Subgroups — C₂×#17989458 #22625608

Order 9344.

488 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₇₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556670	C₂	1	1	yes	maximal, characteristic
3	#17989458	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	C₄	1	2	yes	complemented
6	D₇₃⋊M₁₆	—	2	1	no	—
7	D₇₃⋊M₁₆	—	2	1	no	—
8	D₇₃⋊M₁₆	C₄	1	2	yes	complemented
9	D₇₃⋊M₁₆	C₂²	1	2	yes	—
10	C₂×#11924037	—	2	1	no	—
11	C₂×#11924036	—	2	1	no	—
12	#14556670	—	2	1	no	—
13	#14556670	—	2	1	no	—
14	#14556670	C₄	1	2	yes	complemented
15	#14556670	C₂²	1	2	yes	—
16	C₂×C₄⋊#64408	C₄	1	2	yes	complemented
17	C₂×C₂²⋊#64408	—	2	1	no	—
18	C₂×C₄×#64408	—	2	1	no	—
19	C₂²×C₄×D₇₃	—	2	1	no	—
20	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Dic₇₃	—	2	1	no	—
22	C₂×C₄×Dic₇₃	C₄	1	1	yes	characteristic, complemented
23	C₄²×D₇₃	—	2	1	no	—
24	C₄²×D₇₃	—	2	1	no	—
25	C₄²×D₇₃	C₄	1	2	yes	complemented
26	C₄²×D₇₃	C₂²	1	2	yes	—
27	C₂×C₄²×C₇₃	C₄	1	1	yes	characteristic, complemented
28	C₂²×#64408	—	2	2	no	—
29	#11924038	—	2	2	no	—
30	#11924038	C₂×C₄	1	4	yes	complemented
31	#11924037	—	2	2	no	—
32	#11924037	—	2	2	no	—
33	#11924036	—	2	2	no	—
34	#11924036	—	2	2	no	—
35	C₂×#10596833	—	2	2	no	—
36	C₄⋊#64408	—	2	2	no	—
37	C₄⋊#64408	C₂×C₄	1	4	yes	complemented
38	C₂²⋊#64408	—	2	2	no	—
39	C₂²⋊#64408	—	2	2	no	—
40	C₄×#64408	—	2	2	no	—
41	C₄×#64408	—	2	2	no	—
42	C₂³×D₇₃	D₄	1	1	yes	characteristic
43	C₂²×Dic₇₃	—	2	1	no	—
44	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
45	C₂×C₄×D₇₃	—	2	8	no	—
46	C₂×C₄×D₇₃	—	2	2	no	—
47	C₂×C₄×D₇₃	—	2	2	no	—
48	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
54	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
55	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
57	C₂×C₄×D₇₃	Q₈	1	1	yes	characteristic
58	C₂×C₄×D₇₃	Q₈	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
60	C₄×Dic₇₃	—	2	1	no	—
61	C₄×Dic₇₃	—	2	2	no	—
62	C₄×Dic₇₃	—	2	2	no	—
63	C₄×Dic₇₃	C₂×C₄	1	2	yes	complemented
64	C₂²×C₄×C₇₃	—	2	1	no	—
65	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₇₃	—	2	1	no	—
67	C₄²×C₇₃	C₂×C₄	1	2	yes	complemented
68	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×D₇₃	—	2	4	no	—
70	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×D₇₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂×#64408	—	2	4	no	—
77	C₂×#64408	—	4	4	no	—
78	C₂×C₄×C₇₃	—	2	4	no	—
79	C₂×C₄×C₇₃	—	2	2	no	—
80	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₄²	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₇₃	—	2	4	no	—
87	C₂×Dic₇₃	—	2	2	no	—
88	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
93	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
94	C₄×D₇₃	—	2	8	no	—
95	C₄×D₇₃	—	2	8	no	—
96	C₄×D₇₃	C₄²	1	1	yes	characteristic
97	C₄×D₇₃	C₄²	1	1	yes	characteristic
98	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
100	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
101	C₄×D₇₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
105	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
