Subgroups — C₄×#16863148 #30747017

Order 17152.

470 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#16863148	C₂	1	2	yes	maximal
4	#22558043	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
9	C₂×C₄×C₆₇⋊C₈	C₄	1	2	yes	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	#16863151	—	2	1	no	—
13	#16863148	C₄	1	8	yes	direct product factor
14	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	—	2	1	no	—
16	C₄³×C₆₇	C₂²	1	1	yes	characteristic
17	#336×C₆₇	—	2	1	no	—
18	C₂²×C₆₇⋊C₈	—	2	2	no	—
19	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
24	C₄×C₆₇⋊C₈	—	2	4	no	—
25	C₄×C₆₇⋊C₈	—	2	2	no	—
26	C₄×C₆₇⋊C₈	—	4	2	no	—
27	C₄×C₆₇⋊C₈	C₈	1	8	yes	complemented
28	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₆₇	—	2	2	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
34	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₈×C₆₇	—	2	2	no	—
36	C₄⋊C₈×C₆₇	C₂×C₄	1	4	yes	—
37	C₄×C₈×C₆₇	—	4	2	no	—
38	C₂×C₆₇⋊C₈	—	2	2	no	—
39	C₂×C₆₇⋊C₈	—	2	2	no	—
40	C₂×C₆₇⋊C₈	—	2	2	no	—
41	C₂×C₆₇⋊C₈	—	2	4	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
45	C₂×C₆₇⋊C₈	C₄²	1	2	yes	—
46	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
47	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₆₇	—	2	4	no	—
54	C₂×C₈×C₆₇	—	4	2	no	—
55	C₂×C₈×C₆₇	—	4	2	no	—
56	C₄²×C₆₇	—	2	4	no	—
57	C₄²×C₆₇	—	2	2	no	—
58	C₄²×C₆₇	C₂×C₈	1	4	yes	—
59	C₄²×C₆₇	M₁₆	1	4	yes	—
60	C₄²×C₆₇	D₈	1	2	yes	—
61	C₄²×C₆₇	Q₁₆	1	2	yes	—
62	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
63	C₄²×C₆₇	#41	1	2	yes	—
64	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₆₇	—	2	2	no	—
66	C₂×C₄×C₆₇	—	2	2	no	—
67	C₂×C₄×C₆₇	—	2	2	no	—
68	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
69	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
70	C₂×C₄×C₆₇	#107	1	4	yes	—
71	C₂×C₄×C₆₇	#108	1	4	yes	—
72	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
74	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
76	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
77	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
78	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#135	1	2	yes	—
81	C₈×C₆₇	—	4	4	no	—
82	C₆₇⋊C₈	—	2	2	no	—
83	C₆₇⋊C₈	—	2	4	no	—
84	C₆₇⋊C₈	—	4	4	no	—
85	C₆₇⋊C₈	C₄×C₈	1	4	yes	complemented
86	C₂²×C₆₇	C₄×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
88	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
89	C₂²×C₆₇	#427	1	1	yes	characteristic
90	C₂²×C₆₇	C₂×#108	1	2	yes	—
91	C₄×C₆₇	—	2	4	no	—
92	C₄×C₆₇	—	2	2	no	—
93	C₄×C₆₇	#335	1	8	yes	complemented
94	C₄×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
95	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
96	C₄×C₆₇	#431	1	2	yes	—
97	C₄×C₆₇	C₄×D₈	1	2	yes	—
98	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
99	C₄×#335	—	67	1	no	maximal, Sylow
100	C₂×C₆₇	C₂×#335	1	2	yes	—
101	C₂×C₆₇	#1564	1	2	yes	—
102	C₂×C₆₇	C₄×C₄⋊C₈	1	1	yes	characteristic
103	C₂×C₆₇	C₄×#107	1	1	yes	characteristic
104	C₂×C₆₇	C₄×#108	1	1	yes	characteristic
105	C₄×C₄⋊C₈	—	67	1	no	—
106	C₄×C₄⋊C₈	D₆₇	1	1	yes	characteristic
107	C₄²×C₈	—	67	1	no	—
108	C₂×#335	—	67	2	no	—
109	#1277	—	67	2	no	—
110	C₆₇	C₄×#335	1	1	yes	characteristic, complemented, Sylow
111	C₂×C₄⋊C₈	—	67	2	no	—
112	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
113	C₂×C₄×C₈	—	2	1	no	—
114	C₂×C₄×C₈	—	67	2	no	—
115	C₂×C₄×C₈	—	67	1	no	—
116	C₂×C₄×C₈	—	67	2	no	—
