Subgroups — #1376×D₆₇ #30770682

Order 17152.

670 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22558291	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558290	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1376	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	C₂×#14523983	C₂²	1	2	yes	—
10	C₂×#14523982	—	2	2	no	—
11	M₃₂×D₆₇	—	2	4	no	—
12	M₃₂×D₆₇	C₄	1	8	yes	complemented
13	C₂×C₁₆×D₆₇	—	2	2	no	—
14	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	C₄×C₈×D₆₇	—	2	4	no	—
18	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄×D₆₇	—	2	1	no	—
21	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₆₇	—	2	2	no	—
25	C₂×C₈×D₆₇	—	2	2	no	—
26	C₂×C₈×D₆₇	C₈	1	2	yes	—
27	C₂×C₈×D₆₇	C₈	1	2	yes	—
28	C₂×C₈×D₆₇	C₂×C₄	1	4	yes	—
29	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
30	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
31	C₂×C₈×D₆₇	D₄	1	4	yes	—
32	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
33	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
34	C₂×C₄×Dic₆₇	—	2	1	no	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×D₆₇	—	2	2	no	—
37	C₄²×D₆₇	C₈	1	4	yes	—
38	#14523983	—	2	2	no	—
39	#14523983	C₂×C₄	1	4	yes	complemented
40	#14523982	—	2	8	no	—
41	C₂×C₆₇⋊C₁₆	—	2	2	no	—
42	C₁₆×D₆₇	—	2	8	no	—
43	C₄×C₆₇⋊C₈	—	2	2	no	—
44	C₈×Dic₆₇	—	2	2	no	—
45	C₈×Dic₆₇	—	2	2	no	—
46	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
49	M₃₂×C₆₇	—	2	2	no	—
50	M₃₂×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂×C₁₆×C₆₇	—	2	2	no	—
52	C₄×C₈×C₆₇	—	2	2	no	—
53	C₂³×D₆₇	M₁₆	1	1	yes	characteristic
54	C₂²×Dic₆₇	—	2	1	no	—
55	C₂²×Dic₆₇	M₁₆	1	1	yes	characteristic
56	C₂×C₄×D₆₇	—	2	8	no	—
57	C₂×C₄×D₆₇	—	2	4	no	—
58	C₂×C₄×D₆₇	C₄²	1	2	yes	—
59	C₂×C₄×D₆₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₆₇	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
66	C₂×C₆₇⋊C₈	—	2	1	no	—
67	C₂×C₆₇⋊C₈	—	2	1	no	—
68	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
69	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	2	yes	—
70	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₆₇⋊C₈	C₂×C₈	1	2	yes	—
73	C₈×D₆₇	—	2	4	no	—
74	C₈×D₆₇	—	2	4	no	—
75	C₈×D₆₇	C₂²⋊C₄	1	4	yes	—
76	C₈×D₆₇	C₂²⋊C₄	1	4	yes	—
77	C₈×D₆₇	C₄⋊C₄	1	4	yes	—
78	C₈×D₆₇	C₄⋊C₄	1	4	yes	—
79	C₄×Dic₆₇	—	2	1	no	—
80	C₄×Dic₆₇	—	2	4	no	—
81	C₄×Dic₆₇	C₂×C₈	1	2	yes	—
82	C₆₇⋊C₁₆	—	2	4	no	—
83	C₂²×C₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₆₇	—	2	1	no	—
93	C₄²×C₆₇	C₂×C₈	1	2	yes	—
94	C₁₆×C₆₇	—	2	4	no	—
95	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂²×D₆₇	—	2	4	no	—
97	C₂²×D₆₇	#97	1	2	yes	—
98	C₂²×D₆₇	C₂²⋊C₈	1	2	yes	—
99	C₂²×D₆₇	C₄⋊C₈	1	1	yes	characteristic
100	C₂²×D₆₇	C₄⋊C₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×Dic₆₇	—	2	4	no	—
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	#97	1	2	yes	—
112	C₂×Dic₆₇	C₂²⋊C₈	1	2	yes	—
113	C₂×Dic₆₇	C₄⋊C₈	1	1	yes	characteristic
114	C₂×Dic₆₇	C₄⋊C₈	1	1	yes	characteristic
115	C₄×D₆₇	—	2	16	no	—
116	C₄×D₆₇	#95	1	2	yes	—
117	C₄×D₆₇	#95	1	2	yes	—
118	C₄×D₆₇	C₄×C₈	1	2	yes	—
119	C₄×D₆₇	C₄×C₈	1	2	yes	—
120	C₄×D₆₇	C₂²⋊C₈	1	2	yes	—
