Subgroups — C₂×C₆₇×#2165 #30712675

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2165	C₂	1	8	yes	maximal, direct product factor
3	C₂×#364×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×#349×C₆₇	C₂	1	2	yes	maximal
5	C₂×#347×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₆₇	C₄	1	1	yes	characteristic
7	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₆₇	C₂²	1	4	yes	—
10	C₄×M₁₆×C₆₇	C₄	1	4	yes	—
11	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
12	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#364×C₆₇	C₂²	1	8	yes	complemented
17	#349×C₆₇	C₂²	1	8	yes	—
18	#347×C₆₇	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₆₇	—	2	1	no	—
20	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆×C₆₇	—	2	2	no	—
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂×M₁₆×C₆₇	—	2	2	no	—
24	C₂×M₁₆×C₆₇	—	2	2	no	—
25	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
26	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	—	2	1	no	—
33	C₂×C₄²×C₆₇	—	2	2	no	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
37	M₃₂×C₆₇	—	2	4	no	—
38	M₃₂×C₆₇	C₂³	1	8	yes	—
39	C₂×C₁₆×C₆₇	—	2	4	no	—
40	#97×C₆₇	C₂×C₄	1	4	yes	—
41	#97×C₆₇	C₂×C₄	1	2	yes	—
42	#97×C₆₇	C₂³	1	2	yes	—
43	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	—	2	1	no	—
50	C₂²×C₄×C₆₇	—	2	1	no	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	—	2	4	no	—
53	C₂²×C₄×C₆₇	—	2	1	no	—
54	C₂²×C₄×C₆₇	—	4	4	no	—
55	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	M₁₆×C₆₇	—	2	4	no	—
60	M₁₆×C₆₇	—	2	8	no	—
61	M₁₆×C₆₇	—	2	4	no	—
62	C₂×C₈×C₆₇	—	2	1	no	—
63	C₂×C₈×C₆₇	—	2	1	no	—
64	C₂×C₈×C₆₇	—	2	1	no	—
65	C₂×C₈×C₆₇	—	2	1	no	—
66	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
67	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₇	#41	1	2	yes	—
73	C₂×C₈×C₆₇	#41	1	2	yes	—
74	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
75	C₄²×C₆₇	—	2	1	no	—
76	C₄²×C₆₇	—	2	4	no	—
77	C₄²×C₆₇	—	2	1	no	—
78	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
80	C₁₆×C₆₇	—	2	8	no	—
81	C₂³×C₆₇	—	2	1	no	—
82	C₂³×C₆₇	—	2	1	no	—
83	C₂³×C₆₇	—	4	2	no	—
84	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
85	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	—	2	1	no	—
89	C₂×C₄×C₆₇	—	2	1	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	4	8	no	—
96	C₂×C₄×C₆₇	—	4	2	no	—
97	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
104	C₈×C₆₇	—	2	2	no	—
105	C₈×C₆₇	—	2	2	no	—
106	C₈×C₆₇	#117	1	2	yes	—
107	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
108	C₈×C₆₇	C₂×#41	1	2	yes	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	—	2	2	no	—
111	C₂²×C₆₇	—	4	2	no	—
112	C₂²×C₆₇	—	4	2	no	—
113	C₂²×C₆₇	—	4	2	no	—
114	C₂²×C₆₇	#432	1	1	yes	characteristic
115	C₂²×C₆₇	#432	1	1	yes	characteristic
116	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
117	C₄×C₆₇	—	2	4	no	—
118	C₄×C₆₇	—	2	2	no	—
119	C₄×C₆₇	—	4	2	no	—
120	C₄×C₆₇	#432	1	1	yes	characteristic
121	C₄×C₆₇	#432	1	1	yes	characteristic
122	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
123	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
124	C₂×#2165	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₆₇	—	2	1	no	—
126	C₂×C₆₇	—	2	1	no	—
127	C₂×C₆₇	—	4	2	no	—
128	C₂×C₆₇	#2165	1	2	yes	direct product factor
129	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
130	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
131	#2165	C₂×C₆₇	1	8	yes	direct product factor
132	C₂×#364	C₂×C₆₇	1	2	yes	complemented
133	C₂×#349	C₂×C₆₇	1	2	yes	—
134	C₂×#347	C₂×C₆₇	1	2	yes	complemented
135	C₆₇	C₂×#2165	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²×M₁₆	C₄×C₆₇	1	1	yes	characteristic
137	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
138	C₂²×C₄²	C₄×C₆₇	1	1	yes	characteristic
139	C₂×M₃₂	C₂²×C₆₇	1	4	yes	—
