Subgroups — C₂²×#16863348 #31719218

Order 17152.

3950 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761611	C₂	1	2	yes	maximal
2	C₂²×C₄²×D₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863348	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#14523954	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₈×Dic₆₇	C₂	1	2	yes	maximal, complemented
6	C₂²×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₄×D₆₇	C₄	1	2	yes	—
8	C₂³×C₄×D₆₇	C₂²	1	1	yes	characteristic
9	C₂²×#11761611	C₄	1	32	yes	complemented
10	C₂²×#11761611	C₂²	1	24	yes	—
11	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₆₇	C₄	1	2	yes	—
13	C₂²×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×D₆₇	C₄	1	12	yes	—
15	C₂×C₄²×D₆₇	C₂²	1	12	yes	—
16	#16863348	C₂²	1	64	yes	direct product factor
17	C₂×#14523954	C₂²	1	12	yes	complemented
18	C₂×C₈×Dic₆₇	C₂²	1	24	yes	complemented
19	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₆₇	C₂²	1	12	yes	complemented
22	C₂⁴×D₆₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂³×Dic₆₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×D₆₇	—	2	16	no	—
26	C₂²×C₄×D₆₇	C₂×C₄	1	24	yes	—
27	C₂²×C₄×D₆₇	C₂×C₄	1	6	yes	—
28	C₂²×C₄×D₆₇	C₂×C₄	1	16	yes	—
29	C₂²×C₄×D₆₇	C₂³	1	6	yes	—
30	C₂×#11761611	C₂×C₄	1	192	yes	complemented
31	C₂×#11761611	C₂³	1	32	yes	—
32	C₂²×C₆₇⋊C₈	C₂×C₄	1	16	yes	complemented
33	C₂²×C₆₇⋊C₈	C₂³	1	12	yes	—
34	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₆₇	C₂×C₄	1	24	yes	—
36	C₂×C₄×Dic₆₇	C₂³	1	6	yes	—
37	C₄²×D₆₇	C₂×C₄	1	48	yes	—
38	C₄²×D₆₇	C₂³	1	16	yes	—
39	#14523954	C₂³	1	16	yes	complemented
40	C₈×Dic₆₇	C₂³	1	32	yes	complemented
41	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	16	yes	complemented
44	C₂²×C₈×C₆₇	C₂³	1	12	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	6	yes	—
46	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
47	#97×C₆₇	C₂³	1	16	yes	complemented
48	C₂³×D₆₇	—	2	8	no	—
49	C₂³×D₆₇	C₄²	1	8	yes	—
50	C₂³×D₆₇	C₂²×C₄	1	6	yes	—
51	C₂²×Dic₆₇	C₄²	1	8	yes	—
52	C₂²×Dic₆₇	M₁₆	1	16	yes	complemented
53	C₂²×Dic₆₇	C₂²×C₄	1	6	yes	—
54	C₂²×Dic₆₇	C₂²×C₄	1	12	yes	—
55	C₂×C₄×D₆₇	—	2	96	no	—
56	C₂×C₄×D₆₇	C₄²	1	48	yes	—
57	C₂×C₄×D₆₇	C₂²×C₄	1	32	yes	—
58	C₂×C₄×D₆₇	C₂²×C₄	1	12	yes	—
59	C₂×C₄×D₆₇	C₂²×C₄	1	48	yes	—
