Subgroups — C₂×C₄×#14523977 #30853875

Order 17152.

1358 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523977	C₂	1	2	yes	maximal
2	C₂×C₄×C₄⋊Dic₆₇	C₂	1	2	yes	maximal
3	C₄×#14523977	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16863159	C₂	1	2	yes	maximal
5	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₆₇	C₂²	1	4	yes	—
7	C₂²×C₄×Dic₆₇	—	2	2	no	—
8	C₂×#14523977	C₄	1	16	yes	direct product factor
9	C₂×#14523977	C₂²	1	8	yes	—
10	C₄×C₄⋊Dic₆₇	—	2	4	no	—
11	C₄×C₄⋊Dic₆₇	C₂²	1	8	yes	—
12	C₂×C₆₇⋊#95	—	2	2	no	—
13	#16863159	C₂²	1	8	yes	—
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₆₇	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₆₇	C₂²	1	4	yes	—
19	C₂³×Dic₆₇	—	2	4	no	—
20	C₂×C₄⋊Dic₆₇	—	2	4	no	—
21	C₂×C₄⋊Dic₆₇	—	2	8	no	—
22	C₂×C₄⋊Dic₆₇	C₂×C₄	1	16	yes	—
23	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
24	C₂×C₄×Dic₆₇	—	2	8	no	—
25	C₂×C₄×Dic₆₇	—	4	8	no	—
26	#14523977	C₂×C₄	1	32	yes	direct product factor
27	C₆₇⋊#95	—	2	8	no	—
28	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	—	2	4	no	—
32	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
41	C₄×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
42	C₂²×Dic₆₇	—	2	16	no	—
43	C₂²×Dic₆₇	—	2	8	no	—
44	C₂²×Dic₆₇	—	4	8	no	—
45	C₂²×Dic₆₇	—	4	8	no	—
46	C₄⋊Dic₆₇	—	2	16	no	—
47	C₄⋊Dic₆₇	—	2	8	no	—
48	C₄⋊Dic₆₇	C₂²×C₄	1	16	yes	—
49	C₄×Dic₆₇	—	4	16	no	—
50	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	—	2	4	no	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
54	C₂²×C₄×C₆₇	D₈	1	4	yes	—
55	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₆₇	#41	1	4	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₆₇	—	2	8	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	C₄²	1	16	yes	complemented
67	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₄²×C₆₇	C₄⋊C₄	1	8	yes	—
71	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
72	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
73	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₆₇	C₄×D₄	1	4	yes	—
75	C₂³×C₆₇	C₂×D₈	1	2	yes	—
76	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
77	C₂³×C₆₇	#135	1	4	yes	—
78	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	8	no	—
81	C₂×C₄×C₆₇	#107	1	16	yes	complemented
82	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
83	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₆₇	#117	1	4	yes	—
87	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
88	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
89	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
90	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
92	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
95	C₂×Dic₆₇	—	2	16	no	—
96	C₂×Dic₆₇	—	4	8	no	—
97	C₂×Dic₆₇	—	4	8	no	—
98	C₂×Dic₆₇	—	4	32	no	—
99	C₈×C₆₇	—	2	4	no	—
100	C₈×C₆₇	C₂×C₄²	1	8	yes	complemented
101	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
102	C₂²×C₆₇	C₂×#107	1	4	yes	—
103	C₂²×C₆₇	#427	1	4	yes	—
104	C₂²×C₆₇	C₄×D₈	1	8	yes	—
105	C₂²×C₆₇	C₄×Q₁₆	1	8	yes	—
106	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
108	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
109	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
110	C₂²×C₆₇	C₂×#135	1	2	yes	—
111	C₄×C₆₇	—	2	4	no	—
112	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
113	C₄×C₆₇	C₂×#107	1	8	yes	complemented
114	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
115	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
116	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
117	Dic₆₇	—	4	16	no	—
118	C₂×C₄×#107	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	C₄×#107	1	8	yes	complemented
120	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
123	C₂×C₆₇	C₂×C₄×D₈	1	2	yes	—
124	C₂×C₆₇	C₂×C₄×Q₁₆	1	2	yes	—
125	C₂²×#107	—	67	2	no	—
126	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
127	C₂×C₄×C₄⋊C₄	—	67	2	no	—
128	C₄×#107	—	67	8	no	—
129	C₂×#340	—	67	2	no	—
130	C₆₇	C₂×C₄×#107	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
133	C₂²×C₄⋊C₄	—	67	4	no	—
134	C₂²×C₄²	—	134	2	no	—
135	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
136	C₂×#107	—	67	16	no	—
137	C₂×#107	—	67	8	no	—
138	C₂×C₄×C₈	Dic₆₇	1	8	yes	complemented
139	C₂×C₄×C₈	C₂×D₆₇	1	4	yes	—
140	C₄×C₄⋊C₄	—	67	8	no	—
141	C₄×C₄⋊C₄	—	134	4	no	—
142	C₂×#95	—	134	2	no	—
143	#340	—	67	8	no	—
144	C₂³×C₄	—	134	4	no	—
145	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
146	C₂³×C₄	D₂₆₈	1	1	yes	characteristic
147	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
148	C₂²×C₈	—	2	4	no	—
149	C₂²×C₈	C₄×D₆₇	1	8	yes	—