140	C₄×M₁₆	C₄×C₆₇	1	4	yes	—
141	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
142	C₂×#97	C₄×C₆₇	1	1	yes	characteristic
143	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
144	C₂×C₄×C₈	C₄×C₆₇	1	1	yes	characteristic
145	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
146	#364	C₂²×C₆₇	1	8	yes	complemented
147	#349	C₂²×C₆₇	1	8	yes	—
148	#347	C₂²×C₆₇	1	8	yes	complemented
149	C₂³×C₄	—	2	1	no	—
150	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
151	C₂×M₁₆	—	2	2	no	—
152	C₂×M₁₆	—	2	2	no	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	—
156	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
157	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
158	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
165	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
167	M₃₂	—	2	4	no	—
168	M₃₂	C₂³×C₆₇	1	8	yes	—
169	C₂×C₁₆	—	2	4	no	—
170	#97	C₂×C₄×C₆₇	1	4	yes	—
171	#97	C₂×C₄×C₆₇	1	2	yes	—
172	#97	C₂³×C₆₇	1	2	yes	—
173	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	—
174	C₄×C₈	C₂×C₄×C₆₇	1	2	yes	—
175	C₄×C₈	C₂³×C₆₇	1	2	yes	—
176	C₂⁴	M₁₆×C₆₇	1	1	yes	characteristic
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	M₁₆×C₆₇	1	1	yes	characteristic
186	C₂²×C₄	M₁₆×C₆₇	1	2	yes	—
187	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
189	M₁₆	—	2	4	no	—
190	M₁₆	—	2	8	no	—
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	M₁₆×C₆₇	1	2	yes	—
197	C₂×C₈	M₁₆×C₆₇	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
199	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
200	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
202	C₂×C₈	#41×C₆₇	1	2	yes	—
203	C₂×C₈	#41×C₆₇	1	2	yes	—
204	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
205	C₄²	—	2	1	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
209	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
210	C₁₆	—	2	8	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂×M₁₆×C₆₇	1	1	yes	characteristic
215	C₂³	C₂×M₁₆×C₆₇	1	2	yes	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	8	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
228	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
229	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
230	C₂×C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
231	C₂×C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
232	C₂×C₄	C₂×M₁₆×C₆₇	1	1	yes	characteristic
233	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	#117×C₆₇	1	2	yes	—
237	C₈	C₂×M₁₆×C₆₇	1	4	yes	—
238	C₈	C₂×#41×C₆₇	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	#432×C₆₇	1	1	yes	characteristic
245	C₂²	#432×C₆₇	1	1	yes	characteristic
246	C₂²	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	#432×C₆₇	1	1	yes	characteristic
251	C₄	#432×C₆₇	1	1	yes	characteristic
252	C₄	C₂×#117×C₆₇	1	1	yes	characteristic
253	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	C₆₇×#2165	1	2	yes	direct product factor
258	C₂	C₂×#432×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#117
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
socle	C₂²×C₆₇	#432
prime radical	C₂⁴×C₆₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2165	C₂	direct (N × H)
3. C₂×#364×C₆₇	C₂	semidirect
5. C₂×#347×C₆₇	C₂	semidirect
16. #364×C₆₇	C₂²	semidirect
18. #347×C₆₇	C₂²	semidirect
124. C₂×#2165	C₆₇	direct (N × H)
128. C₂×C₆₇	#2165	direct (N × H)
131. #2165	C₂×C₆₇	direct (N × H)
132. C₂×#364	C₂×C₆₇	semidirect
134. C₂×#347	C₂×C₆₇	semidirect
135. C₆₇	C₂×#2165	direct (N × H)
146. #364	C₂²×C₆₇	semidirect
148. #347	C₂²×C₆₇	semidirect
257. C₂	C₆₇×#2165	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×M₁₆×C₆₇	2
C₆₇×#2165	2
C₂×#364×C₆₇	2
C₂×#349×C₆₇	2
C₂×#347×C₆₇	2
C₂×#2165	67

Subgroups — C2×C67×#2165 #30712675

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2165 #30712675