Subgroups — C₂×#22612138 #33240761

Order 18688.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988825	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988624	C₂	1	1	yes	maximal, characteristic
3	#22612138	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17988414	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#17988413	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
8	C₂×#358×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₂×#14556567	—	2	1	no	—
11	C₂×#14556549	—	2	1	no	—
12	C₂²×#11924003	C₂²	1	1	yes	characteristic
13	#17988825	—	2	2	no	—
14	#17988825	C₂²	1	4	yes	—
15	#17988624	—	2	2	no	—
16	#17988624	C₂²	1	4	yes	—
17	C₂×#14556525	C₂²	1	1	yes	characteristic
18	C₂×#14556525	C₂²	1	1	yes	characteristic
19	C₂×#14556522	—	2	1	no	—
20	C₂×#14556515	—	2	1	no	—
21	#17988414	C₂²	1	4	yes	complemented
22	#17988413	C₂²	1	4	yes	complemented
23	#17988391	C₂²	1	4	yes	complemented
24	#17988366	C₂²	1	4	yes	—
25	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
28	#358×C₇₃	C₂²	1	4	yes	complemented
29	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂²×#10596831	D₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	—	2	2	no	—
32	C₂×C₈×D₇₃	—	2	2	no	—
33	C₂×C₈×D₇₃	D₄	1	2	yes	—
34	C₂×C₈×D₇₃	C₂³	1	2	yes	—
35	#14556567	—	2	4	no	—
36	#14556567	—	4	2	no	—
37	#14556549	—	2	4	no	—
38	#14556549	—	4	2	no	—
39	C₂×C₄⋊Dic₇₃	—	2	1	no	—
40	C₂×C₄⋊Dic₇₃	—	2	1	no	—
41	C₂×#11924007	—	4	1	no	—
42	C₂×#11924005	—	4	1	no	—
43	C₂×#11924005	—	4	1	no	—
44	C₂×C₄×Dic₇₃	—	4	1	no	—
45	C₂×#11924003	—	2	2	no	—
46	C₂×#11924003	C₂³	1	2	yes	—
47	#14556525	—	2	1	no	—
48	#14556525	—	2	1	no	—
49	#14556525	C₂³	1	2	yes	—
50	#14556525	C₂³	1	2	yes	—
51	#14556522	—	2	4	no	—
52	#14556515	—	2	4	no	—
53	C₂²×Q₈×C₇₃	—	2	1	no	—
54	C₂×Q₁₆×C₇₃	—	2	1	no	—
55	C₂×Q₁₆×C₇₃	—	2	2	no	—
56	C₂×Q₁₆×C₇₃	—	2	2	no	—
57	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
60	C₂×C₁₆×C₇₃	—	2	2	no	—
61	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
62	#107×C₇₃	—	2	1	no	—
63	#107×C₇₃	C₂³	1	2	yes	—
64	C₂³×D₇₃	D₈	1	1	yes	characteristic
65	C₂²×Dic₇₃	—	4	1	no	—
66	C₂²×Dic₇₃	—	4	1	no	—
67	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
68	C₂×C₄×D₇₃	—	2	2	no	—
69	C₂×C₄×D₇₃	—	2	2	no	—
70	C₂×C₄×D₇₃	D₈	1	2	yes	—
71	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
72	C₂×#10596831	—	2	2	no	—
73	C₂×#10596831	C₂×D₄	1	2	yes	—
74	C₈×D₇₃	—	2	4	no	—
75	C₈×D₇₃	—	2	2	no	—
76	C₈×D₇₃	—	2	2	no	—
77	C₄⋊Dic₇₃	—	2	2	no	—
78	C₄⋊Dic₇₃	—	2	2	no	—
79	C₄⋊Dic₇₃	—	4	1	no	—
80	C₄⋊Dic₇₃	—	4	1	no	—
81	#11924007	—	4	4	no	—
82	#11924005	—	4	4	no	—
83	#11924005	—	4	4	no	—
84	C₄×Dic₇₃	—	4	4	no	—
85	#11924003	—	2	2	no	—
86	C₂×Q₈×C₇₃	—	2	2	no	—
87	C₂×Q₈×C₇₃	—	4	1	no	—
88	C₂×Q₈×C₇₃	—	4	2	no	—
89	C₂×Q₈×C₇₃	—	4	2	no	—
90	C₂²×C₄×C₇₃	—	4	1	no	—
91	C₂²×C₄×C₇₃	—	4	1	no	—
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₇₃	—	2	2	no	—
94	Q₁₆×C₇₃	—	2	2	no	—
95	Q₁₆×C₇₃	—	4	2	no	—
96	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	#41	1	2	yes	—
98	C₂×C₈×C₇₃	#41	1	2	yes	—
99	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
100	C₄⋊C₄×C₇₃	—	2	2	no	—
101	C₄⋊C₄×C₇₃	—	4	1	no	—
102	C₁₆×C₇₃	—	2	2	no	—
103	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
