Subgroups — C₂³×#14556611 #33999769

Order 18688.

11422 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	C₂³×#11924028	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14556611	C₂	1	56	yes	maximal, direct product factor
3	C₂³×#11924018	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×#11924016	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11924015	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11924013	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×#41×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924028	—	2	14	no	—
10	C₂²×#11924028	C₂²	1	28	yes	complemented
11	C₂³×Dic₁₄₆	C₂²	1	1	yes	characteristic
12	C₂³×D₂₉₂	C₂²	1	1	yes	characteristic
13	C₂³×C₂²⋊D₇₃	—	2	1	no	—
14	C₂³×C₄×D₇₃	—	2	1	no	—
15	C₂×#14556611	C₂²	1	448	yes	direct product factor
16	C₂²×#11924018	C₂²	1	28	yes	complemented
17	C₂²×#11924016	C₂²	1	28	yes	complemented
18	C₂²×#11924015	C₂²	1	28	yes	complemented
19	C₂²×#11924013	C₂²	1	28	yes	complemented
20	C₂³×#10596831	C₂²	1	1	yes	characteristic
21	C₂³×#10596831	C₂²	1	1	yes	characteristic
22	C₂³×#10596831	C₂²	1	28	yes	complemented
23	C₂²×#41×C₇₃	—	2	14	no	—
24	C₂²×#41×C₇₃	C₂²	1	28	yes	complemented
25	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₂³×D₄×C₇₃	—	2	1	no	—
27	C₂³×D₄×C₇₃	C₂²	1	1	yes	characteristic
28	C₂⁴×C₄×C₇₃	—	2	1	no	—
29	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
30	C₂⁴×D₇₃	—	2	1	no	—
31	C₂×#11924028	—	2	168	no	—
32	C₂×#11924028	C₂³	1	112	yes	complemented
33	C₂²×Dic₁₄₆	—	2	7	no	—
34	C₂²×Dic₁₄₆	C₂³	1	14	yes	—
35	C₂²×D₂₉₂	—	2	7	no	—
36	C₂²×D₂₉₂	C₂³	1	14	yes	—
37	C₂²×C₂²⋊D₇₃	—	2	28	no	—
38	C₂³×Dic₇₃	—	2	1	no	—
39	C₂²×C₄×D₇₃	—	2	28	no	—
40	#14556611	C₂³	1	512	yes	direct product factor
41	C₂×#11924018	C₂³	1	112	yes	complemented
42	C₂×#11924016	C₂³	1	112	yes	complemented
43	C₂×#11924015	C₂³	1	112	yes	complemented
44	C₂×#11924013	C₂³	1	112	yes	complemented
45	C₂²×#10596831	C₂³	1	14	yes	—
46	C₂²×#10596831	C₂³	1	14	yes	—
47	C₂²×#10596831	C₂³	1	112	yes	complemented
48	C₂⁵×C₇₃	—	2	1	no	—
49	C₂⁵×C₇₃	D₄	1	1	yes	characteristic
50	C₂×#41×C₇₃	—	2	168	no	—
51	C₂×#41×C₇₃	C₂³	1	112	yes	complemented
52	C₂²×Q₈×C₇₃	—	2	7	no	—
53	C₂²×Q₈×C₇₃	C₂³	1	14	yes	—
