Subgroups — C₂²×#16863420 #31719443

Order 17152.

3438 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761612	C₂	1	2	yes	maximal
2	C₂×#16863420	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	16	no	—
6	C₂²×#11761612	C₂²	1	24	yes	—
7	C₂³×C₆₇⋊C₈	C₂²	1	4	yes	—
8	#16863420	C₂²	1	64	yes	direct product factor
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	48	yes	complemented
10	C₂⁴×C₄×C₆₇	C₄	1	2	yes	—
11	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₆₇	C₄	1	12	yes	—
14	C₂²×C₄²×C₆₇	C₄	1	1	yes	characteristic
15	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
16	C₂²×C₄²×C₆₇	C₂²	1	12	yes	—
17	C₂×#11761612	—	2	96	no	—
18	C₂×#11761612	C₂³	1	32	yes	—
19	C₂²×C₆₇⋊C₈	—	2	16	no	—
20	C₂²×C₆₇⋊C₈	C₂³	1	24	yes	—
21	C₄⋊(C₆₇⋊C₈)	C₂³	1	64	yes	complemented
22	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₆₇	—	2	16	no	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₆₇	C₂×C₄	1	24	yes	—
26	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
29	C₂³×C₄×C₆₇	D₄	1	8	yes	—
30	C₂³×C₄×C₆₇	Q₈	1	8	yes	—
31	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	48	yes	—
35	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
37	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
38	C₂×C₄²×C₆₇	C₂³	1	16	yes	—
39	#11761612	—	2	64	no	—
40	C₂×C₆₇⋊C₈	—	2	48	no	—
41	C₂×C₆₇⋊C₈	C₂⁴	1	16	yes	—
42	C₂⁴×C₆₇	—	2	8	no	—
43	C₂⁴×C₆₇	C₄⋊C₄	1	8	yes	—
44	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
45	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₆₇	—	2	8	no	—
47	C₂²×C₄×C₆₇	—	2	96	no	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	48	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	8	yes	—
50	C₂²×C₄×C₆₇	M₁₆	1	16	yes	complemented
51	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	32	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
54	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	24	yes	—
59	C₂²×C₄×C₆₇	C₂×Q₈	1	24	yes	—
60	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
63	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
64	C₄²×C₆₇	C₂²×C₄	1	24	yes	—
65	C₄²×C₆₇	C₂²×C₄	1	16	yes	—
66	C₄²×C₆₇	C₂⁴	1	8	yes	—
67	C₂³×C₆₇	—	2	24	no	—
68	C₂³×C₆₇	—	2	32	no	—
69	C₂³×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
71	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
72	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
73	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
74	C₂×C₄×C₆₇	—	2	24	no	—
75	C₂×C₄×C₆₇	—	2	64	no	—
76	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	48	yes	—
77	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
78	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
79	C₂×C₄×C₆₇	C₂×M₁₆	1	48	yes	complemented
80	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
81	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
82	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
83	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
84	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
85	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
86	C₂×C₄×C₆₇	C₂²×Q₈	1	8	yes	—
87	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
88	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
89	C₆₇⋊C₈	—	2	16	no	—
90	C₂²×C₆₇	—	2	48	no	—
91	C₂²×C₆₇	—	2	8	no	—
92	C₂²×C₆₇	#423	1	16	yes	complemented
93	C₂²×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
94	C₂²×C₆₇	C₂²×C₄⋊C₄	1	6	yes	—
95	C₂²×C₆₇	C₂²×M₁₆	1	12	yes	—
96	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
97	C₄×C₆₇	—	2	8	no	—
98	C₄×C₆₇	C₂²×C₄⋊C₄	1	6	yes	—
99	C₄×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
100	C₄×C₆₇	C₂²×M₁₆	1	16	yes	complemented
101	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
102	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
103	C₂²×#423	—	67	1	no	maximal, Sylow
104	C₂×C₆₇	—	2	8	no	—
105	C₂×C₆₇	C₂×#423	1	12	yes	complemented
106	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₆₇	C₂³×M₁₆	1	2	yes	—
108	C₂³×M₁₆	—	67	2	no	—
109	C₂³×C₄²	D₆₇	1	1	yes	characteristic
110	C₂×#423	—	67	24	no	—
111	C₂²×C₄⋊C₈	—	67	4	no	—
112	C₆₇	C₂²×#423	1	1	yes	characteristic, complemented, Sylow
113	C₂⁴×C₄	Dic₆₇	1	2	yes	—
114	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
115	C₂²×M₁₆	—	67	24	no	—
116	C₂²×M₁₆	—	134	16	no	—
117	C₂³×C₈	—	67	4	no	—
118	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
119	C₂²×C₄²	Dic₆₇	1	12	yes	—
120	C₂²×C₄²	Dic₆₇	1	1	yes	characteristic
121	C₂²×C₄²	C₂×D₆₇	1	12	yes	—
122	C₂²×C₄²	C₂×D₆₇	1	2	yes	—
123	#423	—	67	64	no	—
