Subgroups — C₄×#16863354 #30853173

Order 17152.

1526 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863354	C₂	1	4	yes	maximal
2	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×#14524002	C₂	1	1	yes	maximal, characteristic
4	#22559897	C₂	1	4	yes	maximal, complemented
5	C₄×#14523953	C₂	1	2	yes	maximal, complemented
6	C₄×C₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#14524002	C₂²	1	2	yes	—
10	#16863354	C₄	1	32	yes	direct product factor
11	C₂×#14523953	C₂²	1	8	yes	—
12	C₂×C₄×C₆₇⋊C₈	C₄	1	8	yes	complemented
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
16	C₂×C₈×Dic₆₇	C₂²	1	8	yes	—
17	C₄×C₄⋊Dic₆₇	C₄	1	4	yes	—
18	C₄×C₂²⋊Dic₆₇	C₄	1	4	yes	—
19	#16863242	C₂²	1	2	yes	—
20	C₄²×Dic₆₇	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×M₁₆×C₆₇	C₄	1	8	yes	complemented
24	C₂×#97×C₆₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
27	C₂²×C₆₇⋊C₈	C₂×C₄	1	16	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
29	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
30	#14524002	C₂×C₄	1	16	yes	—
31	C₂×C₄⋊Dic₆₇	C₂×C₄	1	4	yes	—
32	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₆₇	C₂×C₄	1	8	yes	—
34	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
35	#14523953	C₂×C₄	1	32	yes	complemented
36	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
37	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
38	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
39	C₈×Dic₆₇	C₂×C₄	1	32	yes	complemented
40	C₆₇⋊#95	C₂×C₄	1	8	yes	—
41	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂×M₁₆×C₆₇	C₂×C₄	1	16	yes	—
44	C₂×M₁₆×C₆₇	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
46	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
51	#97×C₆₇	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₆₇	C₂×C₄	1	8	yes	—
53	C₂²×Dic₆₇	C₂²×C₄	1	8	yes	—
54	C₂×C₆₇⋊C₈	C₄²	1	32	yes	complemented
55	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
56	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
57	C₂×C₆₇⋊C₈	C₂²×C₄	1	4	yes	—
58	C₂×C₆₇⋊C₈	C₂²×C₄	1	8	yes	—
59	C₄⋊Dic₆₇	C₄²	1	16	yes	—
60	C₂²⋊Dic₆₇	C₄²	1	16	yes	—
61	C₄×Dic₆₇	—	2	16	no	—
62	C₄×Dic₆₇	C₂²×C₄	1	16	yes	—
63	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	—	2	8	no	—
65	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
73	M₁₆×C₆₇	C₄²	1	32	yes	complemented
74	C₂×C₈×C₆₇	C₂²×C₄	1	16	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	8	yes	—
76	C₄²×C₆₇	C₄²	1	8	yes	—
77	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
79	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
80	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂³×C₆₇	—	2	4	no	—
82	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
83	C₂³×C₆₇	C₂×C₄²	1	2	yes	—
84	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	—	2	4	no	—
