Subgroups — C₂×C₆₇×#1594 #30697140

Order 17152.

806 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#436×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#1594	C₂	1	8	yes	maximal, direct product factor
4	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄⋊C₈×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
8	C₂²×M₁₆×C₆₇	—	2	2	no	—
9	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
10	C₄×D₈×C₆₇	C₄	1	4	yes	—
11	C₄×D₈×C₆₇	C₂²	1	4	yes	—
12	#436×C₆₇	—	2	4	no	—
13	#436×C₆₇	C₂²	1	8	yes	complemented
14	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×#102×C₆₇	C₄	1	2	yes	—
17	C₂×C₂²⋊C₈×C₆₇	—	2	2	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#334×C₆₇	C₂²	1	4	yes	complemented
22	D₄⋊C₈×C₆₇	C₂²	1	8	yes	complemented
23	#322×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₇	—	2	2	no	—
26	C₂×D₈×C₆₇	—	2	4	no	—
27	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₆₇	—	2	8	no	—
29	C₂×M₁₆×C₆₇	—	4	8	no	—
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	—	2	1	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₆₇	D₄	1	2	yes	—
35	C₄×D₄×C₆₇	—	2	2	no	—
36	C₄×D₄×C₆₇	—	2	2	no	—
37	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
38	C₄×D₄×C₆₇	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	#107×C₆₇	C₂×C₄	1	4	yes	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
45	#102×C₆₇	C₂×C₄	1	8	yes	—
46	C₂²⋊C₈×C₆₇	—	2	8	no	—
47	C₄×C₈×C₆₇	—	2	1	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₄×C₈×C₆₇	C₂³	1	2	yes	—
51	C₄×C₈×C₆₇	C₂³	1	2	yes	—
52	C₄×C₈×C₆₇	C₂³	1	2	yes	—
53	C₂⁴×C₆₇	—	2	2	no	—
54	C₂×D₄×C₆₇	—	2	2	no	—
55	C₂×D₄×C₆₇	—	2	4	no	—
56	C₂×D₄×C₆₇	—	2	4	no	—
57	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	—	2	4	no	—
60	C₂²×C₄×C₆₇	—	2	4	no	—
61	C₂²×C₄×C₆₇	—	4	4	no	—
62	C₂²×C₄×C₆₇	—	4	4	no	—
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
66	D₈×C₆₇	—	2	8	no	—
67	M₁₆×C₆₇	—	4	16	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	—	4	2	no	—
73	C₂×C₈×C₆₇	—	4	4	no	—
74	C₂×C₈×C₆₇	M₁₆	1	4	yes	complemented
75	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₆₇	—	2	2	no	—
79	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
80	C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₂²⋊C₄×C₆₇	—	2	4	no	—
82	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₆₇	#41	1	1	yes	characteristic
85	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₂³×C₆₇	—	2	4	no	—
87	C₂³×C₆₇	—	4	4	no	—
88	C₂³×C₆₇	—	4	4	no	—
89	C₂³×C₆₇	—	4	4	no	—
90	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
91	D₄×C₆₇	—	2	4	no	—
92	D₄×C₆₇	—	2	4	no	—
93	D₄×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	4	no	—
96	C₂×C₄×C₆₇	—	4	4	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	—	4	4	no	—
99	C₂×C₄×C₆₇	—	4	8	no	—
100	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₆₇	#131	1	2	yes	—
105	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₈×C₆₇	—	2	2	no	—
110	C₈×C₆₇	—	2	4	no	—
111	C₈×C₆₇	—	4	2	no	—
112	C₈×C₆₇	—	4	4	no	—
113	C₈×C₆₇	C₂×M₁₆	1	4	yes	complemented
114	C₂²×C₆₇	—	4	4	no	—
115	C₂²×C₆₇	—	4	4	no	—
116	C₂²×C₆₇	—	4	4	no	—
117	C₂²×C₆₇	—	8	8	no	—
118	C₂²×C₆₇	#436	1	2	yes	—
119	C₂²×C₆₇	#442	1	2	yes	—
120	C₂²×C₆₇	#444	1	2	yes	—
121	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
122	C₄×C₆₇	—	2	1	no	—
123	C₄×C₆₇	—	2	1	no	—
124	C₄×C₆₇	—	4	4	no	—
125	C₄×C₆₇	#436	1	2	yes	—
126	C₄×C₆₇	#442	1	2	yes	—
127	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
128	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
129	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
130	C₂×#1594	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂×C₆₇	—	8	4	no	—
132	C₂×C₆₇	#1594	1	4	yes	direct product factor
133	C₂×C₆₇	C₂×#436	1	1	yes	characteristic
134	C₂×C₆₇	C₂×#442	1	1	yes	characteristic
135	C₂×C₆₇	C₂×#444	1	1	yes	characteristic
136	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
137	C₂×#436	C₂×C₆₇	1	2	yes	complemented
138	#1594	C₂×C₆₇	1	8	yes	direct product factor
139	C₂×#334	C₂×C₆₇	1	1	yes	characteristic, complemented
140	C₂×D₄⋊C₈	C₂×C₆₇	1	2	yes	complemented
141	C₂×#322	C₂×C₆₇	1	1	yes	characteristic, complemented