124	C₂×C₄⋊C₈	—	67	48	no	—
125	C₂⁵	C₂×Dic₆₇	1	1	yes	characteristic
126	C₂³×C₄	—	2	16	no	—
127	C₂³×C₄	Dic₁₃₄	1	8	yes	—
128	C₂³×C₄	D₂₆₈	1	8	yes	—
129	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
130	C₂³×C₄	C₂×Dic₆₇	1	24	yes	—
131	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
132	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
133	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
134	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
135	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
136	C₂×M₁₆	—	67	32	no	—
137	C₂×M₁₆	—	134	96	no	—
138	C₂²×C₈	—	67	24	no	—
139	C₂²×C₈	—	134	16	no	—
140	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
141	C₂×C₄²	C₂×Dic₆₇	1	48	yes	—
142	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
143	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
144	C₂×C₄²	C₂²×D₆₇	1	12	yes	—
145	C₂×C₄²	C₂²×D₆₇	1	16	yes	—
146	C₄⋊C₈	—	67	64	no	—
147	C₂⁴	—	2	8	no	—
148	C₂⁴	C₄⋊Dic₆₇	1	8	yes	—
149	C₂⁴	C₂²×Dic₆₇	1	6	yes	—
150	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
151	C₂²×C₄	—	2	8	no	—
152	C₂²×C₄	—	2	96	no	—
153	C₂²×C₄	C₄⋊Dic₆₇	1	48	yes	—
154	C₂²×C₄	C₄⋊Dic₆₇	1	8	yes	—
155	C₂²×C₄	#11761612	1	16	yes	complemented
156	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
157	C₂²×C₄	C₂²×Dic₆₇	1	32	yes	—
158	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
159	C₂²×C₄	C₂²×Dic₆₇	1	6	yes	—
160	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
161	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
162	C₂²×C₄	C₂×D₂₆₈	1	4	yes	—
163	C₂²×C₄	C₂×D₂₆₈	1	24	yes	—
164	C₂²×C₄	C₂×Dic₁₃₄	1	24	yes	—
165	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
166	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
167	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
168	M₁₆	—	134	64	no	—
169	C₂×C₈	—	67	16	no	—
170	C₂×C₈	—	134	48	no	—
171	C₄²	C₂²×Dic₆₇	1	16	yes	—
172	C₄²	C₂²×Dic₆₇	1	24	yes	—
173	C₄²	C₂²×Dic₆₇	1	16	yes	—
174	C₄²	C₂³×D₆₇	1	8	yes	—
175	C₂³	—	2	24	no	—
176	C₂³	—	2	32	no	—
177	C₂³	C₂×C₄⋊Dic₆₇	1	24	yes	—
178	C₂³	C₂×C₄⋊Dic₆₇	1	4	yes	—
179	C₂³	C₂×#11761612	1	8	yes	—
180	C₂³	C₂³×Dic₆₇	1	4	yes	—
181	C₂³	C₂³×Dic₆₇	1	3	yes	—
182	C₂×C₄	—	2	24	no	—
183	C₂×C₄	—	2	64	no	—
184	C₂×C₄	C₂×C₄⋊Dic₆₇	1	48	yes	—
185	C₂×C₄	C₂×C₄⋊Dic₆₇	1	24	yes	—
186	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
187	C₂×C₄	C₂×#11761612	1	48	yes	complemented
188	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
189	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
190	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
191	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
192	C₂×C₄	C₂²×D₂₆₈	1	6	yes	—
193	C₂×C₄	C₂²×D₂₆₈	1	8	yes	—
194	C₂×C₄	C₂²×Dic₁₃₄	1	8	yes	—
195	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
196	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
197	C₈	—	134	16	no	—
198	C₂²	—	2	48	no	—
199	C₂²	—	2	8	no	—
200	C₂²	#16863420	1	16	yes	direct product factor
201	C₂²	C₂²×C₄⋊Dic₆₇	1	8	yes	—
202	C₂²	C₂²×C₄⋊Dic₆₇	1	6	yes	—
203	C₂²	C₂²×#11761612	1	12	yes	—
204	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
205	C₄	—	2	8	no	—
206	C₄	C₂²×C₄⋊Dic₆₇	1	6	yes	—
207	C₄	C₂²×C₄⋊Dic₆₇	1	8	yes	—
208	C₄	C₂²×#11761612	1	16	yes	complemented
209	C₄	C₂³×D₂₆₈	1	1	yes	characteristic
210	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
211	C₂	—	2	8	no	—
212	C₂	C₂×#16863420	1	12	yes	direct product factor
213	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic
214	C₂	C₂³×#11761612	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂³×C₄²×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂³×C₄²	D₆₇
hypocenter	C₆₇	C₂²×#423
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×#16863420	C₂	direct (N × H)
3. C₂²×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
8. #16863420	C₂²	direct (N × H)
9. C₂×C₄⋊(C₆₇⋊C₈)	C₂²	semidirect
21. C₄⋊(C₆₇⋊C₈)	C₂³	semidirect
50. C₂²×C₄×C₆₇	M₁₆	semidirect
79. C₂×C₄×C₆₇	C₂×M₁₆	semidirect
92. C₂²×C₆₇	#423	semidirect
100. C₄×C₆₇	C₂²×M₁₆	semidirect
105. C₂×C₆₇	C₂×#423	semidirect
112. C₆₇	C₂²×#423	semidirect
155. C₂²×C₄	#11761612	semidirect
187. C₂×C₄	C₂×#11761612	semidirect
200. C₂²	#16863420	direct (N × H)
208. C₄	C₂²×#11761612	semidirect
212. C₂	C₂×#16863420	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×#11761612	2
C₂×#16863420	2
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂³×C₄²×C₆₇	2
C₂²×#423	67

Subgroups — C22×#16863420 #31719443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863420 #31719443