Subgroups — #1374×D₆₇ #30770666

Order 17152.

726 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558287	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558286	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₆₇	C₄	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₁₆×D₆₇	—	2	4	no	—
11	C₂×C₁₆×D₆₇	—	2	4	no	—
12	C₂×C₁₆×D₆₇	C₄	1	8	yes	complemented
13	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×D₆₇	—	2	4	no	—
17	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄×D₆₇	C₈	1	2	yes	—
20	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×D₆₇	—	2	2	no	—
24	C₂×C₈×D₆₇	—	2	2	no	—
25	C₂×C₈×D₆₇	C₈	1	4	yes	—
26	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₈×D₆₇	D₄	1	2	yes	—
28	C₂×C₈×D₆₇	D₄	1	4	yes	—
29	C₂×C₈×D₆₇	Q₈	1	2	yes	—
30	C₂×C₄×Dic₆₇	C₈	1	2	yes	—
31	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄²×D₆₇	—	2	2	no	—
33	C₄²×D₆₇	—	2	2	no	—
34	C₂×C₆₇⋊C₁₆	—	2	2	no	—
35	C₂×C₆₇⋊C₁₆	—	2	2	no	—
36	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	complemented
37	C₁₆×D₆₇	—	2	16	no	—
38	C₄×C₆₇⋊C₈	—	2	2	no	—
39	C₈×Dic₆₇	—	2	2	no	—
40	C₈×Dic₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₆×C₆₇	—	2	2	no	—
45	C₂×C₁₆×C₆₇	—	2	2	no	—
46	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
47	C₄×C₈×C₆₇	—	2	2	no	—
48	C₂³×D₆₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
50	C₂²×Dic₆₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×D₆₇	—	2	4	no	—
52	C₂×C₄×D₆₇	—	2	4	no	—
53	C₂×C₄×D₆₇	C₁₆	1	8	yes	complemented
54	C₂×C₄×D₆₇	C₄²	1	2	yes	—
55	C₂×C₄×D₆₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
58	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
59	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
60	C₂×C₆₇⋊C₈	—	2	1	no	—
61	C₂×C₆₇⋊C₈	—	2	1	no	—
62	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
63	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
65	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
66	C₈×D₆₇	—	2	8	no	—
67	C₈×D₆₇	C₂²⋊C₄	1	8	yes	—
68	C₈×D₆₇	C₄⋊C₄	1	8	yes	—
69	C₄×Dic₆₇	—	2	1	no	—
70	C₄×Dic₆₇	—	2	1	no	—
71	C₄×Dic₆₇	—	2	4	no	—
72	C₆₇⋊C₁₆	—	2	4	no	—
73	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	2	1	no	—
77	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₆₇	—	2	1	no	—
83	C₄²×C₆₇	—	2	1	no	—
84	C₁₆×C₆₇	—	2	4	no	—
85	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
86	C₂²×D₆₇	C₄×C₈	1	2	yes	—
87	C₂²×D₆₇	C₂²⋊C₈	1	2	yes	—
88	C₂²×D₆₇	C₄⋊C₈	1	2	yes	—
89	C₂²×D₆₇	C₂×C₁₆	1	4	yes	—
90	C₂²×D₆₇	M₃₂	1	4	yes	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	C₂×C₁₆	1	4	yes	complemented
94	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	2	no	—
101	C₂×Dic₆₇	—	2	2	no	—
102	C₂×Dic₆₇	C₄×C₈	1	2	yes	—
103	C₂×Dic₆₇	C₂²⋊C₈	1	2	yes	—
104	C₂×Dic₆₇	C₄⋊C₈	1	2	yes	—
105	C₂×Dic₆₇	C₂×C₁₆	1	4	yes	complemented
106	C₄×D₆₇	—	2	16	no	—
107	C₄×D₆₇	#95	1	4	yes	—
108	C₄×D₆₇	#97	1	4	yes	—
109	C₄×D₆₇	C₂²⋊C₈	1	4	yes	—
