Subgroups — C₂³×#14556615 #33999774

Order 18688.

13310 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924027	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14556615	C₂	1	56	yes	maximal, direct product factor
4	C₂³×#11924017	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11924016	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11924013	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₇₃	—	2	1	no	—
10	C₂²×#11924027	—	2	14	no	—
11	C₂²×#11924027	C₂²	1	28	yes	complemented
12	C₂³×Dic₁₄₆	C₂²	1	1	yes	characteristic, complemented
13	C₂³×D₂₉₂	C₂²	1	1	yes	characteristic
14	C₂³×C₂²⋊D₇₃	—	2	1	no	—
15	C₂³×C₂²⋊D₇₃	—	2	1	no	—
16	C₂²×D₄×D₇₃	—	2	14	no	—
17	C₂²×D₄×D₇₃	C₂²	1	28	yes	complemented
18	C₂⁴×Dic₇₃	—	2	1	no	—
19	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
20	C₂×#14556615	C₂²	1	448	yes	direct product factor
21	C₂²×#11924017	C₂²	1	28	yes	complemented
22	C₂²×#11924016	C₂²	1	28	yes	complemented
23	C₂²×#11924013	C₂²	1	28	yes	complemented
24	C₂²×#11924011	C₂²	1	28	yes	complemented
25	C₂³×#10596831	C₂²	1	1	yes	characteristic, complemented
26	C₂³×D₄×C₇₃	C₂²	1	1	yes	characteristic
27	C₂³×D₄×C₇₃	C₂²	1	1	yes	characteristic, complemented
28	C₂²×D₈×C₇₃	C₂²	1	28	yes	complemented
29	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic, complemented
30	C₂⁴×D₇₃	—	2	28	no	—
31	C₂⁴×D₇₃	—	2	1	no	—
32	C₂⁴×D₇₃	—	4	16	no	—
33	C₂⁴×D₇₃	D₄	1	1	yes	characteristic
34	C₂×#11924027	—	2	168	no	—
35	C₂×#11924027	C₂³	1	112	yes	complemented
36	C₂²×Dic₁₄₆	—	2	7	no	—
37	C₂²×Dic₁₄₆	C₂³	1	14	yes	complemented
38	C₂²×D₂₉₂	—	2	7	no	—
39	C₂²×D₂₉₂	C₂³	1	14	yes	—
40	C₂²×C₂²⋊D₇₃	—	2	28	no	—
41	C₂²×C₂²⋊D₇₃	—	2	28	no	—
42	C₂×D₄×D₇₃	—	2	168	no	—
43	C₂×D₄×D₇₃	C₂³	1	112	yes	complemented
44	C₂³×Dic₇₃	—	2	28	no	—
45	C₂³×Dic₇₃	—	2	1	no	—
46	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
47	C₂²×C₄×D₇₃	D₄	1	14	yes	—
48	C₂²×C₄×D₇₃	C₂³	1	14	yes	—
49	#14556615	C₂³	1	512	yes	direct product factor
50	C₂×#11924017	C₂³	1	112	yes	complemented
51	C₂×#11924016	C₂³	1	112	yes	complemented
52	C₂×#11924013	C₂³	1	112	yes	complemented
53	C₂×#11924011	C₂³	1	112	yes	complemented
54	C₂²×#10596831	C₂³	1	14	yes	complemented
55	C₂⁵×C₇₃	—	2	1	no	—
56	C₂⁵×C₇₃	—	2	1	no	—
57	C₂²×D₄×C₇₃	—	2	7	no	—
58	C₂²×D₄×C₇₃	—	2	7	no	—
59	C₂²×D₄×C₇₃	C₂³	1	14	yes	—
60	C₂²×D₄×C₇₃	C₂³	1	14	yes	complemented
61	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
62	C₂×D₈×C₇₃	C₂³	1	112	yes	complemented
