Subgroups — C₂²×#16863225 #31008882

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14523979	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863225	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#108×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	—	2	1	no	—
6	C₂²×C₆₇⋊C₁₆	C₄	1	6	yes	—
7	C₂²×C₆₇⋊C₁₆	C₄	1	2	yes	—
8	C₂²×C₆₇⋊C₁₆	C₂²	1	6	yes	—
9	C₂×#14523979	—	2	3	no	—
10	C₂×#14523979	C₂²	1	6	yes	—
11	#16863225	C₂²	1	16	yes	direct product factor
12	C₂²×M₁₆×C₆₇	—	2	1	no	—
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₇	—	2	3	no	—
15	C₂×#108×C₆₇	C₂²	1	6	yes	—
16	C₂×#11761612	—	2	6	no	—
17	C₂×#11761612	—	4	3	no	—
18	C₂²×C₆₇⋊C₈	—	4	1	no	—
19	C₂×C₆₇⋊C₁₆	C₂×C₄	1	12	yes	—
20	C₂×C₆₇⋊C₁₆	C₂×C₄	1	12	yes	—
21	C₂×C₆₇⋊C₁₆	C₂³	1	4	yes	—
22	#14523979	—	2	6	no	—
23	#14523979	C₂³	1	4	yes	—
24	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
25	C₂×M₁₆×C₆₇	—	2	6	no	—
26	C₂×M₁₆×C₆₇	—	4	3	no	—
27	C₂²×C₈×C₆₇	—	4	1	no	—
28	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	D₄	1	3	yes	—
31	C₂²×C₈×C₆₇	Q₈	1	3	yes	—
32	C₂²×C₈×C₆₇	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₆₇	C₂³	1	3	yes	—
34	#108×C₆₇	—	2	6	no	—
35	#108×C₆₇	C₂³	1	4	yes	—
36	#11761612	—	2	4	no	—
37	#11761612	—	4	6	no	—
38	C₂×C₆₇⋊C₈	—	4	6	no	—
39	C₆₇⋊C₁₆	C₂²×C₄	1	8	yes	—
40	C₂⁴×C₆₇	Q₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	C₄⋊C₄	1	3	yes	—
42	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	D₈	1	3	yes	—
44	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	Q₁₆	1	3	yes	—
46	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
47	M₁₆×C₆₇	—	2	4	no	—
48	M₁₆×C₆₇	—	4	6	no	—
49	C₂×C₈×C₆₇	—	4	6	no	—
50	C₂×C₈×C₆₇	C₄⋊C₄	1	6	yes	—
51	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
52	C₂×C₈×C₆₇	C₄⋊C₄	1	3	yes	—
53	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
54	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
55	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
56	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
57	C₂×C₈×C₆₇	C₂×Q₈	1	3	yes	—
58	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
59	C₂³×C₆₇	—	2	4	no	—
60	C₂³×C₆₇	#107	1	3	yes	—
61	C₂³×C₆₇	#107	1	1	yes	characteristic
62	C₂³×C₆₇	C₂×Q₁₆	1	3	yes	—
63	C₂×C₄×C₆₇	#107	1	6	yes	—
64	C₂×C₄×C₆₇	#107	1	3	yes	—
65	C₂×C₄×C₆₇	#107	1	3	yes	—
66	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
67	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
69	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
70	C₂×C₄×C₆₇	C₂×Q₁₆	1	3	yes	—
71	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
72	C₈×C₆₇	—	4	4	no	—
73	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
74	C₈×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
75	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
76	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
77	C₆₇⋊C₈	—	4	4	no	—
78	C₂²×C₆₇	—	2	6	no	—
79	C₂²×C₆₇	—	2	6	no	—
80	C₂²×C₆₇	#368	1	4	yes	complemented
81	C₂²×C₆₇	C₂×#107	1	3	yes	—
82	C₂²×C₆₇	C₂×#107	1	3	yes	—
83	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
84	C₄×C₆₇	C₂×#107	1	3	yes	—
85	C₄×C₆₇	C₂×#107	1	3	yes	—
86	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
87	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
88	C₂²×#368	—	67	1	no	maximal, Sylow
89	C₂×C₆₇	—	2	1	no	—
90	C₂×C₆₇	—	2	3	no	—
91	C₂×C₆₇	C₂×#368	1	6	yes	complemented
92	C₂×C₆₇	C₂²×#107	1	1	yes	characteristic
93	C₂³×C₁₆	—	67	1	no	—
94	C₂²×#108	—	67	1	no	—
95	C₂²×#108	D₆₇	1	1	yes	characteristic
96	C₂×#368	—	67	12	no	—