150	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
151	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
152	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
153	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
154	C₂×C₄⋊C₄	—	67	16	no	—
155	C₂×C₄⋊C₄	—	67	8	no	—
156	C₂×C₄⋊C₄	—	134	4	no	—
157	C₂×C₄⋊C₄	—	134	8	no	—
158	C₂×C₄²	—	268	8	no	—
159	C₂×C₄²	—	134	8	no	—
160	C₂×C₄²	Dic₁₃₄	1	4	yes	—
161	C₂×C₄²	D₂₆₈	1	2	yes	—
162	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
163	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
164	#107	—	67	32	no	—
165	C₄×C₈	C₂×Dic₆₇	1	16	yes	complemented
166	#95	—	134	8	no	—
167	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	268	8	no	—
170	C₂²×C₄	—	268	8	no	—
171	C₂²×C₄	—	134	16	no	—
172	C₂²×C₄	—	134	8	no	—
173	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
174	C₂²×C₄	C₄⋊Dic₆₇	1	4	yes	—
175	C₂²×C₄	D₅₃₆	1	4	yes	—
176	C₂²×C₄	Dic₂₆₈	1	4	yes	—
177	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
178	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
179	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
180	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
182	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
183	C₂²×C₄	#11761628	1	4	yes	—
184	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	C₄×Dic₆₇	1	16	yes	complemented
188	C₂×C₈	C₂×C₄×D₆₇	1	8	yes	—
189	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
190	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
191	C₄⋊C₄	—	67	16	no	—
192	C₄⋊C₄	—	134	16	no	—
193	C₄⋊C₄	—	134	8	no	—
194	C₄²	—	268	16	no	—
195	C₄²	C₄⋊Dic₆₇	1	8	yes	—
196	C₄²	C₂²×Dic₆₇	1	4	yes	—
197	C₄²	C₂×Dic₁₃₄	1	4	yes	—
198	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
199	C₂³	C₄×D₂₆₈	1	4	yes	—
200	C₂³	C₂×D₅₃₆	1	2	yes	—
201	C₂³	C₂×Dic₂₆₈	1	2	yes	—
202	C₂³	#14524065	1	4	yes	—
203	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	268	8	no	—
207	C₂×C₄	—	268	8	no	—
208	C₂×C₄	—	268	32	no	—
209	C₂×C₄	—	134	16	no	—
210	C₂×C₄	#14523977	1	16	yes	direct product factor
211	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
212	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
213	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
214	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
215	C₂×C₄	#14524002	1	4	yes	—
216	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
217	C₂×C₄	C₄×Dic₁₃₄	1	8	yes	—
218	C₂×C₄	C₂×D₅₃₆	1	4	yes	—
219	C₂×C₄	C₂×Dic₂₆₈	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
221	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
222	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
223	C₂×C₄	C₂×#11761628	1	2	yes	—
224	C₈	—	2	4	no	—
225	C₈	C₂×C₄×Dic₆₇	1	8	yes	complemented
226	C₂²	C₄×C₄⋊Dic₆₇	1	4	yes	—
227	C₂²	C₂×#14523977	1	4	yes	—
228	C₂²	#16863434	1	4	yes	—
229	C₂²	C₄×D₅₃₆	1	8	yes	—
230	C₂²	C₄×Dic₂₆₈	1	8	yes	—
231	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
232	C₂²	C₂×C₄×D₂₆₈	1	2	yes	—
233	C₂²	C₂²×D₅₃₆	1	1	yes	characteristic
234	C₂²	C₂²×Dic₂₆₈	1	1	yes	characteristic
235	C₂²	C₂×#14524065	1	2	yes	—
236	C₄	—	2	4	no	—
237	C₄	—	268	16	no	—
238	C₄	C₄×C₄⋊Dic₆₇	1	4	yes	—
239	C₄	C₂×#14523977	1	8	yes	direct product factor
240	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
241	C₄	C₂×#14524002	1	1	yes	characteristic
242	C₄	C₂×C₄×Dic₁₃₄	1	2	yes	—
243	C₂	C₄×#14523977	1	8	yes	direct product factor
244	C₂	C₂×C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
245	C₂	C₂²×#14523977	1	1	yes	characteristic
246	C₂	C₂×#16863434	1	1	yes	characteristic
247	C₂	C₂×C₄×D₅₃₆	1	2	yes	—
248	C₂	C₂×C₄×Dic₂₆₈	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×C₄×#107
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#14523977	C₂	direct (N × H)
8. C₂×#14523977	C₄	direct (N × H)
17. C₂×C₄×C₈×C₆₇	C₄	semidirect
26. #14523977	C₂×C₄	direct (N × H)
41. C₄×C₈×C₆₇	C₂×C₄	semidirect
66. C₂×C₈×C₆₇	C₄²	semidirect
81. C₂×C₄×C₆₇	#107	semidirect
100. C₈×C₆₇	C₂×C₄²	semidirect
113. C₄×C₆₇	C₂×#107	semidirect
119. C₂×C₆₇	C₄×#107	semidirect
130. C₆₇	C₂×C₄×#107	semidirect
138. C₂×C₄×C₈	Dic₆₇	semidirect
165. C₄×C₈	C₂×Dic₆₇	semidirect
187. C₂×C₈	C₄×Dic₆₇	semidirect
210. C₂×C₄	#14523977	direct (N × H)
225. C₈	C₂×C₄×Dic₆₇	semidirect
239. C₄	C₂×#14523977	direct (N × H)
243. C₂	C₄×#14523977	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×#14523977	2
C₂×C₄×C₄⋊Dic₆₇	2
C₄×#14523977	2
C₂×#16863159	2
C₂²×C₄×C₈×C₆₇	2
C₂×C₄×#107	67

Subgroups — C2×C4×#14523977 #30853875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14523977 #30853875