Subgroups — C₂×#22561929 #31008723

Order 17152.

1150 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
3	#22561929	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863200	C₂	1	4	yes	maximal, complemented
5	C₂³×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524002	—	2	1	no	—
7	C₂×#14523983	—	2	4	no	—
8	C₂×#14523983	C₂²	1	4	yes	—
9	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	#16863434	—	2	2	no	—
11	#16863434	C₂²	1	4	yes	—
12	C₂×#14523977	C₂²	1	2	yes	—
13	C₂×#14523975	—	2	1	no	—
14	#16863200	C₂²	1	16	yes	complemented
15	C₂³×Q₈×C₆₇	—	2	1	no	—
16	C₂²×Q₁₆×C₆₇	—	2	1	no	—
17	C₂²×Q₁₆×C₆₇	—	2	2	no	—
18	C₂²×Q₁₆×C₆₇	—	2	4	no	—
19	C₂²×Q₁₆×C₆₇	C₄	1	8	yes	complemented
20	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	—
21	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#14524002	—	2	4	no	—
24	#14524002	—	4	2	no	—
25	C₂×C₄⋊Dic₆₇	—	2	2	no	—
26	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
27	C₂×C₄×Dic₆₇	—	4	1	no	—
28	#14523983	—	2	8	no	—
29	C₂×C₆₇⋊C₁₆	—	2	4	no	—
30	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
31	#14523977	—	2	2	no	—
32	#14523977	C₂³	1	4	yes	—
33	#14523975	—	2	4	no	—
34	C₂²×Q₈×C₆₇	—	2	8	no	—
35	C₂²×Q₈×C₆₇	—	2	2	no	—
36	C₂²×Q₈×C₆₇	—	2	4	no	—
37	C₂²×Q₈×C₆₇	—	4	1	no	—
38	C₂²×Q₈×C₆₇	—	4	2	no	—
39	C₂²×Q₈×C₆₇	—	4	4	no	—
40	C₂³×C₄×C₆₇	—	4	1	no	—
41	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
42	C₂×Q₁₆×C₆₇	—	2	16	no	—
43	C₂×Q₁₆×C₆₇	—	2	8	no	—
44	C₂×Q₁₆×C₆₇	—	2	4	no	—
45	C₂×Q₁₆×C₆₇	—	2	2	no	—
46	C₂×Q₁₆×C₆₇	—	2	4	no	—
47	C₂×Q₁₆×C₆₇	—	2	8	no	—
48	C₂×Q₁₆×C₆₇	C₂×C₄	1	8	yes	—
49	C₂×Q₁₆×C₆₇	C₂×C₄	1	16	yes	complemented
50	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
51	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
52	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₆₇	D₄	1	2	yes	—
54	C₂²×C₈×C₆₇	D₄	1	4	yes	—
55	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	2	4	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	C₂²⋊Dic₆₇	—	4	4	no	—
61	C₄×Dic₆₇	—	4	4	no	—
62	C₆₇⋊C₁₆	—	2	4	no	—
63	C₂⁴×C₆₇	D₈	1	1	yes	characteristic
64	C₂×Q₈×C₆₇	—	2	8	no	—
65	C₂×Q₈×C₆₇	—	2	16	no	—
66	C₂×Q₈×C₆₇	—	2	4	no	—
67	C₂×Q₈×C₆₇	—	4	16	no	—
68	C₂×Q₈×C₆₇	—	4	8	no	—
69	C₂×Q₈×C₆₇	—	4	4	no	—
70	C₂×Q₈×C₆₇	—	4	2	no	—
71	C₂×Q₈×C₆₇	—	4	8	no	—
72	C₂×Q₈×C₆₇	—	4	4	no	—
73	C₂²×C₄×C₆₇	—	4	8	no	—
74	C₂²×C₄×C₆₇	—	4	4	no	—
75	C₂²×C₄×C₆₇	—	4	2	no	—
76	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	D₈	1	2	yes	—
79	C₂²×C₄×C₆₇	SD₁₆	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
82	Q₁₆×C₆₇	—	2	8	no	—
83	Q₁₆×C₆₇	—	2	16	no	—
84	Q₁₆×C₆₇	—	2	4	no	—
85	Q₁₆×C₆₇	C₂²×C₄	1	8	yes	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	8	yes	—
88	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
94	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
95	C₂³×C₆₇	—	2	4	no	—
96	C₂³×C₆₇	#102	1	4	yes	—
97	C₂³×C₆₇	C₂×D₈	1	2	yes	—
98	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₆₇	—	2	8	no	—
100	Q₈×C₆₇	—	4	4	no	—
101	Q₈×C₆₇	—	4	8	no	—
102	Q₈×C₆₇	—	4	16	no	—
