Subgroups — #30774350

Order 17152.

398 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	#22558286	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
7	C₈×Dic₁₃₄	—	2	2	no	—
8	C₈×Dic₁₃₄	—	2	2	no	—
9	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
10	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
11	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
12	C₄⋊(C₆₇⋊C₁₆)	—	2	2	no	—
13	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₆₇	—	2	2	no	—
16	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
17	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄×Dic₁₃₄	—	2	2	no	—
19	C₄×Dic₁₃₄	—	2	2	no	—
20	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
21	C₂×Dic₆₇⋊C₄	C₈	1	2	yes	—
22	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
23	C₂×C₆₇⋊C₁₆	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	4	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
26	Dic₆₇⋊C₈	—	2	4	no	—
27	C₈×Dic₆₇	—	2	2	no	—
28	C₈×Dic₆₇	—	2	2	no	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₁₆×C₆₇	—	2	2	no	—
33	C₂×C₁₆×C₆₇	—	2	2	no	—
34	C₂×C₁₆×C₆₇	Q₈	1	4	yes	complemented
35	C₄×C₈×C₆₇	D₄	1	2	yes	—
36	C₄×C₈×C₆₇	D₄	1	2	yes	—
37	C₂×Dic₁₃₄	—	2	2	no	—
38	C₂×Dic₁₃₄	—	2	2	no	—
39	C₂×Dic₁₃₄	C₁₆	1	4	yes	complemented
40	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
41	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
42	C₂×C₆₇⋊C₈	—	2	2	no	—
43	C₂×C₆₇⋊C₈	—	2	4	no	—
44	C₄⋊Dic₆₇	—	2	2	no	—
45	Dic₆₇⋊C₄	—	2	4	no	—
46	C₄×Dic₆₇	—	2	2	no	—
47	C₄×Dic₆₇	—	2	2	no	—
48	C₆₇⋊C₁₆	—	4	2	no	—
49	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₇	—	2	2	no	—
53	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₆₇	#41	1	2	yes	—
58	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
59	C₄²×C₆₇	M₁₆	1	2	yes	—
60	C₁₆×C₆₇	—	2	4	no	—
61	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
62	C₂×C₄×C₆₇	—	2	2	no	—
63	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
64	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
65	C₂×C₄×C₆₇	C₂×C₁₆	1	2	yes	—
66	C₂×C₄×C₆₇	M₃₂	1	2	yes	—
67	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
69	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
70	C₂×C₄×C₆₇	#131	1	2	yes	—
71	C₂×Dic₆₇	—	2	2	no	—
72	C₂×Dic₆₇	—	2	2	no	—
73	C₂×Dic₆₇	—	2	2	no	—
74	C₂×Dic₆₇	—	2	4	no	—
75	C₂×Dic₆₇	C₂×C₁₆	1	4	yes	—
76	Dic₁₃₄	—	2	8	no	—
77	C₈×C₆₇	—	2	2	no	—
78	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
79	C₈×C₆₇	#124	1	1	yes	characteristic
80	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
81	C₆₇⋊C₈	—	4	2	no	—
82	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
84	C₂²×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
85	C₂²×C₆₇	C₂×M₃₂	1	1	yes	characteristic
86	C₂²×C₆₇	#504	1	2	yes	—
87	C₄×C₆₇	—	2	2	no	—
88	C₄×C₆₇	C₂²⋊C₁₆	1	4	yes	—
89	C₄×C₆₇	#391	1	1	yes	characteristic
90	C₄×C₆₇	#408	1	1	yes	characteristic
91	C₄×C₆₇	#446	1	2	yes	—
92	Dic₆₇	—	2	4	no	—
93	Dic₆₇	—	4	2	no	—
94	#13263	—	67	1	no	maximal, Sylow
95	C₂×C₆₇	#1984	1	1	yes	characteristic
96	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂×C₆₇	#2113	1	1	yes	characteristic
98	C₂×C₆₇	Q₈×C₁₆	1	2	yes	—
99	C₂×C₆₇	#2183	1	2	yes	—
100	C₂×C₈×Q₈	—	67	1	no	—
101	C₂×C₄⋊C₁₆	—	67	1	no	—
102	C₂×C₄×C₁₆	D₆₇	1	1	yes	characteristic
103	#1374	—	67	4	no	—
104	C₆₇	#13263	1	1	yes	characteristic, complemented, Sylow
105	C₂×C₄×Q₈	—	67	1	no	—
