Subgroups — C₂×#368×D₆₇ #31008881

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 294 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	1	yes	maximal, characteristic
2	C₂×#108×D₆₇	C₂	1	2	yes	maximal
3	C₂×#16863226	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863225	C₂	1	2	yes	maximal, complemented
5	#368×D₆₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#368×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₆₇	—	2	2	no	—
8	C₂²×C₈×D₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₆₇	C₄	1	4	yes	—
11	C₂×C₁₆×D₆₇	C₄	1	2	yes	—
12	C₂×C₁₆×D₆₇	C₄	1	2	yes	—
13	C₂×C₁₆×D₆₇	C₂²	1	4	yes	—
14	C₂×#14523979	C₂²	1	2	yes	—
15	C₂×#14523978	—	2	2	no	—
16	#108×D₆₇	—	2	4	no	—
17	#108×D₆₇	C₂²	1	8	yes	—
18	#16863226	C₂²	1	4	yes	complemented
19	#16863225	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#108×C₆₇	C₂²	1	2	yes	—
22	#368×C₆₇	C₂²	1	4	yes	complemented
23	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761611	—	4	2	no	—
26	M₁₆×D₆₇	—	2	8	no	—
27	M₁₆×D₆₇	—	4	4	no	—
28	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₈×D₆₇	—	4	2	no	—
30	C₂×C₈×D₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₆₇	D₄	1	2	yes	—
34	C₂×C₈×D₆₇	Q₈	1	2	yes	—
35	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
36	C₂×C₈×D₆₇	Q₈	1	1	yes	characteristic
37	C₂×C₈×D₆₇	C₂³	1	2	yes	—
38	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
41	C₁₆×D₆₇	C₂×C₄	1	8	yes	—
42	C₁₆×D₆₇	C₂×C₄	1	8	yes	—
43	#14523979	—	2	2	no	—
44	#14523979	C₂³	1	4	yes	—
45	#14523978	—	2	8	no	—
46	C₂×M₁₆×C₆₇	—	2	2	no	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
51	#108×C₆₇	—	2	2	no	—
52	#108×C₆₇	C₂³	1	4	yes	—
53	C₂³×D₆₇	Q₁₆	1	1	yes	characteristic
54	C₂²×Dic₆₇	Q₁₆	1	1	yes	characteristic
55	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×D₆₇	C₄⋊C₄	1	2	yes	—
57	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
59	C₂×C₄×D₆₇	D₈	1	2	yes	—
60	C₂×C₄×D₆₇	D₈	1	1	yes	characteristic
61	C₂×C₄×D₆₇	Q₁₆	1	2	yes	—
62	C₂×C₄×D₆₇	C₂×D₄	1	2	yes	—
63	#11761612	—	2	4	no	—
64	#11761612	—	4	2	no	—
65	#11761611	—	4	8	no	—
66	C₂×C₆₇⋊C₈	—	4	2	no	—
67	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₆₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
71	C₈×D₆₇	—	4	8	no	—
72	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
73	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
74	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
75	C₈×D₆₇	C₄⋊C₄	1	2	yes	—
76	C₈×D₆₇	C₂²×C₄	1	2	yes	—
77	C₈×D₆₇	C₂²×C₄	1	2	yes	—
78	C₈×D₆₇	C₂×Q₈	1	2	yes	—
79	C₈×D₆₇	C₂×Q₈	1	2	yes	—
80	C₆₇⋊C₁₆	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	M₁₆×C₆₇	—	2	4	no	—
83	M₁₆×C₆₇	—	4	2	no	—
84	C₂×C₈×C₆₇	—	4	2	no	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
92	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
93	C₂²×D₆₇	—	2	4	no	—
94	C₂²×D₆₇	#107	1	1	yes	characteristic
95	C₂²×D₆₇	#107	1	2	yes	—
96	C₂²×D₆₇	#107	1	1	yes	characteristic
97	C₂²×D₆₇	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×Dic₆₇	#107	1	2	yes	—
105	C₂×Dic₆₇	#107	1	1	yes	characteristic
106	C₂×Dic₆₇	#107	1	1	yes	characteristic
107	C₂×Dic₆₇	C₂×Q₁₆	1	2	yes	—
108	C₄×D₆₇	#107	1	2	yes	—
109	C₄×D₆₇	#107	1	2	yes	—
110	C₄×D₆₇	#107	1	2	yes	—
