Subgroups — C₄×#16863434 #30853881

Order 17152.

1182 conjugacy classes of subgroups.

The subgroup lattice has 234 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	2	yes	maximal
2	C₄×#14524002	C₂	1	2	yes	maximal
3	C₄×#14523977	C₂	1	4	yes	maximal, complemented
4	C₄×#14523975	C₂	1	4	yes	maximal, complemented
5	#22559926	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂×#14524002	C₂²	1	4	yes	—
8	#16863434	C₄	1	16	yes	direct product factor
9	C₂×#14523977	C₂²	1	4	yes	—
10	C₂×#14523975	C₂²	1	4	yes	—
11	C₄×C₄⋊Dic₆₇	C₂²	1	8	yes	—
12	C₄×C₂²⋊Dic₆₇	—	2	4	no	—
13	#16863242	—	2	2	no	—
14	C₄²×Dic₆₇	—	2	2	no	—
15	#16863159	C₂²	1	8	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₄	1	8	yes	complemented
20	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	#14524002	—	2	8	no	—
23	#14524002	C₂×C₄	1	16	yes	—
24	C₂×C₄⋊Dic₆₇	C₂³	1	8	yes	—
25	C₂×C₂²⋊Dic₆₇	—	2	4	no	—
26	C₂×C₄×Dic₆₇	—	2	4	no	—
27	C₂×C₄×Dic₆₇	—	2	8	no	—
28	#14523977	C₂×C₄	1	16	yes	complemented
29	#14523975	C₂×C₄	1	16	yes	complemented
30	C₆₇⋊#95	—	2	8	no	—
31	C₂³×C₄×C₆₇	D₄	1	2	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	—	2	4	no	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	Q₈	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
46	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
47	C₂²×Dic₆₇	—	2	8	no	—
48	C₄⋊Dic₆₇	—	2	8	no	—
49	C₄⋊Dic₆₇	C₂²×C₄	1	16	yes	—
50	C₂²⋊Dic₆₇	—	2	16	no	—
51	C₄×Dic₆₇	—	2	16	no	—
52	C₄×Dic₆₇	—	4	16	no	—
53	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	—	2	8	no	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	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	4	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
73	C₄²×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
75	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
76	C₂³×C₆₇	—	2	4	no	—
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
78	C₂³×C₆₇	C₄×D₄	1	4	yes	—
79	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	16	no	—
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	#135	1	8	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
94	C₂×Dic₆₇	—	2	16	no	—
95	C₂×Dic₆₇	—	4	16	no	—
96	C₈×C₆₇	—	2	4	no	—
97	C₈×C₆₇	C₂×C₄²	1	4	yes	—
98	C₈×C₆₇	C₂×C₄²	1	4	yes	—
99	C₂²×C₆₇	—	2	4	no	—
100	C₂²×C₆₇	—	2	8	no	—
101	C₂²×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
102	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×C₄×D₄	1	2	yes	—
104	C₂²×C₆₇	C₂×#135	1	4	yes	—
105	C₄×C₆₇	—	2	4	no	—
106	C₄×C₆₇	C₄×C₄⋊C₄	1	4	yes	—
107	C₄×C₆₇	#427	1	8	yes	complemented
108	C₄×C₆₇	C₂²×C₄²	1	1	yes	characteristic
109	C₄×C₆₇	C₂×#117	1	1	yes	characteristic
110	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
111	Dic₆₇	—	4	16	no	—
112	C₄×#427	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	2	4	no	—
114	C₂×C₆₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#427	1	2	yes	—
116	C₂×C₆₇	C₄×#135	1	4	yes	—
117	C₂×#427	—	67	2	no	—
118	C₂²×C₄×C₈	D₆₇	1	1	yes	characteristic
119	C₄×#117	—	67	2	no	—
120	C₄×#107	—	67	4	no	—
121	C₄×#106	—	67	4	no	—
122	#1735	—	67	2	no	—
123	C₆₇	C₄×#427	1	1	yes	characteristic, complemented, Sylow
124	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
125	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
126	C₂×#117	—	67	4	no	—
127	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
128	#427	—	67	16	no	—
129	C₂×#107	—	67	4	no	—
130	C₂×#106	—	67	4	no	—
131	C₂×C₄×C₈	Dic₆₇	1	8	yes	complemented
132	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
133	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
134	C₄×C₄⋊C₄	—	67	8	no	—
135	C₄×C₂²⋊C₄	—	134	4	no	—
136	#376	—	134	2	no	—
137	C₄³	—	134	2	no	—
138	#340	—	67	8	no	—
139	C₂³×C₄	D₂₆₈	1	2	yes	—
140	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
141	C₂²×C₈	—	2	4	no	—
142	C₂²×C₈	C₄×D₆₇	1	8	yes	—
143	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
144	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
145	C₂²×C₈	C₂²×D₆₇	1	2	yes	—
146	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
