Subgroups — C₂²×#16863354 #31719225

Order 17152.

3518 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	C₂⁴×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14524002	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863354	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14523953	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₄	1	16	yes	complemented
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₆₇⋊C₈	C₂²	1	12	yes	—
11	C₂×#14524002	C₄	1	12	yes	—
12	C₂×#14524002	C₂²	1	12	yes	—
13	C₂²×C₄⋊Dic₆₇	C₄	1	2	yes	—
14	C₂²×C₂²⋊Dic₆₇	C₄	1	2	yes	—
15	C₂²×C₄×Dic₆₇	C₂²	1	2	yes	—
16	#16863354	C₂²	1	64	yes	direct product factor
17	C₂×#14523953	C₂²	1	24	yes	complemented
18	C₂×C₈×Dic₆₇	C₂²	1	24	yes	complemented
19	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₆₇	C₄	1	16	yes	complemented
21	C₂²×M₁₆×C₆₇	C₂²	1	12	yes	—
22	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
23	C₂³×Dic₆₇	C₂×C₄	1	4	yes	—
24	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
25	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
26	C₂²×C₆₇⋊C₈	C₂×C₄	1	96	yes	complemented
27	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
28	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
29	C₂²×C₆₇⋊C₈	C₂³	1	16	yes	—
30	#14524002	C₂×C₄	1	48	yes	—
31	#14524002	C₂³	1	16	yes	—
32	C₂×C₄⋊Dic₆₇	C₂×C₄	1	24	yes	—
33	C₂×C₂²⋊Dic₆₇	C₂×C₄	1	24	yes	—
34	C₂×C₄×Dic₆₇	C₂×C₄	1	12	yes	—
35	C₂×C₄×Dic₆₇	C₂³	1	12	yes	—
36	#14523953	C₂³	1	32	yes	complemented
37	C₈×Dic₆₇	C₂³	1	32	yes	complemented
38	C₂⁵×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	C₂×C₄	1	6	yes	—
41	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
42	C₂³×C₄×C₆₇	C₂×C₄	1	8	yes	—
43	C₂³×C₄×C₆₇	C₂³	1	6	yes	—
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₆₇	C₂×C₄	1	96	yes	complemented
46	C₂×M₁₆×C₆₇	C₂³	1	16	yes	—
47	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	—
48	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
49	C₂²×Dic₆₇	—	2	16	no	—
50	C₂²×Dic₆₇	C₂²×C₄	1	24	yes	—
51	C₂×C₆₇⋊C₈	C₂²×C₄	1	24	yes	—
52	C₂×C₆₇⋊C₈	C₂²×C₄	1	24	yes	—
53	C₂×C₆₇⋊C₈	C₂²×C₄	1	64	yes	complemented
54	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
55	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
56	C₄⋊Dic₆₇	C₂²×C₄	1	32	yes	—
57	C₂²⋊Dic₆₇	C₂²×C₄	1	32	yes	—
58	C₄×Dic₆₇	C₂²×C₄	1	24	yes	—
59	C₄×Dic₆₇	C₂⁴	1	8	yes	—
60	C₂⁴×C₆₇	—	2	8	no	—
61	C₂⁴×C₆₇	C₄²	1	8	yes	—
62	C₂⁴×C₆₇	C₂²×C₄	1	6	yes	—
63	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₄²	1	48	yes	—
65	C₂²×C₄×C₆₇	C₄²	1	8	yes	—
66	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	6	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	24	yes	—
73	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
74	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
75	M₁₆×C₆₇	C₂²×C₄	1	64	yes	complemented
76	C₂×C₈×C₆₇	C₂²×C₄	1	48	yes	—
77	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
78	C₂³×C₆₇	—	2	24	no	—
79	C₂³×C₆₇	—	2	32	no	—
80	C₂³×C₆₇	C₂×C₄²	1	24	yes	—
81	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
82	C₂³×C₆₇	#131	1	8	yes	—