121	C₄×D₆₇	C₂²⋊C₈	1	2	yes	—
122	C₄×D₆₇	C₄⋊C₈	1	2	yes	—
123	C₄×D₆₇	C₄⋊C₈	1	2	yes	—
124	C₈×C₆₇	—	2	2	no	—
125	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
126	C₈×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
127	C₆₇⋊C₈	—	2	2	no	—
128	C₆₇⋊C₈	#95	1	2	yes	—
129	C₆₇⋊C₈	#95	1	2	yes	—
130	C₂²×C₆₇	—	2	2	no	—
131	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
132	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
134	C₂×D₆₇	—	2	2	no	—
135	C₂×D₆₇	—	2	4	no	—
136	C₂×D₆₇	—	2	4	no	—
137	C₂×D₆₇	#336	1	2	yes	—
138	C₂×D₆₇	#336	1	2	yes	—
139	C₂×D₆₇	#364	1	4	yes	—
140	C₄×C₆₇	—	2	4	no	—
141	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
142	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
143	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
145	Dic₆₇	—	2	4	no	—
146	Dic₆₇	#336	1	2	yes	—
147	Dic₆₇	#336	1	2	yes	—
148	C₂×#1376	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	2	1	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#364	1	2	yes	—
153	D₆₇	—	2	2	no	—
154	D₆₇	#1376	1	4	yes	direct product factor
155	C₂²×M₃₂	—	67	2	no	—
156	C₂²×C₄×C₈	—	67	1	no	—
157	#1376	—	67	2	no	—
158	#1376	—	67	1	no	—
159	#1376	D₆₇	1	1	yes	characteristic, direct product factor
160	C₆₇	C₂×#1376	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₈	—	67	1	no	—
162	C₂³×C₈	—	67	1	no	—
163	C₂²×C₄²	—	67	1	no	—
164	C₂×M₃₂	—	67	2	no	—
165	C₂×M₃₂	—	67	8	no	—
166	C₂×M₃₂	—	134	2	no	—
167	C₂×M₃₂	—	134	4	no	—
168	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
169	C₂²×C₁₆	—	134	2	no	—
170	C₂×C₄×C₈	—	67	1	no	—
171	C₂×C₄×C₈	—	67	1	no	—
172	C₂×C₄×C₈	—	67	1	no	—
173	C₂×C₄×C₈	—	134	4	no	—
174	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
175	C₂³×C₄	—	67	1	no	—
176	C₂³×C₄	—	134	1	no	—
177	C₂²×C₈	—	67	1	no	—
178	C₂²×C₈	—	67	4	no	—
179	C₂²×C₈	—	67	4	no	—
180	C₂²×C₈	—	67	1	no	—
181	C₂²×C₈	—	67	1	no	—
182	C₂²×C₈	—	67	1	no	—
183	C₂²×C₈	—	67	1	no	—
184	C₂²×C₈	—	67	1	no	—
185	C₂²×C₈	—	67	2	no	—
186	C₂²×C₈	—	67	2	no	—
187	C₂²×C₈	—	134	2	no	—
188	C₂²×C₈	—	134	2	no	—
189	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
190	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
191	C₂×C₄²	—	67	1	no	—
192	C₂×C₄²	—	67	4	no	—
193	C₂×C₄²	—	134	2	no	—
194	C₂×C₄²	—	134	1	no	—
195	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
196	M₃₂	—	2	2	no	—
197	M₃₂	—	67	4	no	—
198	M₃₂	—	134	8	no	—
199	M₃₂	—	134	2	no	—
200	M₃₂	C₄×D₆₇	1	4	yes	complemented
201	C₂×C₁₆	—	2	2	no	—
202	C₂×C₁₆	—	134	8	no	—
203	C₂×C₁₆	—	134	2	no	—
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	—	134	2	no	—
206	C₄×C₈	—	134	2	no	—
207	C₄×C₈	—	134	2	no	—
208	C₂⁴	—	67	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	67	1	no	—
211	C₂²×C₄	—	67	2	no	—
212	C₂²×C₄	—	67	2	no	—
213	C₂²×C₄	—	67	1	no	—
214	C₂²×C₄	—	67	1	no	—
215	C₂²×C₄	—	67	2	no	—
216	C₂²×C₄	—	67	1	no	—
217	C₂²×C₄	—	67	1	no	—
218	C₂²×C₄	—	67	2	no	—
219	C₂²×C₄	—	134	4	no	—
220	C₂²×C₄	—	134	8	no	—
221	C₂²×C₄	—	134	1	no	—
222	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	67	2	no	—
226	C₂×C₈	—	67	2	no	—
227	C₂×C₈	—	67	1	no	—
228	C₂×C₈	—	67	1	no	—