142	C₆₇	C₂×#1594	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×D₈	C₄×C₆₇	1	1	yes	characteristic
144	C₂²×M₁₆	—	2	2	no	—
145	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
146	C₄×D₈	C₄×C₆₇	1	4	yes	—
147	C₄×D₈	C₂²×C₆₇	1	4	yes	—
148	#436	—	2	4	no	—
149	#436	C₂²×C₆₇	1	8	yes	complemented
150	C₂×#107	C₄×C₆₇	1	1	yes	characteristic
151	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
152	C₂×#102	C₄×C₆₇	1	2	yes	—
153	C₂×C₂²⋊C₈	—	2	2	no	—
154	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
157	#334	C₂²×C₆₇	1	4	yes	complemented
158	D₄⋊C₈	C₂²×C₆₇	1	8	yes	complemented
159	#322	C₂²×C₆₇	1	4	yes	complemented
160	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
161	C₂³×C₄	—	2	2	no	—
162	C₂×D₈	—	2	4	no	—
163	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
164	C₂×M₁₆	—	2	8	no	—
165	C₂×M₁₆	—	4	8	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	—	2	1	no	—
168	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₆₇	1	2	yes	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	—	2	2	no	—
173	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
174	C₄×D₄	C₂³×C₆₇	1	4	yes	—
175	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
176	C₂×C₂²⋊C₄	—	2	2	no	—
177	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
178	#107	C₂×C₄×C₆₇	1	4	yes	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
181	#102	C₂×C₄×C₆₇	1	8	yes	—
182	C₂²⋊C₈	—	2	8	no	—
183	C₄×C₈	—	2	1	no	—
184	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
185	C₄×C₈	D₄×C₆₇	1	2	yes	—
186	C₄×C₈	C₂³×C₆₇	1	2	yes	—
187	C₄×C₈	C₂³×C₆₇	1	2	yes	—
188	C₄×C₈	C₂³×C₆₇	1	2	yes	—
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
201	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
202	D₈	—	2	8	no	—
203	M₁₆	—	4	16	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	M₁₆×C₆₇	1	4	yes	complemented
211	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
220	C₄²	#41×C₆₇	1	1	yes	characteristic
221	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
239	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
240	C₂×C₄	#131×C₆₇	1	2	yes	—
241	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
242	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	4	no	—
247	C₈	—	4	2	no	—
248	C₈	—	4	4	no	—
249	C₈	C₂×M₁₆×C₆₇	1	4	yes	complemented
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	8	8	no	—
254	C₂²	#436×C₆₇	1	2	yes	—
255	C₂²	#442×C₆₇	1	2	yes	—
256	C₂²	#444×C₆₇	1	2	yes	—
257	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	4	no	—
261	C₄	#436×C₆₇	1	2	yes	—
262	C₄	#442×C₆₇	1	2	yes	—
263	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
264	C₄	C₂×#131×C₆₇	1	1	yes	characteristic
265	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
266	C₂	—	8	4	no	—
267	C₂	C₆₇×#1594	1	4	yes	direct product factor
268	C₂	C₂×#436×C₆₇	1	1	yes	characteristic
269	C₂	C₂×#442×C₆₇	1	1	yes	characteristic
270	C₂	C₂×#444×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₆₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂³×C₆₇	C₄×D₄
prime radical	C₂²×D₈×C₆₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#436×C₆₇	C₂	semidirect
3. C₆₇×#1594	C₂	direct (N × H)
4. C₂×#334×C₆₇	C₂	semidirect
5. C₂×D₄⋊C₈×C₆₇	C₂	semidirect
6. C₂×#322×C₆₇	C₂	semidirect
13. #436×C₆₇	C₂²	semidirect
21. #334×C₆₇	C₂²	semidirect
22. D₄⋊C₈×C₆₇	C₂²	semidirect
23. #322×C₆₇	C₂²	semidirect
74. C₂×C₈×C₆₇	M₁₆	semidirect
113. C₈×C₆₇	C₂×M₁₆	semidirect
130. C₂×#1594	C₆₇	direct (N × H)
132. C₂×C₆₇	#1594	direct (N × H)
137. C₂×#436	C₂×C₆₇	semidirect
138. #1594	C₂×C₆₇	direct (N × H)
139. C₂×#334	C₂×C₆₇	semidirect
140. C₂×D₄⋊C₈	C₂×C₆₇	semidirect
141. C₂×#322	C₂×C₆₇	semidirect
142. C₆₇	C₂×#1594	direct (N × H)
149. #436	C₂²×C₆₇	semidirect
157. #334	C₂²×C₆₇	semidirect
158. D₄⋊C₈	C₂²×C₆₇	semidirect
159. #322	C₂²×C₆₇	semidirect
210. C₂×C₈	M₁₆×C₆₇	semidirect
249. C₈	C₂×M₁₆×C₆₇	semidirect
267. C₂	C₆₇×#1594	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₆₇	2
C₂×#436×C₆₇	2
C₆₇×#1594	2
C₂×#334×C₆₇	2
C₂×D₄⋊C₈×C₆₇	2
C₂×#322×C₆₇	2
C₂×#1594	67

Subgroups — C2×C67×#1594 #30697140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#1594 #30697140