Subgroups — C₂×#22562018 #31008950

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	#22562018	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863216	C₂	1	2	yes	maximal, complemented
4	C₂×#16863178	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863173	C₂	1	2	yes	maximal, complemented
6	C₂×#349×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11761612	C₄	1	1	yes	characteristic
8	C₂²×C₄×Dic₆₇	C₄	1	1	yes	characteristic
9	C₂×#14523983	C₂²	1	2	yes	—
10	M₁₆×Dic₆₇	C₄	1	4	yes	—
11	M₁₆×Dic₆₇	C₂²	1	4	yes	—
12	C₂×#14523953	C₄	1	1	yes	characteristic
13	C₂×#14523953	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₆₇	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	#16863216	C₂²	1	8	yes	complemented
17	#16863178	C₂²	1	4	yes	—
18	#16863173	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
21	#349×C₆₇	C₂²	1	4	yes	—
22	C₂³×Dic₆₇	—	2	1	no	—
23	C₂×#11761612	—	2	2	no	—
24	C₂×#11761612	—	2	2	no	—
25	C₂×#11761612	C₂×C₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₆₇	—	2	1	no	—
29	C₂×C₄×Dic₆₇	—	2	2	no	—
30	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₆₇	C₂×C₄	1	4	yes	—
33	#14523983	—	2	2	no	—
34	#14523983	C₂³	1	4	yes	—
35	C₂×C₆₇⋊C₁₆	—	2	2	no	—
36	#14523953	C₂×C₄	1	4	yes	—
37	#14523953	C₂×C₄	1	2	yes	—
38	#14523953	C₂³	1	2	yes	—
39	C₈×Dic₆₇	C₂×C₄	1	4	yes	—
40	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
41	C₈×Dic₆₇	C₂³	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂×M₁₆×C₆₇	—	2	2	no	—
44	C₂×M₁₆×C₆₇	—	2	2	no	—
45	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	M₃₂×C₆₇	—	2	2	no	—
50	M₃₂×C₆₇	C₂³	1	4	yes	—
51	C₂×C₁₆×C₆₇	—	2	2	no	—
52	C₂²×Dic₆₇	—	2	1	no	—
53	C₂²×Dic₆₇	—	2	4	no	—
54	C₂²×Dic₆₇	—	2	1	no	—
55	C₂²×Dic₆₇	—	4	4	no	—
56	#11761612	—	2	8	no	—
57	C₂×C₆₇⋊C₈	—	2	1	no	—
58	C₂×C₆₇⋊C₈	—	2	1	no	—
59	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
60	C₂×C₆₇⋊C₈	M₁₆	1	2	yes	—
61	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
62	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
63	C₄×Dic₆₇	—	2	1	no	—
64	C₄×Dic₆₇	—	2	4	no	—
65	C₄×Dic₆₇	—	2	1	no	—
66	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
67	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
68	C₆₇⋊C₁₆	—	2	4	no	—
69	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	—	2	1	no	—
74	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	M₁₆×C₆₇	—	2	4	no	—
79	M₁₆×C₆₇	—	2	4	no	—
80	C₂×C₈×C₆₇	—	2	1	no	—
81	C₂×C₈×C₆₇	—	2	1	no	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₆₇	#41	1	2	yes	—
85	C₂×C₈×C₆₇	#41	1	2	yes	—
86	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
87	C₁₆×C₆₇	—	2	4	no	—
88	C₂³×C₆₇	—	2	1	no	—
89	C₂³×C₆₇	—	2	1	no	—
90	C₂³×C₆₇	—	4	2	no	—
91	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂³×C₆₇	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₆₇	—	2	1	no	—
94	C₂×C₄×C₆₇	—	2	1	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
100	C₂×C₄×C₆₇	#117	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	C₂×M₁₆	1	2	yes	—
104	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₆₇	—	2	4	no	—
107	C₂×Dic₆₇	—	2	2	no	—
108	C₂×Dic₆₇	—	2	2	no	—
109	C₂×Dic₆₇	—	4	8	no	—
110	C₂×Dic₆₇	—	4	2	no	—
111	C₈×C₆₇	—	2	2	no	—
112	C₈×C₆₇	#117	1	2	yes	—
113	C₈×C₆₇	C₂×#41	1	2	yes	—
114	C₆₇⋊C₈	—	2	2	no	—
115	C₆₇⋊C₈	C₂×M₁₆	1	4	yes	—
116	C₂²×C₆₇	—	2	2	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	4	2	no	—
119	C₂²×C₆₇	—	4	2	no	—
120	C₂²×C₆₇	—	4	2	no	—
121	C₂²×C₆₇	#432	1	1	yes	characteristic
122	C₂²×C₆₇	#432	1	1	yes	characteristic
123	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
124	C₄×C₆₇	—	2	2	no	—
125	C₄×C₆₇	#432	1	1	yes	characteristic
126	C₄×C₆₇	#432	1	1	yes	characteristic
127	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
128	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
129	Dic₆₇	—	2	4	no	—
130	Dic₆₇	—	4	2	no	—
131	C₂×#2165	—	67	1	no	maximal, Sylow
132	C₂×C₆₇	—	2	1	no	—
133	C₂×C₆₇	—	2	1	no	—
134	C₂×C₆₇	—	4	2	no	—
135	C₂×C₆₇	#2165	1	2	yes	complemented
136	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
137	C₂×C₄×M₁₆	—	67	1	no	—
138	#2165	—	67	8	no	—
