Subgroups — C₂²×#16863795 #31725422

Order 17152.

1966 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524035	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863795	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×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂²×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14524035	—	2	6	no	—
10	C₂×#14524035	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
12	C₂²×Dic₆₇⋊C₄	—	2	2	no	—
13	#16863795	C₂²	1	64	yes	direct product factor
14	C₂×#14523976	C₂²	1	24	yes	—
15	C₂×#14523974	C₂²	1	24	yes	—
16	C₂×C₄×C₆₇⋊C₈	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	2	no	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
22	C₂³×Dic₆₇	—	2	2	no	—
23	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	—
24	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	—
25	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
26	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
27	#14524035	—	2	24	no	—
28	#14524035	C₂³	1	16	yes	—
29	C₂×C₄⋊Dic₆₇	—	2	6	no	—
30	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
31	C₂×Dic₆₇⋊C₄	—	2	24	no	—
32	#14523976	C₂³	1	32	yes	—
33	#14523974	C₂³	1	32	yes	—
34	C₄×C₆₇⋊C₈	C₂³	1	16	yes	—
35	C₂³×C₄×C₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	D₄	1	2	yes	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	#125×C₆₇	—	2	24	no	—
39	#125×C₆₇	C₂³	1	16	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	24	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
42	C₂×C₄⋊C₄×C₆₇	C₂³	1	12	yes	—
43	C₂×C₄²×C₆₇	D₄	1	6	yes	—
44	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
45	C₂²×Dic₆₇	—	2	12	no	—
46	C₂²×Dic₆₇	—	4	8	no	—
47	C₂×C₆₇⋊C₈	C₂×Q₈	1	24	yes	—
48	C₂×C₆₇⋊C₈	C₂×Q₈	1	24	yes	—
49	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
50	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
51	C₄⋊Dic₆₇	—	2	12	no	—
52	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
53	Dic₆₇⋊C₄	—	2	32	no	—
54	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	—	2	8	no	—
56	C₂²×C₄×C₆₇	—	2	12	no	—
57	C₂²×C₄×C₆₇	—	4	8	no	—
58	C₂²×C₄×C₆₇	C₂×D₄	1	12	yes	—
59	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
60	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
61	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
63	C₄⋊C₄×C₆₇	—	2	32	no	—
64	C₄⋊C₄×C₆₇	—	2	12	no	—
65	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
66	C₄²×C₆₇	C₂×D₄	1	12	yes	—
67	C₄²×C₆₇	C₂⁴	1	4	yes	—
68	C₂³×C₆₇	C₄⋊Q₈	1	4	yes	—
69	C₂³×C₆₇	#135	1	8	yes	—
70	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
71	C₂×C₄×C₆₇	—	2	24	no	—
72	C₂×C₄×C₆₇	—	2	8	no	—
73	C₂×C₄×C₆₇	—	4	24	no	—
74	C₂×C₄×C₆₇	C₄⋊Q₈	1	12	yes	—
75	C₂×C₄×C₆₇	C₂²×D₄	1	8	yes	—
76	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
77	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
78	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
79	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
80	C₂×Dic₆₇	—	2	8	no	—
81	C₂×Dic₆₇	—	4	24	no	—
82	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	—
83	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	—
84	C₂²×C₆₇	#499	1	16	yes	complemented
85	C₂²×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
86	C₂²×C₆₇	C₂×#135	1	12	yes	—
87	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
88	C₄×C₆₇	—	2	8	no	—
89	C₄×C₆₇	—	4	8	no	—
90	C₄×C₆₇	C₂×C₄⋊Q₈	1	6	yes	—
91	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
92	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
93	Dic₆₇	—	4	8	no	—
94	C₂²×#499	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	C₂×#499	1	12	yes	complemented
96	C₂×C₆₇	C₂²×C₄⋊Q₈	1	1	yes	characteristic
97	C₂×C₆₇	C₂²×#135	1	2	yes	—
98	C₂²×#125	—	67	1	no	—
