Subgroups — C₂×C₆₇×#2483 #30713997

Order 17152.

4382 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2483	C₂	1	16	yes	maximal, direct product factor
3	C₂×#397×C₆₇	C₂	1	4	yes	maximal, complemented
4	C₂×#385×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#381×C₆₇	C₂	1	4	yes	maximal, complemented
6	C₂×#379×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₆₇	C₂²	1	4	yes	complemented
11	C₂²×C₂²⋊C₄×C₆₇	—	2	16	no	—
12	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
14	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
15	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	16	yes	—
16	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
17	#397×C₆₇	C₂²	1	32	yes	complemented
18	#385×C₆₇	C₂²	1	32	yes	complemented
19	#381×C₆₇	C₂²	1	32	yes	complemented
20	#379×C₆₇	C₂²	1	8	yes	—
21	C₂×#95×C₆₇	C₂²	1	4	yes	complemented
22	C₂×#95×C₆₇	C₂²	1	8	yes	complemented
23	C₂⁵×C₆₇	—	2	16	no	—
24	C₂⁵×C₆₇	—	4	4	no	—
25	C₂⁵×C₆₇	D₄	1	8	yes	—
26	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
27	C₂⁵×C₆₇	C₂³	1	4	yes	—
28	C₂⁵×C₆₇	C₂³	1	2	yes	—
29	C₂³×C₄×C₆₇	—	2	8	no	—
30	C₂³×C₄×C₆₇	—	2	8	no	—
31	C₂³×C₄×C₆₇	—	2	16	no	—
32	C₂³×C₄×C₆₇	—	4	8	no	—
33	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
34	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
36	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
38	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	complemented
40	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
41	C₂×C₂²⋊C₄×C₆₇	—	2	64	no	—
42	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
43	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
44	C₂×C₂²⋊C₄×C₆₇	—	4	4	no	—
45	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
46	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
47	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
48	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
49	#95×C₆₇	C₂³	1	16	yes	complemented
50	#95×C₆₇	C₂³	1	32	yes	complemented
51	C₂⁴×C₆₇	—	2	8	no	—
52	C₂⁴×C₆₇	—	2	8	no	—
53	C₂⁴×C₆₇	—	2	64	no	—
54	C₂⁴×C₆₇	—	2	32	no	—
55	C₂⁴×C₆₇	—	2	8	no	—
56	C₂⁴×C₆₇	—	2	4	no	—
57	C₂⁴×C₆₇	—	4	16	no	—
58	C₂⁴×C₆₇	—	4	8	no	—
59	C₂⁴×C₆₇	—	4	64	no	—
60	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
61	C₂⁴×C₆₇	C₂×D₄	1	16	yes	—
62	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
63	C₂⁴×C₆₇	#41	1	16	yes	—
64	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
65	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
66	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	16	no	—
68	C₂²×C₄×C₆₇	—	2	16	no	—
69	C₂²×C₄×C₆₇	—	2	8	no	—
70	C₂²×C₄×C₆₇	—	2	8	no	—
71	C₂²×C₄×C₆₇	—	2	16	no	—
72	C₂²×C₄×C₆₇	—	2	32	no	—
73	C₂²×C₄×C₆₇	—	2	16	no	—
74	C₂²×C₄×C₆₇	—	2	8	no	—
75	C₂²×C₄×C₆₇	—	2	16	no	—
76	C₂²×C₄×C₆₇	—	4	16	no	—
77	C₂²×C₄×C₆₇	—	4	32	no	—
78	C₂²×C₄×C₆₇	—	4	32	no	—
79	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
82	C₄⋊C₄×C₆₇	—	4	16	no	—
83	C₂²⋊C₄×C₆₇	—	2	32	no	—
84	C₂²⋊C₄×C₆₇	—	2	32	no	—
85	C₂²⋊C₄×C₆₇	—	2	32	no	—
86	C₂²⋊C₄×C₆₇	—	4	8	no	—
87	C₂³×C₆₇	—	2	16	no	—
88	C₂³×C₆₇	—	2	16	no	—
89	C₂³×C₆₇	—	2	64	no	—
90	C₂³×C₆₇	—	2	8	no	—
91	C₂³×C₆₇	—	2	8	no	—
92	C₂³×C₆₇	—	2	64	no	—
93	C₂³×C₆₇	—	2	16	no	—
94	C₂³×C₆₇	—	2	8	no	—
95	C₂³×C₆₇	—	2	8	no	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	—	4	16	no	—
98	C₂³×C₆₇	—	4	64	no	—
99	C₂³×C₆₇	—	4	128	no	—
100	C₂³×C₆₇	—	4	16	no	—
101	C₂³×C₆₇	#123	1	32	yes	—
102	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
103	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
104	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
105	C₂³×C₆₇	C₂×#41	1	16	yes	—
106	C₂³×C₆₇	C₂×#41	1	8	yes	—
107	C₂³×C₆₇	H(5,𝔽₂)	1	4	yes	—