86	C₂×C₄×C₆₇	—	2	16	no	—
87	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
89	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
90	C₂×C₄×C₆₇	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×C₄²	1	8	yes	—
92	C₂×C₄×C₆₇	C₂×C₄²	1	4	yes	—
93	C₂×C₄×C₆₇	#131	1	8	yes	—
94	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
96	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₆₇	—	2	16	no	—
98	C₂×Dic₆₇	C₂×C₄²	1	16	yes	—
99	C₈×C₆₇	C₂×C₄²	1	16	yes	—
100	C₆₇⋊C₈	C₂×C₄²	1	8	yes	—
101	C₆₇⋊C₈	C₂×C₄²	1	8	yes	—
102	C₂²×C₆₇	—	2	4	no	—
103	C₂²×C₆₇	—	2	8	no	—
104	C₂²×C₆₇	C₄³	1	4	yes	—
105	C₂²×C₆₇	C₂²×C₄²	1	2	yes	—
106	C₂²×C₆₇	C₂²×C₄²	1	1	yes	characteristic
107	C₂²×C₆₇	C₂×#131	1	4	yes	—
108	C₄×C₆₇	—	2	4	no	—
109	C₄×C₆₇	C₄³	1	4	yes	—
110	C₄×C₆₇	#405	1	8	yes	complemented
111	C₄×C₆₇	C₂²×C₄²	1	2	yes	—
112	C₄×C₆₇	C₂²×C₄²	1	2	yes	—
113	Dic₆₇	—	2	16	no	—
114	C₄×#405	—	67	1	no	maximal, Sylow
115	C₂×C₆₇	—	2	4	no	—
116	C₂×C₆₇	C₂×C₄³	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#405	1	2	yes	—
118	C₂×C₆₇	C₄×#131	1	4	yes	—
119	C₂×#405	—	67	4	no	—
120	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
121	C₂²×C₄×C₈	—	67	1	no	—
122	C₄×#117	—	67	1	no	—
123	#1726	—	67	4	no	—
124	C₄×#97	—	67	2	no	—
125	C₄²×C₈	—	67	2	no	—
126	C₆₇	C₄×#405	1	1	yes	characteristic, complemented, Sylow
127	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
128	C₂³×C₈	—	67	2	no	—
129	C₂×#117	—	67	2	no	—
130	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
131	#405	—	67	32	no	—
132	C₄×M₁₆	Dic₆₇	1	8	yes	complemented
133	C₂×#97	—	67	8	no	—
134	C₂×#97	C₂×D₆₇	1	2	yes	—
135	C₂×C₄×C₈	—	67	8	no	—
136	C₂×C₄×C₈	—	67	8	no	—
137	C₂×C₄×C₈	—	67	2	no	—
138	C₂×C₄×C₈	—	67	1	no	—
139	C₂×C₄×C₈	—	67	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
141	C₄×C₄⋊C₄	—	67	4	no	—
142	C₄×C₂²⋊C₄	—	67	4	no	—
143	#376	—	67	2	no	—
144	C₄³	—	67	2	no	—
145	C₂³×C₄	C₄×D₆₇	1	2	yes	—
146	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
147	C₂×M₁₆	C₄×D₆₇	1	16	yes	—
148	C₂×M₁₆	C₂×Dic₆₇	1	8	yes	—
149	C₂²×C₈	—	67	8	no	—
150	C₂²×C₈	—	67	16	no	—
151	C₂²×C₈	—	67	2	no	—
152	C₂²×C₈	—	67	2	no	—
153	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
154	#117	—	67	16	no	—
155	C₂×C₄⋊C₄	—	67	4	no	—
156	C₂×C₂²⋊C₄	—	67	4	no	—
157	C₂×C₄²	—	67	4	no	—
158	C₂×C₄²	—	67	8	no	—
159	C₂×C₄²	C₄×D₆₇	1	2	yes	—
160	C₂×C₄²	C₄×D₆₇	1	4	yes	—
161	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
162	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
164	#97	—	67	32	no	—
165	#97	C₂×Dic₆₇	1	8	yes	—
166	C₄×C₈	—	67	32	no	—
167	C₄×C₈	—	67	8	no	—
168	C₄×C₈	—	67	4	no	—
169	C₄×C₈	—	67	4	no	—
170	C₄×C₈	C₂×Dic₆₇	1	8	yes	—
171	#95	—	67	8	no	—
172	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	—	67	8	no	—
175	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
