Subgroups — #137×Dic₆₇ #22569030

Order 8576.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863503	C₂	1	2	yes	maximal, complemented
4	#16863497	C₂	1	2	yes	maximal, complemented
5	Q₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
6	SD₁₆×Dic₆₇	C₂	1	2	yes	maximal, complemented
7	#16863438	C₂	1	1	yes	maximal, characteristic, complemented
8	#16863405	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863397	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#137×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#11761612	C₂²	1	1	yes	characteristic
13	Q₈×Dic₆₇	—	2	1	no	—
14	Q₈×Dic₆₇	—	2	1	no	—
15	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	Q₈×Dic₆₇	C₂²	1	2	yes	—
17	Q₈×Dic₆₇	C₂²	1	2	yes	—
18	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic
19	D₄×Dic₆₇	—	2	1	no	—
20	D₄×Dic₆₇	—	2	1	no	—
21	D₄×Dic₆₇	C₂²	1	1	yes	characteristic
22	D₄×Dic₆₇	C₂²	1	1	yes	characteristic
23	#14524002	—	2	1	no	—
24	#14524002	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Dic₆₇	—	2	1	no	—
26	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₆₇	—	2	1	no	—
28	C₂×C₄×Dic₆₇	—	2	1	no	—
29	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
30	#14523977	C₂²	1	2	yes	—
31	#14523975	C₂²	1	2	yes	—
32	#14523967	C₂²	1	2	yes	—
33	#14523967	C₂²	1	2	yes	—
34	#14523967	C₂²	1	2	yes	—
35	#14523964	C₂²	1	2	yes	—
36	#14523953	C₂²	1	2	yes	—
37	C₈×Dic₆₇	C₂²	1	2	yes	—
38	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₆₇	C₂²	1	1	yes	characteristic
40	#137×C₆₇	C₄	1	8	yes	complemented
41	C₂×Q₁₆×C₆₇	C₂²	1	2	yes	—
42	C₂×SD₁₆×C₆₇	C₂²	1	2	yes	—
43	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×Dic₆₇	—	2	1	no	—
45	C₂²×Dic₆₇	—	2	1	no	—
46	C₂²×Dic₆₇	—	2	1	no	—
47	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
48	C₂²×Dic₆₇	D₄	1	1	yes	characteristic
49	#11761612	—	2	2	no	—
50	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₄⋊Dic₆₇	—	2	1	no	—
52	C₄⋊Dic₆₇	—	2	2	no	—
53	C₄⋊Dic₆₇	—	2	2	no	—
54	C₄⋊Dic₆₇	—	2	1	no	—
55	C₄⋊Dic₆₇	—	2	1	no	—
56	C₄⋊Dic₆₇	C₂³	1	2	yes	—
57	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
59	C₂²⋊Dic₆₇	—	2	1	no	—
60	C₂²⋊Dic₆₇	—	2	1	no	—
61	C₂²⋊Dic₆₇	—	2	1	no	—
62	C₄×Dic₆₇	—	2	1	no	—
63	C₄×Dic₆₇	—	2	1	no	—
64	C₄×Dic₆₇	—	2	1	no	—
65	C₄×Dic₆₇	—	2	2	no	—
66	C₄×Dic₆₇	—	2	2	no	—
67	C₄×Dic₆₇	D₄	1	1	yes	characteristic
68	C₄×Dic₆₇	D₄	1	1	yes	characteristic
69	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
70	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
71	#41×C₆₇	—	2	2	no	—
72	#41×C₆₇	C₂×C₄	1	4	yes	—
73	C₂×Q₈×C₆₇	—	2	1	no	—
74	C₂×Q₈×C₆₇	—	2	2	no	—
75	C₂×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
76	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
77	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×D₄×C₆₇	—	2	1	no	—
79	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	—	2	1	no	—
81	C₂²×C₄×C₆₇	—	2	1	no	—
82	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
83	Q₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
84	SD₁₆×C₆₇	C₂×C₄	1	8	yes	complemented
85	M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
