Subgroups — C₂×#22562414 #31010606

Order 17152.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
2	#22562414	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16863223	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16863221	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16863200	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16863200	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
8	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#14524002	—	2	1	no	—
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16863434	—	2	2	no	—
13	#16863434	C₂²	1	4	yes	—
14	C₂×#14523977	C₂²	1	1	yes	characteristic
15	C₂×#14523977	C₂²	1	1	yes	characteristic
16	C₂×#14523975	—	2	1	no	—
17	#16863223	C₂²	1	4	yes	complemented
18	#16863221	C₂²	1	4	yes	complemented
19	#16863200	C₂²	1	4	yes	complemented
20	#16863200	C₂²	1	4	yes	complemented
21	C₂²⋊(C₆₇⋊C₁₆)	C₂²	1	4	yes	—
22	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
25	#467×C₆₇	—	2	2	no	—
26	#467×C₆₇	C₂²	1	4	yes	—
27	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×#103×C₆₇	—	2	1	no	—
29	#14524002	—	2	4	no	—
30	#14524002	—	4	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₄⋊Dic₆₇	—	2	1	no	—
33	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
34	C₂×C₄×Dic₆₇	—	4	1	no	—
35	C₂×C₆₇⋊C₁₆	—	2	2	no	—
36	C₂×C₆₇⋊C₁₆	—	2	2	no	—
37	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
38	C₂×C₆₇⋊C₁₆	C₂³	1	2	yes	—
39	#14523977	—	2	1	no	—
40	#14523977	—	2	1	no	—
41	#14523977	C₂³	1	2	yes	—
42	#14523977	C₂³	1	2	yes	—
43	#14523975	—	2	4	no	—
44	C₂²×Q₈×C₆₇	—	2	1	no	—
45	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
46	C₂×Q₁₆×C₆₇	—	2	1	no	—
47	C₂×Q₁₆×C₆₇	—	2	2	no	—
48	C₂×Q₁₆×C₆₇	—	2	2	no	—
49	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	—	2	2	no	—
52	C₂²×C₈×C₆₇	D₄	1	2	yes	—
53	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²⋊Q₈×C₆₇	—	2	4	no	—
57	C₂²⋊Q₈×C₆₇	—	4	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
60	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
61	#107×C₆₇	—	2	1	no	—
62	#107×C₆₇	C₂³	1	2	yes	—
63	#103×C₆₇	—	2	4	no	—
64	C₂²×Dic₆₇	—	4	1	no	—
65	C₂²×Dic₆₇	—	4	1	no	—
66	C₄⋊Dic₆₇	—	2	2	no	—
67	C₄⋊Dic₆₇	—	2	2	no	—
68	C₄⋊Dic₆₇	—	4	1	no	—
69	C₄⋊Dic₆₇	—	4	1	no	—
70	C₂²⋊Dic₆₇	—	4	4	no	—
71	C₄×Dic₆₇	—	4	4	no	—
72	C₆₇⋊C₁₆	—	2	2	no	—
73	C₆₇⋊C₁₆	—	2	2	no	—
74	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
75	C₂×Q₈×C₆₇	—	2	2	no	—
76	C₂×Q₈×C₆₇	—	4	1	no	—
77	C₂×Q₈×C₆₇	—	4	2	no	—
78	C₂×Q₈×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	—	2	2	no	—
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	4	1	no	—
82	C₂²×C₄×C₆₇	—	4	1	no	—
83	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	D₈	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
87	Q₁₆×C₆₇	—	2	2	no	—
88	Q₁₆×C₆₇	—	2	2	no	—
89	Q₁₆×C₆₇	—	4	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	—	2	4	no	—
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	2	no	—
94	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₆₇	#41	1	2	yes	—
97	C₂×C₈×C₆₇	#41	1	2	yes	—
98	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₆₇	—	2	2	no	—
100	C₄⋊C₄×C₆₇	—	4	1	no	—
