Subgroups — #1844×D₆₇ #30856226

Order 17152.

1358 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#106×D₆₇	C₂	1	4	yes	maximal
2	C₂×C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#384×D₆₇	C₂	1	2	yes	maximal
4	#22560366	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560365	C₂	1	2	yes	maximal, complemented
6	#22560364	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊C₄×D₆₇	—	2	4	no	—
10	C₂×C₄⋊C₄×D₆₇	C₂²	1	4	yes	—
11	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
12	C₂×#14523975	C₂²	1	4	yes	—
13	C₂×#14523974	C₂²	1	8	yes	—
14	#106×D₆₇	—	2	16	no	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
17	C₄×C₈×D₆₇	C₄	1	8	yes	complemented
18	#16863266	C₂²	1	2	yes	—
19	#16863265	—	2	4	no	—
20	#16863265	C₂²	1	4	yes	—
21	C₂×#106×C₆₇	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#384×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄×D₆₇	—	2	4	no	—
25	C₂²×C₄×D₆₇	D₄	1	2	yes	—
26	C₂²×C₄×D₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
28	C₂×C₈×D₆₇	C₂×C₄	1	8	yes	—
29	C₂×C₈×D₆₇	D₄	1	8	yes	—
30	C₂×C₈×D₆₇	Q₈	1	8	yes	—
31	C₂×C₄⋊Dic₆₇	—	2	4	no	—
32	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
33	C₂×Dic₆₇⋊C₄	—	2	4	no	—
34	C₂×Dic₆₇⋊C₄	—	2	8	no	—
35	C₄⋊C₄×D₆₇	—	2	16	no	—
36	C₄⋊C₄×D₆₇	—	4	16	no	—
37	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
38	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
39	C₄²×D₆₇	C₂×C₄	1	4	yes	—
40	C₄²×D₆₇	Q₈	1	4	yes	—
41	#14523975	—	2	8	no	—
42	#14523974	—	2	16	no	—
43	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	complemented
44	C₈×Dic₆₇	C₂×C₄	1	8	yes	complemented
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	#106×C₆₇	—	2	8	no	—
50	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂³×D₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×Dic₆₇	—	2	4	no	—
53	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×D₆₇	—	4	8	no	—
56	C₂×C₄×D₆₇	—	4	16	no	—
57	C₂×C₄×D₆₇	C₄⋊C₄	1	8	yes	—
58	C₂×C₄×D₆₇	SD₁₆	1	16	yes	—
59	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×D₆₇	C₂×D₄	1	4	yes	—
61	C₂×C₄×D₆₇	C₂×Q₈	1	4	yes	—
62	C₂×C₄×D₆₇	C₂×Q₈	1	2	yes	—
63	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
64	C₂×C₆₇⋊C₈	C₂×D₄	1	4	yes	—
65	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
66	C₈×D₆₇	C₄⋊C₄	1	32	yes	complemented
67	C₄⋊Dic₆₇	—	2	8	no	—
68	C₄⋊Dic₆₇	—	4	8	no	—
69	Dic₆₇⋊C₄	—	4	8	no	—
70	Dic₆₇⋊C₄	—	4	16	no	—
71	C₄×Dic₆₇	C₄⋊C₄	1	8	yes	—
72	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
73	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
78	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
80	C₄⋊C₄×C₆₇	—	2	8	no	—
81	C₄⋊C₄×C₆₇	—	4	8	no	—
82	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
83	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
84	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
85	C₂²×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×D₆₇	#127	1	2	yes	—
87	C₂²×D₆₇	C₄⋊Q₈	1	2	yes	—
88	C₂²×D₆₇	C₂×SD₁₆	1	8	yes	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	4	8	no	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×SD₁₆	1	8	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