63	C₂²×C₈×C₇₃	C₂³	1	14	yes	complemented
64	C₂³×D₇₃	—	2	8	no	—
65	C₂³×D₇₃	—	2	14	no	—
66	C₂³×D₇₃	—	2	112	no	—
67	C₂³×D₇₃	—	4	224	no	—
68	C₂³×D₇₃	—	4	8	no	—
69	C₂³×D₇₃	C₂×D₄	1	14	yes	—
70	#11924027	—	2	224	no	—
71	#11924027	C₂⁴	1	64	yes	complemented
72	C₂×Dic₁₄₆	—	2	42	no	—
73	C₂×Dic₁₄₆	C₂⁴	1	28	yes	complemented
74	C₂×D₂₉₂	—	2	42	no	—
75	C₂×D₂₉₂	C₂⁴	1	28	yes	—
76	C₂×C₂²⋊D₇₃	—	2	112	no	—
77	C₂×C₂²⋊D₇₃	—	2	112	no	—
78	D₄×D₇₃	—	2	224	no	—
79	D₄×D₇₃	C₂⁴	1	64	yes	complemented
80	C₂²×Dic₇₃	—	2	14	no	—
81	C₂²×Dic₇₃	—	2	112	no	—
82	C₂²×Dic₇₃	C₂×D₄	1	14	yes	—
83	C₂×C₄×D₇₃	C₂×D₄	1	84	yes	—
84	C₂×C₄×D₇₃	C₂⁴	1	28	yes	—
85	#11924017	C₂⁴	1	64	yes	complemented
86	#11924016	C₂⁴	1	64	yes	complemented
87	#11924013	C₂⁴	1	64	yes	complemented
88	#11924011	C₂⁴	1	64	yes	complemented
89	C₂×#10596831	C₂⁴	1	28	yes	complemented
90	C₂⁴×C₇₃	—	2	14	no	—
91	C₂⁴×C₇₃	—	2	14	no	—
92	C₂⁴×C₇₃	—	4	8	no	—
93	C₂⁴×C₇₃	—	4	8	no	—
94	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₇₃	—	2	42	no	—
96	C₂×D₄×C₇₃	—	2	42	no	—
97	C₂×D₄×C₇₃	C₂⁴	1	28	yes	—
98	C₂×D₄×C₇₃	C₂⁴	1	28	yes	complemented
99	C₂²×C₄×C₇₃	C₂×D₄	1	7	yes	—
100	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
101	D₈×C₇₃	C₂⁴	1	64	yes	complemented
102	C₂×C₈×C₇₃	C₂⁴	1	28	yes	complemented
103	C₂³×C₇₃	—	2	28	no	—
104	C₂³×C₇₃	—	2	28	no	—
105	C₂³×C₇₃	—	4	56	no	—
106	C₂³×C₇₃	—	4	56	no	—
107	C₂³×C₇₃	AGL(1,ℤ/8)	1	8	yes	complemented
108	C₂³×C₇₃	C₂²×D₄	1	7	yes	—
109	C₂²×D₇₃	—	2	56	no	—
110	C₂²×D₇₃	—	2	28	no	—
111	C₂²×D₇₃	—	2	64	no	—
112	C₂²×D₇₃	—	4	448	no	—
113	C₂²×D₇₃	—	4	56	no	—
114	C₂²×D₇₃	C₂²×D₄	1	28	yes	—
115	D₄×C₇₃	—	2	28	no	—
116	D₄×C₇₃	—	2	28	no	—
117	D₄×C₇₃	C₂⁵	1	8	yes	—
118	D₄×C₇₃	C₂⁵	1	8	yes	complemented
119	C₂×C₄×C₇₃	C₂²×D₄	1	21	yes	—
120	C₂×C₄×C₇₃	C₂⁵	1	7	yes	—
121	C₂²⋊D₇₃	—	2	64	no	—
122	C₂²⋊D₇₃	—	2	64	no	—
123	C₂×Dic₇₃	—	2	28	no	—
124	C₂×Dic₇₃	—	2	64	no	—
125	C₂×Dic₇₃	C₂²×D₄	1	28	yes	—
126	D₂₉₂	—	2	28	no	—
127	D₂₉₂	C₂⁵	1	8	yes	—
128	C₄×D₇₃	C₂²×D₄	1	56	yes	—
129	C₄×D₇₃	C₂⁵	1	8	yes	—
130	Dic₁₄₆	—	2	28	no	—
131	Dic₁₄₆	C₂⁵	1	8	yes	complemented
132	C₈×C₇₃	C₂⁵	1	8	yes	complemented
133	#10596831	C₂⁵	1	8	yes	complemented
134	C₂²×C₇₃	—	2	8	no	—
135	C₂²×C₇₃	—	2	8	no	—
136	C₂²×C₇₃	—	4	56	no	—
137	C₂²×C₇₃	—	4	56	no	—
138	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	28	yes	complemented
139	C₂²×C₇₃	C₂³×D₄	1	7	yes	—
140	C₂×D₇₃	—	2	56	no	—
141	C₂×D₇₃	—	2	8	no	—
142	C₂×D₇₃	—	4	128	no	—
143	C₂×D₇₃	—	4	56	no	—
144	C₂×D₇₃	C₂³×D₄	1	8	yes	—
