Subgroups — C₂²×#16863445 #31719524

Order 17152.

2718 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#16863445	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523979	C₂	1	4	yes	maximal, complemented
4	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	2	no	—
6	C₂²×#11761612	—	2	12	no	—
7	C₂²×#11761612	C₂²	1	4	yes	—
8	C₂²×#11761612	C₂²	1	24	yes	—
9	C₂³×C₆₇⋊C₈	—	2	2	no	—
10	#16863445	C₂²	1	64	yes	direct product factor
11	C₂×#14523979	C₂²	1	48	yes	complemented
12	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₄	1	4	yes	—
14	C₂²×M₁₆×C₆₇	C₄	1	12	yes	—
15	C₂²×M₁₆×C₆₇	C₂²	1	12	yes	—
16	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#11761612	—	2	24	no	—
18	C₂×#11761612	—	2	12	no	—
19	C₂×#11761612	—	2	48	no	—
20	C₂×#11761612	C₂³	1	24	yes	—
21	C₂×#11761612	C₂³	1	32	yes	—
22	C₂²×C₆₇⋊C₈	—	2	24	no	—
23	C₂²×C₆₇⋊C₈	—	2	4	no	—
24	#14523979	C₂³	1	64	yes	complemented
25	C₂⁵×C₆₇	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
28	C₂³×C₄×C₆₇	D₄	1	2	yes	—
29	C₂³×C₄×C₆₇	D₄	1	6	yes	—
30	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
31	C₂³×C₄×C₆₇	Q₈	1	6	yes	—
32	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
34	C₂×M₁₆×C₆₇	C₂×C₄	1	48	yes	—
35	C₂×M₁₆×C₆₇	C₂×C₄	1	48	yes	—
36	C₂×M₁₆×C₆₇	C₂³	1	16	yes	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	12	yes	—
38	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
40	#11761612	—	2	32	no	—
41	#11761612	—	2	24	no	—
42	#11761612	C₂⁴	1	16	yes	—
43	C₂×C₆₇⋊C₈	—	2	24	no	—
44	C₂×C₆₇⋊C₈	—	2	32	no	—
45	C₂⁴×C₆₇	—	4	4	no	—
46	C₂⁴×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₂⁴×C₆₇	C₄⋊C₄	1	6	yes	—
48	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
49	C₂⁴×C₆₇	C₂×Q₈	1	6	yes	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	12	yes	—
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	12	yes	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	6	yes	—
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	24	yes	—
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
61	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
64	C₂²×C₄×C₆₇	C₂×Q₈	1	6	yes	—
65	C₂²×C₄×C₆₇	C₂×Q₈	1	12	yes	—
66	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
67	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
68	M₁₆×C₆₇	C₂²×C₄	1	64	yes	—
69	C₂×C₈×C₆₇	C₂²×C₄	1	24	yes	—
70	C₂×C₈×C₆₇	C₂²×C₄	1	24	yes	—
71	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
72	C₂³×C₆₇	—	2	8	no	—
73	C₂³×C₆₇	—	2	4	no	—
74	C₂³×C₆₇	—	4	12	no	—
75	C₂³×C₆₇	—	4	12	no	—
76	C₂³×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
77	C₂³×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
78	C₂³×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
79	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₆₇	C₂²×Q₈	1	3	yes	—
81	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
82	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
83	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	24	yes	—
85	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
87	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	12	yes	—
88	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