229	C₂×C₈	—	67	4	no	—
230	C₂×C₈	—	67	4	no	—
231	C₂×C₈	—	67	4	no	—
232	C₂×C₈	—	67	4	no	—
233	C₂×C₈	—	67	2	no	—
234	C₂×C₈	—	134	4	no	—
235	C₂×C₈	—	134	4	no	—
236	C₂×C₈	—	134	1	no	—
237	C₂×C₈	—	134	1	no	—
238	C₂×C₈	C₈×D₆₇	1	2	yes	—
239	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
240	C₂×C₈	D₄×D₆₇	1	2	yes	—
241	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
242	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
243	C₄²	—	2	1	no	—
244	C₄²	—	67	2	no	—
245	C₄²	—	134	4	no	—
246	C₄²	—	134	1	no	—
247	C₄²	C₈×D₆₇	1	2	yes	—
248	C₁₆	—	2	4	no	—
249	C₁₆	—	134	4	no	—
250	C₂³	—	67	1	no	—
251	C₂³	—	67	2	no	—
252	C₂³	—	67	2	no	—
253	C₂³	—	67	1	no	—
254	C₂³	—	134	4	no	—
255	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	67	2	no	—
259	C₂×C₄	—	67	2	no	—
260	C₂×C₄	—	67	1	no	—
261	C₂×C₄	—	67	1	no	—
262	C₂×C₄	—	67	2	no	—
263	C₂×C₄	—	67	2	no	—
264	C₂×C₄	—	67	2	no	—
265	C₂×C₄	—	67	2	no	—
266	C₂×C₄	—	67	2	no	—
267	C₂×C₄	—	67	2	no	—
268	C₂×C₄	—	67	2	no	—
269	C₂×C₄	—	67	2	no	—
270	C₂×C₄	—	134	16	no	—
271	C₂×C₄	—	134	4	no	—
272	C₂×C₄	—	134	2	no	—
273	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
274	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
275	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
276	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
277	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
278	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
279	C₈	—	2	2	no	—
280	C₈	—	67	2	no	—
281	C₈	—	67	2	no	—
282	C₈	—	134	2	no	—
283	C₈	C₂²⋊C₄×D₆₇	1	2	yes	—
284	C₈	C₄⋊C₄×D₆₇	1	2	yes	—
285	C₂²	—	2	2	no	—
286	C₂²	—	67	2	no	—
287	C₂²	—	67	4	no	—
288	C₂²	—	67	2	no	—
289	C₂²	—	134	4	no	—
290	C₂²	—	134	4	no	—
291	C₂²	—	134	2	no	—
292	C₂²	#97×D₆₇	1	1	yes	characteristic
293	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
294	C₂²	C₄⋊C₈×D₆₇	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	67	2	no	—
297	C₄	—	67	2	no	—
298	C₄	—	134	4	no	—
299	C₄	#95×D₆₇	1	1	yes	characteristic
300	C₄	C₄×C₈×D₆₇	1	1	yes	characteristic
301	C₄	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
302	C₄	C₄⋊C₈×D₆₇	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	67	4	no	—
306	C₂	—	134	2	no	—
307	C₂	#336×D₆₇	1	1	yes	characteristic
308	C₂	#364×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1376	C₂
socle	C₂²×C₆₇	C₂×C₄⋊C₈
prime radical	C₂³×D₆₇	M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1376	D₆₇
hypocenter	C₆₇	C₂×#1376
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1376

Complements

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

N	H ≅ G/N	type
3. #22558291	C₂	semidirect
4. #22558290	C₂	semidirect
5. C₆₇×#1376	C₂	semidirect
12. M₃₂×D₆₇	C₄	semidirect
39. #14523983	C₂×C₄	semidirect
50. M₃₂×C₆₇	C₂×C₄	semidirect
154. D₆₇	#1376	direct (N × H)
159. #1376	D₆₇	direct (N × H)
160. C₆₇	C₂×#1376	semidirect
200. M₃₂	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₆₇	2
C₂×C₄×C₈×D₆₇	2
#22558291	2
#22558290	2
C₆₇×#1376	2
C₂×#1376	67

Subgroups — #1376×D67 #30770682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1376×D₆₇ #30770682