Subgroups — C₂×#22562020 #31008952

Order 17152.

570 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11761611	C₂	1	1	yes	maximal, characteristic
2	#22562020	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863217	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863177	C₂	1	2	yes	maximal
5	C₂×#16863173	C₂	1	2	yes	maximal, complemented
6	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11761611	C₄	1	1	yes	characteristic
8	C₂²×#11761611	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₆₇	C₄	1	1	yes	characteristic
10	C₂×#14523983	C₂²	1	2	yes	—
11	C₄×#11761611	C₄	1	4	yes	—
12	C₄×#11761611	C₂²	1	4	yes	—
13	C₂×#14523953	C₄	1	1	yes	characteristic
14	C₂×#14523953	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₆₇⋊C₈	C₄	1	1	yes	characteristic
16	#16863217	C₂²	1	4	yes	complemented
17	#16863177	C₂²	1	8	yes	—
18	#16863173	C₂²	1	8	yes	complemented
19	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#364×C₆₇	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₆₇	—	2	1	no	—
23	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×#11761611	—	2	2	no	—
25	C₂×#11761611	—	2	2	no	—
26	C₂×#11761611	—	2	1	no	—
27	C₂×#11761611	—	2	1	no	—
28	C₂×#11761611	—	2	1	no	—
29	C₂×#11761611	—	2	1	no	—
30	C₂×#11761611	C₂×C₄	1	2	yes	—
31	C₂×#11761611	C₂×C₄	1	4	yes	—
32	C₂×#11761611	C₂³	1	2	yes	—
33	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₆₇	—	2	1	no	—
36	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×D₆₇	—	2	2	no	—
38	C₄²×D₆₇	C₂×C₄	1	4	yes	—
39	#14523983	—	2	2	no	—
40	#14523983	C₂³	1	4	yes	—
41	C₂×C₆₇⋊C₁₆	—	2	2	no	—
42	#14523953	C₂×C₄	1	4	yes	—
43	#14523953	C₂×C₄	1	2	yes	—
44	#14523953	C₂³	1	2	yes	—
45	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
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	C₂³	1	4	yes	—
51	C₂×C₁₆×C₆₇	—	2	2	no	—
52	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₆₇	C₂³	1	2	yes	—
54	C₂³×D₆₇	M₁₆	1	1	yes	characteristic
55	C₂²×Dic₆₇	—	2	1	no	—
56	C₂²×Dic₆₇	M₁₆	1	1	yes	characteristic
57	C₂×C₄×D₆₇	—	2	1	no	—
58	C₂×C₄×D₆₇	—	2	1	no	—
59	C₂×C₄×D₆₇	—	2	1	no	—
60	C₂×C₄×D₆₇	—	2	1	no	—
61	C₂×C₄×D₆₇	—	2	4	no	—
62	C₂×C₄×D₆₇	—	4	4	no	—
63	C₂×C₄×D₆₇	M₁₆	1	2	yes	—
64	C₂×C₄×D₆₇	C₂²×C₄	1	2	yes	—
65	#11761611	—	2	2	no	—
66	#11761611	—	2	2	no	—
67	#11761611	—	2	8	no	—
68	#11761611	—	2	2	no	—
69	#11761611	—	2	2	no	—
70	C₂×C₆₇⋊C₈	—	2	1	no	—
71	C₂×C₆₇⋊C₈	—	2	1	no	—
72	C₂×C₆₇⋊C₈	—	2	1	no	—
73	C₂×C₆₇⋊C₈	—	2	1	no	—
74	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
76	C₄×Dic₆₇	—	2	1	no	—
77	C₄×Dic₆₇	—	2	4	no	—
78	C₄×Dic₆₇	C₂²×C₄	1	2	yes	—
79	C₆₇⋊C₁₆	—	2	4	no	—
80	C₂²×C₄×C₆₇	—	2	1	no	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
83	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
84	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
87	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
89	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
90	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₄²×C₆₇	—	2	1	no	—
92	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
93	C₁₆×C₆₇	—	2	4	no	—
94	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂²×D₆₇	—	2	1	no	—
