Subgroups — C₂×#22626896 #34032681

Order 18688.

2118 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×#17989646	C₂	1	6	yes	maximal, complemented
2	#22626896	C₂	1	16	yes	maximal, direct product factor
3	C₂×#17989645	C₂	1	2	yes	maximal, complemented
4	C₂×#17989643	C₂	1	3	yes	maximal, complemented
5	C₂²×#14556664	C₂	1	3	yes	maximal, complemented
6	C₂×#131×D₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989646	C₂²	1	48	yes	complemented
8	#17989645	C₂²	1	16	yes	complemented
9	C₂×#14556666	C₂²	1	6	yes	complemented
10	#17989643	C₂²	1	24	yes	complemented
11	C₂×#14556665	C₂²	1	6	yes	complemented
12	C₂×#14556664	C₂²	1	1	yes	characteristic
13	C₂×#14556664	C₂²	1	3	yes	—
14	C₂×#14556664	C₂²	1	6	yes	complemented
15	C₂×#14556664	C₂²	1	24	yes	complemented
16	C₂²×#11924031	C₂²	1	6	yes	complemented
17	C₂×#41×D₇₃	C₄	1	1	yes	characteristic
18	C₂×#14556599	C₂²	1	3	yes	—
19	C₂×#14556598	C₂²	1	1	yes	characteristic
20	C₂×#14556597	C₄	1	3	yes	—
21	#131×D₇₃	C₄	1	8	yes	—
22	#131×D₇₃	C₂²	1	8	yes	—
23	C₂×M₁₆×D₇₃	C₄	1	3	yes	—
24	C₂²×C₈×D₇₃	C₂²	1	3	yes	—
25	C₂×#131×C₇₃	C₄	1	1	yes	characteristic
26	#14556666	C₂³	1	24	yes	complemented
27	#14556665	C₂³	1	24	yes	complemented
28	#14556664	C₂³	1	4	yes	—
29	#14556664	C₂³	1	12	yes	—
30	#14556664	C₂³	1	24	yes	complemented
31	C₂×#11924031	C₂³	1	2	yes	—
32	C₂×#11924031	C₂³	1	6	yes	—
33	C₂×#11924031	C₂³	1	24	yes	complemented
34	C₂×#11924029	C₈	1	1	yes	characteristic
35	C₂×#11924028	C₂×C₄	1	3	yes	—
36	C₂×#11924027	C₈	1	3	yes	—
37	#41×D₇₃	C₈	1	8	yes	—
38	#41×D₇₃	C₂×C₄	1	8	yes	—
39	C₂×Q₈×D₇₃	C₈	1	1	yes	characteristic
40	C₂×D₄×D₇₃	C₈	1	3	yes	—
41	C₂²×C₄×D₇₃	C₂×C₄	1	3	yes	—
42	#14556599	C₂×C₄	1	12	yes	—
43	#14556599	C₂³	1	12	yes	—
44	#14556598	C₂×C₄	1	4	yes	—
45	#14556598	C₂³	1	4	yes	—
46	#14556597	C₂×C₄	1	24	yes	—
47	C₂×#11924012	C₂×C₄	1	3	yes	—
48	C₂×#11924011	C₂×C₄	1	6	yes	—
49	M₁₆×D₇₃	C₂×C₄	1	24	yes	—
50	C₂²×#10596831	C₂³	1	3	yes	—
51	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
52	C₂×C₈×D₇₃	C₂×C₄	1	12	yes	—
53	C₂×C₈×D₇₃	C₂×C₄	1	3	yes	—
54	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×D₇₃	C₂³	1	12	yes	—
56	C₂×C₈×D₇₃	C₂³	1	3	yes	—
57	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
58	#131×C₇₃	C₂×C₄	1	8	yes	—
59	C₂×M₁₆×C₇₃	C₂×C₄	1	3	yes	—
60	C₂²×C₈×C₇₃	C₂×C₄	1	3	yes	—
61	#11924031	C₂⁴	1	4	yes	—
62	#11924031	C₂⁴	1	12	yes	—
63	C₂³×D₇₃	C₂×C₈	1	3	yes	—
64	#11924029	C₂×C₈	1	8	yes	—
65	#11924028	C₂×C₈	1	12	yes	—
66	#11924028	C₂²×C₄	1	12	yes	—
67	#11924027	C₂×C₈	1	24	yes	—
68	C₂×Dic₁₄₆	C₂×C₈	1	3	yes	—
69	Q₈×D₇₃	C₂×C₈	1	8	yes	—
70	C₂×D₂₉₂	C₂×C₈	1	3	yes	—
