Subgroups — C₂×#22626282 #34030093

Order 18688.

694 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×#17989597	C₂	1	2	yes	maximal
2	#22626282	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17989514	C₂	1	2	yes	maximal, complemented
4	C₂×#17989513	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556670	—	2	2	no	—
7	#17989597	—	2	4	no	—
8	#17989597	C₂²	1	8	yes	—
9	C₂×#14556660	C₂²	1	4	yes	—
10	C₂×#14556659	—	2	2	no	—
11	#17989514	C₂²	1	8	yes	complemented
12	#17989513	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	C₂²×#11924003	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂×C₁₆×D₇₃	C₄	1	4	yes	complemented
18	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₇₃	C₂²	1	4	yes	—
21	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
22	#14556670	—	2	8	no	—
23	#14556670	—	4	4	no	—
24	C₂×C₄⋊#64408	—	2	4	no	—
25	C₂×C₂²⋊#64408	—	4	2	no	—
26	C₂×C₄×#64408	—	4	2	no	—
27	#14556660	—	2	4	no	—
28	#14556660	C₂³	1	8	yes	—
29	#14556659	—	2	8	no	—
30	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₂²×#10596831	Q₈	1	1	yes	characteristic
32	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
35	C₂×C₈×D₇₃	D₄	1	2	yes	—
36	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₇₃	Q₈	1	2	yes	—
38	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₇₃	C₂³	1	2	yes	—
40	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
42	C₂×#11924003	C₂×C₄	1	4	yes	complemented
43	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
44	C₁₆×D₇₃	C₂×C₄	1	2	yes	—
45	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
46	C₁₆×D₇₃	C₂×C₄	1	4	yes	complemented
47	C₁₆×D₇₃	C₂³	1	2	yes	—
48	C₁₆×D₇₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
53	C₂²×#64408	—	4	4	no	—
54	C₄⋊#64408	—	2	8	no	—
55	C₄⋊#64408	—	4	4	no	—
56	C₂²⋊#64408	—	4	8	no	—
57	C₄×#64408	—	4	8	no	—
58	C₂³×D₇₃	D₈	1	1	yes	characteristic
59	C₂²×Dic₇₃	Q₁₆	1	1	yes	characteristic
60	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
63	C₂×C₄×D₇₃	D₈	1	2	yes	—
64	C₂×C₄×D₇₃	Q₁₆	1	2	yes	—
65	C₂×C₄×D₇₃	Q₁₆	1	1	yes	characteristic
66	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
68	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
69	C₂×#10596831	C₄⋊C₄	1	2	yes	—
70	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
71	C₂×#10596831	C₂×Q₈	1	2	yes	—
72	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
73	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
74	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
75	C₈×D₇₃	C₂²×C₄	1	1	yes	characteristic
76	C₈×D₇₃	C₂²×C₄	1	2	yes	—
77	C₈×D₇₃	C₂×D₄	1	2	yes	—
78	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
79	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₈×D₇₃	C₂×Q₈	1	2	yes	—
81	C₈×D₇₃	C₂⁴	1	1	yes	characteristic
82	#11924003	C₂²×C₄	1	2	yes	—
83	#11924003	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
89	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
90	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂³×C₇₃	#107	1	1	yes	characteristic
92	C₂²×D₇₃	—	2	4	no	—
93	C₂²×D₇₃	#107	1	1	yes	characteristic
94	C₂²×D₇₃	#107	1	2	yes	—
95	C₂²×D₇₃	C₂×D₈	1	2	yes	—
96	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
97	C₂×#64408	—	4	8	no	—
98	C₂×#64408	—	8	8	no	—
99	C₂×C₄×C₇₃	#107	1	2	yes	—
100	C₂×C₄×C₇₃	#107	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×Dic₇₃	#107	1	2	yes	—