60	C₂×C₄×D₆₇	C₂⁴	1	4	yes	—
61	#11761611	C₂²×C₄	1	128	yes	complemented
62	C₂×C₆₇⋊C₈	C₂²×C₄	1	48	yes	complemented
63	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
64	C₄×Dic₆₇	C₂²×C₄	1	12	yes	—
65	C₄×Dic₆₇	C₂²×C₄	1	32	yes	—
66	C₄×Dic₆₇	C₂⁴	1	4	yes	—
67	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	—	2	8	no	—
69	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	48	yes	complemented
74	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
75	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
76	C₄²×C₆₇	C₂⁴	1	4	yes	—
77	C₂³×C₆₇	C₂×C₄²	1	4	yes	—
78	C₂³×C₆₇	C₂×M₁₆	1	8	yes	—
79	C₂³×C₆₇	C₂³×C₄	1	3	yes	—
80	C₂²×D₆₇	—	2	24	no	—
81	C₂²×D₆₇	—	2	32	no	—
82	C₂²×D₆₇	C₂×C₄²	1	24	yes	—
83	C₂²×D₆₇	C₂³×C₄	1	4	yes	—
84	C₂×C₄×C₆₇	—	2	24	no	—
85	C₂×C₄×C₆₇	C₂×C₄²	1	12	yes	—
86	C₂×C₄×C₆₇	C₂³×C₄	1	8	yes	—
87	C₂×C₄×C₆₇	C₂³×C₄	1	12	yes	—
88	C₂×C₄×C₆₇	C₂³×C₄	1	3	yes	—
89	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
90	C₂×Dic₆₇	C₂×C₄²	1	24	yes	—
91	C₂×Dic₆₇	C₂×M₁₆	1	48	yes	complemented
92	C₂×Dic₆₇	C₂³×C₄	1	8	yes	—
93	C₂×Dic₆₇	C₂³×C₄	1	4	yes	—
94	C₄×D₆₇	—	2	64	no	—
95	C₄×D₆₇	C₂×C₄²	1	48	yes	—
96	C₄×D₆₇	C₂³×C₄	1	16	yes	—
97	C₈×C₆₇	C₂³×C₄	1	16	yes	complemented
98	C₆₇⋊C₈	C₂³×C₄	1	16	yes	complemented
99	C₂²×C₆₇	C₄×M₁₆	1	16	yes	complemented
100	C₂²×C₆₇	C₂²×C₄²	1	6	yes	—
101	C₂²×C₆₇	C₂²×M₁₆	1	12	yes	—
102	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
103	C₂×D₆₇	—	2	48	no	—
104	C₂×D₆₇	—	2	8	no	—
105	C₂×D₆₇	C₂²×C₄²	1	8	yes	—
106	C₄×C₆₇	—	2	8	no	—
107	C₄×C₆₇	C₂²×C₄²	1	6	yes	—
108	C₄×C₆₇	C₂⁴×C₄	1	2	yes	—
109	Dic₆₇	C₂²×C₄²	1	8	yes	—
110	Dic₆₇	C₂²×M₁₆	1	16	yes	complemented
111	C₂²×C₄×M₁₆	—	67	1	no	maximal, Sylow
112	C₂×C₆₇	C₂×C₄×M₁₆	1	12	yes	complemented
113	C₂×C₆₇	C₂³×C₄²	1	1	yes	characteristic
114	C₂×C₆₇	C₂³×M₁₆	1	2	yes	—
115	D₆₇	—	2	8	no	—
116	C₂³×M₁₆	—	67	2	no	—
117	C₂³×C₄²	—	67	1	no	—
118	C₂×C₄×M₁₆	—	67	24	no	—
119	C₂²×#97	—	67	1	no	—
120	C₂²×#97	D₆₇	1	1	yes	characteristic, complemented
121	C₂²×C₄×C₈	—	67	2	no	—
122	C₆₇	C₂²×C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
123	C₂⁴×C₄	—	67	1	no	—
124	C₂⁴×C₄	—	67	2	no	—
125	C₂²×M₁₆	—	67	32	no	—
126	C₂²×M₁₆	—	67	24	no	—
127	C₂³×C₈	—	67	2	no	—
128	C₂³×C₈	C₂×D₆₇	1	2	yes	—
129	C₂²×C₄²	—	67	2	no	—
130	C₂²×C₄²	—	67	1	no	—