99	C₂²×#125	D₆₇	1	1	yes	characteristic
100	C₂×#499	—	67	24	no	—
101	C₂²×#107	—	67	2	no	—
102	C₂²×#106	—	67	2	no	—
103	C₂²×C₄×C₈	—	67	1	no	—
104	C₆₇	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
105	C₂³×C₈	—	67	1	no	—
106	C₂³×C₈	—	67	1	no	—
107	C₂×#125	—	2	6	no	—
108	C₂×#125	—	67	12	no	—
109	C₂×#125	—	134	6	no	—
110	C₂×#125	C₂×D₆₇	1	12	yes	—
111	C₂²×C₄⋊C₄	—	2	2	no	—
112	C₂²×C₄⋊C₄	—	67	2	no	—
113	C₂²×C₄⋊C₄	—	134	2	no	—
114	C₂²×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
115	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
116	#499	—	67	64	no	—
117	C₂×#107	—	67	24	no	—
118	C₂×#106	—	67	24	no	—
119	C₂×C₄×C₈	—	67	12	no	—
120	C₂³×C₄	—	2	2	no	—
121	C₂³×C₄	—	134	2	no	—
122	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
123	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
124	C₂²×C₈	—	67	8	no	—
125	C₂²×C₈	—	67	8	no	—
126	C₂²×C₈	—	67	6	no	—
127	C₂²×C₈	—	67	6	no	—
128	#125	—	2	24	no	—
129	#125	—	67	16	no	—
130	#125	—	134	24	no	—
131	#125	C₂²×D₆₇	1	16	yes	—
132	C₂×C₄⋊C₄	—	2	24	no	—
133	C₂×C₄⋊C₄	—	2	6	no	—
134	C₂×C₄⋊C₄	—	67	12	no	—
135	C₂×C₄⋊C₄	—	134	6	no	—
136	C₂×C₄⋊C₄	—	134	24	no	—
137	C₂×C₄⋊C₄	C₂²×D₆₇	1	12	yes	—
138	C₂×C₄²	C₂²⋊D₆₇	1	6	yes	—
139	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
140	#107	—	67	32	no	—
141	#106	—	67	32	no	—
142	C₄×C₈	—	67	16	no	—
143	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
144	C₂²×C₄	—	2	8	no	—
145	C₂²×C₄	—	2	12	no	—
146	C₂²×C₄	—	4	8	no	—
147	C₂²×C₄	—	268	8	no	—
148	C₂²×C₄	—	134	12	no	—
149	C₂²×C₄	C₂×C₂²⋊D₆₇	1	12	yes	—
150	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
151	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
152	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
153	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
154	C₂×C₈	—	67	24	no	—
155	C₂×C₈	—	67	24	no	—
156	C₂×C₈	—	67	4	no	—
157	C₂×C₈	—	67	4	no	—
158	C₄⋊C₄	—	2	32	no	—
159	C₄⋊C₄	—	2	12	no	—
160	C₄⋊C₄	—	67	8	no	—
161	C₄⋊C₄	—	134	12	no	—
162	C₄⋊C₄	—	134	32	no	—
163	C₄⋊C₄	C₂³×D₆₇	1	8	yes	—
164	C₄²	C₂×C₂²⋊D₆₇	1	12	yes	—
165	C₄²	C₂³×D₆₇	1	4	yes	—
166	C₂³	#14524041	1	4	yes	—
167	C₂³	#14524063	1	8	yes	—
168	C₂³	C₂²×C₂²⋊D₆₇	1	3	yes	—
169	C₂×C₄	—	2	24	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	4	24	no	—
172	C₂×C₄	—	268	24	no	—
173	C₂×C₄	—	134	8	no	—
174	C₂×C₄	#14524041	1	12	yes	—
175	C₂×C₄	C₂²×C₂²⋊D₆₇	1	8	yes	—
176	C₂×C₄	C₂²×C₂²⋊D₆₇	1	3	yes	—
177	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
178	C₂×C₄	C₂×Q₈×D₆₇	1	6	yes	—
179	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
180	C₈	—	67	8	no	—
181	C₈	—	67	8	no	—
182	C₂²	#16863795	1	16	yes	direct product factor
183	C₂²	C₂×#14524041	1	6	yes	—
184	C₂²	C₂×#14524063	1	12	yes	—
185	C₂²	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
186	C₄	—	2	8	no	—
187	C₄	—	4	8	no	—
188	C₄	—	268	8	no	—
189	C₄	C₂×#14524041	1	6	yes	—
190	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
191	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
192	C₂	C₂×#16863795	1	12	yes	direct product factor
193	C₂	C₂²×#14524041	1	1	yes	characteristic
194	C₂	C₂²×#14524063	1	2	yes	—

	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×#499
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×#16863795	C₂	direct (N × H)
13. #16863795	C₂²	direct (N × H)
84. C₂²×C₆₇	#499	semidirect
95. C₂×C₆₇	C₂×#499	semidirect
104. C₆₇	C₂²×#499	semidirect
182. C₂²	#16863795	direct (N × H)
192. C₂	C₂×#16863795	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×#14524035	2
C₂×#16863795	2
C₂²×#14523976	2
C₂²×#14523974	2
C₂²×C₄×C₆₇⋊C₈	2
C₂²×#125×C₆₇	2
C₂²×#499	67

Subgroups — C22×#16863795 #31725422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863795 #31725422