86	C₂×C₈×C₆₇	C₂³	1	2	yes	—
87	C₂³×C₆₇	—	2	1	no	—
88	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
89	Q₈×C₆₇	—	2	4	no	—
90	Q₈×C₆₇	—	2	2	no	—
91	Q₈×C₆₇	—	2	1	no	—
92	Q₈×C₆₇	C₂²×C₄	1	4	yes	—
93	Q₈×C₆₇	C₂²×C₄	1	2	yes	—
94	D₄×C₆₇	—	2	4	no	—
95	D₄×C₆₇	—	2	1	no	—
96	D₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	2	no	—
100	C₂×C₄×C₆₇	—	2	4	no	—
101	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#41	1	2	yes	—
104	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
105	C₂×Dic₆₇	—	2	2	no	—
106	C₂×Dic₆₇	—	2	1	no	—
107	C₂×Dic₆₇	—	2	1	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	4	1	no	—
110	C₂×Dic₆₇	—	4	2	no	—
111	C₂×Dic₆₇	—	4	2	no	—
112	C₂×Dic₆₇	—	4	2	no	—
113	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
114	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
117	C₈×C₆₇	C₂²×C₄	1	4	yes	—
118	C₆₇⋊C₈	—	2	2	no	—
119	C₂²×C₆₇	—	2	1	no	—
120	C₂²×C₆₇	—	2	1	no	—
121	C₂²×C₆₇	—	2	2	no	—
122	C₂²×C₆₇	—	4	1	no	—
123	C₂²×C₆₇	—	4	1	no	—
124	C₂²×C₆₇	C₄×D₄	1	2	yes	—
125	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₆₇	—	2	2	no	—
127	C₄×C₆₇	—	2	4	no	—
128	C₄×C₆₇	C₄×D₄	1	2	yes	—
129	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
131	Dic₆₇	—	2	1	no	—
132	Dic₆₇	—	2	1	no	—
133	Dic₆₇	—	2	1	no	—
134	Dic₆₇	—	4	1	no	—
135	Dic₆₇	—	4	1	no	—
136	Dic₆₇	—	4	2	no	—
137	Dic₆₇	#137	1	2	yes	direct product factor
138	C₂×C₆₇	—	2	2	no	—
139	C₂×C₆₇	—	4	2	no	—
140	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₆₇	C₂×#137	1	1	yes	characteristic
142	C₂×C₆₇	#575	1	1	yes	characteristic
143	C₄×#137	—	67	1	no	maximal, Sylow
144	C₆₇	C₄×#137	1	1	yes	characteristic, complemented, Sylow
145	C₂×#137	D₆₇	1	1	yes	characteristic
146	C₄×#41	—	67	1	no	—
147	C₂×C₄×Q₈	—	67	1	no	—
148	#441	—	67	2	no	—
149	#440	—	67	2	no	—
150	C₄×Q₁₆	—	67	2	no	—
151	C₄×SD₁₆	—	67	2	no	—
152	#428	—	67	1	no	—
153	#419	—	67	1	no	—
154	#417	—	67	1	no	—
155	C₄×M₁₆	—	67	1	no	—
156	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
157	C₂²×Q₈	C₂×D₆₇	1	1	yes	characteristic
158	#137	Dic₆₇	1	8	yes	direct product factor
159	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
160	C₂×SD₁₆	C₂×D₆₇	1	2	yes	—
161	C₂×M₁₆	—	67	1	no	—
162	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
163	C₄×Q₈	—	67	2	no	—
164	C₄×Q₈	—	67	2	no	—
165	C₄×Q₈	—	67	1	no	—
166	C₄×Q₈	—	67	1	no	—
167	C₄×Q₈	—	134	1	no	—
168	C₄×Q₈	—	134	1	no	—
169	C₄×D₄	—	67	1	no	—
170	C₄×D₄	—	67	1	no	—
171	C₄×D₄	—	134	1	no	—
172	C₄×D₄	—	134	1	no	—
173	#117	—	67	1	no	—
174	#117	—	134	1	no	—
175	C₂×C₄⋊C₄	—	67	1	no	—
176	C₂×C₄⋊C₄	—	134	1	no	—
177	C₂×C₄²	—	67	1	no	—
178	C₂×C₄²	—	134	1	no	—
179	C₂×C₄²	—	134	1	no	—
180	#107	—	67	2	no	—
181	#106	—	67	2	no	—
182	#103	—	67	2	no	—
183	#103	—	67	2	no	—
184	#103	—	67	2	no	—
185	#102	—	67	2	no	—
186	#97	—	67	2	no	—
187	C₄×C₈	—	67	2	no	—
188	#41	—	2	2	no	—
189	#41	C₂×Dic₆₇	1	4	yes	—
190	C₂×Q₈	—	2	1	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	C₂×Dic₆₇	1	4	yes	complemented
193	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
194	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₂×D₄	—	2	1	no	—