101	C₄⋊C₄×C₆₇	—	4	4	no	—
102	C₂²⋊C₄×C₆₇	—	4	4	no	—
103	C₂³×C₆₇	—	2	2	no	—
104	C₂³×C₆₇	—	2	2	no	—
105	C₂³×C₆₇	—	2	2	no	—
106	C₂³×C₆₇	C₂×D₈	1	2	yes	—
107	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
108	Q₈×C₆₇	—	4	2	no	—
109	Q₈×C₆₇	—	4	2	no	—
110	Q₈×C₆₇	—	8	2	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	4	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	8	2	no	—
118	C₂×C₄×C₆₇	—	8	2	no	—
119	C₂×C₄×C₆₇	#123	1	2	yes	—
120	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	#137	1	2	yes	—
123	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₆₇	—	4	2	no	—
125	C₂×Dic₆₇	—	4	2	no	—
126	C₂×Dic₆₇	—	8	2	no	—
127	C₂×Dic₆₇	—	8	2	no	—
128	C₈×C₆₇	—	2	2	no	—
129	C₈×C₆₇	#123	1	2	yes	—
130	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
131	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
132	C₂²×C₆₇	—	2	2	no	—
133	C₂²×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	4	4	no	—
136	C₂²×C₆₇	#480	1	2	yes	—
137	C₂²×C₆₇	#508	1	2	yes	—
138	C₂²×C₆₇	#510	1	2	yes	—
139	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	8	2	no	—
142	C₄×C₆₇	—	8	2	no	—
143	C₄×C₆₇	#480	1	2	yes	—
144	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
145	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
146	Dic₆₇	—	8	2	no	—
147	Dic₆₇	—	8	2	no	—
148	C₂×#2237	—	67	1	no	maximal, Sylow
149	C₂×C₆₇	—	4	2	no	—
150	C₂×C₆₇	#2237	1	4	yes	complemented
151	C₂×C₆₇	C₂×#480	1	1	yes	characteristic
152	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
153	C₂×C₆₇	C₂×#510	1	1	yes	characteristic
154	C₂×#467	D₆₇	1	1	yes	characteristic
155	C₂×#427	—	67	1	no	—
156	#2237	—	67	8	no	—
157	C₂×#367	—	67	1	no	—
158	C₂×#366	—	67	1	no	—
159	C₂×#358	—	67	1	no	—
160	C₂×#358	—	67	1	no	—
161	C₂×C₂²⋊C₁₆	—	67	1	no	—
162	C₆₇	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
164	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
165	C₂×C₂²⋊Q₈	—	2	1	no	—
166	C₂×#117	—	134	1	no	—
167	C₂²×C₁₆	—	67	1	no	—
168	C₂²×C₁₆	—	67	1	no	—
169	#467	—	2	2	no	—
170	#467	C₂×D₆₇	1	4	yes	—
171	#427	—	67	4	no	—
172	#427	—	134	2	no	—
173	C₂×#107	—	67	1	no	—
174	C₂×#107	—	67	1	no	—
175	C₂×#107	C₂×D₆₇	1	1	yes	characteristic
176	C₂×#106	—	134	1	no	—
177	C₂×#103	—	2	1	no	—
178	#367	—	67	4	no	—
179	#366	—	67	4	no	—
180	#358	—	67	4	no	—
181	#358	—	67	4	no	—
182	C₂²⋊C₁₆	—	67	4	no	—
183	C₂²×Q₈	—	2	1	no	—
184	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
185	C₂×Q₁₆	—	2	1	no	—
186	C₂×Q₁₆	—	2	2	no	—
187	C₂×Q₁₆	—	2	2	no	—
188	C₂×Q₁₆	C₂²×D₆₇	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
193	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
194	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
195	C₂²⋊Q₈	—	2	4	no	—
196	C₂²⋊Q₈	—	4	2	no	—
197	#117	—	268	2	no	—
198	#117	—	134	4	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	4	1	no	—
201	C₂×C₄⋊C₄	—	134	1	no	—
202	C₂×C₄⋊C₄	—	134	1	no	—
203	C₂×C₂²⋊C₄	—	4	1	no	—
204	C₂×C₂²⋊C₄	—	268	1	no	—
205	C₂×C₄²	—	268	1	no	—
206	C₂×C₁₆	—	67	2	no	—
207	C₂×C₁₆	—	67	2	no	—
208	C₂×C₁₆	—	134	2	no	—
209	C₂×C₁₆	—	134	2	no	—
210	#107	—	2	1	no	—
211	#107	—	67	2	no	—
212	#107	—	67	2	no	—
213	#107	—	134	1	no	—
214	#107	—	134	1	no	—