97	C₆₇	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
98	C₂²×M₁₆	—	2	1	no	—
99	C₂²×M₁₆	—	134	1	no	—
100	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
101	C₂²×C₁₆	—	67	2	no	—
102	C₂²×C₁₆	—	67	6	no	—
103	C₂²×C₁₆	—	67	6	no	—
104	C₂×#108	—	2	3	no	—
105	C₂×#108	—	67	6	no	—
106	C₂×#108	—	134	3	no	—
107	C₂×#108	C₂×D₆₇	1	6	yes	—
108	#368	—	67	16	no	—
109	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂×M₁₆	—	2	6	no	—
111	C₂×M₁₆	—	4	3	no	—
112	C₂×M₁₆	—	268	3	no	—
113	C₂×M₁₆	—	134	6	no	—
114	C₂²×C₈	—	4	1	no	—
115	C₂²×C₈	—	268	1	no	—
116	C₂²×C₈	Dic₁₃₄	1	1	yes	characteristic
117	C₂²×C₈	Dic₁₃₄	1	3	yes	—
118	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
119	C₂²×C₈	C₄×D₆₇	1	3	yes	—
120	C₂²×C₈	C₂²⋊D₆₇	1	3	yes	—
121	C₂²×C₈	C₂²×D₆₇	1	3	yes	—
122	C₂×C₁₆	—	67	12	no	—
123	C₂×C₁₆	—	67	12	no	—
124	C₂×C₁₆	—	67	4	no	—
125	#108	—	2	6	no	—
126	#108	—	67	4	no	—
127	#108	—	134	6	no	—
128	#108	C₂²×D₆₇	1	4	yes	—
129	C₂⁴	#11761618	1	1	yes	characteristic
130	C₂²×C₄	Dic₆₇⋊C₄	1	1	yes	characteristic
131	C₂²×C₄	Dic₆₇⋊C₄	1	3	yes	—
132	C₂²×C₄	#11761613	1	3	yes	—
133	C₂²×C₄	#11761613	1	1	yes	characteristic
134	C₂²×C₄	#11761618	1	3	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
136	M₁₆	—	2	4	no	—
137	M₁₆	—	4	6	no	—
138	M₁₆	—	268	6	no	—
139	M₁₆	—	134	4	no	—
140	C₂×C₈	—	4	6	no	—
141	C₂×C₈	—	268	6	no	—
142	C₂×C₈	Dic₆₇⋊C₄	1	6	yes	—
143	C₂×C₈	Dic₆₇⋊C₄	1	3	yes	—
144	C₂×C₈	Dic₆₇⋊C₄	1	3	yes	—
145	C₂×C₈	C₂×C₄×D₆₇	1	3	yes	—
146	C₂×C₈	C₂×C₄×D₆₇	1	3	yes	—
147	C₂×C₈	C₂×C₂²⋊D₆₇	1	3	yes	—
148	C₂×C₈	C₂×Dic₁₃₄	1	3	yes	—
149	C₂×C₈	C₂×Dic₁₃₄	1	3	yes	—
150	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
151	C₁₆	—	67	8	no	—
152	C₂³	—	2	4	no	—
153	C₂³	#14523976	1	3	yes	—
154	C₂³	#14523976	1	1	yes	characteristic
155	C₂³	C₂×#11761618	1	3	yes	—
156	C₂×C₄	#14523976	1	6	yes	—
157	C₂×C₄	#14523976	1	3	yes	—
158	C₂×C₄	#14523976	1	3	yes	—
159	C₂×C₄	C₂×Dic₆₇⋊C₄	1	3	yes	—
160	C₂×C₄	C₂×Dic₆₇⋊C₄	1	3	yes	—
161	C₂×C₄	C₂×#11761613	1	3	yes	—
162	C₂×C₄	C₂×#11761613	1	3	yes	—
163	C₂×C₄	C₂×#11761618	1	3	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
165	C₈	—	4	4	no	—
166	C₈	—	268	4	no	—
167	C₈	C₂×Dic₆₇⋊C₄	1	3	yes	—
168	C₈	C₂×Dic₆₇⋊C₄	1	3	yes	—
169	C₈	C₂²×C₄×D₆₇	1	1	yes	characteristic
170	C₈	C₂²×Dic₁₃₄	1	1	yes	characteristic
171	C₂²	—	2	6	no	—
172	C₂²	—	2	6	no	—
173	C₂²	#16863225	1	4	yes	direct product factor
174	C₂²	C₂×#14523976	1	3	yes	—
175	C₂²	C₂×#14523976	1	3	yes	—
176	C₂²	C₂²×#11761618	1	1	yes	characteristic
177	C₄	C₂×#14523976	1	3	yes	—
178	C₄	C₂×#14523976	1	3	yes	—
179	C₄	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
180	C₄	C₂²×#11761613	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	3	no	—
183	C₂	C₂×#16863225	1	6	yes	direct product factor
184	C₂	C₂²×#14523976	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11761613
commutator	C₈×C₆₇	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₂²×#108×C₆₇	C₂
socle	C₂³×C₆₇	#107
prime radical	C₂⁴×C₆₇	Q₁₆
cosocle	C₂×C₈×C₆₇	C₂⁴
hypercenter	C₂²×#108	D₆₇
hypocenter	C₆₇	C₂²×#368
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#16863225	C₂	direct (N × H)
11. #16863225	C₂²	direct (N × H)
80. C₂²×C₆₇	#368	semidirect
91. C₂×C₆₇	C₂×#368	semidirect
97. C₆₇	C₂²×#368	semidirect
173. C₂²	#16863225	direct (N × H)
183. C₂	C₂×#16863225	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×C₆₇⋊C₁₆	2
C₂²×#14523979	2
C₂×#16863225	2
C₂²×#108×C₆₇	2
C₂²×#368	67

Subgroups — C22×#16863225 #31008882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863225 #31008882