Subgroups — C₂²×#16863256 #31014091

Order 17152.

8670 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊Dic₆₇	C₂	1	2	yes	maximal, complemented
2	C₂³×C₂²⋊Dic₆₇	C₂	1	2	yes	maximal
3	C₂×#16863256	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₆₇⋊#95	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₆₇	C₂²	1	4	yes	—
7	C₂²×C₄⋊Dic₆₇	—	2	16	no	—
8	C₂²×C₄⋊Dic₆₇	C₂²	1	24	yes	complemented
9	C₂²×C₂²⋊Dic₆₇	—	2	16	no	—
10	C₂²×C₂²⋊Dic₆₇	C₂²	1	24	yes	—
11	#16863256	C₂²	1	64	yes	direct product factor
12	C₂×C₆₇⋊#95	C₂²	1	24	yes	complemented
13	C₂⁶×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	—	2	16	no	—
15	C₂⁴×C₄×C₆₇	C₄	1	16	yes	complemented
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄×C₆₇	C₂²	1	12	yes	—
18	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂³×Dic₆₇	—	2	16	no	—
20	C₂³×Dic₆₇	—	2	16	no	—
21	C₂³×Dic₆₇	—	2	16	no	—
22	C₂³×Dic₆₇	C₂³	1	24	yes	—
23	C₂×C₄⋊Dic₆₇	—	2	96	no	—
24	C₂×C₄⋊Dic₆₇	C₂³	1	32	yes	complemented
25	C₂×C₂²⋊Dic₆₇	—	2	96	no	—
26	C₂×C₂²⋊Dic₆₇	C₂³	1	32	yes	—
27	C₆₇⋊#95	C₂³	1	32	yes	complemented
28	C₂⁵×C₆₇	—	2	8	no	—
29	C₂⁵×C₆₇	—	2	8	no	—
30	C₂⁵×C₆₇	C₂×C₄	1	8	yes	—
31	C₂⁵×C₆₇	D₄	1	8	yes	—
32	C₂⁵×C₆₇	Q₈	1	8	yes	—
33	C₂⁵×C₆₇	C₂³	1	6	yes	—
34	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	—	2	8	no	—
36	C₂³×C₄×C₆₇	—	2	128	no	—
37	C₂³×C₄×C₆₇	—	2	96	no	—
38	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
40	C₂³×C₄×C₆₇	C₂×C₄	1	96	yes	complemented
41	C₂³×C₄×C₆₇	D₄	1	16	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	16	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
45	C₂²×Dic₆₇	—	2	48	no	—
46	C₂²×Dic₆₇	—	2	48	no	—
47	C₂²×Dic₆₇	—	2	48	no	—
48	C₂²×Dic₆₇	—	4	64	no	—
49	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
50	C₄⋊Dic₆₇	—	2	64	no	—
51	C₂²⋊Dic₆₇	—	2	64	no	—
52	C₂⁴×C₆₇	—	2	32	no	—
53	C₂⁴×C₆₇	—	2	32	no	—
54	C₂⁴×C₆₇	—	2	64	no	—
55	C₂⁴×C₆₇	—	2	64	no	—
56	C₂⁴×C₆₇	—	2	24	no	—
57	C₂⁴×C₆₇	—	2	24	no	—
58	C₂⁴×C₆₇	C₄⋊C₄	1	64	yes	complemented
59	C₂⁴×C₆₇	C₂²×C₄	1	24	yes	—
60	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
61	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
62	C₂⁴×C₆₇	C₂×D₄	1	24	yes	—
63	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
64	C₂⁴×C₆₇	C₂×Q₈	1	24	yes	—
65	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
66	C₂⁴×C₆₇	#41	1	8	yes	—
67	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
68	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
69	C₂²×C₄×C₆₇	—	2	32	no	—
70	C₂²×C₄×C₆₇	—	2	24	no	—
71	C₂²×C₄×C₆₇	—	2	384	no	—
72	C₂²×C₄×C₆₇	—	2	64	no	—
73	C₂²×C₄×C₆₇	C₂²⋊C₄	1	64	yes	complemented
74	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	64	yes	complemented
77	C₂²×C₄×C₆₇	C₂×D₄	1	48	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