215	#107	C₂²×D₆₇	1	2	yes	—
216	#106	—	134	4	no	—
217	#103	—	2	4	no	—
218	C₂⁴	#11761613	1	1	yes	characteristic
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×Q₈	—	4	2	no	—
222	C₂×Q₈	—	4	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	4	1	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	268	1	no	—
228	C₂²×C₄	—	268	1	no	—
229	C₂²×C₄	#11761613	1	2	yes	—
230	C₂²×C₄	#11761613	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
232	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
233	Q₁₆	—	2	2	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	4	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
241	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂×C₈	#11761627	1	2	yes	—
243	C₂×C₈	#11761627	1	2	yes	—
244	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	268	1	no	—
249	C₄⋊C₄	—	268	1	no	—
250	C₄⋊C₄	—	134	2	no	—
251	C₄⋊C₄	—	134	2	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₂²⋊C₄	—	268	4	no	—
254	C₄²	—	268	4	no	—
255	C₁₆	—	134	2	no	—
256	C₁₆	—	134	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×#11761613	1	2	yes	—
261	C₂³	C₂×#11761613	1	1	yes	characteristic
262	Q₈	—	4	2	no	—
263	Q₈	—	4	2	no	—
264	Q₈	—	8	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	2	no	—
272	C₂×C₄	—	8	2	no	—
273	C₂×C₄	—	268	2	no	—
274	C₂×C₄	—	268	2	no	—
275	C₂×C₄	—	536	2	no	—
276	C₂×C₄	—	536	2	no	—
277	C₂×C₄	#14524024	1	2	yes	—
278	C₂×C₄	C₂×#11761613	1	2	yes	—
279	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
280	C₂×C₄	#14524073	1	2	yes	—
281	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	#14524024	1	2	yes	—
284	C₈	C₂×#11761627	1	1	yes	characteristic
285	C₈	C₂×#11761627	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	#16863707	1	2	yes	—
291	C₂²	#16863826	1	2	yes	—
292	C₂²	#16863837	1	2	yes	—
293	C₂²	C₂²×#11761613	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	2	no	—
297	C₄	—	536	2	no	—
298	C₄	—	536	2	no	—
299	C₄	#16863707	1	2	yes	—
300	C₄	C₂×#14524024	1	1	yes	characteristic
301	C₄	C₂×#14524073	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	#22562414	1	4	yes	direct product factor
304	C₂	C₂×#16863707	1	1	yes	characteristic
305	C₂	C₂×#16863826	1	1	yes	characteristic
306	C₂	C₂×#16863837	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂×#467×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
hypercenter	C₂×#467	D₆₇
hypocenter	C₆₇	C₂×#2237
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#467

Complements

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

N	H ≅ G/N	type
2. #22562414	C₂	direct (N × H)
3. C₂×#16863223	C₂	semidirect
4. C₂×#16863221	C₂	semidirect
5. C₂×#16863200	C₂	semidirect
6. C₂×#16863200	C₂	semidirect
17. #16863223	C₂²	semidirect
18. #16863221	C₂²	semidirect
19. #16863200	C₂²	semidirect
20. #16863200	C₂²	semidirect
150. C₂×C₆₇	#2237	semidirect
162. C₆₇	C₂×#2237	semidirect
303. C₂	#22562414	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×#16863434	2
#22562414	2
C₂×#16863223	2
C₂×#16863221	2
C₂×#16863200	2
C₂×C₂²⋊(C₆₇⋊C₁₆)	2
C₂×#467×C₆₇	2
C₂×#2237	67

Subgroups — C2×#22562414 #31010606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22562414 #31010606