104	C₂×Dic₇₃	#107	1	1	yes	characteristic
105	C₂×Dic₇₃	C₂×Q₁₆	1	2	yes	—
106	C₂×Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
107	C₄×D₇₃	#107	1	2	yes	—
108	C₄×D₇₃	#107	1	2	yes	—
109	C₄×D₇₃	#107	1	2	yes	—
110	C₄×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₄×D₇₃	C₂×D₈	1	2	yes	—
113	C₄×D₇₃	C₂×D₈	1	1	yes	characteristic
114	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
115	C₄×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
116	C₄×D₇₃	C₂²×D₄	1	1	yes	characteristic
117	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₈×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
120	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
121	#10596831	C₂×C₄⋊C₄	1	2	yes	—
122	#10596831	C₂²×Q₈	1	1	yes	characteristic
123	C₂²×C₇₃	—	2	2	no	—
124	C₂²×C₇₃	C₂×#107	1	2	yes	—
125	C₂²×C₇₃	C₂×#107	1	1	yes	characteristic
126	C₂×D₇₃	—	2	2	no	—
127	C₂×D₇₃	—	2	4	no	—
128	C₂×D₇₃	—	2	4	no	—
129	C₂×D₇₃	C₂×#107	1	2	yes	—
130	C₂×D₇₃	C₂×#107	1	1	yes	characteristic
131	C₂×D₇₃	#508	1	4	yes	—
132	C₂×D₇₃	C₂²×D₈	1	1	yes	characteristic
133	#64408	—	8	8	no	—
134	C₄×C₇₃	C₂×#107	1	1	yes	characteristic
135	C₄×C₇₃	C₂×#107	1	2	yes	—
136	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	Dic₇₃	C₂×#107	1	2	yes	—
138	Dic₇₃	C₂×#107	1	1	yes	characteristic
139	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂×#2158	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	2	no	—
142	C₂×C₇₃	#2158	1	2	yes	complemented
143	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
144	D₇₃	—	2	2	no	—
145	D₇₃	#2158	1	2	yes	complemented
146	D₇₃	C₂×#508	1	2	yes	—
147	C₂³×C₁₆	—	73	1	no	—
148	C₂×#427	—	73	2	no	—
149	#2158	—	73	8	no	—
150	C₂×#367	—	73	2	no	—
151	C₂×#366	—	73	2	no	—
152	C₇₃	C₂×#2158	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₈	—	73	1	no	—
154	C₂×#117	—	146	2	no	—
155	C₂²×C₁₆	—	73	1	no	—
156	C₂²×C₁₆	—	73	1	no	—
157	C₂²×C₁₆	—	73	1	no	—
158	C₂²×C₁₆	—	73	1	no	—
159	C₂²×C₁₆	—	73	1	no	—
160	C₂²×C₁₆	—	73	4	no	—
161	C₂²×C₁₆	—	73	4	no	—
162	C₂²×C₁₆	#64408	1	1	yes	characteristic, complemented
163	#427	—	73	8	no	—
164	#427	—	146	4	no	—
165	C₂×#107	—	73	4	no	—
166	C₂×#106	—	146	2	no	—
167	#367	—	73	8	no	—
168	#366	—	73	8	no	—
169	C₂³×C₄	—	73	1	no	—
170	C₂²×C₈	—	73	1	no	—
171	C₂²×C₈	—	73	1	no	—
172	C₂²×C₈	—	73	2	no	—
173	C₂²×C₈	—	73	1	no	—
174	C₂²×C₈	—	73	2	no	—
175	C₂²×C₈	—	73	1	no	—
176	C₂²×C₈	—	73	2	no	—
177	C₂²×C₈	—	73	1	no	—
178	C₂²×C₈	—	73	2	no	—
179	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
180	#117	—	292	4	no	—
181	#117	—	146	8	no	—
182	C₂×C₄⋊C₄	—	146	4	no	—
183	C₂×C₂²⋊C₄	—	292	2	no	—
184	C₂×C₄²	—	292	2	no	—
185	C₂×C₁₆	—	73	1	no	—
186	C₂×C₁₆	—	73	1	no	—
187	C₂×C₁₆	—	73	4	no	—
188	C₂×C₁₆	—	73	2	no	—
189	C₂×C₁₆	—	73	2	no	—
190	C₂×C₁₆	—	73	2	no	—
191	C₂×C₁₆	—	73	2	no	—
192	C₂×C₁₆	—	73	4	no	—
193	C₂×C₁₆	—	73	4	no	—
194	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
195	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
196	C₂×C₁₆	C₂×#64408	1	4	yes	complemented
197	#107	—	73	8	no	—
198	#107	—	146	4	no	—
199	#106	—	146	8	no	—
200	C₂⁴	—	73	1	no	—
201	C₂²×C₄	—	292	4	no	—
202	C₂²×C₄	—	73	1	no	—
203	C₂²×C₄	—	73	2	no	—
204	C₂²×C₄	—	73	2	no	—
205	C₂²×C₄	—	73	1	no	—
206	C₂²×C₄	—	73	1	no	—
207	C₂²×C₄	—	73	2	no	—
208	C₂²×C₄	—	73	1	no	—
209	C₂²×C₄	—	73	1	no	—
210	C₂²×C₄	—	73	2	no	—