147	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
148	#117	—	67	16	no	—
149	#117	—	134	8	no	—
150	C₂×C₄⋊C₄	—	67	8	no	—
151	C₂×C₂²⋊C₄	—	134	4	no	—
152	C₂×C₄²	—	134	8	no	—
153	C₂×C₄²	—	134	4	no	—
154	C₂×C₄²	Dic₁₃₄	1	4	yes	—
155	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
156	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
157	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
158	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
159	#107	—	67	16	no	—
160	#106	—	67	16	no	—
161	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
162	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
163	#95	—	134	8	no	—
164	C₂⁴	C₂×D₂₆₈	1	1	yes	characteristic
165	C₂²×C₄	—	2	8	no	—
166	C₂²×C₄	—	134	8	no	—
167	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
168	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
169	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
170	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
171	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
173	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
174	C₂²×C₄	#11761628	1	4	yes	—
175	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
176	C₂×C₈	—	2	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₄×Dic₆₇	1	16	yes	complemented
179	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
180	C₂×C₈	C₂×C₄×D₆₇	1	4	yes	—
181	C₂×C₈	C₂²×Dic₆₇	1	4	yes	—
182	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
183	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
184	C₄⋊C₄	—	67	16	no	—
185	C₄⋊C₄	—	134	8	no	—
186	C₂²⋊C₄	—	134	16	no	—
187	C₄²	—	268	16	no	—
188	C₄²	—	134	16	no	—
189	C₄²	C₄⋊Dic₆₇	1	4	yes	—
190	C₄²	C₄⋊Dic₆₇	1	4	yes	—
191	C₄²	C₂²×Dic₆₇	1	4	yes	—
192	C₄²	C₂×Dic₁₃₄	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	C₂×C₄⋊Dic₆₇	1	2	yes	—
195	C₂³	C₄×D₂₆₈	1	4	yes	—
196	C₂³	C₂²×D₂₆₈	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	268	16	no	—
200	C₂×C₄	—	134	16	no	—
201	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
205	C₂×C₄	#14524002	1	4	yes	—
206	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
207	C₂×C₄	C₄×Dic₁₃₄	1	8	yes	—
208	C₂×C₄	#14524065	1	8	yes	—
209	C₂×C₄	C₂²×C₄×D₆₇	1	2	yes	—
210	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
212	C₂×C₄	C₂×#11761628	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
215	C₈	C₂×C₄×Dic₆₇	1	4	yes	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	4	no	—
218	C₂²	C₄×C₄⋊Dic₆₇	1	4	yes	—
219	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
220	C₂²	C₂×C₄×D₂₆₈	1	2	yes	—
221	C₂²	C₂×#14524065	1	4	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	268	16	no	—
224	C₄	C₄×C₄⋊Dic₆₇	1	4	yes	—
225	C₄	#16863434	1	8	yes	direct product factor
226	C₄	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
227	C₄	C₂×#14524002	1	1	yes	characteristic
228	C₄	C₂×C₄×Dic₁₃₄	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	C₂×C₄×C₄⋊Dic₆₇	1	1	yes	characteristic
231	C₂	C₂×#16863434	1	2	yes	—
232	C₂	C₄×#14524065	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	D₂₆₈
commutator	C₄×C₆₇	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂²×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×D₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂²×C₄×C₈	D₆₇
hypocenter	C₆₇	C₄×#427
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ 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	semidirect
4. C₄×#14523975	C₂	semidirect
8. #16863434	C₄	direct (N × H)
19. C₂×C₄×C₈×C₆₇	C₄	semidirect
28. #14523977	C₂×C₄	semidirect
29. #14523975	C₂×C₄	semidirect
66. C₂×C₈×C₆₇	C₄²	semidirect
107. C₄×C₆₇	#427	semidirect
123. C₆₇	C₄×#427	semidirect
131. C₂×C₄×C₈	Dic₆₇	semidirect
178. C₂×C₈	C₄×Dic₆₇	semidirect
225. C₄	#16863434	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
C₄×#14524002	2
C₄×#14523977	2
C₄×#14523975	2
#22559926	2
C₂²×C₄×C₈×C₆₇	2
C₄×#427	67

Subgroups — C4×#16863434 #30853881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863434 #30853881