106	C₂²×C₁₆	—	67	2	no	—
107	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
108	C₈×Q₈	—	134	2	no	—
109	C₈×Q₈	—	134	2	no	—
110	C₂×C₄⋊C₈	—	67	2	no	—
111	C₂×C₄⋊C₈	—	67	1	no	—
112	C₂×C₄×C₈	—	67	2	no	—
113	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
114	C₄⋊C₁₆	—	134	2	no	—
115	C₄×C₁₆	—	2	2	no	—
116	C₂²×Q₈	—	67	1	no	—
117	C₂²×C₈	—	67	2	no	—
118	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
120	C₄×Q₈	—	134	2	no	—
121	C₄×Q₈	—	134	2	no	—
122	C₂×C₄⋊C₄	—	67	2	no	—
123	C₂×C₄⋊C₄	—	67	1	no	—
124	C₂×C₄²	—	67	2	no	—
125	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	134	2	no	—
129	C₂×C₁₆	—	134	4	no	—
130	C₂×C₁₆	Dic₁₃₄	1	4	yes	complemented
131	C₄⋊C₈	—	134	4	no	—
132	C₄⋊C₈	—	134	2	no	—
133	C₄×C₈	—	134	2	no	—
134	C₄×C₈	—	134	2	no	—
135	C₄×C₈	D₂₆₈	1	2	yes	—
136	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
137	C₂×Q₈	—	67	4	no	—
138	C₂×Q₈	—	134	2	no	—
139	C₂×Q₈	—	134	2	no	—
140	C₂²×C₄	—	67	2	no	—
141	C₂²×C₄	—	67	2	no	—
142	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
143	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	134	2	no	—
147	C₂×C₈	—	134	4	no	—
148	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
149	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
150	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
151	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
152	C₂×C₈	#11761628	1	2	yes	—
153	C₄⋊C₄	—	134	4	no	—
154	C₄⋊C₄	—	134	2	no	—
155	C₄²	—	134	2	no	—
156	C₄²	—	134	2	no	—
157	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
158	C₄²	#11761611	1	2	yes	—
159	C₁₆	—	2	4	no	—
160	C₁₆	—	268	2	no	—
161	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
162	Q₈	—	134	8	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	67	4	no	—
165	C₂×C₄	—	134	4	no	—
166	C₂×C₄	—	134	2	no	—
167	C₂×C₄	—	134	2	no	—
168	C₂×C₄	—	134	2	no	—
169	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
170	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
171	C₂×C₄	C₁₆×D₆₇	1	2	yes	—
172	C₂×C₄	#14523982	1	2	yes	—
173	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
175	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
176	C₂×C₄	#14524049	1	2	yes	—
177	C₈	—	2	2	no	—
178	C₈	—	268	2	no	—
179	C₈	#14524022	1	2	yes	—
180	C₈	#14524031	1	1	yes	characteristic
181	C₈	#14524043	1	1	yes	characteristic
182	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
183	C₂²	C₈×Dic₁₃₄	1	2	yes	—
184	C₂²	C₂×C₁₆×D₆₇	1	1	yes	characteristic
185	C₂²	C₂×#14523982	1	1	yes	characteristic
186	C₂²	#16863812	1	2	yes	—
187	C₄	—	2	2	no	—
188	C₄	—	268	2	no	—
189	C₄	—	134	4	no	—
190	C₄	(C₂×D₆₇)⋊C₁₆	1	4	yes	—
191	C₄	#16863302	1	1	yes	characteristic
192	C₄	#16863362	1	1	yes	characteristic
193	C₄	#16863526	1	2	yes	—
194	C₂	#22561145	1	1	yes	characteristic
195	C₂	C₂×(C₂×D₆₇)⋊C₁₆	1	1	yes	characteristic
196	C₂	#22561822	1	1	yes	characteristic
197	C₂	C₁₆×Dic₁₃₄	1	2	yes	—
198	C₂	#22562117	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂²×C₁₆
Fitting	C₂×C₄×C₁₆×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₆₇
hypocenter	C₆₇	#13263
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
34. C₂×C₁₆×C₆₇	Q₈	semidirect
39. C₂×Dic₁₃₄	C₁₆	semidirect
104. C₆₇	#13263	semidirect
130. C₂×C₁₆	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₃₄	2
C₂×C₄⋊(C₆₇⋊C₁₆)	2
#22558286	2
C₂×C₄×C₁₆×C₆₇	2
#13263	67