96	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
97	C₂×Dic₆₇	—	4	4	no	—
98	C₂×Dic₆₇	—	4	8	no	—
99	C₂×Dic₆₇	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×Dic₆₇	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×Dic₆₇	#127	1	2	yes	—
102	C₂×Dic₆₇	C₄⋊Q₈	1	2	yes	—
103	C₂×Dic₆₇	C₂×SD₁₆	1	8	yes	—
104	C₄×D₆₇	—	8	16	no	—
105	C₄×D₆₇	#106	1	32	yes	complemented
106	C₄×D₆₇	C₂×C₄⋊C₄	1	8	yes	—
107	C₄×D₆₇	C₄⋊Q₈	1	8	yes	—
108	C₈×C₆₇	C₂×C₄⋊C₄	1	8	yes	complemented
109	C₆₇⋊C₈	C₂×C₄⋊C₄	1	8	yes	complemented
110	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
112	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
113	C₂²×C₆₇	C₂²×SD₁₆	1	4	yes	—
114	C₂×D₆₇	#384	1	4	yes	—
115	C₂×D₆₇	C₂×#106	1	8	yes	—
116	C₂×D₆₇	#492	1	8	yes	—
117	C₂×D₆₇	#498	1	8	yes	—
118	C₄×C₆₇	—	8	4	no	—
119	C₄×C₆₇	C₂×#106	1	8	yes	complemented
120	C₄×C₆₇	C₂²×C₄⋊C₄	1	2	yes	—
121	C₄×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
122	Dic₆₇	—	8	4	no	—
123	Dic₆₇	#384	1	4	yes	—
124	Dic₆₇	C₂×#106	1	8	yes	complemented
125	C₂×#1844	—	67	1	no	maximal, Sylow
126	C₂×C₆₇	C₂×#384	1	1	yes	characteristic
127	C₂×C₆₇	C₂²×#106	1	2	yes	—
128	C₂×C₆₇	C₂×#492	1	2	yes	—
129	C₂×C₆₇	C₂×#498	1	2	yes	—
130	D₆₇	#1844	1	8	yes	direct product factor
131	C₂²×#106	—	67	4	no	—
132	C₂²×C₄×C₈	—	67	1	no	—
133	C₂×#384	—	67	2	no	—
134	#1844	—	67	4	no	—
135	#1844	—	67	1	no	—
136	#1844	—	67	2	no	—
137	#1844	D₆₇	1	1	yes	characteristic, direct product factor
138	C₆₇	C₂×#1844	1	1	yes	characteristic, complemented, Sylow
139	C₂³×C₈	—	67	2	no	—
140	C₂²×C₄⋊C₄	—	67	4	no	—
141	C₂²×C₄⋊C₄	—	134	4	no	—
142	C₂²×C₄²	—	67	1	no	—
143	C₂×#106	—	67	4	no	—
144	C₂×#106	—	67	8	no	—
145	C₂×#106	—	134	16	no	—
146	C₂×#106	C₂×D₆₇	1	4	yes	—
147	C₂×C₄×C₈	—	67	2	no	—
148	C₂×C₄×C₈	—	67	1	no	—
149	C₂×C₄×C₈	—	67	8	no	—
150	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
151	#384	—	67	4	no	—
152	#384	—	67	2	no	—
153	#384	—	134	4	no	—
154	#384	C₂×D₆₇	1	2	yes	—
155	C₂³×C₄	—	67	2	no	—
156	C₂³×C₄	—	67	1	no	—
157	C₂³×C₄	—	134	4	no	—
158	C₂²×C₈	—	67	2	no	—
159	C₂²×C₈	—	67	8	no	—
160	C₂²×C₈	—	67	8	no	—
161	C₂²×C₈	—	67	8	no	—
162	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
163	C₂×C₄⋊C₄	—	2	4	no	—
164	C₂×C₄⋊C₄	—	268	16	no	—
165	C₂×C₄⋊C₄	—	67	4	no	—
166	C₂×C₄⋊C₄	—	134	16	no	—
167	C₂×C₄⋊C₄	—	134	4	no	—
168	C₂×C₄⋊C₄	—	134	8	no	—
169	C₂×C₄⋊C₄	—	134	4	no	—
170	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
171	C₂×C₄²	—	67	2	no	—
172	C₂×C₄²	—	67	1	no	—
173	C₂×C₄²	—	67	4	no	—
174	C₂×C₄²	—	67	4	no	—
175	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	#106	—	2	8	no	—
177	#106	—	134	16	no	—
178	#106	—	134	8	no	—
179	C₄×C₈	—	67	8	no	—
180	C₄×C₈	—	67	4	no	—
181	C₄×C₈	C₄×D₆₇	1	4	yes	complemented
182	C₂⁴	—	67	1	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	268	16	no	—
185	C₂²×C₄	—	268	8	no	—
186	C₂²×C₄	—	67	2	no	—
187	C₂²×C₄	—	67	1	no	—
188	C₂²×C₄	—	67	8	no	—
189	C₂²×C₄	—	67	16	no	—
190	C₂²×C₄	—	67	2	no	—
191	C₂²×C₄	—	67	4	no	—
192	C₂²×C₄	—	67	2	no	—