131	C₂²×C₄²	—	67	12	no	—
132	C₂²×C₄²	—	67	12	no	—
133	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
134	C₄×M₁₆	—	67	64	no	—
135	C₂×#97	—	67	12	no	—
136	C₂×#97	C₂×D₆₇	1	12	yes	complemented
137	C₂×C₄×C₈	—	67	24	no	—
138	C₂⁵	—	67	1	no	—
139	C₂³×C₄	—	67	1	no	—
140	C₂³×C₄	—	67	2	no	—
141	C₂³×C₄	—	67	16	no	—
142	C₂³×C₄	—	67	6	no	—
143	C₂³×C₄	—	67	6	no	—
144	C₂³×C₄	—	67	24	no	—
145	C₂³×C₄	—	134	16	no	—
146	C₂³×C₄	C₄×D₆₇	1	2	yes	—
147	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
148	C₂×M₁₆	—	67	192	no	—
149	C₂×M₁₆	—	67	32	no	—
150	C₂²×C₈	—	67	16	no	—
151	C₂²×C₈	—	67	12	no	—
152	C₂²×C₈	C₄×D₆₇	1	16	yes	complemented
153	C₂²×C₈	C₂²×D₆₇	1	12	yes	—
154	C₂×C₄²	—	67	24	no	—
155	C₂×C₄²	—	67	6	no	—
156	C₂×C₄²	—	67	6	no	—
157	C₂×C₄²	—	67	48	no	—
158	C₂×C₄²	—	67	16	no	—
159	C₂×C₄²	C₄×D₆₇	1	6	yes	—
160	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
161	#97	—	67	16	no	—
162	#97	C₂²×D₆₇	1	16	yes	complemented
163	C₄×C₈	—	67	32	no	—
164	C₂⁴	—	67	8	no	—
165	C₂⁴	—	67	6	no	—
166	C₂⁴	—	134	8	no	—
167	C₂⁴	C₂×C₄×D₆₇	1	1	yes	characteristic
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	67	8	no	—
170	C₂²×C₄	—	67	16	no	—
171	C₂²×C₄	—	67	6	no	—
172	C₂²×C₄	—	67	12	no	—
173	C₂²×C₄	—	67	48	no	—
174	C₂²×C₄	—	67	12	no	—
175	C₂²×C₄	—	67	48	no	—
176	C₂²×C₄	—	67	4	no	—
177	C₂²×C₄	—	67	32	no	—
178	C₂²×C₄	—	134	96	no	—
179	C₂²×C₄	C₂×C₄×D₆₇	1	12	yes	—
180	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
181	C₂²×C₄	C₂×C₄×D₆₇	1	8	yes	—
182	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
183	M₁₆	—	67	128	no	—
184	C₂×C₈	—	67	48	no	—
185	C₂×C₈	—	67	8	no	—
186	C₂×C₈	C₂×C₄×D₆₇	1	48	yes	complemented
187	C₂×C₈	C₂³×D₆₇	1	8	yes	—
188	C₄²	—	67	32	no	—
189	C₄²	—	67	12	no	—
190	C₄²	—	67	4	no	—
191	C₄²	C₂×C₄×D₆₇	1	12	yes	—
192	C₄²	C₂³×D₆₇	1	4	yes	—
193	C₂³	—	67	24	no	—
194	C₂³	—	67	4	no	—
195	C₂³	—	134	32	no	—
196	C₂³	—	134	24	no	—
197	C₂³	C₄²×D₆₇	1	4	yes	—
198	C₂³	M₁₆×D₆₇	1	8	yes	—
199	C₂³	C₂²×C₄×D₆₇	1	3	yes	—
200	C₂×C₄	—	2	24	no	—
201	C₂×C₄	—	67	24	no	—
202	C₂×C₄	—	67	48	no	—
203	C₂×C₄	—	67	4	no	—
204	C₂×C₄	—	67	8	no	—
205	C₂×C₄	—	67	48	no	—
206	C₂×C₄	—	67	16	no	—
207	C₂×C₄	—	134	64	no	—
208	C₂×C₄	C₄²×D₆₇	1	12	yes	—
209	C₂×C₄	C₂²×C₄×D₆₇	1	8	yes	—
210	C₂×C₄	C₂²×C₄×D₆₇	1	3	yes	—
211	C₂×C₄	C₂²×C₄×D₆₇	1	12	yes	—