110	C₄×D₆₇	C₄⋊C₈	1	4	yes	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
113	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₆₇⋊C₈	—	2	2	no	—
115	C₆₇⋊C₈	#95	1	4	yes	—
116	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
117	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₆₇	C₂²×C₁₆	1	2	yes	—
120	C₂²×C₆₇	C₂×M₃₂	1	2	yes	—
121	C₂×D₆₇	#336	1	4	yes	—
122	C₂×D₆₇	C₄×C₁₆	1	4	yes	—
123	C₂×D₆₇	#346	1	4	yes	—
124	C₂×D₆₇	C₂²⋊C₁₆	1	8	yes	—
125	C₂×D₆₇	C₄⋊C₁₆	1	8	yes	—
126	C₄×C₆₇	—	2	4	no	—
127	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
128	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
129	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
131	Dic₆₇	—	2	4	no	—
132	Dic₆₇	#336	1	4	yes	—
133	C₂×#1374	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
135	C₂×C₆₇	C₂×C₄×C₁₆	1	1	yes	characteristic
136	C₂×C₆₇	C₂×#346	1	1	yes	characteristic
137	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	2	yes	—
138	C₂×C₆₇	C₂×C₄⋊C₁₆	1	2	yes	—
139	D₆₇	#1374	1	8	yes	direct product factor
140	C₂³×C₁₆	—	67	2	no	—
141	C₂²×C₄×C₈	—	67	1	no	—
142	#1374	—	67	2	no	—
143	#1374	—	67	1	no	—
144	#1374	D₆₇	1	1	yes	characteristic, direct product factor
145	C₆₇	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
146	C₂³×C₈	—	67	1	no	—
147	C₂³×C₈	—	67	1	no	—
148	C₂²×C₄²	—	67	1	no	—
149	C₂²×C₁₆	—	67	2	no	—
150	C₂²×C₁₆	—	67	8	no	—
151	C₂²×C₁₆	—	134	4	no	—
152	C₂²×C₁₆	—	134	4	no	—
153	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
154	C₂×C₄×C₈	—	67	1	no	—
155	C₂×C₄×C₈	—	67	1	no	—
156	C₂×C₄×C₈	—	67	1	no	—
157	C₂×C₄×C₈	—	134	4	no	—
158	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
159	C₂³×C₄	—	67	1	no	—
160	C₂³×C₄	—	67	2	no	—
161	C₂²×C₈	—	67	1	no	—
162	C₂²×C₈	—	67	4	no	—
163	C₂²×C₈	—	67	4	no	—
164	C₂²×C₈	—	67	2	no	—
165	C₂²×C₈	—	67	2	no	—
166	C₂²×C₈	—	67	1	no	—
167	C₂²×C₈	—	67	4	no	—
168	C₂²×C₈	—	134	2	no	—
169	C₂²×C₈	—	134	2	no	—
170	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×C₄²	—	67	2	no	—
173	C₂×C₄²	—	67	1	no	—
174	C₂×C₄²	—	134	2	no	—
175	C₂×C₄²	—	134	2	no	—
176	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	2	2	no	—
179	C₂×C₁₆	—	67	4	no	—
180	C₂×C₁₆	—	134	16	no	—
181	C₂×C₁₆	—	134	2	no	—
182	C₂×C₁₆	—	134	2	no	—
183	C₂×C₁₆	C₄×D₆₇	1	4	yes	complemented
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	134	2	no	—
186	C₄×C₈	—	134	2	no	—
187	C₄×C₈	—	134	2	no	—
188	C₂⁴	—	67	1	no	—
189	C₂²×C₄	—	67	1	no	—
190	C₂²×C₄	—	67	2	no	—
191	C₂²×C₄	—	67	2	no	—
192	C₂²×C₄	—	67	2	no	—
193	C₂²×C₄	—	67	2	no	—
194	C₂²×C₄	—	67	2	no	—
195	C₂²×C₄	—	67	2	no	—
196	C₂²×C₄	—	67	2	no	—
197	C₂²×C₄	—	67	8	no	—
198	C₂²×C₄	—	134	4	no	—
199	C₂²×C₄	—	134	4	no	—
200	C₂²×C₄	C₈×D₆₇	1	2	yes	—
201	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	67	2	no	—
205	C₂×C₈	—	67	2	no	—
206	C₂×C₈	—	67	2	no	—
207	C₂×C₈	—	67	8	no	—
208	C₂×C₈	—	67	8	no	—
209	C₂×C₈	—	67	2	no	—
210	C₂×C₈	—	134	8	no	—