108	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
109	C₂×C₄×C₆₇	—	2	8	no	—
110	C₂×C₄×C₆₇	—	2	8	no	—
111	C₂×C₄×C₆₇	—	2	16	no	—
112	C₂×C₄×C₆₇	—	2	16	no	—
113	C₂×C₄×C₆₇	—	2	8	no	—
114	C₂×C₄×C₆₇	—	2	16	no	—
115	C₂×C₄×C₆₇	—	4	32	no	—
116	C₂×C₄×C₆₇	—	4	32	no	—
117	C₂×C₄×C₆₇	—	4	16	no	—
118	C₂×C₄×C₆₇	—	4	16	no	—
119	C₂×C₄×C₆₇	—	4	16	no	—
120	C₂²×C₆₇	—	2	8	no	—
121	C₂²×C₆₇	—	2	8	no	—
122	C₂²×C₆₇	—	2	32	no	—
123	C₂²×C₆₇	—	2	64	no	—
124	C₂²×C₆₇	—	2	8	no	—
125	C₂²×C₆₇	—	2	4	no	—
126	C₂²×C₆₇	—	4	64	no	—
127	C₂²×C₆₇	—	4	8	no	—
128	C₂²×C₆₇	—	4	16	no	—
129	C₂²×C₆₇	C₂×#123	1	16	yes	—
130	C₂²×C₆₇	C₂×#123	1	8	yes	—
131	C₂²×C₆₇	#534	1	4	yes	—
132	C₂²×C₆₇	#551	1	16	yes	—
133	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
134	C₂²×C₆₇	C₂²×#41	1	4	yes	—
135	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
136	C₄×C₆₇	—	4	8	no	—
137	C₄×C₆₇	—	4	8	no	—
138	C₄×C₆₇	—	4	8	no	—
139	C₂×#2483	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₆₇	—	2	16	no	—
141	C₂×C₆₇	—	4	4	no	—
142	C₂×C₆₇	#2483	1	8	yes	direct product factor
143	C₂×C₆₇	C₂²×#123	1	2	yes	—
144	C₂×C₆₇	C₂×#534	1	1	yes	characteristic
145	C₂×C₆₇	C₂×#551	1	4	yes	—
146	C₂³×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
147	#2483	C₂×C₆₇	1	16	yes	direct product factor
148	C₂×#397	C₂×C₆₇	1	4	yes	complemented
149	C₂×#385	C₂×C₆₇	1	4	yes	complemented
150	C₂×#381	C₂×C₆₇	1	4	yes	complemented
151	C₂×#379	C₂×C₆₇	1	1	yes	characteristic
152	C₆₇	C₂×#2483	1	1	yes	characteristic, direct product factor, Sylow
153	C₂⁶	C₂²×C₆₇	1	1	yes	characteristic
154	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
155	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
156	C₂²×C₄⋊C₄	C₂²×C₆₇	1	4	yes	complemented
157	C₂²×C₂²⋊C₄	—	2	16	no	—
158	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
159	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	16	yes	—
160	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
161	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
162	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
163	#397	C₂²×C₆₇	1	32	yes	complemented
164	#385	C₂²×C₆₇	1	32	yes	complemented
165	#381	C₂²×C₆₇	1	32	yes	complemented
166	#379	C₂²×C₆₇	1	8	yes	—
167	C₂×#95	C₂²×C₆₇	1	4	yes	complemented
168	C₂×#95	C₂²×C₆₇	1	8	yes	complemented
169	C₂⁵	—	2	16	no	—
170	C₂⁵	—	4	4	no	—
171	C₂⁵	D₄×C₆₇	1	8	yes	—
172	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
173	C₂⁵	C₂³×C₆₇	1	4	yes	—
174	C₂⁵	C₂³×C₆₇	1	2	yes	—
175	C₂³×C₄	—	2	8	no	—
176	C₂³×C₄	—	2	8	no	—
177	C₂³×C₄	—	2	16	no	—
178	C₂³×C₄	—	4	8	no	—
179	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
180	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
181	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
182	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
183	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
184	C₂×C₄⋊C₄	—	4	8	no	—
185	C₂×C₄⋊C₄	C₂³×C₆₇	1	16	yes	complemented
186	C₂×C₂²⋊C₄	—	2	16	no	—
187	C₂×C₂²⋊C₄	—	2	64	no	—
188	C₂×C₂²⋊C₄	—	2	16	no	—
189	C₂×C₂²⋊C₄	—	2	16	no	—
190	C₂×C₂²⋊C₄	—	4	4	no	—
191	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
192	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
193	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
194	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
195	#95	C₂³×C₆₇	1	16	yes	complemented
196	#95	C₂³×C₆₇	1	32	yes	complemented
197	C₂⁴	—	2	8	no	—
198	C₂⁴	—	2	8	no	—
199	C₂⁴	—	2	64	no	—
200	C₂⁴	—	2	32	no	—
201	C₂⁴	—	2	8	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	4	16	no	—
204	C₂⁴	—	4	8	no	—
205	C₂⁴	—	4	64	no	—
206	C₂⁴	C₂×D₄×C₆₇	1	4	yes	—
207	C₂⁴	C₂×D₄×C₆₇	1	16	yes	—
208	C₂⁴	C₂×D₄×C₆₇	1	8	yes	—
209	C₂⁴	#41×C₆₇	1	16	yes	—