96	C₂²×D₆₇	—	2	1	no	—
97	C₂²×D₆₇	—	4	2	no	—
98	C₂²×D₆₇	C₂×M₁₆	1	2	yes	—
99	C₂×C₄×C₆₇	—	2	4	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
102	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₆₇	—	2	1	no	—
108	C₂×Dic₆₇	—	2	1	no	—
109	C₂×Dic₆₇	—	2	2	no	—
110	C₂×Dic₆₇	—	4	2	no	—
111	C₂×Dic₆₇	C₂×M₁₆	1	2	yes	—
112	C₄×D₆₇	—	2	2	no	—
113	C₄×D₆₇	—	2	2	no	—
114	C₄×D₆₇	—	2	2	no	—
115	C₄×D₆₇	—	2	2	no	—
116	C₄×D₆₇	—	4	8	no	—
117	C₈×C₆₇	#117	1	1	yes	characteristic
118	C₈×C₆₇	#117	1	1	yes	characteristic
119	C₈×C₆₇	C₂×M₁₆	1	4	yes	—
120	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
121	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
122	C₆₇⋊C₈	—	2	2	no	—
123	C₆₇⋊C₈	—	2	2	no	—
124	C₂²×C₆₇	—	2	2	no	—
125	C₂²×C₆₇	#432	1	1	yes	characteristic
126	C₂²×C₆₇	#432	1	1	yes	characteristic
127	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
128	C₂×D₆₇	—	2	2	no	—
129	C₂×D₆₇	—	4	2	no	—
130	C₂×D₆₇	—	4	2	no	—
131	C₂×D₆₇	—	4	2	no	—
132	C₄×C₆₇	—	2	4	no	—
133	C₄×C₆₇	#432	1	1	yes	characteristic
134	C₄×C₆₇	#432	1	1	yes	characteristic
135	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
136	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
137	Dic₆₇	—	2	2	no	—
138	Dic₆₇	—	4	2	no	—
139	C₂×#2165	—	67	1	no	maximal, Sylow
140	C₂×C₆₇	—	2	1	no	—
141	C₂×C₆₇	—	2	1	no	—
142	C₂×C₆₇	#2165	1	2	yes	complemented
143	C₂×C₆₇	C₂×#432	1	1	yes	characteristic
144	D₆₇	—	4	2	no	—
145	C₂×C₄×M₁₆	—	67	1	no	—
146	#2165	—	67	8	no	—
147	C₂×#364	—	67	1	no	—
148	C₂×#364	D₆₇	1	1	yes	characteristic, complemented
149	C₂×#349	—	67	2	no	—
150	C₂×#347	—	67	2	no	—
151	C₆₇	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
152	C₂²×M₁₆	—	67	1	no	—
153	C₂²×M₁₆	—	67	1	no	—
154	C₂²×C₄²	—	67	1	no	—
155	C₂×M₃₂	—	67	2	no	—
156	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
157	C₄×M₁₆	—	67	4	no	—
158	C₄×M₁₆	—	67	4	no	—
159	C₂×#97	—	67	1	no	—
160	C₂×#97	—	67	1	no	—
161	C₂×C₄×C₈	—	67	1	no	—
162	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
163	#364	—	67	4	no	—
164	#364	C₂×D₆₇	1	4	yes	complemented
165	#349	—	67	8	no	—
166	#347	—	67	8	no	—
167	C₂³×C₄	—	67	1	no	—
168	C₂³×C₄	—	134	1	no	—
169	C₂×M₁₆	—	67	2	no	—
170	C₂×M₁₆	—	67	4	no	—
171	C₂×M₁₆	—	67	2	no	—
172	C₂×M₁₆	—	134	2	no	—
173	C₂×M₁₆	—	134	2	no	—
174	C₂×M₁₆	—	134	1	no	—
175	C₂×M₁₆	—	134	1	no	—
176	C₂×M₁₆	—	134	1	no	—
177	C₂×M₁₆	—	134	1	no	—
178	C₂²×C₈	—	67	1	no	—
179	C₂²×C₈	—	67	1	no	—
180	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
182	C₂×C₄²	—	67	1	no	—
183	C₂×C₄²	—	67	4	no	—
184	C₂×C₄²	—	134	2	no	—
185	C₂×C₄²	—	134	1	no	—
186	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
187	M₃₂	—	2	2	no	—
188	M₃₂	—	67	4	no	—
189	M₃₂	—	134	2	no	—
190	M₃₂	C₂²×D₆₇	1	4	yes	—
191	C₂×C₁₆	—	2	2	no	—
192	C₂×C₁₆	—	134	2	no	—
193	#97	—	67	2	no	—
194	#97	—	67	4	no	—
195	#97	—	67	2	no	—
196	C₄×C₈	—	67	4	no	—
197	C₄×C₈	C₄×D₆₇	1	2	yes	—
198	C₄×C₈	C₂²×D₆₇	1	2	yes	—
199	C₂⁴	—	67	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	268	4	no	—
202	C₂²×C₄	—	67	1	no	—
203	C₂²×C₄	—	67	2	no	—
204	C₂²×C₄	—	67	2	no	—
205	C₂²×C₄	—	134	4	no	—
206	C₂²×C₄	—	134	1	no	—
207	C₂²×C₄	—	134	1	no	—
208	C₂²×C₄	—	134	1	no	—
