Subgroups — C₄×#16863874 #31014082

Order 17152.

3402 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863874	C₂	1	2	yes	maximal
2	#22562574	C₂	1	8	yes	maximal, complemented
3	#22562566	C₂	1	6	yes	maximal
4	C₄×Q₈×Dic₆₇	C₂	1	8	yes	maximal, complemented
5	C₄×#14524002	C₂	1	6	yes	maximal
6	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863874	C₄	1	32	yes	direct product factor
8	C₂×Q₈×Dic₆₇	C₂²	1	16	yes	—
9	C₂×#14524002	C₂²	1	12	yes	—
10	C₄⋊C₄×Dic₆₇	C₂²	1	48	yes	—
11	C₄×C₄⋊Dic₆₇	C₂²	1	12	yes	—
12	C₄×C₄⋊Dic₆₇	C₂²	1	12	yes	—
13	C₄×C₂²⋊Dic₆₇	C₂²	1	12	yes	—
14	C₄×C₂²⋊Dic₆₇	C₂²	1	4	yes	—
15	#16863242	C₂²	1	12	yes	—
16	C₄²×Dic₆₇	C₂²	1	12	yes	—
17	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	C₄	1	16	yes	complemented
19	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
22	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
23	Q₈×Dic₆₇	C₂×C₄	1	128	yes	complemented
24	#14524002	C₂×C₄	1	96	yes	—
25	C₂×C₄⋊Dic₆₇	C₂³	1	24	yes	—
26	C₂×C₂²⋊Dic₆₇	C₂³	1	8	yes	—
27	C₂×C₄×Dic₆₇	C₂³	1	8	yes	—
28	C₂×C₄×Dic₆₇	C₂³	1	24	yes	—
29	C₂×C₄×Dic₆₇	C₂³	1	24	yes	—
30	C₆₇⋊#95	C₂³	1	8	yes	—
31	C₆₇⋊#95	C₂³	1	24	yes	—
32	C₂²×Q₈×C₆₇	C₂×C₄	1	16	yes	—
33	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
34	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
35	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	3	yes	—
38	C₄×Q₈×C₆₇	C₂×C₄	1	32	yes	—
39	C₄×Q₈×C₆₇	C₂×C₄	1	32	yes	—
40	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
41	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
42	C₂×C₄²×C₆₇	C₂×C₄	1	24	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
45	C₂²×Dic₆₇	C₂⁴	1	16	yes	—
46	C₄⋊Dic₆₇	C₂²×C₄	1	96	yes	—
47	C₂²⋊Dic₆₇	C₂²×C₄	1	32	yes	—
48	C₄×Dic₆₇	—	2	16	no	—
49	C₄×Dic₆₇	—	4	24	no	—
50	C₄×Dic₆₇	C₂²×C₄	1	96	yes	—
51	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₆₇	C₄²	1	64	yes	complemented
53	C₂×Q₈×C₆₇	C₂²×C₄	1	32	yes	—
54	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
55	C₂²×C₄×C₆₇	—	2	4	no	—
56	C₂²×C₄×C₆₇	—	4	6	no	—
57	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
60	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
63	C₂²×C₄×C₆₇	C₂⁴	1	6	yes	—
64	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	C₂²×C₄	1	48	yes	—
66	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
67	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
68	C₂³×C₆₇	—	2	2	no	—
69	C₂³×C₆₇	—	2	2	no	—
70	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
71	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
72	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
73	Q₈×C₆₇	C₂×C₄²	1	64	yes	—
74	C₂×C₄×C₆₇	—	2	2	no	—
75	C₂×C₄×C₆₇	—	2	12	no	—
76	C₂×C₄×C₆₇	—	2	8	no	—
77	C₂×C₄×C₆₇	—	4	12	no	—
78	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	—
79	C₂×C₄×C₆₇	C₂³×C₄	1	24	yes	—
80	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
82	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
84	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
85	C₂×Dic₆₇	—	2	16	no	—
86	C₂×Dic₆₇	—	2	16	no	—
87	C₂×Dic₆₇	C₂³×C₄	1	32	yes	—
88	C₂²×C₆₇	—	2	2	no	—
89	C₂²×C₆₇	—	2	2	no	—
90	C₂²×C₆₇	—	2	8	no	—
91	C₂²×C₆₇	C₂²×C₄²	1	4	yes	—
92	C₂²×C₆₇	C₂⁴×C₄	1	2	yes	—
93	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
94	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
95	C₂²×C₆₇	#585	1	2	yes	—
96	C₄×C₆₇	—	2	2	no	—
97	C₄×C₆₇	—	2	12	no	—
98	C₄×C₆₇	C₂²×C₄²	1	24	yes	—
99	C₄×C₆₇	#519	1	4	yes	complemented
100	Dic₆₇	—	2	32	no	—
101	C₄×#519	—	67	1	no	maximal, Sylow
102	C₂×C₆₇	—	2	4	no	—
103	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
104	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
105	C₂×C₆₇	#3428	1	1	yes	characteristic
106	C₂×C₆₇	C₄×2₋¹⁺⁴	1	4	yes	—
107	C₂×#519	—	67	2	no	—
108	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
109	#2276	—	67	8	no	—
110	#2274	—	67	6	no	—
111	C₄²×Q₈	—	67	8	no	—
112	C₄×#117	—	67	6	no	—
113	C₆₇	C₄×#519	1	1	yes	characteristic, complemented, Sylow
114	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic
115	#519	—	67	32	no	—
116	C₂×C₄×Q₈	—	67	16	no	—
117	C₂×C₄×Q₈	Dic₆₇	1	16	yes	complemented