193	C₂²×C₄	—	67	4	no	—
194	C₂²×C₄	—	134	4	no	—
195	C₂²×C₄	D₄×D₆₇	1	2	yes	—
196	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
197	C₂×C₈	—	67	4	no	—
198	C₂×C₈	—	67	4	no	—
199	C₂×C₈	—	67	4	no	—
200	C₂×C₈	—	67	32	no	—
201	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
202	C₂×C₈	D₄×D₆₇	1	4	yes	—
203	C₂×C₈	Q₈×D₆₇	1	4	yes	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄⋊C₄	—	4	8	no	—
206	C₄⋊C₄	—	268	16	no	—
207	C₄⋊C₄	—	268	8	no	—
208	C₄⋊C₄	—	268	8	no	—
209	C₄⋊C₄	—	134	8	no	—
210	C₄²	—	67	8	no	—
211	C₄²	—	67	2	no	—
212	C₄²	—	67	2	no	—
213	C₄²	C₂×C₄×D₆₇	1	2	yes	—
214	C₄²	Q₈×D₆₇	1	2	yes	—
215	C₂³	—	67	2	no	—
216	C₂³	—	67	8	no	—
217	C₂³	—	67	2	no	—
218	C₂³	—	67	2	no	—
219	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
220	C₂×C₄	—	4	8	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	268	8	no	—
223	C₂×C₄	—	268	4	no	—
224	C₂×C₄	—	536	16	no	—
225	C₂×C₄	—	67	2	no	—
226	C₂×C₄	—	67	4	no	—
227	C₂×C₄	—	67	8	no	—
228	C₂×C₄	—	67	2	no	—
229	C₂×C₄	—	67	2	no	—
230	C₂×C₄	—	67	8	no	—
231	C₂×C₄	—	67	32	no	—
232	C₂×C₄	—	67	8	no	—
233	C₂×C₄	C₄⋊C₄×D₆₇	1	4	yes	—
234	C₂×C₄	SD₁₆×D₆₇	1	8	yes	—
235	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
236	C₂×C₄	C₂×D₄×D₆₇	1	2	yes	—
237	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
238	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
239	C₈	—	67	8	no	—
240	C₈	C₄⋊C₄×D₆₇	1	8	yes	complemented
241	C₂²	—	67	4	no	—
242	C₂²	—	67	8	no	—
243	C₂²	—	67	8	no	—
244	C₂²	—	67	8	no	—
245	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
246	C₂²	#127×D₆₇	1	1	yes	characteristic
247	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
248	C₂²	C₂×SD₁₆×D₆₇	1	4	yes	—
249	C₄	—	8	4	no	—
250	C₄	—	536	4	no	—
251	C₄	—	67	4	no	—
252	C₄	—	67	8	no	—
253	C₄	#106×D₆₇	1	8	yes	complemented
254	C₄	C₂×C₄⋊C₄×D₆₇	1	2	yes	—
255	C₄	C₄⋊Q₈×D₆₇	1	2	yes	—
256	C₂	—	67	8	no	—
257	C₂	#384×D₆₇	1	1	yes	characteristic
258	C₂	C₂×#106×D₆₇	1	2	yes	—
259	C₂	#492×D₆₇	1	2	yes	—
260	C₂	#498×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₆₇×#1844	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂³×D₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1844	D₆₇
hypocenter	C₆₇	C₂×#1844
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
4. #22560366	C₂	semidirect
5. #22560365	C₂	semidirect
6. #22560364	C₂	semidirect
7. C₆₇×#1844	C₂	semidirect
17. C₄×C₈×D₆₇	C₄	semidirect
43. C₄×C₆₇⋊C₈	C₂×C₄	semidirect
44. C₈×Dic₆₇	C₂×C₄	semidirect
50. C₄×C₈×C₆₇	C₂×C₄	semidirect
66. C₈×D₆₇	C₄⋊C₄	semidirect
105. C₄×D₆₇	#106	semidirect
108. C₈×C₆₇	C₂×C₄⋊C₄	semidirect
109. C₆₇⋊C₈	C₂×C₄⋊C₄	semidirect
119. C₄×C₆₇	C₂×#106	semidirect
124. Dic₆₇	C₂×#106	semidirect
130. D₆₇	#1844	direct (N × H)
137. #1844	D₆₇	direct (N × H)
138. C₆₇	C₂×#1844	semidirect
181. C₄×C₈	C₄×D₆₇	semidirect
240. C₈	C₄⋊C₄×D₆₇	semidirect
253. C₄	#106×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#106×D₆₇	2
C₂×C₄×C₈×D₆₇	2
#384×D₆₇	2
#22560366	2
#22560365	2
#22560364	2
C₆₇×#1844	2
C₂×#1844	67

Subgroups — #1844×D67 #30856226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1844×D₆₇ #30856226