209	C₂²×C₄	—	134	1	no	—
210	C₂²×C₄	—	134	1	no	—
211	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
212	M₁₆	—	134	2	no	—
213	M₁₆	—	134	2	no	—
214	M₁₆	—	134	8	no	—
215	M₁₆	—	134	2	no	—
216	M₁₆	—	134	2	no	—
217	C₂×C₈	—	67	2	no	—
218	C₂×C₈	—	67	2	no	—
219	C₂×C₈	—	134	1	no	—
220	C₂×C₈	—	134	1	no	—
221	C₂×C₈	—	134	1	no	—
222	C₂×C₈	—	134	1	no	—
223	C₂×C₈	#11761611	1	2	yes	—
224	C₂×C₈	#11761611	1	2	yes	—
225	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
227	C₂×C₈	#11761627	1	1	yes	characteristic
228	C₂×C₈	#11761627	1	1	yes	characteristic
229	C₂×C₈	#11761629	1	1	yes	characteristic
230	C₂×C₈	#11761629	1	1	yes	characteristic
231	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
232	C₄²	—	2	1	no	—
233	C₄²	—	67	2	no	—
234	C₄²	—	134	4	no	—
235	C₄²	—	134	1	no	—
236	C₄²	C₂×C₄×D₆₇	1	2	yes	—
237	C₁₆	—	2	4	no	—
238	C₁₆	—	134	4	no	—
239	C₂³	—	268	2	no	—
240	C₂³	—	67	2	no	—
241	C₂³	—	134	1	no	—
242	C₂³	—	134	1	no	—
243	C₂³	M₁₆×D₆₇	1	1	yes	characteristic
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	268	8	no	—
247	C₂×C₄	—	268	2	no	—
248	C₂×C₄	—	67	2	no	—
249	C₂×C₄	—	134	2	no	—
250	C₂×C₄	—	134	1	no	—
251	C₂×C₄	—	134	1	no	—
252	C₂×C₄	—	134	2	no	—
253	C₂×C₄	—	134	2	no	—
254	C₂×C₄	—	134	2	no	—
255	C₂×C₄	—	134	2	no	—
256	C₂×C₄	#14524001	1	1	yes	characteristic
257	C₂×C₄	#14524001	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
260	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
262	C₈	—	134	2	no	—
263	C₈	—	134	2	no	—
264	C₈	#14524001	1	1	yes	characteristic
265	C₈	#14524001	1	1	yes	characteristic
266	C₈	C₂×#11761611	1	4	yes	—
267	C₈	C₂×#11761627	1	1	yes	characteristic
268	C₈	C₂×#11761629	1	1	yes	characteristic
269	C₂²	—	2	2	no	—
270	C₂²	—	268	2	no	—
271	C₂²	—	268	2	no	—
272	C₂²	—	268	2	no	—
273	C₂²	—	134	2	no	—
274	C₂²	#16863452	1	1	yes	characteristic
275	C₂²	#16863452	1	1	yes	characteristic
276	C₂²	C₂×M₁₆×D₆₇	1	1	yes	characteristic
277	C₄	—	2	4	no	—
278	C₄	—	268	2	no	—
279	C₄	—	134	2	no	—
280	C₄	#16863452	1	1	yes	characteristic
281	C₄	#16863452	1	1	yes	characteristic
282	C₄	C₂×#14524001	1	1	yes	characteristic
283	C₄	C₂²×#11761611	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	268	2	no	—
287	C₂	#22562020	1	2	yes	direct product factor
288	C₂	C₂×#16863452	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524001
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#432
prime radical	C₂³×D₆₇	M₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	C₂×#2165
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #22562020	C₂	direct (N × H)
3. C₂×#16863217	C₂	semidirect
5. C₂×#16863173	C₂	semidirect
6. C₂×#364×C₆₇	C₂	semidirect
16. #16863217	C₂²	semidirect
18. #16863173	C₂²	semidirect
21. #364×C₆₇	C₂²	semidirect
142. C₂×C₆₇	#2165	semidirect
148. C₂×#364	D₆₇	semidirect
151. C₆₇	C₂×#2165	semidirect
164. #364	C₂×D₆₇	semidirect
287. C₂	#22562020	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×#11761611	2
#22562020	2
C₂×#16863217	2
C₂×#16863177	2
C₂×#16863173	2
C₂×#364×C₆₇	2
C₂×#2165	67

Subgroups — C2×#22562020 #31008952

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562020 #31008952