Subgroups — C₂²×#16863265 #31014120

Order 17152.

5502 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	C₂³×C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₆₇⋊C₄	C₂	1	4	yes	maximal
3	C₂³×C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16863265	C₂	1	24	yes	maximal, direct product factor
5	C₂³×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×Dic₆₇	C₂²	1	2	yes	—
7	C₂⁴×Dic₆₇	C₂²	1	2	yes	—
8	C₂²×C₄⋊Dic₆₇	—	2	8	no	—
9	C₂²×C₄⋊Dic₆₇	C₂²	1	12	yes	—
10	C₂²×Dic₆₇⋊C₄	—	2	32	no	—
11	C₂²×Dic₆₇⋊C₄	C₂²	1	48	yes	—
12	C₂²×C₄×Dic₆₇	C₄	1	16	yes	complemented
13	C₂²×C₄×Dic₆₇	C₂²	1	12	yes	—
14	#16863265	C₂²	1	64	yes	direct product factor
15	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
16	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₆₇	—	2	8	no	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	12	yes	—
19	C₂³×Dic₆₇	—	2	16	no	—
20	C₂³×Dic₆₇	—	2	8	no	—
21	C₂³×Dic₆₇	C₂×C₄	1	16	yes	—
22	C₂³×Dic₆₇	D₄	1	16	yes	—
23	C₂³×Dic₆₇	Q₈	1	16	yes	—
24	C₂³×Dic₆₇	C₂³	1	12	yes	—
25	C₂³×Dic₆₇	C₂³	1	12	yes	—
26	C₂×C₄⋊Dic₆₇	—	2	48	no	—
27	C₂×C₄⋊Dic₆₇	C₂³	1	16	yes	—
28	C₂×Dic₆₇⋊C₄	—	2	192	no	—
29	C₂×Dic₆₇⋊C₄	C₂³	1	64	yes	—
30	C₂×C₄×Dic₆₇	C₂×C₄	1	96	yes	complemented
31	C₂×C₄×Dic₆₇	C₂³	1	16	yes	—
32	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	—	2	8	no	—
34	C₂³×C₄×C₆₇	—	2	16	no	—
35	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
36	C₂³×C₄×C₆₇	D₄	1	8	yes	—
37	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
38	C₂³×C₄×C₆₇	C₂³	1	12	yes	—
39	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	48	no	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	16	yes	—
42	C₂²×Dic₆₇	—	2	48	no	—
43	C₂²×Dic₆₇	—	2	24	no	—
44	C₂²×Dic₆₇	—	4	32	no	—
45	C₂²×Dic₆₇	C₄⋊C₄	1	128	yes	complemented
46	C₂²×Dic₆₇	C₂²×C₄	1	48	yes	—
47	C₂²×Dic₆₇	C₂×D₄	1	48	yes	—
48	C₂²×Dic₆₇	C₂×Q₈	1	48	yes	—
49	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
50	C₂²×Dic₆₇	C₂⁴	1	8	yes	—
51	C₄⋊Dic₆₇	—	2	32	no	—
52	Dic₆₇⋊C₄	—	2	128	no	—
53	C₄×Dic₆₇	C₂²×C₄	1	64	yes	complemented
54	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
55	C₂⁴×C₆₇	C₂×D₄	1	8	yes	—
56	C₂⁴×C₆₇	C₂×D₄	1	4	yes	—
57	C₂⁴×C₆₇	C₂×Q₈	1	8	yes	—
58	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
59	C₂⁴×C₆₇	C₂⁴	1	3	yes	—
60	C₂²×C₄×C₆₇	—	2	24	no	—