176	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
179	C₂²×C₄	C₂×C₄×D₆₇	1	8	yes	—
180	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
181	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
182	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
183	M₁₆	C₄×Dic₆₇	1	32	yes	complemented
184	C₂×C₈	—	67	32	no	—
185	C₂×C₈	—	67	4	no	—
186	C₂×C₈	—	67	4	no	—
187	C₂×C₈	—	67	8	no	—
188	C₂×C₈	—	67	8	no	—
189	C₂×C₈	C₂×C₄×D₆₇	1	16	yes	—
190	C₂×C₈	C₂²×Dic₆₇	1	8	yes	—
191	C₄⋊C₄	—	67	16	no	—
192	C₂²⋊C₄	—	67	16	no	—
193	C₄²	—	67	16	no	—
194	C₄²	—	134	16	no	—
195	C₄²	C₄×Dic₆₇	1	8	yes	—
196	C₄²	C₂×C₄×D₆₇	1	2	yes	—
197	C₄²	C₂×C₄×D₆₇	1	2	yes	—
198	C₄²	C₂²×Dic₆₇	1	2	yes	—
199	C₄²	C₂²×Dic₆₇	1	2	yes	—
200	C₂³	—	2	4	no	—
201	C₂³	C₄²×D₆₇	1	4	yes	—
202	C₂³	C₂×C₄×Dic₆₇	1	2	yes	—
203	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	67	16	no	—
207	C₂×C₄	—	134	16	no	—
208	C₂×C₄	C₄²×D₆₇	1	4	yes	—
209	C₂×C₄	C₄²×D₆₇	1	8	yes	—
210	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
211	C₂×C₄	C₂×C₄×Dic₆₇	1	2	yes	—
212	C₂×C₄	C₂×C₄×Dic₆₇	1	8	yes	—
213	C₂×C₄	C₂×C₄×Dic₆₇	1	4	yes	—
214	C₂×C₄	#14524051	1	8	yes	—
215	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
216	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
217	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
218	C₈	—	67	8	no	—
219	C₈	—	67	8	no	—
220	C₈	C₂×C₄×Dic₆₇	1	16	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	8	no	—
223	C₂²	C₄²×Dic₆₇	1	4	yes	—
224	C₂²	C₂×C₄²×D₆₇	1	2	yes	—
225	C₂²	C₂²×C₄×Dic₆₇	1	1	yes	characteristic
226	C₂²	C₂×#14524051	1	4	yes	—
227	C₄	—	2	4	no	—
228	C₄	—	134	16	no	—
229	C₄	C₄²×Dic₆₇	1	4	yes	—
230	C₄	#16863354	1	8	yes	direct product factor
231	C₄	C₂×C₄²×D₆₇	1	2	yes	—
232	C₄	C₂²×C₄×Dic₆₇	1	2	yes	—
233	C₂	—	2	4	no	—
234	C₂	C₂×C₄²×Dic₆₇	1	1	yes	characteristic
235	C₂	C₂×#16863354	1	2	yes	—
236	C₂	C₄×#14524051	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₆₇
commutator	C₂×C₆₇	C₂×C₄³
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₄×#405
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. #22559897	C₂	semidirect
5. C₄×#14523953	C₂	semidirect
6. C₄×C₈×Dic₆₇	C₂	semidirect
10. #16863354	C₄	direct (N × H)
12. C₂×C₄×C₆₇⋊C₈	C₄	semidirect
23. C₄×M₁₆×C₆₇	C₄	semidirect
35. #14523953	C₂×C₄	semidirect
39. C₈×Dic₆₇	C₂×C₄	semidirect
54. C₂×C₆₇⋊C₈	C₄²	semidirect
73. M₁₆×C₆₇	C₄²	semidirect
110. C₄×C₆₇	#405	semidirect
126. C₆₇	C₄×#405	semidirect
132. C₄×M₁₆	Dic₆₇	semidirect
183. M₁₆	C₄×Dic₆₇	semidirect
230. C₄	#16863354	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×#16863354	2
C₂²×C₄×C₆₇⋊C₈	2
C₄×#14524002	2
#22559897	2
C₄×#14523953	2
C₄×C₈×Dic₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₄×#405	67

Subgroups — C4×#16863354 #30853173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863354 #30853173