104	C₂²×D₇₃	—	2	2	no	—
105	C₂²×D₇₃	—	2	2	no	—
106	C₂²×D₇₃	—	2	2	no	—
107	C₂²×D₇₃	C₂×D₈	1	2	yes	—
108	Q₈×C₇₃	—	4	2	no	—
109	Q₈×C₇₃	—	4	2	no	—
110	Q₈×C₇₃	—	8	2	no	—
111	C₂×C₄×C₇₃	—	4	2	no	—
112	C₂×C₄×C₇₃	—	4	2	no	—
113	C₂×C₄×C₇₃	—	8	2	no	—
114	C₂×C₄×C₇₃	—	8	2	no	—
115	C₂×C₄×C₇₃	#123	1	2	yes	—
116	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#137	1	2	yes	—
118	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₇₃	—	2	2	no	—
120	C₂×Dic₇₃	—	4	2	no	—
121	C₂×Dic₇₃	—	4	2	no	—
122	C₂×Dic₇₃	—	8	2	no	—
123	C₂×Dic₇₃	—	8	2	no	—
124	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
125	C₄×D₇₃	—	2	2	no	—
126	C₄×D₇₃	—	2	2	no	—
127	C₄×D₇₃	—	2	4	no	—
128	C₈×C₇₃	#123	1	2	yes	—
129	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
130	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
131	#10596831	—	2	2	no	—
132	C₂²×C₇₃	#480	1	2	yes	—
133	C₂²×C₇₃	#508	1	2	yes	—
134	C₂²×C₇₃	#510	1	2	yes	—
135	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
136	C₂×D₇₃	—	2	2	no	—
137	C₂×D₇₃	—	2	2	no	—
138	C₂×D₇₃	—	2	2	no	—
139	C₂×D₇₃	—	4	4	no	—
140	C₄×C₇₃	—	8	2	no	—
141	C₄×C₇₃	—	8	2	no	—
142	C₄×C₇₃	#480	1	2	yes	—
143	C₄×C₇₃	C₂×#123	1	1	yes	characteristic
144	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
145	Dic₇₃	—	2	2	no	—
146	Dic₇₃	—	8	2	no	—
147	Dic₇₃	—	8	2	no	—
148	C₂×#2237	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	#2237	1	4	yes	complemented
150	C₂×C₇₃	C₂×#480	1	1	yes	characteristic
151	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
152	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
153	D₇₃	—	4	2	no	—
154	C₂×#467	—	73	1	no	—
155	C₂×#427	—	73	1	no	—
156	#2237	—	73	8	no	—
157	C₂×#367	—	73	1	no	—
158	C₂×#366	—	73	1	no	—
159	C₂×#358	—	73	1	no	—
160	C₂×#358	D₇₃	1	1	yes	characteristic, complemented
161	C₂×C₂²⋊C₁₆	—	73	1	no	—
162	C₇₃	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
164	C₂³×C₈	—	73	1	no	—
165	C₂×C₂²⋊Q₈	—	146	1	no	—
166	C₂×#117	—	146	1	no	—
167	C₂²×C₁₆	—	73	1	no	—
168	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
169	#467	—	73	4	no	—
170	#467	—	146	2	no	—
171	#427	—	73	4	no	—
172	#427	—	146	2	no	—
173	C₂×#107	—	73	1	no	—
174	C₂×#107	—	73	1	no	—
175	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
176	C₂×#106	—	146	1	no	—
177	C₂×#103	—	146	1	no	—
178	#367	—	73	4	no	—
179	#366	—	73	4	no	—
180	#358	—	73	4	no	—
181	#358	C₂×D₇₃	1	4	yes	complemented
182	C₂²⋊C₁₆	—	73	4	no	—
183	C₂²×Q₈	—	2	1	no	—
184	C₂³×C₄	—	73	1	no	—
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
189	C₂²×C₈	—	73	1	no	—
190	C₂²×C₈	—	73	2	no	—
191	C₂²×C₈	—	73	2	no	—
192	C₂²×C₈	—	146	2	no	—
193	C₂²×C₈	—	146	2	no	—
194	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
195	C₂²⋊Q₈	—	292	2	no	—
196	C₂²⋊Q₈	—	146	4	no	—
197	#117	—	292	2	no	—
198	#117	—	146	4	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	292	1	no	—
201	C₂×C₄⋊C₄	—	146	1	no	—
202	C₂×C₄⋊C₄	—	146	1	no	—
203	C₂×C₂²⋊C₄	—	292	1	no	—
204	C₂×C₂²⋊C₄	—	292	1	no	—
205	C₂×C₄²	—	292	1	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	73	2	no	—
208	C₂×C₁₆	—	146	2	no	—
209	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
210	#107	—	2	1	no	—
211	#107	—	73	2	no	—
212	#107	—	73	2	no	—
213	#107	—	146	1	no	—