145	C₄×C₇₃	C₂³×D₄	1	7	yes	—
146	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
147	Dic₇₃	—	2	8	no	—
148	Dic₇₃	C₂³×D₄	1	8	yes	—
149	C₂³×AGL(1,ℤ/8)	—	73	1	no	maximal, Sylow
150	C₂×C₇₃	—	4	8	no	—
151	C₂×C₇₃	—	4	8	no	—
152	C₂×C₇₃	C₂²×AGL(1,ℤ/8)	1	14	yes	complemented
153	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
154	D₇₃	—	2	8	no	—
155	D₇₃	—	4	8	no	—
156	C₂³×#41	—	73	1	no	—
157	C₂⁴×D₄	—	73	1	no	—
158	C₂²×AGL(1,ℤ/8)	—	73	56	no	—
159	C₂³×SD₁₆	—	73	1	no	—
160	C₂³×SD₁₆	—	73	1	no	—
161	C₂³×D₈	—	73	1	no	—
162	C₂³×D₈	D₇₃	1	1	yes	characteristic, complemented
163	C₂³×M₁₆	—	73	1	no	—
164	C₇₃	C₂³×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
165	C₂⁶	—	146	1	no	—
166	C₂²×#41	—	73	28	no	—
167	C₂²×#41	—	146	14	no	—
168	C₂³×Q₈	—	73	1	no	—
169	C₂³×D₄	—	73	28	no	—
170	C₂³×D₄	—	73	1	no	—
171	C₂³×D₄	—	146	14	no	—
172	C₂³×D₄	—	146	1	no	—
173	C₂³×D₄	—	146	1	no	—
174	C₂³×D₄	C₂×D₇₃	1	1	yes	characteristic
175	C₂³×D₄	C₂×D₇₃	1	1	yes	characteristic, complemented
176	C₂⁴×C₄	—	73	1	no	—
177	C₂⁴×C₄	—	146	1	no	—
178	C₂×AGL(1,ℤ/8)	—	73	448	no	—
179	C₂²×SD₁₆	—	73	28	no	—
180	C₂²×SD₁₆	—	73	28	no	—
181	C₂²×D₈	—	73	28	no	—
182	C₂²×D₈	C₂×D₇₃	1	28	yes	complemented
183	C₂²×M₁₆	—	73	28	no	—
184	C₂³×C₈	—	73	1	no	—
185	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic, complemented
186	C₂⁵	—	2	1	no	—
187	C₂⁵	—	2	1	no	—
188	C₂⁵	—	292	16	no	—
189	C₂⁵	—	73	1	no	—
190	C₂⁵	—	146	28	no	—
191	C₂⁵	—	146	1	no	—
192	C₂×#41	—	73	112	no	—
193	C₂×#41	—	146	168	no	—
194	C₂²×Q₈	—	73	14	no	—
195	C₂²×Q₈	—	146	7	no	—
196	C₂²×D₄	—	2	7	no	—
197	C₂²×D₄	—	2	7	no	—
198	C₂²×D₄	—	73	112	no	—
199	C₂²×D₄	—	73	14	no	—
200	C₂²×D₄	—	146	168	no	—
201	C₂²×D₄	—	146	28	no	—
202	C₂²×D₄	—	146	28	no	—
203	C₂²×D₄	—	146	7	no	—
204	C₂²×D₄	C₂²×D₇₃	1	14	yes	—
205	C₂²×D₄	C₂²×D₇₃	1	14	yes	complemented
206	C₂³×C₄	—	73	1	no	—
207	C₂³×C₄	—	73	14	no	—
208	C₂³×C₄	—	73	14	no	—
209	C₂³×C₄	—	146	28	no	—
210	C₂³×C₄	—	146	1	no	—
211	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
212	AGL(1,ℤ/8)	—	73	512	no	—
213	C₂×SD₁₆	—	73	112	no	—
214	C₂×SD₁₆	—	73	112	no	—
215	C₂×D₈	—	73	112	no	—
216	C₂×D₈	C₂²×D₇₃	1	112	yes	complemented
217	C₂×M₁₆	—	73	112	no	—
218	C₂²×C₈	—	73	14	no	—
219	C₂²×C₈	C₂²×D₇₃	1	14	yes	complemented
220	C₂⁴	—	2	14	no	—
221	C₂⁴	—	2	14	no	—
222	C₂⁴	—	4	8	no	—
223	C₂⁴	—	4	8	no	—
224	C₂⁴	—	292	224	no	—
225	C₂⁴	—	292	8	no	—
226	C₂⁴	—	73	14	no	—
227	C₂⁴	—	146	112	no	—
228	C₂⁴	—	146	14	no	—
229	C₂⁴	—	146	8	no	—