212	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
213	C₈	—	67	16	no	—
214	C₈	C₂²×C₄×D₆₇	1	16	yes	complemented
215	C₂²	—	67	8	no	—
216	C₂²	—	134	8	no	—
217	C₂²	—	134	48	no	—
218	C₂²	#16863348	1	16	yes	direct product factor
219	C₂²	C₂×C₄²×D₆₇	1	6	yes	—
220	C₂²	C₂×M₁₆×D₆₇	1	12	yes	—
221	C₂²	C₂³×C₄×D₆₇	1	1	yes	characteristic
222	C₄	—	2	8	no	—
223	C₄	—	67	8	no	—
224	C₄	—	67	16	no	—
225	C₄	C₂×C₄²×D₆₇	1	6	yes	—
226	C₄	C₂³×C₄×D₆₇	1	2	yes	—
227	C₂	—	134	8	no	—
228	C₂	C₂×#16863348	1	12	yes	direct product factor
229	C₂	C₂²×C₄²×D₆₇	1	1	yes	characteristic
230	C₂	C₂²×M₁₆×D₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₆₇
commutator	C₂×C₆₇	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×#97×C₆₇	C₂
socle	C₂⁴×C₆₇	C₂²×C₄
prime radical	C₂⁴×D₆₇	C₂×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×#97	D₆₇
hypocenter	C₆₇	C₂²×C₄×M₁₆
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. C₂×#16863348	C₂	direct (N × H)
4. C₂²×#14523954	C₂	semidirect
5. C₂²×C₈×Dic₆₇	C₂	semidirect
6. C₂²×#97×C₆₇	C₂	semidirect
9. C₂²×#11761611	C₄	semidirect
16. #16863348	C₂²	direct (N × H)
17. C₂×#14523954	C₂²	semidirect
18. C₂×C₈×Dic₆₇	C₂²	semidirect
21. C₂×#97×C₆₇	C₂²	semidirect
30. C₂×#11761611	C₂×C₄	semidirect
32. C₂²×C₆₇⋊C₈	C₂×C₄	semidirect
39. #14523954	C₂³	semidirect
40. C₈×Dic₆₇	C₂³	semidirect
43. C₂²×C₈×C₆₇	C₂×C₄	semidirect
47. #97×C₆₇	C₂³	semidirect
52. C₂²×Dic₆₇	M₁₆	semidirect
61. #11761611	C₂²×C₄	semidirect
62. C₂×C₆₇⋊C₈	C₂²×C₄	semidirect
73. C₂×C₈×C₆₇	C₂²×C₄	semidirect
91. C₂×Dic₆₇	C₂×M₁₆	semidirect
97. C₈×C₆₇	C₂³×C₄	semidirect
98. C₆₇⋊C₈	C₂³×C₄	semidirect
99. C₂²×C₆₇	C₄×M₁₆	semidirect
110. Dic₆₇	C₂²×M₁₆	semidirect
112. C₂×C₆₇	C₂×C₄×M₁₆	semidirect
120. C₂²×#97	D₆₇	semidirect
122. C₆₇	C₂²×C₄×M₁₆	semidirect
136. C₂×#97	C₂×D₆₇	semidirect
152. C₂²×C₈	C₄×D₆₇	semidirect
162. #97	C₂²×D₆₇	semidirect
186. C₂×C₈	C₂×C₄×D₆₇	semidirect
214. C₈	C₂²×C₄×D₆₇	semidirect
218. C₂²	#16863348	direct (N × H)
228. C₂	C₂×#16863348	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×#11761611	2
C₂²×C₄²×D₆₇	2
C₂×#16863348	2
C₂²×#14523954	2
C₂²×C₈×Dic₆₇	2
C₂²×#97×C₆₇	2
C₂²×C₄×M₁₆	67

Subgroups — C22×#16863348 #31719218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863348 #31719218