106	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
107	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
108	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
111	C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
112	#10596833	—	2	4	no	—
113	C₂²×C₇₃	—	2	2	no	—
114	C₂²×C₇₃	#95	1	1	yes	characteristic
115	C₂²×C₇₃	#95	1	1	yes	characteristic
116	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×D₇₃	—	2	2	no	—
118	C₂×D₇₃	—	2	2	no	—
119	C₂×D₇₃	—	2	2	no	—
120	C₂×D₇₃	—	2	2	no	—
121	C₂×D₇₃	—	2	2	no	—
122	C₂×D₇₃	#95	1	1	yes	characteristic
123	C₂×D₇₃	#95	1	1	yes	characteristic
124	C₂×D₇₃	#104	1	4	yes	—
125	C₂×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	#64408	—	4	4	no	—
128	C₄×C₇₃	—	2	4	no	—
129	C₄×C₇₃	#95	1	1	yes	characteristic
130	C₄×C₇₃	#95	1	1	yes	characteristic
131	C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
132	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	Dic₇₃	—	2	4	no	—
134	Dic₇₃	#95	1	1	yes	characteristic
135	Dic₇₃	#95	1	1	yes	characteristic
136	Dic₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₇₃	—	2	1	no	—
139	C₂×C₇₃	—	2	1	no	—
140	C₂×C₇₃	#339	1	2	yes	complemented
141	C₂×C₇₃	C₂×#95	1	1	yes	characteristic
142	D₇₃	—	2	1	no	—
143	D₇₃	—	2	1	no	—
144	D₇₃	#339	1	2	yes	complemented
145	D₇₃	C₂×#104	1	2	yes	—
146	C₂×#339	—	73	1	no	maximal, Sylow
147	C₇₃	C₂×#339	1	1	yes	characteristic, complemented, Sylow
148	C₂²×M₁₆	—	73	1	no	—
149	C₂×#117	—	73	1	no	—
150	C₂²×C₄²	—	73	1	no	—
151	#339	—	73	4	no	—
152	C₂³×C₄	—	73	1	no	—
153	C₂³×C₄	—	146	1	no	—
154	C₂×M₁₆	—	73	2	no	—
155	C₂×M₁₆	—	73	2	no	—
156	C₂×M₁₆	—	73	2	no	—
157	C₂×M₁₆	—	146	1	no	—
158	C₂×M₁₆	—	146	1	no	—
159	C₂×M₁₆	—	146	1	no	—
160	C₂²×C₈	—	146	1	no	—
161	#117	—	73	2	no	—
162	#117	—	73	2	no	—
163	#117	—	146	1	no	—
164	#117	—	146	1	no	—
165	C₂×C₄⋊C₄	—	73	2	no	—
166	C₂×C₂²⋊C₄	—	146	1	no	—
167	C₂×C₄²	—	73	1	no	—
168	C₂×C₄²	—	73	2	no	—
169	C₂×C₄²	—	73	2	no	—
170	C₂×C₄²	—	146	1	no	—
171	C₂×C₄²	—	146	1	no	—
172	C₂×C₄²	—	146	1	no	—
173	C₂×C₄²	—	146	1	no	—
174	C₂×C₄²	#64408	1	1	yes	characteristic, complemented
175	C₂⁴	—	73	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	73	1	no	—
178	C₂²×C₄	—	73	1	no	—
179	C₂²×C₄	—	73	1	no	—
180	C₂²×C₄	—	73	1	no	—
181	C₂²×C₄	—	73	1	no	—
182	C₂²×C₄	—	73	1	no	—
183	C₂²×C₄	—	73	1	no	—
184	C₂²×C₄	—	73	1	no	—
185	C₂²×C₄	—	73	1	no	—
186	C₂²×C₄	—	73	1	no	—
187	C₂²×C₄	—	73	1	no	—
188	C₂²×C₄	—	73	1	no	—
189	C₂²×C₄	—	73	1	no	—
190	C₂²×C₄	—	146	2	no	—
191	C₂²×C₄	—	146	2	no	—
192	C₂²×C₄	—	146	8	no	—
193	C₂²×C₄	—	146	2	no	—
194	C₂²×C₄	—	146	1	no	—
195	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
196	M₁₆	—	73	4	no	—
197	M₁₆	—	146	2	no	—
198	M₁₆	—	146	2	no	—
199	M₁₆	—	146	2	no	—
200	C₂×C₈	—	146	2	no	—
201	C₂×C₈	—	146	2	no	—
202	C₂×C₈	—	146	2	no	—
203	C₄⋊C₄	—	73	4	no	—
204	C₄⋊C₄	—	146	2	no	—
205	C₂²⋊C₄	—	146	2	no	—
206	C₂²⋊C₄	—	146	2	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	73	2	no	—
209	C₄²	—	146	2	no	—
210	C₄²	—	146	2	no	—
211	C₄²	—	146	2	no	—
212	C₄²	—	146	2	no	—
213	C₄²	—	146	1	no	—