80	C₂³×C₆₇	—	2	48	no	—
81	C₂³×C₆₇	—	2	48	no	—
82	C₂³×C₆₇	—	2	96	no	—
83	C₂³×C₆₇	—	2	96	no	—
84	C₂³×C₆₇	—	2	8	no	—
85	C₂³×C₆₇	—	2	8	no	—
86	C₂³×C₆₇	—	2	256	no	—
87	C₂³×C₆₇	C₂×C₂²⋊C₄	1	16	yes	—
88	C₂³×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
89	C₂³×C₆₇	C₂×C₄⋊C₄	1	96	yes	complemented
90	C₂³×C₆₇	#117	1	16	yes	—
91	C₂³×C₆₇	#121	1	32	yes	—
92	C₂³×C₆₇	C₂²⋊Q₈	1	32	yes	—
93	C₂³×C₆₇	C₂³×C₄	1	8	yes	—
94	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
95	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
96	C₂³×C₆₇	C₂²×D₄	1	8	yes	—
97	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
98	C₂³×C₆₇	C₂²×Q₈	1	8	yes	—
99	C₂³×C₆₇	C₂²×Q₈	1	6	yes	—
100	C₂³×C₆₇	C₂×#41	1	12	yes	—
101	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
102	C₂×C₄×C₆₇	—	2	48	no	—
103	C₂×C₄×C₆₇	—	2	8	no	—
104	C₂×C₄×C₆₇	—	2	128	no	—
105	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	96	yes	complemented
106	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
108	C₂×C₄×C₆₇	C₂²×D₄	1	16	yes	—
109	C₂×Dic₆₇	—	2	16	no	—
110	C₂×Dic₆₇	—	2	16	no	—
111	C₂×Dic₆₇	—	2	16	no	—
112	C₂×Dic₆₇	—	4	96	no	—
113	C₂²×C₆₇	—	2	8	no	—
114	C₂²×C₆₇	—	2	8	no	—
115	C₂²×C₆₇	—	2	16	no	—
116	C₂²×C₆₇	—	2	16	no	—
117	C₂²×C₆₇	—	2	192	no	—
118	C₂²×C₆₇	#380	1	64	yes	complemented
119	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	12	yes	—
120	C₂²×C₆₇	C₂²×C₄⋊C₄	1	12	yes	—
121	C₂²×C₆₇	C₂²×C₄⋊C₄	1	16	yes	complemented
122	C₂²×C₆₇	C₂×#117	1	12	yes	—
123	C₂²×C₆₇	C₂×#121	1	24	yes	—
124	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	24	yes	—
125	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
126	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
127	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
128	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
129	C₂²×C₆₇	C₂²×#41	1	2	yes	—
130	C₄×C₆₇	—	2	8	no	—
131	C₄×C₆₇	C₂²×C₂²⋊C₄	1	16	yes	complemented
132	Dic₆₇	—	4	16	no	—
133	C₂²×#380	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	—	2	16	no	—
135	C₂×C₆₇	C₂×#380	1	24	yes	complemented
136	C₂×C₆₇	C₂³×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
138	C₂×C₆₇	C₂²×#117	1	1	yes	characteristic
139	C₂×C₆₇	C₂²×#121	1	2	yes	—
140	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	2	yes	—
141	C₂⁵×C₄	D₆₇	1	1	yes	characteristic
142	C₂³×C₄⋊C₄	—	67	2	no	—
143	C₂³×C₂²⋊C₄	—	67	2	no	—
144	C₂×#380	—	67	24	no	—
145	C₂²×#95	—	67	2	no	—
146	C₆₇	C₂²×#380	1	1	yes	characteristic, complemented, Sylow
147	C₂⁶	C₂×D₆₇	1	1	yes	characteristic
148	C₂⁴×C₄	—	2	16	no	—
149	C₂⁴×C₄	—	67	4	no	—
150	C₂⁴×C₄	Dic₆₇	1	16	yes	complemented
151	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
152	C₂⁴×C₄	C₂×D₆₇	1	12	yes	—
153	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