211	C₂×C₈	—	134	1	no	—
212	C₂×C₈	—	134	1	no	—
213	C₂×C₈	C₈×D₆₇	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
215	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
216	C₂×C₈	D₄×D₆₇	1	2	yes	—
217	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
218	C₄²	—	2	1	no	—
219	C₄²	—	2	1	no	—
220	C₄²	—	134	4	no	—
221	C₄²	—	134	1	no	—
222	C₄²	—	134	1	no	—
223	C₁₆	—	2	4	no	—
224	C₁₆	—	134	4	no	—
225	C₂³	—	67	4	no	—
226	C₂³	—	67	4	no	—
227	C₂³	—	67	2	no	—
228	C₂³	—	67	2	no	—
229	C₂³	—	67	2	no	—
230	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	67	2	no	—
234	C₂×C₄	—	67	2	no	—
235	C₂×C₄	—	67	2	no	—
236	C₂×C₄	—	67	4	no	—
237	C₂×C₄	—	67	4	no	—
238	C₂×C₄	—	67	4	no	—
239	C₂×C₄	—	67	4	no	—
240	C₂×C₄	—	67	4	no	—
241	C₂×C₄	—	134	16	no	—
242	C₂×C₄	—	134	2	no	—
243	C₂×C₄	—	134	2	no	—
244	C₂×C₄	C₁₆×D₆₇	1	4	yes	complemented
245	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
246	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
247	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
249	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	67	4	no	—
253	C₈	—	134	2	no	—
254	C₈	C₂²⋊C₄×D₆₇	1	2	yes	—
255	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
256	C₂²	—	67	4	no	—
257	C₂²	—	67	4	no	—
258	C₂²	—	67	8	no	—
259	C₂²	—	67	4	no	—
260	C₂²	—	67	8	no	—
261	C₂²	C₄×C₈×D₆₇	1	1	yes	characteristic
262	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
263	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
264	C₂²	C₂×C₁₆×D₆₇	1	2	yes	—
265	C₂²	M₃₂×D₆₇	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	67	4	no	—
268	C₄	—	134	4	no	—
269	C₄	#95×D₆₇	1	1	yes	characteristic
270	C₄	#97×D₆₇	1	1	yes	characteristic
271	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
272	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
273	C₂	—	67	8	no	—
274	C₂	#336×D₆₇	1	1	yes	characteristic
275	C₂	C₄×C₁₆×D₆₇	1	1	yes	characteristic
276	C₂	#346×D₆₇	1	1	yes	characteristic
277	C₂	C₂²⋊C₁₆×D₆₇	1	2	yes	—
278	C₂	C₄⋊C₁₆×D₆₇	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄×C₁₆
Fitting	C₆₇×#1374	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂³×D₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1374	D₆₇
hypocenter	C₆₇	C₂×#1374
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. #22558287	C₂	semidirect
4. #22558286	C₂	semidirect
5. C₆₇×#1374	C₂	semidirect
12. C₂×C₁₆×D₆₇	C₄	semidirect
36. C₂×C₆₇⋊C₁₆	C₂×C₄	semidirect
46. C₂×C₁₆×C₆₇	C₂×C₄	semidirect
53. C₂×C₄×D₆₇	C₁₆	semidirect
93. C₂×C₄×C₆₇	C₂×C₁₆	semidirect
105. C₂×Dic₆₇	C₂×C₁₆	semidirect
139. D₆₇	#1374	direct (N × H)
144. #1374	D₆₇	direct (N × H)
145. C₆₇	C₂×#1374	semidirect
183. C₂×C₁₆	C₄×D₆₇	semidirect
244. C₂×C₄	C₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×D₆₇	2
C₂×C₄×C₈×D₆₇	2
#22558287	2
#22558286	2
C₆₇×#1374	2
C₂×#1374	67

Subgroups — #1374×D67 #30770666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1374×D₆₇ #30770666