71	C₂×C₂²⋊D₇₃	C₂×C₈	1	6	yes	—
72	D₄×D₇₃	C₂×C₈	1	24	yes	—
73	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
74	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
75	C₂×C₄×D₇₃	C₂×C₈	1	12	yes	—
76	C₂×C₄×D₇₃	C₂×C₈	1	3	yes	—
77	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×D₇₃	C₂²×C₄	1	12	yes	—
79	C₂×C₄×D₇₃	C₂²×C₄	1	3	yes	—
80	#11924012	C₂²×C₄	1	12	yes	—
81	#11924011	C₂²×C₄	1	24	yes	—
82	C₂×#10596831	C₂²×C₄	1	3	yes	—
83	C₂×#10596831	C₂²×C₄	1	6	yes	—
84	C₂×#10596831	C₂⁴	1	1	yes	characteristic
85	C₂×#10596831	C₂⁴	1	6	yes	—
86	C₈×D₇₃	C₂²×C₄	1	2	yes	—
87	C₈×D₇₃	C₂²×C₄	1	12	yes	—
88	C₈×D₇₃	C₂²×C₄	1	12	yes	—
89	C₈×D₇₃	C₂²×C₄	1	6	yes	—
90	C₈×D₇₃	C₂⁴	1	2	yes	—
91	C₈×D₇₃	C₂⁴	1	6	yes	—
92	#41×C₇₃	C₂×C₈	1	4	yes	—
93	#41×C₇₃	C₂²×C₄	1	4	yes	—
94	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
95	C₂×D₄×C₇₃	C₂×C₈	1	3	yes	—
96	C₂²×C₄×C₇₃	C₂²×C₄	1	3	yes	—
97	M₁₆×C₇₃	C₂²×C₄	1	12	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	3	yes	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	12	yes	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₇₃	C₂²×C₈	1	3	yes	—
102	C₂²×D₇₃	—	2	12	no	—
103	C₂²×D₇₃	C₂²×C₈	1	12	yes	—
104	C₂²×D₇₃	C₂²×C₈	1	3	yes	—
105	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
106	Q₈×C₇₃	C₂²×C₈	1	4	yes	—
107	D₄×C₇₃	C₂²×C₈	1	12	yes	—
108	C₂×C₄×C₇₃	C₂²×C₈	1	3	yes	—
109	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
110	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂³×C₄	1	6	yes	—
112	C₂²⋊D₇₃	C₂²×C₈	1	24	yes	—
113	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×Dic₇₃	C₂²×C₈	1	3	yes	—
115	C₂×Dic₇₃	C₂²×C₈	1	12	yes	—
116	D₂₉₂	C₂²×C₈	1	12	yes	—
117	C₄×D₇₃	C₂²×C₈	1	2	yes	—
118	C₄×D₇₃	C₂²×C₈	1	12	yes	—
119	C₄×D₇₃	C₂²×C₈	1	12	yes	—
120	C₄×D₇₃	C₂²×C₈	1	6	yes	—
121	C₄×D₇₃	C₂³×C₄	1	2	yes	—
122	C₄×D₇₃	C₂³×C₄	1	6	yes	—
123	Dic₁₄₆	C₂²×C₈	1	12	yes	—
124	C₈×C₇₃	C₂³×C₄	1	2	yes	—
125	C₈×C₇₃	C₂³×C₄	1	6	yes	—
126	#10596831	C₂³×C₄	1	1	yes	characteristic
127	#10596831	C₂³×C₄	1	6	yes	—
128	#10596831	C₂⁵	1	1	yes	characteristic
129	C₂²×C₇₃	—	2	6	no	—
130	C₂²×C₇₃	C₂³×C₈	1	6	yes	—
131	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
132	C₂×D₇₃	—	2	6	no	—
133	C₂×D₇₃	—	2	24	no	—
134	C₂×D₇₃	#504	1	4	yes	complemented
135	C₂×D₇₃	C₂³×C₈	1	6	yes	—
136	C₂×D₇₃	C₂³×C₈	1	2	yes	—
137	C₄×C₇₃	C₂³×C₈	1	6	yes	—
138	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
139	C₄×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
140	Dic₇₃	C₂³×C₈	1	6	yes	—
141	Dic₇₃	C₂³×C₈	1	2	yes	—
142	C₂²×#504	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	6	no	—