214	C₄²	C₂×#64408	1	2	yes	complemented
215	C₂³	—	73	1	no	—
216	C₂³	—	73	1	no	—
217	C₂³	—	73	1	no	—
218	C₂³	—	73	1	no	—
219	C₂³	—	73	1	no	—
220	C₂³	—	73	1	no	—
221	C₂³	—	146	4	no	—
222	C₂³	C₂²⋊#64408	1	1	yes	characteristic
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	292	4	no	—
226	C₂×C₄	—	73	1	no	—
227	C₂×C₄	—	73	1	no	—
228	C₂×C₄	—	73	1	no	—
229	C₂×C₄	—	73	1	no	—
230	C₂×C₄	—	73	1	no	—
231	C₂×C₄	—	73	1	no	—
232	C₂×C₄	—	73	1	no	—
233	C₂×C₄	—	73	1	no	—
234	C₂×C₄	—	73	1	no	—
235	C₂×C₄	—	73	1	no	—
236	C₂×C₄	—	73	1	no	—
237	C₂×C₄	—	73	1	no	—
238	C₂×C₄	—	73	1	no	—
239	C₂×C₄	—	73	1	no	—
240	C₂×C₄	—	73	1	no	—
241	C₂×C₄	—	73	1	no	—
242	C₂×C₄	—	73	1	no	—
243	C₂×C₄	—	73	1	no	—
244	C₂×C₄	—	73	1	no	—
245	C₂×C₄	—	73	1	no	—
246	C₂×C₄	—	73	1	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	146	8	no	—
249	C₂×C₄	—	146	8	no	—
250	C₂×C₄	—	146	4	no	—
251	C₂×C₄	—	146	4	no	—
252	C₂×C₄	—	146	2	no	—
253	C₂×C₄	C₄×#64408	1	1	yes	characteristic
254	C₂×C₄	C₄×#64408	1	1	yes	characteristic
255	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
256	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
257	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
258	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
259	C₈	—	146	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	73	4	no	—
262	C₂²	—	73	1	no	—
263	C₂²	—	73	1	no	—
264	C₂²	—	73	1	no	—
265	C₂²	—	73	1	no	—
266	C₂²	—	146	2	no	—
267	C₂²	—	146	2	no	—
268	C₂²	—	146	2	no	—
269	C₂²	—	146	2	no	—
270	C₂²	—	146	2	no	—
271	C₂²	(C₂×#64408)⋊C₄	1	1	yes	characteristic
272	C₂²	(C₂×#64408)⋊C₄	1	1	yes	characteristic
273	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	292	4	no	—
276	C₄	—	73	1	no	—
277	C₄	—	73	1	no	—
278	C₄	—	73	1	no	—
279	C₄	—	73	1	no	—
280	C₄	—	146	4	no	—
281	C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
282	C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
283	C₄	C₂×C₄×#64408	1	1	yes	characteristic
284	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	73	2	no	—
288	C₂	—	73	2	no	—
289	C₂	—	146	1	no	—
290	C₂	—	146	1	no	—
291	C₂	#17989458	1	2	yes	direct product factor
292	C₂	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄⋊#64408
commutator	C₄×C₇₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×C₄²×C₇₃	C₄
socle	C₂²×C₇₃	#95
hypercenter	C₂×C₄²	#64408
hypocenter	C₇₃	C₂×#339
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. #17989458	C₂	direct (N × H)
5. C₂×#11924038	C₄	semidirect
8. D₇₃⋊M₁₆	C₄	semidirect
14. #14556670	C₄	semidirect
16. C₂×C₄⋊#64408	C₄	semidirect
22. C₂×C₄×Dic₇₃	C₄	semidirect
25. C₄²×D₇₃	C₄	semidirect
27. C₂×C₄²×C₇₃	C₄	semidirect
30. #11924038	C₂×C₄	semidirect
37. C₄⋊#64408	C₂×C₄	semidirect
63. C₄×Dic₇₃	C₂×C₄	semidirect
67. C₄²×C₇₃	C₂×C₄	semidirect
140. C₂×C₇₃	#339	semidirect
144. D₇₃	#339	semidirect
147. C₇₃	C₂×#339	semidirect
174. C₂×C₄²	#64408	semidirect
214. C₄²	C₂×#64408	semidirect
291. C₂	#17989458	direct (N × H)

Subgroups — C2×#17989458 #22625608

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989458 #22625608