154	C₂²×C₄⋊C₄	—	67	24	no	—
155	C₂²×C₄⋊C₄	—	134	16	no	—
156	C₂²×C₂²⋊C₄	—	67	24	no	—
157	C₂²×C₂²⋊C₄	—	134	16	no	—
158	#380	—	67	64	no	—
159	C₂×#95	—	67	24	no	—
160	C₂⁵	—	2	8	no	—
161	C₂⁵	—	2	8	no	—
162	C₂⁵	Dic₁₃₄	1	8	yes	—
163	C₂⁵	D₂₆₈	1	8	yes	—
164	C₂⁵	C₂×Dic₆₇	1	8	yes	—
165	C₂⁵	C₂²×D₆₇	1	6	yes	—
166	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
167	C₂³×C₄	—	2	8	no	—
168	C₂³×C₄	—	2	128	no	—
169	C₂³×C₄	—	2	96	no	—
170	C₂³×C₄	—	67	24	no	—
171	C₂³×C₄	—	134	16	no	—
172	C₂³×C₄	—	134	16	no	—
173	C₂³×C₄	—	134	16	no	—
174	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
175	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
176	C₂³×C₄	C₂×Dic₆₇	1	96	yes	complemented
177	C₂³×C₄	C₂²⋊D₆₇	1	16	yes	—
178	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
179	C₂³×C₄	C₂²×D₆₇	1	16	yes	—
180	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
181	C₂×C₄⋊C₄	—	67	32	no	—
182	C₂×C₄⋊C₄	—	134	96	no	—
183	C₂×C₂²⋊C₄	—	67	32	no	—
184	C₂×C₂²⋊C₄	—	134	96	no	—
185	#95	—	67	32	no	—
186	C₂⁴	—	2	32	no	—
187	C₂⁴	—	2	32	no	—
188	C₂⁴	—	2	64	no	—
189	C₂⁴	—	2	64	no	—
190	C₂⁴	—	2	24	no	—
191	C₂⁴	—	2	24	no	—
192	C₂⁴	C₄⋊Dic₆₇	1	64	yes	complemented
193	C₂⁴	C₂²×Dic₆₇	1	24	yes	—
194	C₂⁴	C₂²×Dic₆₇	1	4	yes	—
195	C₂⁴	C₂×C₂²⋊D₆₇	1	8	yes	—
196	C₂⁴	C₂×D₂₆₈	1	24	yes	—
197	C₂⁴	C₂×D₂₆₈	1	4	yes	—
198	C₂⁴	C₂×Dic₁₃₄	1	24	yes	—
199	C₂⁴	C₂×Dic₁₃₄	1	4	yes	—
200	C₂⁴	#11761628	1	8	yes	—
201	C₂⁴	C₂³×D₆₇	1	4	yes	—
202	C₂⁴	C₂³×D₆₇	1	3	yes	—
203	C₂²×C₄	—	2	32	no	—
204	C₂²×C₄	—	2	24	no	—
205	C₂²×C₄	—	2	384	no	—
206	C₂²×C₄	—	2	64	no	—
207	C₂²×C₄	—	268	64	no	—
208	C₂²×C₄	—	67	16	no	—
209	C₂²×C₄	—	134	48	no	—
210	C₂²×C₄	—	134	48	no	—
211	C₂²×C₄	—	134	48	no	—
212	C₂²×C₄	C₂²⋊Dic₆₇	1	64	yes	complemented
213	C₂²×C₄	C₂²×Dic₆₇	1	24	yes	—
214	C₂²×C₄	C₂²×Dic₆₇	1	24	yes	—
215	C₂²×C₄	C₂²×Dic₆₇	1	64	yes	complemented
216	C₂²×C₄	C₂×C₂²⋊D₆₇	1	48	yes	—
217	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
218	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
219	C₄⋊C₄	—	134	64	no	—
220	C₂²⋊C₄	—	134	64	no	—
221	C₂³	—	2	256	no	—
222	C₂³	—	2	48	no	—
223	C₂³	—	2	48	no	—
224	C₂³	—	2	96	no	—
225	C₂³	—	2	96	no	—
226	C₂³	—	2	8	no	—
227	C₂³	—	2	8	no	—
228	C₂³	C₂×C₂²⋊Dic₆₇	1	16	yes	—
229	C₂³	C₂×C₄⋊Dic₆₇	1	16	yes	—
230	C₂³	C₂×C₄⋊Dic₆₇	1	96	yes	complemented
231	C₂³	#14524002	1	16	yes	—
232	C₂³	#14524017	1	32	yes	—
233	C₂³	#14524022	1	32	yes	—
234	C₂³	C₂³×Dic₆₇	1	8	yes	—
235	C₂³	C₂³×Dic₆₇	1	6	yes	—
236	C₂³	C₂²×C₂²⋊D₆₇	1	12	yes	—
237	C₂³	C₂²×D₂₆₈	1	8	yes	—
238	C₂³	C₂²×D₂₆₈	1	6	yes	—
239	C₂³	C₂²×Dic₁₃₄	1	8	yes	—
240	C₂³	C₂²×Dic₁₃₄	1	6	yes	—
241	C₂³	C₂×#11761628	1	12	yes	—
242	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
243	C₂×C₄	—	2	48	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	128	no	—