211	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
212	C₂×C₈	—	73	1	no	—
213	C₂×C₈	—	73	2	no	—
214	C₂×C₈	—	73	1	no	—
215	C₂×C₈	—	73	2	no	—
216	C₂×C₈	—	73	1	no	—
217	C₂×C₈	—	73	2	no	—
218	C₂×C₈	—	73	1	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	1	no	—
224	C₂×C₈	—	73	1	no	—
225	C₂×C₈	—	73	2	no	—
226	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
227	C₂×C₈	C₄⋊#64408	1	2	yes	—
228	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
229	C₂×C₈	C₂²×#64408	1	2	yes	—
230	C₄⋊C₄	—	292	4	no	—
231	C₄⋊C₄	—	146	8	no	—
232	C₂²⋊C₄	—	292	8	no	—
233	C₄²	—	292	8	no	—
234	C₁₆	—	73	2	no	—
235	C₁₆	—	73	2	no	—
236	C₁₆	C₂²×#64408	1	2	yes	—
237	C₁₆	C₂²×#64408	1	2	yes	—
238	C₂³	—	73	1	no	—
239	C₂³	—	73	2	no	—
240	C₂³	—	73	2	no	—
241	C₂³	—	73	1	no	—
242	C₂³	—	146	4	no	—
243	C₂³	#14556660	1	1	yes	characteristic
244	C₂×C₄	—	292	8	no	—
245	C₂×C₄	—	584	8	no	—
246	C₂×C₄	—	73	2	no	—
247	C₂×C₄	—	73	1	no	—
248	C₂×C₄	—	73	2	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	73	1	no	—
251	C₂×C₄	—	73	1	no	—
252	C₂×C₄	—	73	2	no	—
253	C₂×C₄	—	73	2	no	—
254	C₂×C₄	—	73	1	no	—
255	C₂×C₄	—	73	2	no	—
256	C₂×C₄	—	73	2	no	—
257	C₂×C₄	—	73	2	no	—
258	C₂×C₄	—	73	1	no	—
259	C₂×C₄	—	73	2	no	—
260	C₂×C₄	#14556660	1	2	yes	—
261	C₂×C₄	#14556660	1	1	yes	characteristic
262	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
264	C₈	—	73	2	no	—
265	C₈	—	73	1	no	—
266	C₈	—	73	1	no	—
267	C₈	C₂×C₄⋊#64408	1	2	yes	—
268	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
269	C₈	C₂³×#64408	1	1	yes	characteristic
270	C₂²	—	2	2	no	—
271	C₂²	—	73	1	no	—
272	C₂²	—	73	1	no	—
273	C₂²	—	73	4	no	—
274	C₂²	—	73	2	no	—
275	C₂²	—	146	4	no	—
276	C₂²	—	146	4	no	—
277	C₂²	—	146	2	no	—
278	C₂²	C₂×#14556660	1	2	yes	—
279	C₂²	C₂×#14556660	1	1	yes	characteristic
280	C₄	—	584	8	no	—
281	C₄	—	73	1	no	—
282	C₄	—	73	2	no	—
283	C₄	—	73	1	no	—
284	C₄	C₂×#14556660	1	1	yes	characteristic
285	C₄	C₂×#14556660	1	2	yes	—
286	C₄	C₂²×C₄⋊#64408	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	73	2	no	—
289	C₂	—	73	2	no	—
290	C₂	—	146	2	no	—
291	C₂	#22626282	1	2	yes	direct product factor
292	C₂	C₂²×#14556660	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556660
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₈	C₂³×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#107
prime radical	C₂³×D₇₃	D₈
cosocle	C₈×D₇₃	C₂⁴
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	C₂×#2158
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #22626282	C₂	direct (N × H)
3. C₂×#17989514	C₂	semidirect
4. C₂×#17989513	C₂	semidirect
11. #17989514	C₂²	semidirect
12. #17989513	C₂²	semidirect
14. C₂²×#11924003	C₄	semidirect
15. C₂×C₁₆×D₇₃	C₄	semidirect
16. C₂×C₁₆×D₇₃	C₄	semidirect
17. C₂×C₁₆×D₇₃	C₄	semidirect
21. C₂²×C₁₆×C₇₃	C₄	semidirect
42. C₂×#11924003	C₂×C₄	semidirect
45. C₁₆×D₇₃	C₂×C₄	semidirect
46. C₁₆×D₇₃	C₂×C₄	semidirect
52. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
142. C₂×C₇₃	#2158	semidirect
145. D₇₃	#2158	semidirect
152. C₇₃	C₂×#2158	semidirect
162. C₂²×C₁₆	#64408	semidirect
196. C₂×C₁₆	C₂×#64408	semidirect
291. C₂	#22626282	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×#17989597	2
#22626282	2
C₂×#17989514	2
C₂×#17989513	2
C₂²×C₁₆×D₇₃	2
C₂×#2158	73

Subgroups — C2×#22626282 #34030093

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22626282 #34030093