230	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
231	#41	—	73	64	no	—
232	#41	—	146	224	no	—
233	C₂×Q₈	—	73	28	no	—
234	C₂×Q₈	—	146	42	no	—
235	C₂×D₄	—	2	42	no	—
236	C₂×D₄	—	2	42	no	—
237	C₂×D₄	—	73	64	no	—
238	C₂×D₄	—	73	28	no	—
239	C₂×D₄	—	146	224	no	—
240	C₂×D₄	—	146	112	no	—
241	C₂×D₄	—	146	112	no	—
242	C₂×D₄	—	146	42	no	—
243	C₂×D₄	C₂³×D₇₃	1	28	yes	—
244	C₂×D₄	C₂³×D₇₃	1	28	yes	complemented
245	C₂²×C₄	—	73	14	no	—
246	C₂²×C₄	—	73	84	no	—
247	C₂²×C₄	—	73	28	no	—
248	C₂²×C₄	—	146	112	no	—
249	C₂²×C₄	—	146	14	no	—
250	C₂²×C₄	D₄×D₇₃	1	7	yes	—
251	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
252	SD₁₆	—	73	64	no	—
253	SD₁₆	—	73	64	no	—
254	D₈	—	73	64	no	—
255	D₈	C₂³×D₇₃	1	64	yes	complemented
256	M₁₆	—	73	64	no	—
257	C₂×C₈	—	73	28	no	—
258	C₂×C₈	C₂³×D₇₃	1	28	yes	complemented
259	C₂³	—	2	28	no	—
260	C₂³	—	2	28	no	—
261	C₂³	—	4	56	no	—
262	C₂³	—	4	56	no	—
263	C₂³	—	292	448	no	—
264	C₂³	—	292	56	no	—
265	C₂³	—	73	28	no	—
266	C₂³	—	146	64	no	—
267	C₂³	—	146	28	no	—
268	C₂³	—	146	56	no	—
269	C₂³	#14556615	1	8	yes	direct product factor
270	C₂³	C₂×D₄×D₇₃	1	7	yes	—
271	Q₈	—	73	8	no	—
272	Q₈	—	146	28	no	—
273	D₄	—	2	28	no	—
274	D₄	—	2	28	no	—
275	D₄	—	73	8	no	—
276	D₄	—	146	64	no	—
277	D₄	—	146	64	no	—
278	D₄	—	146	28	no	—
279	D₄	C₂⁴×D₇₃	1	8	yes	—
280	D₄	C₂⁴×D₇₃	1	8	yes	complemented
281	C₂×C₄	—	73	28	no	—
282	C₂×C₄	—	73	56	no	—
283	C₂×C₄	—	73	8	no	—
284	C₂×C₄	—	146	64	no	—
285	C₂×C₄	—	146	28	no	—
286	C₂×C₄	C₂×D₄×D₇₃	1	21	yes	—
287	C₂×C₄	C₂⁴×D₇₃	1	7	yes	—
288	C₈	—	73	8	no	—
289	C₈	C₂⁴×D₇₃	1	8	yes	complemented
290	C₂²	—	2	8	no	—
291	C₂²	—	2	8	no	—
292	C₂²	—	4	56	no	—
293	C₂²	—	4	56	no	—
294	C₂²	—	292	128	no	—
295	C₂²	—	292	56	no	—
296	C₂²	—	73	8	no	—
297	C₂²	—	146	8	no	—
298	C₂²	—	146	56	no	—
299	C₂²	C₂×#14556615	1	28	yes	direct product factor
300	C₂²	C₂²×D₄×D₇₃	1	7	yes	—
301	C₄	—	73	8	no	—
302	C₄	—	146	8	no	—
303	C₄	C₂²×D₄×D₇₃	1	7	yes	—
304	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
305	C₂	—	4	8	no	—
306	C₂	—	4	8	no	—
307	C₂	—	292	8	no	—
308	C₂	—	146	8	no	—
309	C₂	C₂²×#14556615	1	14	yes	direct product factor
310	C₂	C₂³×D₄×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₃
commutator, cosocle	C₄×C₇₃	C₂⁶
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂³×D₈×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
hypercenter	C₂³×D₈	D₇₃
hypocenter	C₇₃	C₂³×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×D₈

Complements