111	C₄×D₆₇	#107	1	2	yes	—
112	C₄×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
113	C₄×D₆₇	C₂×C₄⋊C₄	1	2	yes	—
114	C₄×D₆₇	C₂×D₈	1	2	yes	—
115	C₄×D₆₇	C₂×D₈	1	2	yes	—
116	C₈×C₆₇	—	4	4	no	—
117	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
121	C₆₇⋊C₈	—	4	4	no	—
122	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
123	C₆₇⋊C₈	C₂×C₄⋊C₄	1	2	yes	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
127	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
128	C₂×D₆₇	—	2	2	no	—
129	C₂×D₆₇	—	2	4	no	—
130	C₂×D₆₇	—	2	4	no	—
131	C₂×D₆₇	#368	1	4	yes	direct product factor
132	C₂×D₆₇	C₂×#107	1	2	yes	—
133	C₂×D₆₇	C₂×#107	1	2	yes	—
134	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#107	1	1	yes	characteristic
136	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
138	Dic₆₇	C₂×#107	1	2	yes	—
139	Dic₆₇	C₂×#107	1	2	yes	—
140	C₂²×#368	—	67	1	no	maximal, Sylow
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	—	2	1	no	—
143	C₂×C₆₇	C₂×#368	1	2	yes	complemented
144	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
145	D₆₇	—	2	2	no	—
146	D₆₇	C₂×#368	1	4	yes	direct product factor
147	C₂³×C₁₆	—	67	1	no	—
148	C₂²×#108	—	67	2	no	—
149	C₂×#368	—	67	1	no	—
150	C₂×#368	—	67	2	no	—
151	C₂×#368	—	67	8	no	—
152	C₂×#368	D₆₇	1	1	yes	characteristic, direct product factor
153	C₆₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	134	2	no	—
155	C₂³×C₈	—	67	1	no	—
156	C₂²×C₁₆	—	67	1	no	—
157	C₂²×C₁₆	—	67	2	no	—
158	C₂²×C₁₆	—	67	2	no	—
159	C₂²×C₁₆	—	67	4	no	—
160	C₂²×C₁₆	—	67	4	no	—
161	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
162	C₂×#108	—	67	2	no	—
163	C₂×#108	—	67	8	no	—
164	C₂×#108	—	134	4	no	—
165	C₂×#108	—	134	2	no	—
166	C₂×#108	C₂×D₆₇	1	2	yes	—
167	#368	—	67	4	no	—
168	#368	—	67	8	no	—
169	#368	C₂×D₆₇	1	4	yes	direct product factor
170	C₂³×C₄	—	67	1	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	268	2	no	—
173	C₂×M₁₆	—	268	4	no	—
174	C₂×M₁₆	—	134	8	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂²×C₈	—	268	2	no	—
177	C₂²×C₈	—	67	1	no	—
178	C₂²×C₈	—	67	1	no	—
179	C₂²×C₈	—	67	1	no	—
180	C₂²×C₈	—	67	2	no	—
181	C₂²×C₈	—	67	2	no	—
182	C₂²×C₈	—	67	1	no	—
183	C₂²×C₈	—	67	1	no	—
184	C₂²×C₈	—	67	2	no	—
185	C₂²×C₈	—	67	2	no	—
186	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
187	C₂×C₁₆	—	67	2	no	—
188	C₂×C₁₆	—	67	2	no	—
189	C₂×C₁₆	—	67	2	no	—
190	C₂×C₁₆	—	67	8	no	—
191	C₂×C₁₆	—	67	8	no	—
192	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
193	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
194	C₂×C₁₆	C₂²×D₆₇	1	2	yes	—
195	#108	—	2	2	no	—
196	#108	—	67	4	no	—
197	#108	—	134	8	no	—
198	#108	—	134	2	no	—
199	#108	C₂²×D₆₇	1	4	yes	—
200	C₂⁴	—	67	1	no	—
201	C₂²×C₄	—	67	1	no	—
202	C₂²×C₄	—	67	1	no	—
203	C₂²×C₄	—	67	2	no	—
204	C₂²×C₄	—	67	1	no	—
205	C₂²×C₄	—	67	2	no	—
206	C₂²×C₄	—	67	1	no	—
207	C₂²×C₄	—	67	1	no	—
208	C₂²×C₄	—	67	2	no	—
209	C₂²×C₄	—	67	2	no	—
210	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	4	2	no	—
213	M₁₆	—	268	8	no	—
214	M₁₆	—	268	2	no	—
215	M₁₆	—	134	4	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	268	8	no	—
218	C₂×C₈	—	268	2	no	—
219	C₂×C₈	—	67	1	no	—
220	C₂×C₈	—	67	1	no	—