54	C₂²×D₄×C₇₃	—	2	28	no	—
55	C₂²×D₄×C₇₃	—	2	7	no	—
56	C₂²×D₄×C₇₃	C₂³	1	14	yes	—
57	C₂³×C₄×C₇₃	—	2	1	no	—
58	C₂³×C₄×C₇₃	—	2	28	no	—
59	C₂³×C₄×C₇₃	D₄	1	14	yes	—
60	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₇₃	C₂³	1	14	yes	—
63	C₂³×D₇₃	—	2	14	no	—
64	C₂³×D₇₃	—	4	8	no	—
65	#11924028	—	2	224	no	—
66	#11924028	C₂⁴	1	64	yes	complemented
67	C₂×Dic₁₄₆	—	2	42	no	—
68	C₂×Dic₁₄₆	C₂⁴	1	28	yes	—
69	C₂×D₂₉₂	—	2	42	no	—
70	C₂×D₂₉₂	C₂⁴	1	28	yes	—
71	C₂×C₂²⋊D₇₃	—	2	112	no	—
72	C₂²×Dic₇₃	—	2	14	no	—
73	C₂×C₄×D₇₃	—	2	112	no	—
74	#11924018	C₂⁴	1	64	yes	complemented
75	#11924016	C₂⁴	1	64	yes	complemented
76	#11924015	C₂⁴	1	64	yes	complemented
77	#11924013	C₂⁴	1	64	yes	complemented
78	C₂×#10596831	C₂⁴	1	28	yes	—
79	C₂×#10596831	C₂⁴	1	28	yes	—
80	C₂×#10596831	C₂⁴	1	64	yes	complemented
81	C₂⁴×C₇₃	—	2	8	no	—
82	C₂⁴×C₇₃	—	2	14	no	—
83	C₂⁴×C₇₃	—	4	8	no	—
84	C₂⁴×C₇₃	C₂×D₄	1	14	yes	—
85	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
86	#41×C₇₃	—	2	224	no	—
87	#41×C₇₃	C₂⁴	1	64	yes	complemented
88	C₂×Q₈×C₇₃	—	2	42	no	—
89	C₂×Q₈×C₇₃	C₂⁴	1	28	yes	—
90	C₂×D₄×C₇₃	—	2	112	no	—
91	C₂×D₄×C₇₃	—	2	42	no	—
92	C₂×D₄×C₇₃	C₂⁴	1	28	yes	—
93	C₂²×C₄×C₇₃	—	2	14	no	—
94	C₂²×C₄×C₇₃	—	2	112	no	—
95	C₂²×C₄×C₇₃	C₂×D₄	1	14	yes	—
96	C₂²×C₄×C₇₃	C₂×D₄	1	7	yes	—
97	C₂²×C₄×C₇₃	C₂×D₄	1	84	yes	—
98	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
99	C₂²×C₄×C₇₃	C₂⁴	1	28	yes	—
100	C₂³×C₇₃	—	2	56	no	—
101	C₂³×C₇₃	—	2	28	no	—
102	C₂³×C₇₃	—	4	56	no	—
103	C₂³×C₇₃	#135	1	8	yes	complemented
104	C₂³×C₇₃	C₂²×D₄	1	28	yes	—
105	C₂³×C₇₃	C₂²×D₄	1	7	yes	—
106	C₂²×D₇₃	—	2	28	no	—
107	C₂²×D₇₃	—	4	56	no	—
108	Q₈×C₇₃	—	2	28	no	—
109	Q₈×C₇₃	C₂⁵	1	8	yes	—
110	D₄×C₇₃	—	2	64	no	—
111	D₄×C₇₃	—	2	28	no	—
112	D₄×C₇₃	C₂⁵	1	8	yes	—
113	C₂×C₄×C₇₃	—	2	28	no	—
114	C₂×C₄×C₇₃	—	2	64	no	—
115	C₂×C₄×C₇₃	C₂²×D₄	1	28	yes	—
116	C₂×C₄×C₇₃	C₂²×D₄	1	21	yes	—
117	C₂×C₄×C₇₃	C₂²×D₄	1	56	yes	—
118	C₂×C₄×C₇₃	C₂⁵	1	7	yes	—
119	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
120	C₂²⋊D₇₃	—	2	64	no	—
121	C₂×Dic₇₃	—	2	28	no	—
122	D₂₉₂	—	2	28	no	—
123	D₂₉₂	C₂⁵	1	8	yes	—
124	C₄×D₇₃	—	2	64	no	—
125	Dic₁₄₆	—	2	28	no	—
126	Dic₁₄₆	C₂⁵	1	8	yes	—
127	#10596831	C₂⁵	1	8	yes	—
128	#10596831	C₂⁵	1	8	yes	—