144	C₂×C₇₃	C₂×#504	1	2	yes	complemented
145	C₂×C₇₃	C₂⁴×C₈	1	1	yes	characteristic
146	D₇₃	—	2	6	no	—
147	D₇₃	C₂×#504	1	4	yes	complemented
148	C₂²×#131	—	73	1	no	—
149	C₂×#504	—	73	2	no	—
150	C₂×#504	—	73	16	no	—
151	C₂×#504	—	73	6	no	—
152	C₂²×M₃₂	—	73	3	no	—
153	C₂³×C₁₆	—	73	3	no	—
154	C₇₃	C₂²×#504	1	1	yes	characteristic, complemented, Sylow
155	C₂²×#41	—	73	1	no	—
156	C₂×#131	—	73	1	no	—
157	C₂×#131	—	73	8	no	—
158	C₂×#131	—	73	8	no	—
159	C₂×#131	—	73	3	no	—
160	C₂×#131	—	73	3	no	—
161	C₂×#131	#64408	1	1	yes	characteristic
162	C₂²×M₁₆	—	73	3	no	—
163	C₂³×C₈	—	73	3	no	—
164	#504	—	73	16	no	—
165	#504	—	73	48	no	—
166	C₂×M₃₂	—	73	6	no	—
167	C₂×M₃₂	—	73	24	no	—
168	C₂×M₃₂	—	73	6	no	—
169	C₂²×C₁₆	—	73	6	no	—
170	C₂²×C₁₆	—	73	1	no	—
171	C₂²×C₁₆	—	73	3	no	—
172	C₂²×C₁₆	—	73	6	no	—
173	C₂²×C₁₆	—	73	24	no	—
174	C₂×#41	—	73	8	no	—
175	C₂×#41	—	73	8	no	—
176	C₂×#41	—	73	1	no	—
177	C₂×#41	—	73	3	no	—
178	C₂×#41	—	73	3	no	—
179	C₂×#41	C₂×#64408	1	1	yes	characteristic
180	C₂²×Q₈	—	73	1	no	—
181	C₂²×D₄	—	73	3	no	—
182	C₂³×C₄	—	73	3	no	—
183	#131	—	73	4	no	—
184	#131	—	73	4	no	—
185	#131	—	73	12	no	—
186	#131	—	73	12	no	—
187	#131	—	73	24	no	—
188	#131	C₂×#64408	1	8	yes	—
189	C₂×M₁₆	—	73	3	no	—
190	C₂×M₁₆	—	73	24	no	—
191	C₂×M₁₆	—	73	6	no	—
192	C₂×M₁₆	C₂×#64408	1	3	yes	—
193	C₂²×C₈	—	73	3	no	—
194	C₂²×C₈	—	73	3	no	—
195	C₂²×C₈	—	73	12	no	—
196	C₂²×C₈	—	73	3	no	—
197	C₂²×C₈	—	73	12	no	—
198	C₂²×C₈	—	73	1	no	—
199	C₂²×C₈	—	73	3	no	—
200	C₂²×C₈	C₂×#64408	1	3	yes	—
201	M₃₂	—	73	24	no	—
202	M₃₂	—	73	24	no	—
203	C₂×C₁₆	—	73	2	no	—
204	C₂×C₁₆	—	73	4	no	—
205	C₂×C₁₆	—	73	6	no	—
206	C₂×C₁₆	—	73	12	no	—
207	C₂×C₁₆	—	73	24	no	—
208	C₂×C₁₆	—	73	24	no	—
209	C₂⁴	—	73	3	no	—
210	#41	—	73	8	no	—
211	#41	—	73	24	no	—
212	#41	—	73	12	no	—
213	#41	—	73	12	no	—
214	#41	#11924036	1	4	yes	—
215	#41	C₂²×#64408	1	4	yes	—
216	C₂×Q₈	—	73	8	no	—
217	C₂×Q₈	—	73	3	no	—
218	C₂×Q₈	#11924036	1	1	yes	characteristic
219	C₂×D₄	—	73	6	no	—
220	C₂×D₄	—	73	24	no	—
221	C₂×D₄	—	73	3	no	—
222	C₂×D₄	#11924036	1	3	yes	—
223	C₂²×C₄	—	73	3	no	—
224	C₂²×C₄	—	73	3	no	—
225	C₂²×C₄	—	73	3	no	—
226	C₂²×C₄	—	73	12	no	—
227	C₂²×C₄	—	73	1	no	—
228	C₂²×C₄	—	73	3	no	—
229	C₂²×C₄	—	73	12	no	—
230	C₂²×C₄	C₂²×#64408	1	3	yes	—
231	M₁₆	—	73	12	no	—
232	M₁₆	—	73	24	no	—
233	M₁₆	C₂²×#64408	1	12	yes	—
234	C₂×C₈	—	73	3	no	—
235	C₂×C₈	—	73	6	no	—
236	C₂×C₈	—	73	1	no	—
237	C₂×C₈	—	73	6	no	—
238	C₂×C₈	—	73	2	no	—
239	C₂×C₈	—	73	12	no	—
240	C₂×C₈	—	73	6	no	—
241	C₂×C₈	—	73	12	no	—
242	C₂×C₈	—	73	2	no	—
243	C₂×C₈	—	73	6	no	—
244	C₂×C₈	C₂²×#64408	1	3	yes	—
245	C₂×C₈	C₂²×#64408	1	12	yes	—