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

N	H ≅ G/N	type
1. C₂³×#11924027	C₂	semidirect
2. C₂³×D₄×D₇₃	C₂	semidirect
3. C₂²×#14556615	C₂	direct (N × H)
4. C₂³×#11924017	C₂	semidirect
5. C₂³×#11924016	C₂	semidirect
6. C₂³×#11924013	C₂	semidirect
7. C₂³×#11924011	C₂	semidirect
8. C₂³×D₈×C₇₃	C₂	semidirect
11. C₂²×#11924027	C₂²	semidirect
12. C₂³×Dic₁₄₆	C₂²	semidirect
17. C₂²×D₄×D₇₃	C₂²	semidirect
20. C₂×#14556615	C₂²	direct (N × H)
21. C₂²×#11924017	C₂²	semidirect
22. C₂²×#11924016	C₂²	semidirect
23. C₂²×#11924013	C₂²	semidirect
24. C₂²×#11924011	C₂²	semidirect
25. C₂³×#10596831	C₂²	semidirect
27. C₂³×D₄×C₇₃	C₂²	semidirect
28. C₂²×D₈×C₇₃	C₂²	semidirect
29. C₂³×C₈×C₇₃	C₂²	semidirect
35. C₂×#11924027	C₂³	semidirect
37. C₂²×Dic₁₄₆	C₂³	semidirect
43. C₂×D₄×D₇₃	C₂³	semidirect
49. #14556615	C₂³	direct (N × H)
50. C₂×#11924017	C₂³	semidirect
51. C₂×#11924016	C₂³	semidirect
52. C₂×#11924013	C₂³	semidirect
53. C₂×#11924011	C₂³	semidirect
54. C₂²×#10596831	C₂³	semidirect
60. C₂²×D₄×C₇₃	C₂³	semidirect
62. C₂×D₈×C₇₃	C₂³	semidirect
63. C₂²×C₈×C₇₃	C₂³	semidirect
71. #11924027	C₂⁴	semidirect
73. C₂×Dic₁₄₆	C₂⁴	semidirect
79. D₄×D₇₃	C₂⁴	semidirect
85. #11924017	C₂⁴	semidirect
86. #11924016	C₂⁴	semidirect
87. #11924013	C₂⁴	semidirect
88. #11924011	C₂⁴	semidirect
89. C₂×#10596831	C₂⁴	semidirect
98. C₂×D₄×C₇₃	C₂⁴	semidirect
101. D₈×C₇₃	C₂⁴	semidirect
102. C₂×C₈×C₇₃	C₂⁴	semidirect
107. C₂³×C₇₃	AGL(1,ℤ/8)	semidirect
118. D₄×C₇₃	C₂⁵	semidirect
131. Dic₁₄₆	C₂⁵	semidirect
132. C₈×C₇₃	C₂⁵	semidirect
133. #10596831	C₂⁵	semidirect
138. C₂²×C₇₃	C₂×AGL(1,ℤ/8)	semidirect
152. C₂×C₇₃	C₂²×AGL(1,ℤ/8)	semidirect
162. C₂³×D₈	D₇₃	semidirect
164. C₇₃	C₂³×AGL(1,ℤ/8)	semidirect
175. C₂³×D₄	C₂×D₇₃	semidirect
182. C₂²×D₈	C₂×D₇₃	semidirect
185. C₂³×C₈	C₂×D₇₃	semidirect
205. C₂²×D₄	C₂²×D₇₃	semidirect
216. C₂×D₈	C₂²×D₇₃	semidirect
219. C₂²×C₈	C₂²×D₇₃	semidirect
244. C₂×D₄	C₂³×D₇₃	semidirect
255. D₈	C₂³×D₇₃	semidirect
258. C₂×C₈	C₂³×D₇₃	semidirect
269. C₂³	#14556615	direct (N × H)
280. D₄	C₂⁴×D₇₃	semidirect
289. C₈	C₂⁴×D₇₃	semidirect
299. C₂²	C₂×#14556615	direct (N × H)
309. C₂	C₂²×#14556615	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×#11924027	2
C₂³×D₄×D₇₃	2
C₂²×#14556615	2
C₂³×#11924017	2
C₂³×#11924016	2
C₂³×#11924013	2
C₂³×#11924011	2
C₂³×D₈×C₇₃	2
C₂³×AGL(1,ℤ/8)	73

Subgroups — C23×#14556615 #33999774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556615 #33999774