61	C₂²×C₄×C₆₇	—	2	48	no	—
62	C₂²×C₄×C₆₇	—	4	32	no	—
63	C₂²×C₄×C₆₇	C₄⋊C₄	1	64	yes	complemented
64	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
65	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
66	C₂²×C₄×C₆₇	C₂×Q₈	1	24	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
68	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
69	C₄⋊C₄×C₆₇	—	2	32	no	—
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	32	yes	—
71	C₂³×C₆₇	C₂×C₄⋊C₄	1	16	yes	—
72	C₂³×C₆₇	#127	1	16	yes	—
73	C₂³×C₆₇	C₄⋊Q₈	1	16	yes	—
74	C₂³×C₆₇	C₄⋊Q₈	1	32	yes	—
75	C₂³×C₆₇	C₂³×C₄	1	6	yes	—
76	C₂³×C₆₇	C₂²×D₄	1	12	yes	—
77	C₂³×C₆₇	C₂²×D₄	1	6	yes	—
78	C₂³×C₆₇	C₂²×Q₈	1	12	yes	—
79	C₂³×C₆₇	C₂²×Q₈	1	6	yes	—
80	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	8	no	—
82	C₂×C₄×C₆₇	—	2	16	no	—
83	C₂×C₄×C₆₇	—	4	48	no	—
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	96	yes	complemented
85	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
86	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
87	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
88	C₂×Dic₆₇	—	2	16	no	—
89	C₂×Dic₆₇	—	2	8	no	—
90	C₂×Dic₆₇	—	4	48	no	—
91	C₂×Dic₆₇	C₂×C₄⋊C₄	1	192	yes	complemented
92	C₂×Dic₆₇	C₂³×C₄	1	16	yes	—
93	C₂×Dic₆₇	C₂²×D₄	1	16	yes	—
94	C₂×Dic₆₇	C₂²×Q₈	1	16	yes	—
95	C₂²×C₆₇	#384	1	64	yes	complemented
96	C₂²×C₆₇	C₂²×C₄⋊C₄	1	24	yes	—
97	C₂²×C₆₇	C₂²×C₄⋊C₄	1	12	yes	—
98	C₂²×C₆₇	C₂×#127	1	12	yes	—
99	C₂²×C₆₇	C₂×C₄⋊Q₈	1	12	yes	—
100	C₂²×C₆₇	C₂×C₄⋊Q₈	1	24	yes	—
101	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
102	C₂²×C₆₇	C₂³×D₄	1	2	yes	—
103	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
104	C₂²×C₆₇	C₂³×Q₈	1	2	yes	—
105	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
106	C₄×C₆₇	—	4	8	no	—
107	C₄×C₆₇	C₂²×C₄⋊C₄	1	16	yes	complemented
108	Dic₆₇	—	4	8	no	—
109	Dic₆₇	C₂²×C₄⋊C₄	1	32	yes	complemented
110	C₂²×#384	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	C₂×#384	1	24	yes	complemented
112	C₂×C₆₇	C₂³×C₄⋊C₄	1	2	yes	—
113	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₆₇	C₂²×#127	1	1	yes	characteristic
115	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₆₇	C₂²×C₄⋊Q₈	1	2	yes	—
117	C₂³×C₄⋊C₄	—	67	4	no	—
118	C₂³×C₄⋊C₄	—	67	1	no	—
119	C₂³×C₄⋊C₄	D₆₇	1	1	yes	characteristic
120	C₂³×C₄²	—	67	1	no	—
121	C₂×#384	—	67	24	no	—