89	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
92	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
93	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
94	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
95	C₂×C₄×C₆₇	C₂²×Q₈	1	3	yes	—
96	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
97	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
98	C₈×C₆₇	C₂³×C₄	1	16	yes	—
99	C₆₇⋊C₈	—	2	16	no	—
100	C₂²×C₆₇	—	2	24	no	—
101	C₂²×C₆₇	—	2	6	no	—
102	C₂²×C₆₇	—	2	6	no	—
103	C₂²×C₆₇	—	4	12	no	—
104	C₂²×C₆₇	—	4	4	no	—
105	C₂²×C₆₇	#430	1	4	yes	complemented
106	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
107	C₂²×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
108	C₂²×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
109	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
110	C₄×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
111	C₄×C₆₇	C₂²×C₄⋊C₄	1	3	yes	—
112	C₄×C₆₇	C₂²×C₄⋊C₄	1	8	yes	—
113	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
114	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
115	C₂²×#430	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	2	8	no	—
117	C₂×C₆₇	—	2	3	no	—
118	C₂×C₆₇	—	2	1	no	—
119	C₂×C₆₇	C₂×#430	1	6	yes	complemented
120	C₂×C₆₇	C₂³×C₄⋊C₄	1	1	yes	characteristic
121	C₂³×M₁₆	—	67	2	no	—
122	C₂³×M₁₆	D₆₇	1	1	yes	characteristic
123	C₂×#430	—	67	24	no	—
124	C₂²×#108	—	67	4	no	—
125	C₆₇	C₂²×#430	1	1	yes	characteristic, complemented, Sylow
126	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
127	C₂²×M₁₆	—	67	4	no	—
128	C₂²×M₁₆	—	67	24	no	—
129	C₂²×M₁₆	—	134	12	no	—
130	C₂²×M₁₆	—	134	2	no	—
131	C₂²×M₁₆	Dic₆₇	1	4	yes	—
132	C₂²×M₁₆	Dic₆₇	1	12	yes	—
133	C₂²×M₁₆	C₂×D₆₇	1	12	yes	—
134	C₂³×C₈	—	134	2	no	—
135	C₂³×C₈	C₂×D₆₇	1	2	yes	—
136	#430	—	67	64	no	—
137	C₂×#108	—	67	48	no	—
138	C₂⁵	Dic₁₃₄	1	1	yes	characteristic
139	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
140	C₂³×C₄	Dic₁₃₄	1	6	yes	—
141	C₂³×C₄	D₂₆₈	1	6	yes	—
142	C₂³×C₄	D₂₆₈	1	2	yes	—
143	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
144	C₂³×C₄	C₂×Dic₆₇	1	6	yes	—
145	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
146	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
147	C₂×M₁₆	—	67	24	no	—
148	C₂×M₁₆	—	67	32	no	—
149	C₂×M₁₆	—	134	24	no	—
150	C₂×M₁₆	—	134	12	no	—
151	C₂×M₁₆	—	134	48	no	—
152	C₂×M₁₆	C₂×Dic₆₇	1	48	yes	—
153	C₂×M₁₆	C₂×Dic₆₇	1	48	yes	—
154	C₂×M₁₆	C₂²×D₆₇	1	16	yes	—
155	C₂²×C₈	—	134	24	no	—
156	C₂²×C₈	—	134	4	no	—
157	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
158	C₂²×C₈	C₂×Dic₆₇	1	12	yes	—
159	C₂²×C₈	C₂²×D₆₇	1	12	yes	—
160	#108	—	67	64	no	—
161	C₂⁴	—	4	4	no	—
162	C₂⁴	C₄⋊Dic₆₇	1	2	yes	—
163	C₂⁴	C₄⋊Dic₆₇	1	6	yes	—
164	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
165	C₂⁴	C₂×Dic₁₃₄	1	6	yes	—
166	C₂²×C₄	C₄⋊Dic₆₇	1	12	yes	—
167	C₂²×C₄	C₄⋊Dic₆₇	1	6	yes	—
168	C₂²×C₄	C₄⋊Dic₆₇	1	24	yes	—
169	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
170	C₂²×C₄	C₄⋊Dic₆₇	1	12	yes	—
171	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
173	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