118	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
119	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
120	C₂×#117	—	67	12	no	—
121	C₂²×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
122	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
123	C₄×C₄⋊C₄	—	67	48	no	—
124	C₄×C₄⋊C₄	—	67	12	no	—
125	C₄×C₄⋊C₄	—	67	12	no	—
126	C₄×C₂²⋊C₄	—	67	4	no	—
127	C₄×C₂²⋊C₄	—	67	12	no	—
128	#376	—	67	12	no	—
129	C₄³	—	67	12	no	—
130	C₂²×Q₈	C₄×D₆₇	1	16	yes	—
131	C₂²×Q₈	C₂×Dic₆₇	1	8	yes	—
132	C₂²×Q₈	C₂²×D₆₇	1	4	yes	—
133	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
134	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
135	C₂³×C₄	C₂²×D₆₇	1	3	yes	—
136	C₄×Q₈	—	67	128	no	—
137	C₄×Q₈	C₂×Dic₆₇	1	32	yes	—
138	C₄×Q₈	C₂×Dic₆₇	1	32	yes	—
139	#117	—	67	96	no	—
140	C₂×C₄⋊C₄	—	67	24	no	—
141	C₂×C₄⋊C₄	C₂×Dic₆₇	1	24	yes	—
142	C₂×C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
143	C₂×C₂²⋊C₄	—	67	8	no	—
144	C₂×C₄²	—	67	24	no	—
145	C₂×C₄²	—	67	24	no	—
146	C₂×C₄²	—	67	8	no	—
147	C₂×C₄²	C₂×Dic₆₇	1	24	yes	—
148	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
149	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
150	#95	—	67	8	no	—
151	#95	—	67	24	no	—
152	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
153	C₂×Q₈	C₄×Dic₆₇	1	64	yes	complemented
154	C₂×Q₈	C₂×C₄×D₆₇	1	32	yes	—
155	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	—	4	6	no	—
158	C₂²×C₄	—	67	16	no	—
159	C₂²×C₄	C₂×C₄×D₆₇	1	24	yes	—
160	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
161	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
162	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
163	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
164	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
165	C₂²×C₄	C₂³×D₆₇	1	6	yes	—
166	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
167	C₄⋊C₄	—	67	96	no	—
168	C₄⋊C₄	C₂²×Dic₆₇	1	48	yes	—
169	C₂²⋊C₄	—	67	32	no	—
170	C₄²	—	268	24	no	—
171	C₄²	—	67	96	no	—
172	C₄²	—	134	16	no	—
173	C₄²	C₂²×Dic₆₇	1	24	yes	—
174	C₄²	C₂²×Dic₆₇	1	24	yes	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	C₂²×C₄×D₆₇	1	4	yes	—
178	C₂³	C₂³×Dic₆₇	1	2	yes	—
179	C₂³	C₂⁴×D₆₇	1	1	yes	characteristic
180	Q₈	C₂×C₄×Dic₆₇	1	64	yes	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	12	no	—
183	C₂×C₄	—	2	8	no	—
184	C₂×C₄	—	4	12	no	—
185	C₂×C₄	—	67	32	no	—
186	C₂×C₄	—	134	16	no	—
187	C₂×C₄	—	134	16	no	—
188	C₂×C₄	C₂×C₄×Dic₆₇	1	48	yes	—
189	C₂×C₄	C₂²×C₄×D₆₇	1	24	yes	—
190	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
191	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
192	C₂×C₄	C₂³×Dic₆₇	1	12	yes	—
193	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
194	C₂×C₄	#14524091	1	4	yes	—
195	C₂²	—	2	2	no	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	8	no	—
198	C₂²	C₂²×C₄×Dic₆₇	1	4	yes	—
199	C₂²	C₂³×C₄×D₆₇	1	2	yes	—
200	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
201	C₂²	C₂×#14524091	1	2	yes	—
202	C₂²	#16864232	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	12	no	—
205	C₄	—	134	32	no	—
206	C₄	C₂²×C₄×Dic₆₇	1	24	yes	—
207	C₄	#16863874	1	4	yes	direct product factor
208	C₂	—	2	4	no	—
209	C₂	C₂³×C₄×Dic₆₇	1	1	yes	characteristic
210	C₂	C₂×#16863874	1	1	yes	characteristic
211	C₂	#22573412	1	1	yes	characteristic
212	C₂	C₄×#14524091	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂×C₆₇	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle, cosocle	C₂³×C₆₇	C₂⁵
prime radical	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₄×#519
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #22562574	C₂	semidirect
4. C₄×Q₈×Dic₆₇	C₂	semidirect
7. #16863874	C₄	direct (N × H)
18. C₂×C₄×Q₈×C₆₇	C₄	semidirect
23. Q₈×Dic₆₇	C₂×C₄	semidirect
52. C₂×Q₈×C₆₇	C₄²	semidirect
99. C₄×C₆₇	#519	semidirect
113. C₆₇	C₄×#519	semidirect
117. C₂×C₄×Q₈	Dic₆₇	semidirect
153. C₂×Q₈	C₄×Dic₆₇	semidirect
207. C₄	#16863874	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×#16863874	2
#22562574	2
#22562566	2
C₄×Q₈×Dic₆₇	2
C₄×#14524002	2
C₂²×C₄×Q₈×C₆₇	2
C₄×#519	67

Subgroups — C4×#16863874 #31014082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863874 #31014082