122	C₆₇	C₂²×#384	1	1	yes	characteristic, complemented, Sylow
123	C₂⁴×C₄	—	67	2	no	—
124	C₂⁴×C₄	—	67	2	no	—
125	C₂⁴×C₄	C₂×D₆₇	1	2	yes	—
126	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
127	C₂²×C₄⋊C₄	—	2	8	no	—
128	C₂²×C₄⋊C₄	—	67	48	no	—
129	C₂²×C₄⋊C₄	—	67	12	no	—
130	C₂²×C₄⋊C₄	—	134	32	no	—
131	C₂²×C₄⋊C₄	—	134	8	no	—
132	C₂²×C₄⋊C₄	C₂×D₆₇	1	12	yes	—
133	C₂²×C₄²	—	67	16	no	—
134	C₂²×C₄²	—	67	12	no	—
135	#384	—	67	64	no	—
136	C₂⁵	C₂²×D₆₇	1	1	yes	characteristic
137	C₂³×C₄	—	2	16	no	—
138	C₂³×C₄	—	2	8	no	—
139	C₂³×C₄	—	67	16	no	—
140	C₂³×C₄	—	67	16	no	—
141	C₂³×C₄	—	67	16	no	—
142	C₂³×C₄	—	67	12	no	—
143	C₂³×C₄	—	67	12	no	—
144	C₂³×C₄	—	134	16	no	—
145	C₂³×C₄	—	134	8	no	—
146	C₂³×C₄	Dic₁₃₄	1	8	yes	—
147	C₂³×C₄	C₄×D₆₇	1	8	yes	—
148	C₂³×C₄	C₂²⋊D₆₇	1	8	yes	—
149	C₂³×C₄	C₂²×D₆₇	1	12	yes	—
150	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
151	C₂×C₄⋊C₄	—	2	48	no	—
152	C₂×C₄⋊C₄	—	67	64	no	—
153	C₂×C₄⋊C₄	—	67	16	no	—
154	C₂×C₄⋊C₄	—	134	192	no	—
155	C₂×C₄⋊C₄	—	134	48	no	—
156	C₂×C₄⋊C₄	C₂²×D₆₇	1	16	yes	—
157	C₂×C₄²	—	67	96	no	—
158	C₂×C₄²	—	67	16	no	—
159	C₂⁴	C₂×C₄×D₆₇	1	4	yes	—
160	C₂⁴	D₄×D₆₇	1	8	yes	—
161	C₂⁴	C₂×C₂²⋊D₆₇	1	4	yes	—
162	C₂⁴	Q₈×D₆₇	1	8	yes	—
163	C₂⁴	C₂×Dic₁₃₄	1	4	yes	—
164	C₂⁴	C₂³×D₆₇	1	3	yes	—
165	C₂²×C₄	—	2	48	no	—
166	C₂²×C₄	—	2	24	no	—
167	C₂²×C₄	—	4	32	no	—
168	C₂²×C₄	—	268	32	no	—
169	C₂²×C₄	—	67	128	no	—
170	C₂²×C₄	—	67	48	no	—
171	C₂²×C₄	—	67	48	no	—
172	C₂²×C₄	—	67	48	no	—
173	C₂²×C₄	—	67	8	no	—
174	C₂²×C₄	—	67	8	no	—
175	C₂²×C₄	—	134	48	no	—
176	C₂²×C₄	—	134	24	no	—
177	C₂²×C₄	Dic₆₇⋊C₄	1	64	yes	complemented
178	C₂²×C₄	C₂×C₄×D₆₇	1	24	yes	—
179	C₂²×C₄	C₂×C₂²⋊D₆₇	1	24	yes	—
180	C₂²×C₄	C₂×Dic₁₃₄	1	24	yes	—
181	C₂²×C₄	C₂³×D₆₇	1	8	yes	—
182	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
183	C₄⋊C₄	—	2	32	no	—
184	C₄⋊C₄	—	134	128	no	—
185	C₄⋊C₄	—	134	32	no	—
186	C₄²	—	67	64	no	—
187	C₂³	C₄⋊C₄×D₆₇	1	32	yes	—
188	C₂³	C₂×Dic₆₇⋊C₄	1	16	yes	—
189	C₂³	#14524039	1	16	yes	—
190	C₂³	#14524041	1	16	yes	—
191	C₂³	#14524042	1	32	yes	—
192	C₂³	C₂²×C₄×D₆₇	1	6	yes	—
193	C₂³	C₂×D₄×D₆₇	1	12	yes	—
194	C₂³	C₂²×C₂²⋊D₆₇	1	6	yes	—
195	C₂³	C₂×Q₈×D₆₇	1	12	yes	—
196	C₂³	C₂²×Dic₁₃₄	1	6	yes	—
197	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