129	C₂²×C₇₃	—	2	56	no	—
130	C₂²×C₇₃	—	2	8	no	—
131	C₂²×C₇₃	—	4	56	no	—
132	C₂²×C₇₃	C₂×#135	1	28	yes	complemented
133	C₂²×C₇₃	C₂³×D₄	1	8	yes	—
134	C₂²×C₇₃	C₂³×D₄	1	7	yes	—
135	C₂×D₇₃	—	2	8	no	—
136	C₂×D₇₃	—	4	56	no	—
137	C₄×C₇₃	—	2	8	no	—
138	C₄×C₇₃	C₂³×D₄	1	7	yes	—
139	C₄×C₇₃	C₂³×D₄	1	8	yes	—
140	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
141	Dic₇₃	—	2	8	no	—
142	C₂³×#135	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	8	no	—
144	C₂×C₇₃	—	4	8	no	—
145	C₂×C₇₃	C₂²×#135	1	14	yes	complemented
146	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
147	D₇₃	—	4	8	no	—
148	C₂³×#41	—	73	1	no	—
149	C₂³×#41	D₇₃	1	1	yes	characteristic, complemented
150	C₂²×#135	—	73	56	no	—
151	C₂³×Q₁₆	—	73	1	no	—
152	C₂³×SD₁₆	—	73	1	no	—
153	C₂³×SD₁₆	—	73	1	no	—
154	C₂³×D₈	—	73	1	no	—
155	C₂⁴×C₈	—	73	1	no	—
156	C₇₃	C₂³×#135	1	1	yes	characteristic, complemented, Sylow
157	C₂²×#41	—	2	14	no	—
158	C₂²×#41	—	73	28	no	—
159	C₂²×#41	—	146	14	no	—
160	C₂²×#41	C₂×D₇₃	1	28	yes	complemented
161	C₂³×Q₈	—	73	1	no	—
162	C₂³×Q₈	C₂×D₇₃	1	1	yes	characteristic
163	C₂³×D₄	—	2	1	no	—
164	C₂³×D₄	—	73	1	no	—
165	C₂³×D₄	—	146	1	no	—
166	C₂³×D₄	C₂×D₇₃	1	1	yes	characteristic
167	C₂⁴×C₄	—	2	1	no	—
168	C₂⁴×C₄	—	146	1	no	—
169	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
170	C₂×#135	—	73	448	no	—
171	C₂²×Q₁₆	—	73	28	no	—
172	C₂²×SD₁₆	—	73	28	no	—
173	C₂²×SD₁₆	—	73	28	no	—
174	C₂²×D₈	—	73	28	no	—
175	C₂³×C₈	—	73	1	no	—
176	C₂³×C₈	—	73	1	no	—
177	C₂³×C₈	—	73	28	no	—
178	C₂⁵	—	2	1	no	—
179	C₂⁵	—	146	1	no	—
180	C₂⁵	C₂²⋊D₇₃	1	1	yes	characteristic
181	C₂×#41	—	2	168	no	—
182	C₂×#41	—	73	112	no	—
183	C₂×#41	—	146	168	no	—
184	C₂×#41	C₂²×D₇₃	1	112	yes	complemented
185	C₂²×Q₈	—	2	7	no	—
186	C₂²×Q₈	—	73	14	no	—
187	C₂²×Q₈	—	146	7	no	—
188	C₂²×Q₈	C₂²×D₇₃	1	14	yes	—
189	C₂²×D₄	—	2	28	no	—
190	C₂²×D₄	—	2	7	no	—
191	C₂²×D₄	—	73	14	no	—
192	C₂²×D₄	—	146	28	no	—
193	C₂²×D₄	—	146	7	no	—
194	C₂²×D₄	C₂²×D₇₃	1	14	yes	—
195	C₂³×C₄	—	2	1	no	—
196	C₂³×C₄	—	2	28	no	—
197	C₂³×C₄	—	146	1	no	—
198	C₂³×C₄	—	146	28	no	—
199	C₂³×C₄	C₂²⋊D₇₃	1	14	yes	—
200	C₂³×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
201	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