83	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
84	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
85	C₂×C₄×C₆₇	C₂×C₄²	1	48	yes	—
86	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
87	C₂×C₄×C₆₇	C₂×C₄²	1	24	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
89	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
90	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
91	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
92	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
94	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
95	C₂×Dic₆₇	—	2	48	no	—
96	C₂×Dic₆₇	C₂³×C₄	1	16	yes	—
97	C₈×C₆₇	C₂³×C₄	1	16	yes	—
98	C₆₇⋊C₈	C₂³×C₄	1	8	yes	—
99	C₆₇⋊C₈	C₂³×C₄	1	8	yes	—
100	C₂²×C₆₇	—	2	48	no	—
101	C₂²×C₆₇	—	2	8	no	—
102	C₂²×C₆₇	#405	1	16	yes	complemented
103	C₂²×C₆₇	C₂²×C₄²	1	8	yes	—
104	C₂²×C₆₇	C₂²×C₄²	1	6	yes	—
105	C₂²×C₆₇	C₂×#131	1	12	yes	—
106	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
107	C₄×C₆₇	C₂²×C₄²	1	6	yes	—
108	C₄×C₆₇	C₂²×C₄²	1	8	yes	—
109	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
110	C₄×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
111	Dic₆₇	—	2	16	no	—
112	C₂²×#405	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	2	8	no	—
114	C₂×C₆₇	C₂×#405	1	12	yes	complemented
115	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
116	C₂×C₆₇	C₂²×#131	1	2	yes	—
117	C₂³×M₁₆	D₆₇	1	1	yes	characteristic
118	C₂⁴×C₈	—	67	1	no	—
119	C₂²×#117	—	67	1	no	—
120	C₂×#405	—	67	24	no	—
121	C₂²×#97	—	67	2	no	—
122	C₂²×C₄×C₈	—	67	2	no	—
123	C₆₇	C₂²×#405	1	1	yes	characteristic, complemented, Sylow
124	C₂⁴×C₄	C₂×D₆₇	1	1	yes	characteristic
125	C₂²×M₁₆	Dic₆₇	1	16	yes	complemented
126	C₂²×M₁₆	C₂×D₆₇	1	12	yes	—
127	C₂³×C₈	—	67	16	no	—
128	C₂³×C₈	—	67	12	no	—
129	C₂³×C₈	—	67	1	no	—
130	C₂³×C₈	—	67	1	no	—
131	C₂³×C₈	C₂×D₆₇	1	2	yes	—
132	C₂×#117	—	67	12	no	—
133	C₂×#117	—	67	12	no	—
134	C₂²×C₄⋊C₄	—	67	2	no	—
135	C₂²×C₂²⋊C₄	—	67	2	no	—
136	C₂²×C₄²	—	67	2	no	—
137	#405	—	67	64	no	—
138	C₂×#97	—	67	24	no	—
139	C₂×C₄×C₈	—	67	24	no	—
140	C₂⁵	C₄×D₆₇	1	1	yes	characteristic
141	C₂³×C₄	—	67	4	no	—
142	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
143	C₂³×C₄	C₄×D₆₇	1	6	yes	—
144	C₂³×C₄	C₄×D₆₇	1	8	yes	—
145	C₂³×C₄	C₂×Dic₆₇	1	8	yes	—
146	C₂³×C₄	C₂²×D₆₇	1	6	yes	—
147	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
148	C₂×M₁₆	C₂×Dic₆₇	1	96	yes	complemented
149	C₂×M₁₆	C₂²×D₆₇	1	16	yes	—
150	C₂²×C₈	—	67	8	no	—
151	C₂²×C₈	—	67	8	no	—
152	C₂²×C₈	—	67	96	no	—
153	C₂²×C₈	—	67	6	no	—
154	C₂²×C₈	—	67	6	no	—
155	C₂²×C₈	—	67	16	no	—
156	C₂²×C₈	C₂×Dic₆₇	1	16	yes	—
157	C₂²×C₈	C₂²×D₆₇	1	12	yes	—
158	#117	—	67	48	no	—
159	#117	—	67	16	no	—
160	C₂×C₄⋊C₄	—	67	24	no	—
161	C₂×C₂²⋊C₄	—	67	24	no	—
162	C₂×C₄²	—	67	12	no	—
163	C₂×C₄²	—	67	12	no	—
164	#97	—	67	32	no	—
165	C₄×C₈	—	67	32	no	—
166	C₂⁴	—	2	8	no	—
167	C₂⁴	C₄×Dic₆₇	1	8	yes	—
168	C₂⁴	C₂×C₄×D₆₇	1	6	yes	—
169	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
170	C₂²×C₄	—	67	24	no	—
171	C₂²×C₄	—	134	16	no	—
172	C₂²×C₄	C₄×Dic₆₇	1	48	yes	—
173	C₂²×C₄	C₄×Dic₆₇	1	8	yes	—
174	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
