Subgroups — #334×Dic₆₇ #30747014

Order 17152.

470 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊C₈×Dic₆₇	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863147	C₂	1	1	yes	maximal, characteristic
4	#22558043	C₂	1	2	yes	maximal
5	C₂×#334×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₆₇	—	2	2	no	—
9	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
10	C₂×C₈×Dic₆₇	C₄	1	2	yes	—
11	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
12	#16863150	—	2	2	no	—
13	#16863147	C₄	1	4	yes	complemented
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#334×C₆₇	C₄	1	4	yes	complemented
17	C₂²×C₆₇⋊C₈	—	2	2	no	—
18	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
21	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
22	C₂×C₄×Dic₆₇	Q₈	1	2	yes	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
25	C₄×C₆₇⋊C₈	—	2	1	no	—
26	C₄×C₆₇⋊C₈	—	2	1	no	—
27	C₈×Dic₆₇	—	2	4	no	—
28	C₈×Dic₆₇	—	4	4	no	—
29	C₈×Dic₆₇	C₈	1	8	yes	complemented
30	C₂²×C₈×C₆₇	—	2	2	no	—
31	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₆₇	—	2	2	no	—
35	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
37	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
38	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
39	C₂²×Dic₆₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₆₇⋊C₈	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	2	no	—
42	C₂×C₆₇⋊C₈	—	2	4	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
46	C₄×Dic₆₇	—	2	2	no	—
47	C₄×Dic₆₇	—	2	4	no	—
48	C₄×Dic₆₇	C₂×C₈	1	4	yes	—
49	C₄×Dic₆₇	M₁₆	1	4	yes	—
50	C₄×Dic₆₇	SD₁₆	1	4	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₆₇	—	2	2	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	—	4	2	no	—
57	C₂×C₈×C₆₇	—	4	2	no	—
58	C₂×C₈×C₆₇	C₄²	1	2	yes	—
59	C₂×C₈×C₆₇	C₄²	1	2	yes	—
60	C₂×C₈×C₆₇	C₂×C₈	1	4	yes	—
61	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₆₇	#41	1	1	yes	characteristic
64	C₄²×C₆₇	#41	1	1	yes	characteristic
65	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
69	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
70	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
74	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
75	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄×C₆₇	#135	1	2	yes	—
77	C₂×Dic₆₇	—	2	2	no	—
78	C₂×Dic₆₇	—	2	2	no	—
79	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
80	C₂×Dic₆₇	C₄⋊C₈	1	4	yes	—
81	C₂×Dic₆₇	#106	1	4	yes	—
82	C₂×Dic₆₇	#108	1	4	yes	—
83	C₈×C₆₇	—	2	2	no	—
84	C₈×C₆₇	—	4	4	no	—
85	C₈×C₆₇	C₄×C₈	1	4	yes	complemented
86	C₆₇⋊C₈	—	2	4	no	—
87	C₆₇⋊C₈	—	4	4	no	—
88	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
89	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
90	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
91	C₂²×C₆₇	#427	1	1	yes	characteristic
92	C₂²×C₆₇	C₂×#108	1	2	yes	—
93	C₄×C₆₇	—	2	1	no	—
94	C₄×C₆₇	—	2	1	no	—
95	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
96	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
97	C₄×C₆₇	#431	1	2	yes	—
98	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
99	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
100	Dic₆₇	—	2	4	no	—
101	Dic₆₇	#334	1	8	yes	direct product factor
102	C₄×#334	—	67	1	no	maximal, Sylow
103	C₂×C₆₇	C₂×#334	1	2	yes	—
104	C₂×C₆₇	#1564	1	2	yes	—
105	C₂×C₆₇	C₄×C₄⋊C₈	1	1	yes	characteristic
106	C₂×C₆₇	C₄×#106	1	1	yes	characteristic
107	C₂×C₆₇	C₄×#108	1	1	yes	characteristic
108	C₄×C₄⋊C₈	—	67	2	no	—
109	C₄²×C₈	—	67	1	no	—
110	C₂×#334	—	67	1	no	—
111	C₂×#334	D₆₇	1	1	yes	characteristic
112	#1277	—	67	2	no	—
113	C₆₇	C₄×#334	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₄⋊C₈	—	67	2	no	—
115	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
116	C₂×C₄×C₈	—	67	2	no	—
117	C₂×C₄×C₈	—	67	2	no	—
118	C₂×C₄×C₈	—	67	1	no	—