117	C₂×C₄×C₈	—	67	1	no	—
118	C₂×C₄×C₈	—	134	1	no	—
119	C₄³	C₂×D₆₇	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#336	—	134	1	no	—
122	#335	—	67	8	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	—	67	1	no	—
125	C₂²×C₈	—	67	1	no	—
126	C₂²×C₈	—	67	2	no	—
127	C₂²×C₈	—	134	2	no	—
128	C₂×C₄²	Dic₁₃₄	1	2	yes	—
129	C₂×C₄²	C₄×D₆₇	1	2	yes	—
130	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
131	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
133	C₄⋊C₈	—	2	2	no	—
134	C₄⋊C₈	—	67	4	no	—
135	C₄⋊C₈	—	134	2	no	—
136	C₄⋊C₈	C₄×D₆₇	1	4	yes	—
137	C₄×C₈	—	4	2	no	—
138	C₄×C₈	—	268	2	no	—
139	C₄×C₈	—	67	8	no	—
140	C₄×C₈	—	67	4	no	—
141	C₄×C₈	—	134	4	no	—
142	C₄×C₈	—	134	2	no	—
143	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
144	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
145	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
147	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	4	2	no	—
150	C₂×C₈	—	4	2	no	—
151	C₂×C₈	—	268	2	no	—
152	C₂×C₈	—	268	2	no	—
153	C₂×C₈	—	67	2	no	—
154	C₂×C₈	—	67	4	no	—
155	C₂×C₈	—	67	2	no	—
156	C₂×C₈	—	67	4	no	—
157	C₂×C₈	—	134	4	no	—
158	C₂×C₈	—	134	2	no	—
159	C₂×C₈	—	134	2	no	—
160	C₂×C₈	—	134	2	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	2	no	—
163	C₄²	C₈×D₆₇	1	4	yes	—
164	C₄²	#11761611	1	4	yes	—
165	C₄²	#11761613	1	2	yes	—
166	C₄²	#11761618	1	2	yes	—
167	C₄²	C₂×C₄×D₆₇	1	2	yes	—
168	C₄²	#11761628	1	2	yes	—
169	C₂³	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
174	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
175	C₂×C₄	#14523976	1	4	yes	—
176	C₂×C₄	#14523978	1	4	yes	—
177	C₂×C₄	C₄²×D₆₇	1	1	yes	characteristic
178	C₂×C₄	#14524000	1	1	yes	characteristic
179	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
180	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
181	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
182	C₂×C₄	C₂×#11761611	1	2	yes	—
183	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
184	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
185	C₂×C₄	#14524063	1	2	yes	—
186	C₈	—	4	4	no	—
187	C₈	—	268	4	no	—
188	C₈	—	67	4	no	—
189	C₈	—	134	2	no	—
190	C₈	—	134	4	no	—
191	C₂²	C₄×Dic₆₇⋊C₄	1	1	yes	characteristic
192	C₂²	C₂×Dic₆₇⋊C₈	1	2	yes	—
193	C₂²	C₂×#14523976	1	1	yes	characteristic
194	C₂²	#16863432	1	1	yes	characteristic
195	C₂²	C₂×#14523978	1	2	yes	—
196	C₄	—	2	2	no	—
197	C₄	—	2	4	no	—
198	C₄	#16863148	1	8	yes	direct product factor
199	C₄	C₄×C₈×D₆₇	1	1	yes	characteristic
200	C₄	C₄×#11761611	1	1	yes	characteristic
201	C₄	#16863447	1	2	yes	—
202	C₄	C₄×#11761613	1	2	yes	—
203	C₄	C₄×#11761618	1	2	yes	—
204	C₂	C₂×#16863148	1	2	yes	—
205	C₂	#22558955	1	2	yes	—
206	C₂	C₄×Dic₆₇⋊C₈	1	1	yes	characteristic
207	C₂	C₄×#14523976	1	1	yes	characteristic
208	C₂	C₄×#14523978	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×C₄⋊C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×C₄⋊C₈	D₆₇
hypocenter	C₆₇	C₄×#335
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
13. #16863148	C₄	direct (N × H)
27. C₄×C₆₇⋊C₈	C₈	semidirect
85. C₆₇⋊C₈	C₄×C₈	semidirect
93. C₄×C₆₇	#335	semidirect
110. C₆₇	C₄×#335	semidirect
198. C₄	#16863148	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄²×C₆₇⋊C₈	2
C₂×#16863148	2
#22558043	2
C₄×C₄⋊C₈×C₆₇	2
C₄×#335	67

Subgroups — C4×#16863148 #30747017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16863148 #30747017