198	C₂×C₄	—	2	16	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	4	48	no	—
201	C₂×C₄	—	268	48	no	—
202	C₂×C₄	—	67	192	no	—
203	C₂×C₄	—	67	16	no	—
204	C₂×C₄	—	67	16	no	—
205	C₂×C₄	—	67	16	no	—
206	C₂×C₄	—	134	16	no	—
207	C₂×C₄	—	134	8	no	—
208	C₂×C₄	C₂×Dic₆₇⋊C₄	1	96	yes	complemented
209	C₂×C₄	C₂²×C₄×D₆₇	1	8	yes	—
210	C₂×C₄	C₂²×C₂²⋊D₆₇	1	8	yes	—
211	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
212	C₂²	#16863265	1	64	yes	direct product factor
213	C₂²	C₂×C₄⋊C₄×D₆₇	1	24	yes	—
214	C₂²	C₂²×Dic₆₇⋊C₄	1	12	yes	—
215	C₂²	C₂×#14524039	1	12	yes	—
216	C₂²	C₂×#14524041	1	12	yes	—
217	C₂²	C₂×#14524042	1	24	yes	—
218	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
219	C₂²	C₂²×D₄×D₆₇	1	2	yes	—
220	C₂²	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
221	C₂²	C₂²×Q₈×D₆₇	1	2	yes	—
222	C₂²	C₂³×Dic₁₃₄	1	1	yes	characteristic
223	C₄	—	4	8	no	—
224	C₄	—	268	8	no	—
225	C₄	—	67	32	no	—
226	C₄	C₂²×Dic₆₇⋊C₄	1	16	yes	complemented
227	C₂	C₂×#16863265	1	24	yes	direct product factor
228	C₂	C₂²×C₄⋊C₄×D₆₇	1	2	yes	—
229	C₂	C₂³×Dic₆₇⋊C₄	1	1	yes	characteristic
230	C₂	C₂²×#14524039	1	1	yes	characteristic
231	C₂	C₂²×#14524041	1	1	yes	characteristic
232	C₂	C₂²×#14524042	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₆₇
Fitting	C₂³×C₄⋊C₄×C₆₇	C₂
socle, prime radical	C₂⁵×C₆₇	C₂³
cosocle	C₂³×C₆₇	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₆₇
hypocenter	C₆₇	C₂²×#384
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂×#16863265	C₂	direct (N × H)
12. C₂²×C₄×Dic₆₇	C₄	semidirect
14. #16863265	C₂²	direct (N × H)
30. C₂×C₄×Dic₆₇	C₂×C₄	semidirect
45. C₂²×Dic₆₇	C₄⋊C₄	semidirect
53. C₄×Dic₆₇	C₂²×C₄	semidirect
63. C₂²×C₄×C₆₇	C₄⋊C₄	semidirect
84. C₂×C₄×C₆₇	C₂×C₄⋊C₄	semidirect
91. C₂×Dic₆₇	C₂×C₄⋊C₄	semidirect
95. C₂²×C₆₇	#384	semidirect
107. C₄×C₆₇	C₂²×C₄⋊C₄	semidirect
109. Dic₆₇	C₂²×C₄⋊C₄	semidirect
111. C₂×C₆₇	C₂×#384	semidirect
122. C₆₇	C₂²×#384	semidirect
177. C₂²×C₄	Dic₆₇⋊C₄	semidirect
208. C₂×C₄	C₂×Dic₆₇⋊C₄	semidirect
212. C₂²	#16863265	direct (N × H)
226. C₄	C₂²×Dic₆₇⋊C₄	semidirect
227. C₂	C₂×#16863265	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄⋊Dic₆₇	2
C₂³×Dic₆₇⋊C₄	2
C₂³×C₄×Dic₆₇	2
C₂×#16863265	2
C₂³×C₄⋊C₄×C₆₇	2
C₂²×#384	67

Subgroups — C22×#16863265 #31014120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863265 #31014120