139	C₂×#364	—	67	2	no	—
140	C₂×#349	—	67	1	no	—
141	C₂×#349	D₆₇	1	1	yes	characteristic
142	C₂×#347	—	67	2	no	—
143	C₆₇	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
144	C₂²×M₁₆	—	67	1	no	—
145	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
146	C₂²×C₄²	—	67	1	no	—
147	C₂×M₃₂	—	67	2	no	—
148	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
149	C₄×M₁₆	—	67	4	no	—
150	C₄×M₁₆	—	67	4	no	—
151	C₂×#97	—	67	1	no	—
152	C₂×#97	—	67	1	no	—
153	C₂×C₄×C₈	—	67	1	no	—
154	C₂×C₄×C₈	—	67	1	no	—
155	#364	—	67	8	no	—
156	#349	—	67	4	no	—
157	#349	C₂×D₆₇	1	4	yes	—
158	#347	—	67	8	no	—
159	C₂³×C₄	—	134	1	no	—
160	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	67	4	no	—
164	C₂×M₁₆	—	134	2	no	—
165	C₂×M₁₆	—	134	2	no	—
166	C₂×M₁₆	C₄×D₆₇	1	2	yes	—
167	C₂×M₁₆	C₂²×D₆₇	1	2	yes	—
168	C₂²×C₈	—	67	1	no	—
169	C₂²×C₈	—	67	1	no	—
170	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
172	C₂×C₄²	—	67	1	no	—
173	C₂×C₄²	—	67	1	no	—
174	C₂×C₄²	—	67	4	no	—
175	C₂×C₄²	—	134	2	no	—
176	C₂×C₄²	—	134	1	no	—
177	M₃₂	—	2	2	no	—
178	M₃₂	—	67	4	no	—
179	M₃₂	—	134	2	no	—
180	M₃₂	C₂²×D₆₇	1	4	yes	—
181	C₂×C₁₆	—	2	2	no	—
182	C₂×C₁₆	—	134	2	no	—
183	#97	—	67	4	no	—
184	#97	—	67	2	no	—
185	#97	—	67	2	no	—
186	C₄×C₈	—	67	2	no	—
187	C₄×C₈	—	67	4	no	—
188	C₄×C₈	—	67	2	no	—
189	C₂⁴	#11761611	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	268	4	no	—
195	C₂²×C₄	—	134	1	no	—
196	C₂²×C₄	—	134	4	no	—
197	C₂²×C₄	—	134	1	no	—
198	C₂²×C₄	#11761611	1	1	yes	characteristic
199	C₂²×C₄	#11761611	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
201	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	134	8	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	67	2	no	—
208	C₂×C₈	—	67	2	no	—
209	C₂×C₈	—	67	2	no	—
210	C₂×C₈	—	67	2	no	—
211	C₂×C₈	—	134	1	no	—
212	C₂×C₈	—	134	1	no	—
213	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
215	C₂×C₈	#11761627	1	2	yes	—
216	C₂×C₈	#11761627	1	2	yes	—
217	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
218	C₄²	—	67	2	no	—
219	C₄²	—	67	2	no	—
220	C₄²	—	134	1	no	—
221	C₄²	—	134	4	no	—
222	C₄²	—	134	1	no	—
223	C₁₆	—	2	4	no	—
224	C₁₆	—	134	4	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×#11761611	1	1	yes	characteristic
229	C₂³	C₂×#11761611	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	268	2	no	—
237	C₂×C₄	—	268	8	no	—
238	C₂×C₄	—	134	2	no	—
239	C₂×C₄	—	134	4	no	—
240	C₂×C₄	—	134	2	no	—
241	C₂×C₄	#14523999	1	1	yes	characteristic
242	C₂×C₄	#14523999	1	1	yes	characteristic
243	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
244	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
245	C₂×C₄	C₂×#11761611	1	2	yes	—
246	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	67	4	no	—
250	C₈	—	134	2	no	—
251	C₈	#14523999	1	2	yes	—
252	C₈	C₂×#11761627	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#16863450	1	1	yes	characteristic
259	C₂²	#16863450	1	1	yes	characteristic
260	C₂²	C₂²×#11761611	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	268	2	no	—
263	C₄	—	134	4	no	—
264	C₄	#16863450	1	1	yes	characteristic
265	C₄	#16863450	1	1	yes	characteristic
266	C₄	C₂×#14523999	1	1	yes	characteristic
267	C₄	C₂×M₁₆×D₆₇	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	4	2	no	—
271	C₂	#22562018	1	2	yes	direct product factor
272	C₂	C₂×#16863450	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14523999
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#349×C₆₇	C₂
socle	C₂²×C₆₇	#432
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#349	D₆₇
hypocenter	C₆₇	C₂×#2165
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. #22562018	C₂	direct (N × H)
3. C₂×#16863216	C₂	semidirect
5. C₂×#16863173	C₂	semidirect
16. #16863216	C₂²	semidirect
18. #16863173	C₂²	semidirect
135. C₂×C₆₇	#2165	semidirect
143. C₆₇	C₂×#2165	semidirect
271. C₂	#22562018	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×M₁₆×Dic₆₇	2
#22562018	2
C₂×#16863216	2
C₂×#16863178	2
C₂×#16863173	2
C₂×#349×C₆₇	2
C₂×#2165	67

Subgroups — C2×#22562018 #31008950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562018 #31008950