221	C₂×C₈	—	67	2	no	—
222	C₂×C₈	—	67	2	no	—
223	C₂×C₈	—	67	2	no	—
224	C₂×C₈	—	67	2	no	—
225	C₂×C₈	—	67	2	no	—
226	C₂×C₈	—	67	2	no	—
227	C₂×C₈	—	67	2	no	—
228	C₂×C₈	—	67	2	no	—
229	C₂×C₈	—	67	2	no	—
230	C₂×C₈	—	67	2	no	—
231	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
232	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
233	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
234	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
235	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
236	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
237	C₁₆	—	67	4	no	—
238	C₁₆	C₂×C₄×D₆₇	1	4	yes	—
239	C₂³	—	67	1	no	—
240	C₂³	—	67	2	no	—
241	C₂³	—	67	2	no	—
242	C₂³	—	67	1	no	—
243	C₂³	—	134	4	no	—
244	C₂³	Q₁₆×D₆₇	1	1	yes	characteristic
245	C₂×C₄	—	67	2	no	—
246	C₂×C₄	—	67	2	no	—
247	C₂×C₄	—	67	1	no	—
248	C₂×C₄	—	67	1	no	—
249	C₂×C₄	—	67	2	no	—
250	C₂×C₄	—	67	2	no	—
251	C₂×C₄	—	67	2	no	—
252	C₂×C₄	—	67	2	no	—
253	C₂×C₄	—	67	2	no	—
254	C₂×C₄	—	67	2	no	—
255	C₂×C₄	—	67	2	no	—
256	C₂×C₄	—	67	2	no	—
257	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
259	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
260	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
261	C₂×C₄	Q₁₆×D₆₇	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
263	C₈	—	4	4	no	—
264	C₈	—	268	4	no	—
265	C₈	—	67	2	no	—
266	C₈	—	67	2	no	—
267	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
268	C₈	C₄⋊C₄×D₆₇	1	1	yes	characteristic
269	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
270	C₈	C₂×Q₈×D₆₇	1	1	yes	characteristic
271	C₂²	—	2	2	no	—
272	C₂²	—	67	2	no	—
273	C₂²	—	67	4	no	—
274	C₂²	—	67	2	no	—
275	C₂²	—	134	4	no	—
276	C₂²	—	134	4	no	—
277	C₂²	—	134	2	no	—
278	C₂²	#107×D₆₇	1	1	yes	characteristic
279	C₂²	#107×D₆₇	1	1	yes	characteristic
280	C₂²	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
281	C₄	—	67	2	no	—
282	C₄	—	67	2	no	—
283	C₄	#107×D₆₇	1	1	yes	characteristic
284	C₄	#107×D₆₇	1	1	yes	characteristic
285	C₄	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
286	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
287	C₂	—	2	1	no	—
288	C₂	—	2	1	no	—
289	C₂	—	67	4	no	—
290	C₂	—	134	2	no	—
291	C₂	#368×D₆₇	1	2	yes	direct product factor
292	C₂	C₂×#107×D₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#368×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×D₆₇	Q₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#368	D₆₇
hypocenter	C₆₇	C₂²×#368
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
3. C₂×#16863226	C₂	semidirect
4. C₂×#16863225	C₂	semidirect
5. #368×D₆₇	C₂	direct (N × H)
6. C₂×#368×C₆₇	C₂	semidirect
18. #16863226	C₂²	semidirect
19. #16863225	C₂²	semidirect
22. #368×C₆₇	C₂²	semidirect
131. C₂×D₆₇	#368	direct (N × H)
143. C₂×C₆₇	C₂×#368	semidirect
146. D₆₇	C₂×#368	direct (N × H)
152. C₂×#368	D₆₇	direct (N × H)
153. C₆₇	C₂²×#368	semidirect
169. #368	C₂×D₆₇	direct (N × H)
291. C₂	#368×D₆₇	direct (N × H)

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×#108×D₆₇	2
C₂×#16863226	2
C₂×#16863225	2
#368×D₆₇	2
C₂×#368×C₆₇	2
C₂²×#368	67

Subgroups — C2×#368×D67 #31008881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#368×D₆₇ #31008881