175	C₂²×C₄	C₂×C₄×D₆₇	1	6	yes	—
176	C₂²×C₄	C₂×C₄×D₆₇	1	12	yes	—
177	C₂²×C₄	C₂×C₄×D₆₇	1	24	yes	—
178	C₂²×C₄	C₂×C₄×D₆₇	1	4	yes	—
179	C₂²×C₄	C₂²×Dic₆₇	1	4	yes	—
180	C₂²×C₄	C₂²×Dic₆₇	1	24	yes	—
181	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
182	C₂²×C₄	C₂³×D₆₇	1	4	yes	—
183	M₁₆	C₂²×Dic₆₇	1	64	yes	complemented
184	C₂×C₈	—	67	24	no	—
185	C₂×C₈	—	67	24	no	—
186	C₂×C₈	—	67	64	no	—
187	C₂×C₈	—	67	4	no	—
188	C₂×C₈	—	67	4	no	—
189	C₂×C₈	C₂²×Dic₆₇	1	48	yes	—
190	C₂×C₈	C₂³×D₆₇	1	8	yes	—
191	C₄⋊C₄	—	67	32	no	—
192	C₂²⋊C₄	—	67	32	no	—
193	C₄²	—	67	24	no	—
194	C₄²	—	67	8	no	—
195	C₂³	—	2	32	no	—
196	C₂³	—	2	24	no	—
197	C₂³	C₂×C₄×Dic₆₇	1	24	yes	—
198	C₂³	C₂×C₄×Dic₆₇	1	4	yes	—
199	C₂³	#14524051	1	8	yes	—
200	C₂³	C₂²×C₄×D₆₇	1	4	yes	—
201	C₂³	C₂²×C₄×D₆₇	1	3	yes	—
202	C₂×C₄	—	67	16	no	—
203	C₂×C₄	—	134	48	no	—
204	C₂×C₄	C₂×C₄×Dic₆₇	1	48	yes	—
205	C₂×C₄	C₂×C₄×Dic₆₇	1	12	yes	—
206	C₂×C₄	C₂×C₄×Dic₆₇	1	24	yes	—
207	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
208	C₂×C₄	C₂²×C₄×D₆₇	1	4	yes	—
209	C₂×C₄	C₂²×C₄×D₆₇	1	8	yes	—
210	C₂×C₄	C₂²×C₄×D₆₇	1	6	yes	—
211	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
212	C₂×C₄	C₂³×Dic₆₇	1	8	yes	—
213	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
214	C₈	—	67	8	no	—
215	C₈	—	67	8	no	—
216	C₈	C₂³×Dic₆₇	1	16	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	48	no	—
219	C₂²	#16863354	1	16	yes	direct product factor
220	C₂²	C₂²×C₄×Dic₆₇	1	8	yes	—
221	C₂²	C₂²×C₄×Dic₆₇	1	6	yes	—
222	C₂²	C₂×#14524051	1	12	yes	—
223	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
224	C₄	—	134	16	no	—
225	C₄	C₂²×C₄×Dic₆₇	1	6	yes	—
226	C₄	C₂²×C₄×Dic₆₇	1	8	yes	—
227	C₄	C₂³×C₄×D₆₇	1	1	yes	characteristic
228	C₄	C₂⁴×Dic₆₇	1	1	yes	characteristic
229	C₂	—	2	8	no	—
230	C₂	C₂×#16863354	1	12	yes	direct product factor
231	C₂	C₂³×C₄×Dic₆₇	1	1	yes	characteristic
232	C₂	C₂²×#14524051	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂³×M₁₆×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁵×C₆₇	C₂×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂³×M₁₆	D₆₇
hypocenter	C₆₇	C₂²×#405
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#16863354	C₂	direct (N × H)
4. C₂²×#14523953	C₂	semidirect
5. C₂²×C₈×Dic₆₇	C₂	semidirect
7. C₂³×C₆₇⋊C₈	C₄	semidirect
16. #16863354	C₂²	direct (N × H)
17. C₂×#14523953	C₂²	semidirect
18. C₂×C₈×Dic₆₇	C₂²	semidirect
20. C₂²×M₁₆×C₆₇	C₄	semidirect
26. C₂²×C₆₇⋊C₈	C₂×C₄	semidirect
36. #14523953	C₂³	semidirect
37. C₈×Dic₆₇	C₂³	semidirect
45. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
53. C₂×C₆₇⋊C₈	C₂²×C₄	semidirect
75. M₁₆×C₆₇	C₂²×C₄	semidirect
102. C₂²×C₆₇	#405	semidirect
114. C₂×C₆₇	C₂×#405	semidirect
123. C₆₇	C₂²×#405	semidirect
125. C₂²×M₁₆	Dic₆₇	semidirect
148. C₂×M₁₆	C₂×Dic₆₇	semidirect
183. M₁₆	C₂²×Dic₆₇	semidirect
219. C₂²	#16863354	direct (N × H)
230. C₂	C₂×#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
C₂⁴×C₆₇⋊C₈	2
C₂²×#14524002	2
C₂×#16863354	2
C₂²×#14523953	2
C₂²×C₈×Dic₆₇	2
C₂³×M₁₆×C₆₇	2
C₂²×#405	67

Subgroups — C22×#16863354 #31719225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863354 #31719225