174	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
175	C₂²×C₄	C₂×D₂₆₈	1	12	yes	—
176	C₂²×C₄	C₂×D₂₆₈	1	3	yes	—
177	C₂²×C₄	C₂×D₂₆₈	1	12	yes	—
178	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×Dic₁₃₄	1	6	yes	—
180	C₂²×C₄	C₂×Dic₁₃₄	1	12	yes	—
181	C₂²×C₄	C₂×Dic₁₃₄	1	3	yes	—
182	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
183	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
184	M₁₆	—	67	16	no	—
185	M₁₆	—	134	32	no	—
186	M₁₆	—	134	24	no	—
187	M₁₆	C₂²×Dic₆₇	1	64	yes	—
188	C₂×C₈	—	134	24	no	—
189	C₂×C₈	—	134	32	no	—
190	C₂×C₈	C₂²×Dic₆₇	1	24	yes	—
191	C₂×C₈	C₂²×Dic₆₇	1	24	yes	—
192	C₂×C₈	C₂³×D₆₇	1	8	yes	—
193	C₂³	—	2	8	no	—
194	C₂³	—	2	4	no	—
195	C₂³	—	4	12	no	—
196	C₂³	—	4	12	no	—
197	C₂³	C₂×C₄⋊Dic₆₇	1	3	yes	—
198	C₂³	C₂×C₄⋊Dic₆₇	1	12	yes	—
199	C₂³	C₂×C₄⋊Dic₆₇	1	12	yes	—
200	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
201	C₂³	C₂²×Dic₁₃₄	1	3	yes	—
202	C₂³	C₂²×Dic₁₃₄	1	4	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
204	C₂×C₄	C₂×C₄⋊Dic₆₇	1	24	yes	—
205	C₂×C₄	C₂×C₄⋊Dic₆₇	1	24	yes	—
206	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
207	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
208	C₂×C₄	C₂×C₄⋊Dic₆₇	1	12	yes	—
209	C₂×C₄	C₂×C₄⋊Dic₆₇	1	3	yes	—
210	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
211	C₂×C₄	C₂³×Dic₆₇	1	3	yes	—
212	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
213	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
214	C₂×C₄	C₂²×D₂₆₈	1	8	yes	—
215	C₂×C₄	C₂²×D₂₆₈	1	3	yes	—
216	C₂×C₄	C₂²×Dic₁₃₄	1	3	yes	—
217	C₂×C₄	C₂²×Dic₁₃₄	1	4	yes	—
218	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
219	C₈	—	134	16	no	—
220	C₈	C₂³×Dic₆₇	1	16	yes	—
221	C₂²	—	2	24	no	—
222	C₂²	—	2	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	4	12	no	—
225	C₂²	—	4	4	no	—
226	C₂²	#16863445	1	4	yes	direct product factor
227	C₂²	C₂²×C₄⋊Dic₆₇	1	3	yes	—
228	C₂²	C₂²×C₄⋊Dic₆₇	1	8	yes	—
229	C₂²	C₂²×C₄⋊Dic₆₇	1	3	yes	—
230	C₂²	C₂³×Dic₁₃₄	1	1	yes	characteristic
231	C₄	C₂²×C₄⋊Dic₆₇	1	3	yes	—
232	C₄	C₂²×C₄⋊Dic₆₇	1	8	yes	—
233	C₄	C₂²×C₄⋊Dic₆₇	1	3	yes	—
234	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
235	C₄	C₂³×D₂₆₈	1	1	yes	characteristic
236	C₂	—	2	8	no	—
237	C₂	—	2	3	no	—
238	C₂	—	2	1	no	—
239	C₂	C₂×#16863445	1	6	yes	direct product factor
240	C₂	C₂³×C₄⋊Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂³×M₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₄⋊C₄
prime radical	C₂⁵×C₆₇	Q₈
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂³×M₁₆	D₆₇
hypocenter	C₆₇	C₂²×#430
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16863445	C₂	direct (N × H)
3. C₂²×#14523979	C₂	semidirect
10. #16863445	C₂²	direct (N × H)
11. C₂×#14523979	C₂²	semidirect
24. #14523979	C₂³	semidirect
105. C₂²×C₆₇	#430	semidirect
119. C₂×C₆₇	C₂×#430	semidirect
125. C₆₇	C₂²×#430	semidirect
226. C₂²	#16863445	direct (N × H)
239. C₂	C₂×#16863445	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×#16863445	2
C₂²×#14523979	2
C₂³×M₁₆×C₆₇	2
C₂²×#430	67

Subgroups — C22×#16863445 #31719524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863445 #31719524