103	C₂×C₄×C₆₇	—	4	4	no	—
104	C₂×C₄×C₆₇	—	4	16	no	—
105	C₂×C₄×C₆₇	—	4	8	no	—
106	C₂×C₄×C₆₇	#102	1	8	yes	—
107	C₂×C₄×C₆₇	#102	1	4	yes	—
108	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
109	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
113	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
114	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₆₇	—	4	4	no	—
116	C₂×Dic₆₇	—	8	4	no	—
117	C₈×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
118	C₈×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
119	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
121	C₂²×C₆₇	—	2	4	no	—
122	C₂²×C₆₇	—	2	8	no	—
123	C₂²×C₆₇	C₂×#102	1	4	yes	—
124	C₂²×C₆₇	C₂×#102	1	2	yes	—
125	C₂²×C₆₇	#510	1	4	yes	—
126	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
127	C₄×C₆₇	—	4	8	no	—
128	C₄×C₆₇	C₂×#102	1	4	yes	—
129	C₄×C₆₇	C₂×#102	1	2	yes	—
130	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
132	Dic₆₇	—	8	4	no	—
133	C₂×#2141	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	—	2	4	no	—
135	C₂×C₆₇	#2141	1	4	yes	complemented
136	C₂×C₆₇	C₂²×#102	1	1	yes	characteristic
137	C₂×C₆₇	C₂×#510	1	2	yes	—
138	C₂³×Q₁₆	D₆₇	1	1	yes	characteristic
139	C₂²×M₃₂	—	67	1	no	—
140	C₂×#427	—	67	1	no	—
141	#2141	—	67	8	no	—
142	C₂×#358	—	67	4	no	—
143	C₆₇	C₂×#2141	1	1	yes	characteristic, complemented, Sylow
144	C₂³×Q₈	—	2	1	no	—
145	C₂²×Q₁₆	—	2	1	no	—
146	C₂²×Q₁₆	—	2	2	no	—
147	C₂²×Q₁₆	—	2	4	no	—
148	C₂²×Q₁₆	Dic₆₇	1	8	yes	complemented
149	C₂²×Q₁₆	C₂×D₆₇	1	2	yes	—
150	C₂²×Q₁₆	C₂×D₆₇	1	4	yes	—
151	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
152	C₂×#117	—	134	1	no	—
153	C₂×M₃₂	—	67	4	no	—
154	C₂×M₃₂	—	134	4	no	—
155	C₂²×C₁₆	—	67	2	no	—
156	#427	—	67	4	no	—
157	#427	—	134	2	no	—
158	C₂×#107	—	67	2	no	—
159	C₂×#106	—	134	1	no	—
160	#358	—	67	16	no	—
161	C₂²×Q₈	—	2	8	no	—
162	C₂²×Q₈	—	2	4	no	—
163	C₂²×Q₈	—	2	2	no	—
164	C₂²×Q₈	—	4	2	no	—
165	C₂²×Q₈	—	4	1	no	—
166	C₂²×Q₈	—	4	4	no	—
167	C₂³×C₄	—	4	1	no	—
168	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
169	C₂×Q₁₆	—	2	16	no	—
170	C₂×Q₁₆	—	2	8	no	—
171	C₂×Q₁₆	—	2	4	no	—
172	C₂×Q₁₆	—	2	2	no	—
173	C₂×Q₁₆	—	2	4	no	—
174	C₂×Q₁₆	—	2	8	no	—
175	C₂×Q₁₆	C₂×Dic₆₇	1	8	yes	—
176	C₂×Q₁₆	C₂×Dic₆₇	1	16	yes	complemented
177	C₂×Q₁₆	C₂²×D₆₇	1	4	yes	—
178	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
179	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
181	C₂²×C₈	C₂²⋊D₆₇	1	4	yes	—
182	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
183	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
184	#117	—	268	2	no	—
185	#117	—	134	4	no	—
186	C₂×C₄⋊C₄	—	134	2	no	—
187	C₂×C₂²⋊C₄	—	268	1	no	—
188	C₂×C₄²	—	268	1	no	—
189	M₃₂	—	134	8	no	—
190	C₂×C₁₆	—	67	4	no	—
191	C₂×C₁₆	—	134	4	no	—
192	#107	—	67	4	no	—
193	#107	—	134	2	no	—
194	#106	—	134	4	no	—
195	C₂⁴	#11761613	1	1	yes	characteristic
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	16	no	—
200	C₂×Q₈	—	4	8	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	8	no	—
204	C₂×Q₈	—	4	4	no	—