119	C₂×C₄×C₈	—	134	2	no	—
120	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
121	C₄³	—	67	1	no	—
122	#336	—	134	2	no	—
123	#334	—	67	4	no	—
124	#334	Dic₆₇	1	4	yes	direct product factor
125	C₂²×C₈	—	2	2	no	—
126	C₂²×C₈	—	67	2	no	—
127	C₂²×C₈	—	134	2	no	—
128	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
129	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
130	C₂×C₄²	—	67	2	no	—
131	C₂×C₄²	—	67	1	no	—
132	C₂×C₄²	—	67	1	no	—
133	C₂×C₄²	—	67	2	no	—
134	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	67	4	no	—
137	C₄⋊C₈	—	134	2	no	—
138	C₄⋊C₈	C₂×Dic₆₇	1	4	yes	—
139	C₄×C₈	—	268	4	no	—
140	C₄×C₈	—	67	8	no	—
141	C₄×C₈	—	134	4	no	—
142	C₄×C₈	—	134	1	no	—
143	C₄×C₈	—	134	1	no	—
144	C₄×C₈	C₄×D₆₇	1	2	yes	—
145	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
146	C₂²×C₄	—	67	2	no	—
147	C₂²×C₄	—	67	2	no	—
148	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
149	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
150	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	2	no	—
155	C₂×C₈	—	268	2	no	—
156	C₂×C₈	—	268	2	no	—
157	C₂×C₈	—	67	4	no	—
158	C₂×C₈	—	134	4	no	—
159	C₂×C₈	—	134	2	no	—
160	C₂×C₈	—	134	2	no	—
161	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
162	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
163	C₂×C₈	C₈×D₆₇	1	4	yes	—
164	C₄²	—	67	4	no	—
165	C₄²	—	67	4	no	—
166	C₄²	—	67	4	no	—
167	C₄²	—	134	4	no	—
168	C₄²	—	134	2	no	—
169	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
170	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
171	C₄²	#11761627	1	1	yes	characteristic
172	C₄²	#11761629	1	1	yes	characteristic
173	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	67	4	no	—
176	C₂×C₄	—	67	4	no	—
177	C₂×C₄	—	67	4	no	—
178	C₂×C₄	—	67	4	no	—
179	C₂×C₄	—	134	2	no	—
180	C₂×C₄	—	134	2	no	—
181	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
182	C₂×C₄	#14524001	1	1	yes	characteristic
183	C₂×C₄	#14524005	1	1	yes	characteristic
184	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
185	C₂×C₄	#14524008	1	1	yes	characteristic
186	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
187	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
188	C₂×C₄	M₁₆×D₆₇	1	2	yes	—
189	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
190	C₂×C₄	#14524064	1	2	yes	—
191	C₈	—	2	2	no	—
192	C₈	—	4	4	no	—
193	C₈	—	268	4	no	—
194	C₈	—	134	4	no	—
195	C₈	C₈×Dic₆₇	1	4	yes	complemented
196	C₂²	C₄⋊C₄×Dic₆₇	1	1	yes	characteristic
197	C₂²	C₄⋊C₈×D₆₇	1	2	yes	—
198	C₂²	#106×D₆₇	1	1	yes	characteristic
199	C₂²	#16863431	1	1	yes	characteristic
200	C₂²	#108×D₆₇	1	2	yes	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	67	8	no	—
204	C₄	—	134	4	no	—
205	C₄	C₂×C₈×Dic₆₇	1	1	yes	characteristic
206	C₄	M₁₆×Dic₆₇	1	1	yes	characteristic
207	C₄	#16863448	1	2	yes	—
208	C₄	#16863483	1	2	yes	—
209	C₄	SD₁₆×Dic₆₇	1	2	yes	—
210	C₂	#334×D₆₇	1	2	yes	—
211	C₂	#22558954	1	2	yes	—
212	C₂	C₄⋊C₈×Dic₆₇	1	1	yes	characteristic
213	C₂	#106×Dic₆₇	1	1	yes	characteristic
214	C₂	#108×Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#334×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#334	D₆₇
hypocenter	C₆₇	C₄×#334
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#334

Complements

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

N	H ≅ G/N	type
13. #16863147	C₄	semidirect
16. #334×C₆₇	C₄	semidirect
29. C₈×Dic₆₇	C₈	semidirect
85. C₈×C₆₇	C₄×C₈	semidirect
101. Dic₆₇	#334	direct (N × H)
113. C₆₇	C₄×#334	semidirect
124. #334	Dic₆₇	direct (N × H)
195. C₈	C₈×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊C₈×Dic₆₇	2
C₄×C₈×Dic₆₇	2
C₂×#16863147	2
#22558043	2
C₂×#334×C₆₇	2
C₄×#334	67

Subgroups — #334×Dic67 #30747014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #334×Dic₆₇ #30747014