Subgroups — C₂²×#16863806 #31725442

Order 17152.

2142 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524043	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863806	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523976	C₂	1	2	yes	maximal
4	C₂²×#14523974	C₂	1	2	yes	maximal
5	C₂²×#14523954	C₂	1	1	yes	maximal, characteristic
6	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₃₄	—	2	1	no	—
8	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#14524043	—	2	6	no	—
10	C₂×#14524043	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
12	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	#16863806	C₂²	1	64	yes	direct product factor
14	C₂×#14523976	C₂²	1	24	yes	—
15	C₂×#14523974	C₂²	1	24	yes	—
16	C₂×#14523954	C₂²	1	12	yes	—
17	C₂×#125×C₆₇	—	2	6	no	—
18	C₂×#125×C₆₇	C₂²	1	12	yes	—
19	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₃₄	—	2	12	no	—
25	C₂²×Dic₁₃₄	—	4	8	no	—
26	C₂³×Dic₆₇	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	Q₈	1	16	yes	complemented
28	C₂²×C₆₇⋊C₈	C₂³	1	12	yes	—
29	#14524043	—	2	24	no	—
30	#14524043	C₂³	1	16	yes	—
31	C₂×C₄⋊Dic₆₇	—	2	12	no	—
32	C₂×C₄⋊Dic₆₇	—	2	6	no	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
34	#14523976	C₂³	1	32	yes	—
35	#14523974	C₂³	1	32	yes	—
36	#14523954	C₂³	1	16	yes	—
37	C₂³×C₄×C₆₇	—	2	1	no	—
38	C₂³×C₄×C₆₇	—	2	1	no	—
39	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
42	#125×C₆₇	—	2	24	no	—
43	#125×C₆₇	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
50	C₂×C₄²×C₆₇	D₄	1	6	yes	—
51	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
52	C₂×Dic₁₃₄	—	2	16	no	—
53	C₂×Dic₁₃₄	—	4	48	no	—
54	C₂²×Dic₆₇	—	2	12	no	—
55	C₂²×Dic₆₇	—	4	8	no	—
56	C₂×C₆₇⋊C₈	C₂×Q₈	1	48	yes	complemented
57	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
58	C₄⋊Dic₆₇	—	2	16	no	—
59	C₄⋊Dic₆₇	—	2	12	no	—
60	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
61	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	2	4	no	—
63	C₂²×C₄×C₆₇	—	2	4	no	—
64	C₂²×C₄×C₆₇	—	2	6	no	—
65	C₂²×C₄×C₆₇	—	2	6	no	—
66	C₂²×C₄×C₆₇	—	4	4	no	—
67	C₂²×C₄×C₆₇	—	4	4	no	—
68	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
69	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
70	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
71	C₂²×C₄×C₆₇	C₂×Q₈	1	8	yes	—
72	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
73	C₄⋊C₄×C₆₇	—	2	16	no	—
74	C₄⋊C₄×C₆₇	—	2	16	no	—
75	C₄⋊C₄×C₆₇	—	2	6	no	—
76	C₄⋊C₄×C₆₇	—	2	6	no	—
77	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
78	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
79	C₄²×C₆₇	C₂×D₄	1	12	yes	—
80	C₄²×C₆₇	C₂⁴	1	4	yes	—
81	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
82	C₂³×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
83	C₂³×C₆₇	#137	1	4	yes	—
84	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
85	C₂×C₄×C₆₇	—	2	12	no	—
86	C₂×C₄×C₆₇	—	2	12	no	—
87	C₂×C₄×C₆₇	—	2	4	no	—
88	C₂×C₄×C₆₇	—	2	4	no	—
89	C₂×C₄×C₆₇	—	4	12	no	—
90	C₂×C₄×C₆₇	—	4	12	no	—
91	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
92	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
93	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
94	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
95	C₂×C₄×C₆₇	C₂²×Q₈	1	12	yes	—
96	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
97	C₂×Dic₆₇	—	2	8	no	—
98	C₂×Dic₆₇	—	4	24	no	—
99	Dic₁₃₄	—	4	32	no	—
100	C₆₇⋊C₈	C₂²×Q₈	1	16	yes	complemented
101	C₂²×C₆₇	#501	1	16	yes	complemented
102	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
103	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	6	yes	—
104	C₂²×C₆₇	C₂×#137	1	6	yes	—
105	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	4	4	no	—
109	C₄×C₆₇	—	4	4	no	—
110	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
111	C₄×C₆₇	C₂³×Q₈	1	2	yes	—
112	Dic₆₇	—	4	8	no	—
113	C₂²×#501	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	C₂×#501	1	12	yes	complemented
115	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
118	C₂²×C₄⋊Q₈	—	67	1	no	—
119	C₂²×#125	D₆₇	1	1	yes	characteristic
120	C₂×#501	—	67	24	no	—
121	C₂²×#107	—	67	2	no	—
122	C₂²×#106	—	67	2	no	—
123	C₂²×#97	—	67	1	no	—
124	C₆₇	C₂²×#501	1	1	yes	characteristic, complemented, Sylow
125	C₂³×Q₈	—	134	1	no	—