214	#107	—	146	1	no	—
215	#107	C₂²×D₇₃	1	2	yes	—
216	#106	—	146	4	no	—
217	#103	—	146	4	no	—
218	C₂⁴	—	73	1	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	292	1	no	—
226	C₂²×C₄	—	292	1	no	—
227	C₂²×C₄	—	73	1	no	—
228	C₂²×C₄	—	73	2	no	—
229	C₂²×C₄	—	73	2	no	—
230	C₂²×C₄	—	146	2	no	—
231	C₂²×C₄	—	146	2	no	—
232	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	2	no	—
236	C₂×C₈	—	73	2	no	—
237	C₂×C₈	—	146	2	no	—
238	C₂×C₈	—	146	4	no	—
239	C₂×C₈	—	146	2	no	—
240	C₂×C₈	—	146	2	no	—
241	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
242	C₂×C₈	#11924027	1	2	yes	—
243	C₂×C₈	#11924028	1	2	yes	—
244	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	292	4	no	—
248	C₄⋊C₄	—	292	1	no	—
249	C₄⋊C₄	—	292	1	no	—
250	C₄⋊C₄	—	146	2	no	—
251	C₄⋊C₄	—	146	2	no	—
252	C₂²⋊C₄	—	292	4	no	—
253	C₂²⋊C₄	—	292	4	no	—
254	C₄²	—	292	4	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	146	2	no	—
257	C₂³	—	73	2	no	—
258	C₂³	—	146	2	no	—
259	C₂³	—	146	2	no	—
260	C₂³	—	146	2	no	—
261	C₂³	D₈×D₇₃	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	2	no	—
268	C₂×C₄	—	8	2	no	—
269	C₂×C₄	—	292	2	no	—
270	C₂×C₄	—	292	2	no	—
271	C₂×C₄	—	584	2	no	—
272	C₂×C₄	—	584	2	no	—
273	C₂×C₄	—	73	2	no	—
274	C₂×C₄	—	146	2	no	—
275	C₂×C₄	—	146	2	no	—
276	C₂×C₄	—	146	4	no	—
277	C₂×C₄	—	146	2	no	—
278	C₂×C₄	#14556573	1	2	yes	—
279	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
280	C₂×C₄	#14556620	1	2	yes	—
281	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
282	C₈	—	146	2	no	—
283	C₈	#14556573	1	2	yes	—
284	C₈	C₂×#11924027	1	1	yes	characteristic
285	C₈	C₂×#11924028	1	1	yes	characteristic
286	C₂²	—	292	4	no	—
287	C₂²	—	146	2	no	—
288	C₂²	—	146	2	no	—
289	C₂²	—	146	2	no	—
290	C₂²	#17988899	1	2	yes	—
291	C₂²	#17989020	1	2	yes	—
292	C₂²	#17989026	1	2	yes	—
293	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
294	C₄	—	8	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	584	2	no	—
297	C₄	—	584	2	no	—
298	C₄	—	146	2	no	—
299	C₄	#17988899	1	2	yes	—
300	C₄	C₂×#14556573	1	1	yes	characteristic
301	C₄	C₂×#14556620	1	1	yes	characteristic
302	C₂	—	292	2	no	—
303	C₂	#22612138	1	4	yes	direct product factor
304	C₂	C₂×#17988899	1	1	yes	characteristic
305	C₂	C₂×#17989020	1	1	yes	characteristic
306	C₂	C₂×#17989026	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#358×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₈
prime radical	C₂³×D₇₃	D₈
hypercenter	C₂×#358	D₇₃
hypocenter	C₇₃	C₂×#2237
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
3. #22612138	C₂	direct (N × H)
4. C₂×#17988414	C₂	semidirect
5. C₂×#17988413	C₂	semidirect
6. C₂×#17988391	C₂	semidirect
8. C₂×#358×C₇₃	C₂	semidirect
21. #17988414	C₂²	semidirect
22. #17988413	C₂²	semidirect
23. #17988391	C₂²	semidirect
28. #358×C₇₃	C₂²	semidirect
149. C₂×C₇₃	#2237	semidirect
160. C₂×#358	D₇₃	semidirect
162. C₇₃	C₂×#2237	semidirect
181. #358	C₂×D₇₃	semidirect
303. C₂	#22612138	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×#17988825	2
C₂×#17988624	2
#22612138	2
C₂×#17988414	2
C₂×#17988413	2
C₂×#17988391	2
C₂×#17988366	2
C₂×#358×C₇₃	2
C₂×#2237	73

Subgroups — C2×#22612138 #33240761

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22612138 #33240761