—
211	C₂²×C₄	—	73	2	no	—
212	C₂²×C₄	—	73	2	no	—
213	C₂²×C₄	—	73	2	no	—
214	C₂²×C₄	—	73	2	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	146	1	no	—
217	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
218	M₁₆	—	73	16	no	—
219	C₂×C₈	—	73	2	no	—
220	C₂×C₈	—	73	2	no	—
221	C₂×C₈	—	73	2	no	—
222	C₂×C₈	—	73	2	no	—
223	C₂×C₈	—	73	2	no	—
224	C₂×C₈	—	73	2	no	—
225	C₂×C₈	—	73	4	no	—
226	C₂×C₈	—	73	4	no	—
227	C₂×C₈	—	73	4	no	—
228	C₂×C₈	—	73	4	no	—
229	C₂×C₈	C₈×D₇₃	1	2	yes	—
230	C₂×C₈	C₈×D₇₃	1	2	yes	—
231	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
233	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
234	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
236	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
238	C₄⋊C₄	—	146	4	no	—
239	C₂²⋊C₄	—	146	4	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	73	2	no	—
242	C₄²	—	146	1	no	—
243	C₄²	C₂×C₄×D₇₃	1	2	yes	—
244	C₁₆	—	2	4	no	—
245	C₁₆	—	146	4	no	—
246	C₂³	—	292	2	no	—
247	C₂³	—	73	2	no	—
248	C₂³	—	73	2	no	—
249	C₂³	—	73	2	no	—
250	C₂³	M₁₆×D₇₃	1	1	yes	characteristic
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	292	2	no	—
254	C₂×C₄	—	73	2	no	—
255	C₂×C₄	—	73	2	no	—
256	C₂×C₄	—	73	2	no	—
257	C₂×C₄	—	73	4	no	—
258	C₂×C₄	—	73	4	no	—
259	C₂×C₄	—	73	4	no	—
260	C₂×C₄	—	73	4	no	—
261	C₂×C₄	—	146	2	no	—
262	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
263	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
266	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
268	C₈	—	73	4	no	—
269	C₈	—	73	4	no	—
270	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
271	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
272	C₈	C₂×C₈×D₇₃	1	4	yes	—
273	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
274	C₈	C₂×Q₈×D₇₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	292	2	no	—
277	C₂²	—	292	2	no	—
278	C₂²	—	73	4	no	—
279	C₂²	—	146	4	no	—
280	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
281	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
282	C₂²	C₂×M₁₆×D₇₃	1	1	yes	characteristic
283	C₄	—	4	2	no	—
284	C₄	—	292	2	no	—
285	C₄	—	73	4	no	—
286	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
287	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
288	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
289	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	146	4	no	—
293	C₂	#22611695	1	2	yes	direct product factor
294	C₂	C₂×C₄⋊C₈×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#364×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄⋊C₈
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#364	D₇₃
hypocenter	C₇₃	C₂×#2153
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
3. #22611695	C₂	direct (N × H)
4. C₂×#17988408	C₂	semidirect
5. C₂×#17988407	C₂	semidirect
6. C₂×#364×C₇₃	C₂	semidirect
20. #17988408	C₂²	semidirect
21. #17988407	C₂²	semidirect
24. #364×C₇₃	C₂²	semidirect
145. C₂×C₇₃	#2153	semidirect
153. C₂×#364	D₇₃	semidirect
154. C₇₃	C₂×#2153	semidirect
172. #364	C₂×D₇₃	semidirect
293. C₂	#22611695	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×M₃₂×D₇₃	2
C₂×#17988541	2
#22611695	2
C₂×#17988408	2
C₂×#17988407	2
C₂×#364×C₇₃	2
C₂×#2153	73

Subgroups — C2×#22611695 #33238982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611695 #33238982