205	C₂²×C₄	—	4	8	no	—
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	268	2	no	—
209	C₂²×C₄	C₂²⋊Dic₆₇	1	4	yes	—
210	C₂²×C₄	#11761613	1	2	yes	—
211	C₂²×C₄	#11761613	1	1	yes	characteristic
212	C₂²×C₄	#11761616	1	4	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
215	Q₁₆	—	2	8	no	—
216	Q₁₆	—	2	16	no	—
217	Q₁₆	—	2	4	no	—
218	Q₁₆	C₂²×Dic₆₇	1	8	yes	—
219	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
220	C₂×C₈	C₂²⋊Dic₆₇	1	8	yes	—
221	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
222	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
223	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
226	C₂×C₈	C₂×C₂²⋊D₆₇	1	4	yes	—
227	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
228	C₄⋊C₄	—	268	2	no	—
229	C₄⋊C₄	—	134	4	no	—
230	C₂²⋊C₄	—	268	4	no	—
231	C₄²	—	268	4	no	—
232	C₁₆	—	134	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	#14523964	1	4	yes	—
235	C₂³	C₂×#11761613	1	2	yes	—
236	C₂³	C₂×#11761613	1	1	yes	characteristic
237	Q₈	—	2	8	no	—
238	Q₈	—	4	8	no	—
239	Q₈	—	4	16	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	268	4	no	—
245	C₂×C₄	—	536	4	no	—
246	C₂×C₄	#14523964	1	8	yes	—
247	C₂×C₄	#14523964	1	4	yes	—
248	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	4	yes	—
249	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	2	yes	—
250	C₂×C₄	C₂×#11761613	1	2	yes	—
251	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
252	C₂×C₄	C₂×#11761616	1	4	yes	—
253	C₂×C₄	C₂×#11761616	1	2	yes	—
254	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
255	C₈	C₂×C₂²⋊Dic₆₇	1	2	yes	—
256	C₈	C₂×C₂²⋊Dic₆₇	1	4	yes	—
257	C₈	C₂³×Dic₆₇	1	1	yes	characteristic
258	C₈	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	C₂×#14523964	1	4	yes	—
262	C₂²	C₂×#14523964	1	2	yes	—
263	C₂²	#16863837	1	4	yes	—
264	C₂²	C₂²×#11761613	1	1	yes	characteristic
265	C₄	—	4	8	no	—
266	C₄	—	536	4	no	—
267	C₄	C₂×#14523964	1	4	yes	—
268	C₄	C₂×#14523964	1	2	yes	—
269	C₄	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
270	C₄	C₂²×#11761616	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#22561929	1	4	yes	direct product factor
273	C₂	C₂²×#14523964	1	1	yes	characteristic
274	C₂	C₂×#16863837	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂³×Q₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₈
prime radical	C₂⁴×C₆₇	D₈
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂³×Q₁₆	D₆₇
hypocenter	C₆₇	C₂×#2141
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #22561929	C₂	direct (N × H)
4. C₂×#16863200	C₂	semidirect
14. #16863200	C₂²	semidirect
19. C₂²×Q₁₆×C₆₇	C₄	semidirect
49. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
135. C₂×C₆₇	#2141	semidirect
143. C₆₇	C₂×#2141	semidirect
148. C₂²×Q₁₆	Dic₆₇	semidirect
176. C₂×Q₁₆	C₂×Dic₆₇	semidirect
272. C₂	#22561929	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂×#16863434	2
#22561929	2
C₂×#16863200	2
C₂³×Q₁₆×C₆₇	2
C₂×#2141	67

Subgroups — C2×#22561929 #31008723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22561929 #31008723