196	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	67	1	no	—
200	C₂²×C₄	—	67	1	no	—
201	C₂²×C₄	—	134	1	no	—
202	C₂²×C₄	—	134	1	no	—
203	C₂²×C₄	—	134	1	no	—
204	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
205	Q₁₆	C₂×Dic₆₇	1	8	yes	complemented
206	SD₁₆	C₂×Dic₆₇	1	8	yes	complemented
207	M₁₆	—	134	2	no	—
208	M₁₆	C₂×Dic₆₇	1	4	yes	complemented
209	C₂×C₈	—	67	2	no	—
210	C₂×C₈	C₂²×D₆₇	1	2	yes	—
211	C₄⋊C₄	—	67	2	no	—
212	C₄⋊C₄	—	67	1	no	—
213	C₄⋊C₄	—	67	1	no	—
214	C₄⋊C₄	—	134	2	no	—
215	C₄⋊C₄	—	134	2	no	—
216	C₄⋊C₄	—	134	1	no	—
217	C₄⋊C₄	—	134	1	no	—
218	C₄⋊C₄	—	134	1	no	—
219	C₂²⋊C₄	—	134	1	no	—
220	C₂²⋊C₄	—	134	1	no	—
221	C₂²⋊C₄	—	134	1	no	—
222	C₄²	—	67	1	no	—
223	C₄²	—	67	1	no	—
224	C₄²	—	67	1	no	—
225	C₄²	—	67	1	no	—
226	C₄²	—	134	1	no	—
227	C₄²	—	134	1	no	—
228	C₄²	—	134	2	no	—
229	C₄²	—	134	2	no	—
230	C₄²	—	134	1	no	—
231	C₂³	—	2	1	no	—
232	C₂³	D₄×D₆₇	1	1	yes	characteristic
233	Q₈	—	2	4	no	—
234	Q₈	—	2	2	no	—
235	Q₈	—	2	1	no	—
236	Q₈	C₂²×Dic₆₇	1	4	yes	—
237	Q₈	C₂²×Dic₆₇	1	2	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	1	no	—
240	D₄	C₂²×Dic₆₇	1	2	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	268	2	no	—
246	C₂×C₄	—	268	2	no	—
247	C₂×C₄	—	268	2	no	—
248	C₂×C₄	—	268	1	no	—
249	C₂×C₄	—	67	1	no	—
250	C₂×C₄	—	67	1	no	—
251	C₂×C₄	—	67	1	no	—
252	C₂×C₄	—	67	1	no	—
253	C₂×C₄	—	134	1	no	—
254	C₂×C₄	—	134	1	no	—
255	C₂×C₄	—	134	2	no	—
256	C₂×C₄	—	134	2	no	—
257	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
258	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
259	C₂×C₄	#11761627	1	2	yes	—
260	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
261	C₈	—	134	2	no	—
262	C₈	C₂²×Dic₆₇	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	D₄×Dic₆₇	1	2	yes	—
269	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	4	no	—
272	C₄	—	268	1	no	—
273	C₄	—	268	2	no	—
274	C₄	—	268	1	no	—
275	C₄	—	67	2	no	—
276	C₄	—	134	1	no	—
277	C₄	—	134	1	no	—
278	C₄	—	134	1	no	—
279	C₄	D₄×Dic₆₇	1	2	yes	—
280	C₄	C₂³×Dic₆₇	1	1	yes	characteristic
281	C₄	C₂×#11761627	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×D₄×Dic₆₇	1	1	yes	characteristic
285	C₂	#137×D₆₇	1	1	yes	characteristic
286	C₂	#16864195	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#137×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	C₂×#137	D₆₇
hypocenter	C₆₇	C₄×#137
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₆₇	C₂	semidirect
3. #16863503	C₂	semidirect
4. #16863497	C₂	semidirect
5. Q₁₆×Dic₆₇	C₂	semidirect
6. SD₁₆×Dic₆₇	C₂	semidirect
7. #16863438	C₂	semidirect
8. #16863405	C₂	semidirect
10. M₁₆×Dic₆₇	C₂	semidirect
40. #137×C₆₇	C₄	semidirect
75. C₂×Q₈×C₆₇	C₂×C₄	semidirect
83. Q₁₆×C₆₇	C₂×C₄	semidirect
84. SD₁₆×C₆₇	C₂×C₄	semidirect
85. M₁₆×C₆₇	C₂×C₄	semidirect
137. Dic₆₇	#137	direct (N × H)
144. C₆₇	C₄×#137	semidirect
158. #137	Dic₆₇	direct (N × H)
192. C₂×Q₈	C₂×Dic₆₇	semidirect
205. Q₁₆	C₂×Dic₆₇	semidirect
206. SD₁₆	C₂×Dic₆₇	semidirect
208. M₁₆	C₂×Dic₆₇	semidirect

Subgroups — #137×Dic67 #22569030

Conjugacy classes of subgroups

Complements

Subgroups — #137×Dic₆₇ #22569030