126	C₂³×C₈	—	67	2	no	—
127	C₂×C₄⋊Q₈	—	67	12	no	—
128	C₂×C₄⋊Q₈	—	134	6	no	—
129	C₂×#125	—	2	6	no	—
130	C₂×#125	C₂×D₆₇	1	12	yes	—
131	C₂²×C₄⋊C₄	—	2	1	no	—
132	C₂²×C₄⋊C₄	—	2	1	no	—
133	C₂²×C₄⋊C₄	—	67	2	no	—
134	C₂²×C₄⋊C₄	—	134	1	no	—
135	C₂²×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
136	C₂²×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
137	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
138	#501	—	67	64	no	—
139	C₂×#107	—	67	24	no	—
140	C₂×#106	—	67	24	no	—
141	C₂×#97	—	67	12	no	—
142	C₂²×Q₈	—	268	8	no	—
143	C₂²×Q₈	—	134	12	no	—
144	C₂³×C₄	—	2	1	no	—
145	C₂³×C₄	—	2	1	no	—
146	C₂³×C₄	—	134	2	no	—
147	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
149	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
150	C₂²×C₈	—	67	16	no	—
151	C₂²×C₈	—	67	12	no	—
152	C₄⋊Q₈	—	67	16	no	—
153	C₄⋊Q₈	—	134	24	no	—
154	#125	—	2	24	no	—
155	#125	C₂²×D₆₇	1	16	yes	—
156	C₂×C₄⋊C₄	—	2	12	no	—
157	C₂×C₄⋊C₄	—	2	12	no	—
158	C₂×C₄⋊C₄	—	2	3	no	—
159	C₂×C₄⋊C₄	—	2	3	no	—
160	C₂×C₄⋊C₄	—	67	12	no	—
161	C₂×C₄⋊C₄	—	134	6	no	—
162	C₂×C₄⋊C₄	—	134	12	no	—
163	C₂×C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
164	C₂×C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
165	C₂×C₄²	C₂²⋊D₆₇	1	6	yes	—
166	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
167	#107	—	67	32	no	—
168	#106	—	67	32	no	—
169	#97	—	67	16	no	—
170	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
171	C₂×Q₈	—	268	48	no	—
172	C₂×Q₈	—	134	16	no	—
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	2	6	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	—	268	8	no	—
180	C₂²×C₄	—	134	12	no	—
181	C₂²×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
183	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
184	C₂²×C₄	Q₈×D₆₇	1	8	yes	—
185	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
186	C₂×C₈	—	67	48	no	—
187	C₂×C₈	—	67	8	no	—
188	C₄⋊C₄	—	2	16	no	—
189	C₄⋊C₄	—	2	16	no	—
190	C₄⋊C₄	—	2	6	no	—
191	C₄⋊C₄	—	2	6	no	—
192	C₄⋊C₄	—	67	8	no	—
193	C₄⋊C₄	—	134	12	no	—
194	C₄⋊C₄	—	134	16	no	—
195	C₄⋊C₄	C₂³×D₆₇	1	4	yes	—
196	C₄⋊C₄	C₂³×D₆₇	1	4	yes	—
197	C₄²	C₂×C₂²⋊D₆₇	1	12	yes	—
198	C₄²	C₂³×D₆₇	1	4	yes	—
199	C₂³	#14524041	1	4	yes	—
200	C₂³	#14524068	1	4	yes	—
201	C₂³	#14524073	1	4	yes	—
202	C₂³	C₂²×C₂²⋊D₆₇	1	3	yes	—
203	Q₈	—	268	32	no	—
204	C₂×C₄	—	2	12	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	12	no	—
209	C₂×C₄	—	4	12	no	—
210	C₂×C₄	—	268	24	no	—
211	C₂×C₄	—	134	8	no	—
212	C₂×C₄	#14524041	1	12	yes	—
213	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
214	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
215	C₂×C₄	C₂²×C₂²⋊D₆₇	1	3	yes	—
216	C₂×C₄	C₂×Q₈×D₆₇	1	12	yes	—
217	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
218	C₈	—	67	16	no	—
219	C₂²	#16863806	1	16	yes	direct product factor
220	C₂²	C₂×#14524041	1	6	yes	—
221	C₂²	C₂×#14524068	1	6	yes	—
222	C₂²	C₂×#14524073	1	6	yes	—
223	C₂²	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
224	C₄	—	2	4	no	—
225	C₄	—	2	4	no	—
226	C₄	—	4	4	no	—
227	C₄	—	4	4	no	—
228	C₄	—	268	8	no	—
229	C₄	C₂×#14524041	1	6	yes	—
230	C₄	C₂²×Q₈×D₆₇	1	2	yes	—
231	C₂	C₂×#16863806	1	12	yes	direct product factor
232	C₂	C₂²×#14524041	1	1	yes	characteristic
233	C₂	C₂²×#14524068	1	1	yes	characteristic
234	C₂	C₂²×#14524073	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₆₇
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×#125×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂²×#125	D₆₇
hypocenter	C₆₇	C₂²×#501
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#125

Complements

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

N	H ≅ G/N	type
2. C₂×#16863806	C₂	direct (N × H)
13. #16863806	C₂²	direct (N × H)
27. C₂²×C₆₇⋊C₈	Q₈	semidirect
56. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
100. C₆₇⋊C₈	C₂²×Q₈	semidirect
101. C₂²×C₆₇	#501	semidirect
114. C₂×C₆₇	C₂×#501	semidirect
124. C₆₇	C₂²×#501	semidirect
219. C₂²	#16863806	direct (N × H)
231. C₂	C₂×#16863806	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#14524043	2
C₂×#16863806	2
C₂²×#14523976	2
C₂²×#14523974	2
C₂²×#14523954	2
C₂²×#125×C₆₇	2
C₂²×#501	67

Subgroups — C22×#16863806 #31725442

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863806 #31725442