246	C₂×C₄	—	268	96	no	—
247	C₂×C₄	—	134	16	no	—
248	C₂×C₄	—	134	16	no	—
249	C₂×C₄	—	134	16	no	—
250	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	96	yes	complemented
251	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
252	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
253	C₂×C₄	C₂²×C₂²⋊D₆₇	1	16	yes	—
254	C₂²	—	2	192	no	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	16	no	—
258	C₂²	—	2	16	no	—
259	C₂²	#16863256	1	64	yes	direct product factor
260	C₂²	C₂²×C₂²⋊Dic₆₇	1	12	yes	—
261	C₂²	C₂²×C₄⋊Dic₆₇	1	12	yes	—
262	C₂²	C₂²×C₄⋊Dic₆₇	1	16	yes	complemented
263	C₂²	C₂×#14524002	1	12	yes	—
264	C₂²	C₂×#14524017	1	24	yes	—
265	C₂²	C₂×#14524022	1	24	yes	—
266	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
267	C₂²	C₂³×C₂²⋊D₆₇	1	2	yes	—
268	C₂²	C₂³×D₂₆₈	1	1	yes	characteristic
269	C₂²	C₂³×Dic₁₃₄	1	1	yes	characteristic
270	C₂²	C₂²×#11761628	1	2	yes	—
271	C₄	—	2	8	no	—
272	C₄	—	268	16	no	—
273	C₄	C₂²×C₂²⋊Dic₆₇	1	16	yes	complemented
274	C₂	—	2	16	no	—
275	C₂	C₂×#16863256	1	24	yes	direct product factor
276	C₂	C₂³×C₂²⋊Dic₆₇	1	1	yes	characteristic
277	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic
278	C₂	C₂²×#14524002	1	1	yes	characteristic
279	C₂	C₂²×#14524017	1	2	yes	—
280	C₂	C₂²×#14524022	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₆₇
Fitting	C₂⁵×C₄×C₆₇	C₂
socle	C₂⁵×C₆₇	C₂³
prime radical	C₂⁶×C₆₇	C₂²
cosocle	C₂³×C₆₇	C₂⁵
hypercenter	C₂⁵×C₄	D₆₇
hypocenter	C₆₇	C₂²×#380
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₄⋊Dic₆₇	C₂	semidirect
3. C₂×#16863256	C₂	direct (N × H)
4. C₂²×C₆₇⋊#95	C₂	semidirect
8. C₂²×C₄⋊Dic₆₇	C₂²	semidirect
11. #16863256	C₂²	direct (N × H)
12. C₂×C₆₇⋊#95	C₂²	semidirect
15. C₂⁴×C₄×C₆₇	C₄	semidirect
24. C₂×C₄⋊Dic₆₇	C₂³	semidirect
27. C₆₇⋊#95	C₂³	semidirect
40. C₂³×C₄×C₆₇	C₂×C₄	semidirect
58. C₂⁴×C₆₇	C₄⋊C₄	semidirect
73. C₂²×C₄×C₆₇	C₂²⋊C₄	semidirect
76. C₂²×C₄×C₆₇	C₂²×C₄	semidirect
89. C₂³×C₆₇	C₂×C₄⋊C₄	semidirect
105. C₂×C₄×C₆₇	C₂×C₂²⋊C₄	semidirect
118. C₂²×C₆₇	#380	semidirect
121. C₂²×C₆₇	C₂²×C₄⋊C₄	semidirect
131. C₄×C₆₇	C₂²×C₂²⋊C₄	semidirect
135. C₂×C₆₇	C₂×#380	semidirect
146. C₆₇	C₂²×#380	semidirect
150. C₂⁴×C₄	Dic₆₇	semidirect
176. C₂³×C₄	C₂×Dic₆₇	semidirect
192. C₂⁴	C₄⋊Dic₆₇	semidirect
212. C₂²×C₄	C₂²⋊Dic₆₇	semidirect
215. C₂²×C₄	C₂²×Dic₆₇	semidirect
230. C₂³	C₂×C₄⋊Dic₆₇	semidirect
250. C₂×C₄	C₂×C₂²⋊Dic₆₇	semidirect
259. C₂²	#16863256	direct (N × H)
262. C₂²	C₂²×C₄⋊Dic₆₇	semidirect
273. C₄	C₂²×C₂²⋊Dic₆₇	semidirect
275. C₂	C₂×#16863256	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×C₄⋊Dic₆₇	2
C₂³×C₂²⋊Dic₆₇	2
C₂×#16863256	2
C₂²×C₆₇⋊#95	2
C₂⁵×C₄×C₆₇	2
C₂²×#380	67

Subgroups — C22×#16863256 #31014091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863256 #31014091