202	C₂³×C₄	C₂²×D₇₃	1	14	yes	—
203	#135	—	73	512	no	—
204	C₂×Q₁₆	—	73	112	no	—
205	C₂×SD₁₆	—	73	112	no	—
206	C₂×SD₁₆	—	73	112	no	—
207	C₂×D₈	—	73	112	no	—
208	C₂²×C₈	—	73	14	no	—
209	C₂²×C₈	—	73	14	no	—
210	C₂²×C₈	—	73	112	no	—
211	C₂⁴	—	2	8	no	—
212	C₂⁴	—	2	14	no	—
213	C₂⁴	—	4	8	no	—
214	C₂⁴	—	292	8	no	—
215	C₂⁴	—	146	14	no	—
216	C₂⁴	C₂×C₂²⋊D₇₃	1	14	yes	—
217	C₂⁴	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
218	#41	—	2	224	no	—
219	#41	—	73	64	no	—
220	#41	—	146	224	no	—
221	#41	C₂³×D₇₃	1	64	yes	complemented
222	C₂×Q₈	—	2	42	no	—
223	C₂×Q₈	—	73	28	no	—
224	C₂×Q₈	—	146	42	no	—
225	C₂×Q₈	C₂³×D₇₃	1	28	yes	—
226	C₂×D₄	—	2	112	no	—
227	C₂×D₄	—	2	42	no	—
228	C₂×D₄	—	73	28	no	—
229	C₂×D₄	—	146	112	no	—
230	C₂×D₄	—	146	42	no	—
231	C₂×D₄	C₂³×D₇₃	1	28	yes	—
232	C₂²×C₄	—	2	14	no	—
233	C₂²×C₄	—	2	112	no	—
234	C₂²×C₄	—	146	14	no	—
235	C₂²×C₄	—	146	112	no	—
236	C₂²×C₄	C₂×C₂²⋊D₇₃	1	14	yes	—
237	C₂²×C₄	C₂×C₂²⋊D₇₃	1	7	yes	—
238	C₂²×C₄	C₂×C₂²⋊D₇₃	1	84	yes	—
239	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
240	C₂²×C₄	C₂³×D₇₃	1	28	yes	—
241	Q₁₆	—	73	64	no	—
242	SD₁₆	—	73	64	no	—
243	SD₁₆	—	73	64	no	—
244	D₈	—	73	64	no	—
245	C₂×C₈	—	73	28	no	—
246	C₂×C₈	—	73	28	no	—
247	C₂×C₈	—	73	64	no	—
248	C₂³	—	2	56	no	—
249	C₂³	—	2	28	no	—
250	C₂³	—	4	56	no	—
251	C₂³	—	292	56	no	—
252	C₂³	—	146	28	no	—
253	C₂³	#14556611	1	8	yes	direct product factor
254	C₂³	C₂²×C₂²⋊D₇₃	1	28	yes	—
255	C₂³	C₂²×C₂²⋊D₇₃	1	7	yes	—
256	Q₈	—	2	28	no	—
257	Q₈	—	73	8	no	—
258	Q₈	—	146	28	no	—
259	Q₈	C₂⁴×D₇₃	1	8	yes	—
260	D₄	—	2	64	no	—
261	D₄	—	2	28	no	—
262	D₄	—	73	8	no	—
263	D₄	—	146	64	no	—
264	D₄	—	146	28	no	—
265	D₄	C₂⁴×D₇₃	1	8	yes	—
266	C₂×C₄	—	2	28	no	—
267	C₂×C₄	—	2	64	no	—
268	C₂×C₄	—	146	28	no	—
269	C₂×C₄	—	146	64	no	—
270	C₂×C₄	C₂²×C₂²⋊D₇₃	1	28	yes	—
271	C₂×C₄	C₂²×C₂²⋊D₇₃	1	21	yes	—
272	C₂×C₄	C₂²×C₂²⋊D₇₃	1	56	yes	—
273	C₂×C₄	C₂⁴×D₇₃	1	7	yes	—
274	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
275	C₈	—	73	8	no	—
276	C₈	—	73	8	no	—
277	C₂²	—	2	56	no	—
278	C₂²	—	2	8	no	—
279	C₂²	—	4	56	no	—
280	C₂²	—	292	56	no	—
281	C₂²	—	146	8	no	—
282	C₂²	C₂×#14556611	1	28	yes	direct product factor
283	C₂²	C₂³×C₂²⋊D₇₃	1	8	yes	—
284	C₂²	C₂³×C₂²⋊D₇₃	1	7	yes	—
285	C₄	—	2	8	no	—