246	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
247	C₁₆	—	73	4	no	—
248	C₁₆	—	73	12	no	—
249	C₂³	—	73	1	no	—
250	C₂³	—	73	12	no	—
251	C₂³	—	73	3	no	—
252	C₂³	—	146	12	no	—
253	C₂³	C₂×#11924036	1	3	yes	—
254	Q₈	—	73	12	no	—
255	Q₈	C₂×#11924036	1	4	yes	—
256	D₄	—	73	24	no	—
257	D₄	—	73	12	no	—
258	D₄	C₂×#11924036	1	12	yes	—
259	C₂×C₄	—	73	1	no	—
260	C₂×C₄	—	73	3	no	—
261	C₂×C₄	—	73	12	no	—
262	C₂×C₄	—	73	2	no	—
263	C₂×C₄	—	73	12	no	—
264	C₂×C₄	—	73	12	no	—
265	C₂×C₄	—	73	6	no	—
266	C₂×C₄	—	73	2	no	—
267	C₂×C₄	—	73	6	no	—
268	C₂×C₄	C₂×#11924036	1	3	yes	—
269	C₂×C₄	C₂×#11924036	1	6	yes	—
270	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
271	C₂×C₄	C₂³×#64408	1	6	yes	—
272	C₈	—	73	1	no	—
273	C₈	—	73	1	no	—
274	C₈	—	73	6	no	—
275	C₈	C₂³×#64408	1	2	yes	—
276	C₈	C₂³×#64408	1	6	yes	—
277	C₂²	—	2	6	no	—
278	C₂²	—	73	4	no	—
279	C₂²	—	73	2	no	—
280	C₂²	—	73	6	no	—
281	C₂²	—	146	24	no	—
282	C₂²	—	146	6	no	—
283	C₂²	C₂²×#11924036	1	6	yes	—
284	C₂²	C₂²×#11924036	1	1	yes	characteristic
285	C₄	—	73	2	no	—
286	C₄	—	73	6	no	—
287	C₄	C₂²×#11924036	1	1	yes	characteristic
288	C₄	C₂²×#11924036	1	6	yes	—
289	C₄	C₂⁴×#64408	1	1	yes	characteristic
290	C₂	—	2	6	no	—
291	C₂	—	73	4	no	—
292	C₂	—	146	6	no	—
293	C₂	#22626896	1	2	yes	direct product factor
294	C₂	C₂³×#11924036	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×#64408
commutator	C₂×C₇₃	C₂⁴×C₈
Frattini	C₄	C₂⁴×#64408
Fitting	C₂×#131×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₈
prime radical	C₂×#41×D₇₃	C₄
cosocle	#10596831	C₂⁵
hypercenter	C₂×#131	#64408
hypocenter	C₇₃	C₂²×#504
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
1. C₂×#17989646	C₂	semidirect
2. #22626896	C₂	direct (N × H)
3. C₂×#17989645	C₂	semidirect
4. C₂×#17989643	C₂	semidirect
5. C₂²×#14556664	C₂	semidirect
7. #17989646	C₂²	semidirect
8. #17989645	C₂²	semidirect
9. C₂×#14556666	C₂²	semidirect
10. #17989643	C₂²	semidirect
11. C₂×#14556665	C₂²	semidirect
14. C₂×#14556664	C₂²	semidirect
15. C₂×#14556664	C₂²	semidirect
16. C₂²×#11924031	C₂²	semidirect
26. #14556666	C₂³	semidirect
27. #14556665	C₂³	semidirect
30. #14556664	C₂³	semidirect
33. C₂×#11924031	C₂³	semidirect
134. C₂×D₇₃	#504	semidirect
144. C₂×C₇₃	C₂×#504	semidirect
147. D₇₃	C₂×#504	semidirect
154. C₇₃	C₂²×#504	semidirect
293. C₂	#22626896	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×#17989646	2
#22626896	2
C₂×#17989645	2
C₂×#17989643	2
C₂²×#14556664	2
C₂×#131×D₇₃	2
C₂²×#504	73

Subgroups — C2×#22626896 #34032681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626896 #34032681