210	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
211	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
212	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
213	C₂²×C₄	—	2	16	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	2	8	no	—
216	C₂²×C₄	—	2	8	no	—
217	C₂²×C₄	—	2	16	no	—
218	C₂²×C₄	—	2	32	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	16	no	—
222	C₂²×C₄	—	4	16	no	—
223	C₂²×C₄	—	4	32	no	—
224	C₂²×C₄	—	4	32	no	—
225	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
226	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
227	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
228	C₄⋊C₄	—	4	16	no	—
229	C₂²⋊C₄	—	2	32	no	—
230	C₂²⋊C₄	—	2	32	no	—
231	C₂²⋊C₄	—	2	32	no	—
232	C₂²⋊C₄	—	4	8	no	—
233	C₂³	—	2	16	no	—
234	C₂³	—	2	16	no	—
235	C₂³	—	2	64	no	—
236	C₂³	—	2	8	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	2	64	no	—
239	C₂³	—	2	16	no	—
240	C₂³	—	2	8	no	—
241	C₂³	—	2	8	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	16	no	—
244	C₂³	—	4	64	no	—
245	C₂³	—	4	128	no	—
246	C₂³	—	4	16	no	—
247	C₂³	#123×C₆₇	1	32	yes	—
248	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
249	C₂³	C₂²×D₄×C₆₇	1	8	yes	—
250	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
251	C₂³	C₂×#41×C₆₇	1	16	yes	—
252	C₂³	C₂×#41×C₆₇	1	8	yes	—
253	C₂³	H(5,𝔽₂)×C₆₇	1	4	yes	—
254	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	16	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	2	16	no	—
261	C₂×C₄	—	4	32	no	—
262	C₂×C₄	—	4	32	no	—
263	C₂×C₄	—	4	16	no	—
264	C₂×C₄	—	4	16	no	—
265	C₂×C₄	—	4	16	no	—
266	C₂²	—	2	8	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	2	32	no	—
269	C₂²	—	2	64	no	—
270	C₂²	—	2	8	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	4	64	no	—
273	C₂²	—	4	8	no	—
274	C₂²	—	4	16	no	—
275	C₂²	C₂×#123×C₆₇	1	16	yes	—
276	C₂²	C₂×#123×C₆₇	1	8	yes	—
277	C₂²	#534×C₆₇	1	4	yes	—
278	C₂²	#551×C₆₇	1	16	yes	—
279	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
280	C₂²	C₂²×#41×C₆₇	1	4	yes	—
281	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
282	C₄	—	4	8	no	—
283	C₄	—	4	8	no	—
284	C₄	—	4	8	no	—
285	C₂	—	2	16	no	—
286	C₂	—	4	4	no	—
287	C₂	C₆₇×#2483	1	8	yes	direct product factor
288	C₂	C₂²×#123×C₆₇	1	2	yes	—
289	C₂	C₂×#534×C₆₇	1	1	yes	characteristic
290	C₂	C₂×#551×C₆₇	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₆₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁶×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2483	C₂	direct (N × H)
3. C₂×#397×C₆₇	C₂	semidirect
4. C₂×#385×C₆₇	C₂	semidirect
5. C₂×#381×C₆₇	C₂	semidirect
10. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
17. #397×C₆₇	C₂²	semidirect
18. #385×C₆₇	C₂²	semidirect
19. #381×C₆₇	C₂²	semidirect
21. C₂×#95×C₆₇	C₂²	semidirect
22. C₂×#95×C₆₇	C₂²	semidirect
39. C₂×C₄⋊C₄×C₆₇	C₂³	semidirect
49. #95×C₆₇	C₂³	semidirect
50. #95×C₆₇	C₂³	semidirect
139. C₂×#2483	C₆₇	direct (N × H)
142. C₂×C₆₇	#2483	direct (N × H)
147. #2483	C₂×C₆₇	direct (N × H)
148. C₂×#397	C₂×C₆₇	semidirect
149. C₂×#385	C₂×C₆₇	semidirect
150. C₂×#381	C₂×C₆₇	semidirect
152. C₆₇	C₂×#2483	direct (N × H)
156. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
163. #397	C₂²×C₆₇	semidirect
164. #385	C₂²×C₆₇	semidirect
165. #381	C₂²×C₆₇	semidirect
167. C₂×#95	C₂²×C₆₇	semidirect
168. C₂×#95	C₂²×C₆₇	semidirect
185. C₂×C₄⋊C₄	C₂³×C₆₇	semidirect
195. #95	C₂³×C₆₇	semidirect
196. #95	C₂³×C₆₇	semidirect
287. C₂	C₆₇×#2483	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×C₂²⋊C₄×C₆₇	2
C₆₇×#2483	2
C₂×#397×C₆₇	2
C₂×#385×C₆₇	2
C₂×#381×C₆₇	2
C₂×#379×C₆₇	2
C₂×#2483	67

Subgroups — C2×C67×#2483 #30713997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2483 #30713997