286	C₄	—	146	8	no	—
287	C₄	C₂³×C₂²⋊D₇₃	1	8	yes	—
288	C₄	C₂³×C₂²⋊D₇₃	1	7	yes	—
289	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
290	C₂	—	2	8	no	—
291	C₂	—	4	8	no	—
292	C₂	—	292	8	no	—
293	C₂	C₂²×#14556611	1	14	yes	direct product factor
294	C₂	C₂⁴×C₂²⋊D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₇₃
commutator, cosocle	C₄×C₇₃	C₂⁶
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂³×#41×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂³×#41	D₇₃
hypocenter	C₇₃	C₂³×#135
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂³×#11924028	C₂	semidirect
2. C₂²×#14556611	C₂	direct (N × H)
3. C₂³×#11924018	C₂	semidirect
4. C₂³×#11924016	C₂	semidirect
5. C₂³×#11924015	C₂	semidirect
6. C₂³×#11924013	C₂	semidirect
7. C₂⁴×#10596831	C₂	semidirect
8. C₂³×#41×C₇₃	C₂	semidirect
10. C₂²×#11924028	C₂²	semidirect
15. C₂×#14556611	C₂²	direct (N × H)
16. C₂²×#11924018	C₂²	semidirect
17. C₂²×#11924016	C₂²	semidirect
18. C₂²×#11924015	C₂²	semidirect
19. C₂²×#11924013	C₂²	semidirect
22. C₂³×#10596831	C₂²	semidirect
24. C₂²×#41×C₇₃	C₂²	semidirect
32. C₂×#11924028	C₂³	semidirect
40. #14556611	C₂³	direct (N × H)
41. C₂×#11924018	C₂³	semidirect
42. C₂×#11924016	C₂³	semidirect
43. C₂×#11924015	C₂³	semidirect
44. C₂×#11924013	C₂³	semidirect
47. C₂²×#10596831	C₂³	semidirect
51. C₂×#41×C₇₃	C₂³	semidirect
66. #11924028	C₂⁴	semidirect
74. #11924018	C₂⁴	semidirect
75. #11924016	C₂⁴	semidirect
76. #11924015	C₂⁴	semidirect
77. #11924013	C₂⁴	semidirect
80. C₂×#10596831	C₂⁴	semidirect
87. #41×C₇₃	C₂⁴	semidirect
103. C₂³×C₇₃	#135	semidirect
132. C₂²×C₇₃	C₂×#135	semidirect
145. C₂×C₇₃	C₂²×#135	semidirect
149. C₂³×#41	D₇₃	semidirect
156. C₇₃	C₂³×#135	semidirect
160. C₂²×#41	C₂×D₇₃	semidirect
184. C₂×#41	C₂²×D₇₃	semidirect
221. #41	C₂³×D₇₃	semidirect
253. C₂³	#14556611	direct (N × H)
282. C₂²	C₂×#14556611	direct (N × H)
293. C₂	C₂²×#14556611	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#11924028	2
C₂²×#14556611	2
C₂³×#11924018	2
C₂³×#11924016	2
C₂³×#11924015	2
C₂³×#11924013	2
C₂⁴×#10596831	2
C₂³×#41×C₇₃	2
C₂³×#135	73

Subgroups — C23×#14556611 #33999769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556611 #33999769